Essentially normal Hilbert modules and K homology III: Homogenous quotient modules of Hardy modules on the bidisk

 Damian Freeman
 10 months ago
 Views:
Transcription
1 Science in China Series A: Mathematics Mar., 2007, Vol. 50, No. 3, Essentially normal Hilbert modules and K homology III: Homogenous quotient modules of Hardy modules on the bidisk Kunyu GUO & Penghui WANG School of Mathematics, Fudan University, Shanghai , China ( Abstract In this paper, we study the homogenous quotient modules of the Hardy module on the bidisk. The essential normality of the homogenous quotient modules is completely characterized. We also describe the essential spectrum for a general quotient module. The paper also considers K homology invariant defined in the case of the homogenous quotient modules on the bidisk. Keywords: MSC(2000): Hardy modules, bidisk, essential normality, Khomology. 47A13, 47A20, 46H25, 46C99 1 Introduction Let T = (T 1,..., T n ) be a tuple of commuting operators acting on a Hilbert space H. Using DouglasPaulsen s Hilbert module language [1], we endow H with a C[z 1,..., z n ]module structure by p x = p(t 1,..., T n )x, p C[z 1,..., z n ], x H. In Arveson s language [2 4], a Hilbert module is called essentially normal if the selfcommutators Tk T j T j Tk of its canonical operators are all compact (one can also refer to such Hilbert modules as essentially reductive, see [1, 5, 6]). Much work has been done along this line [1 11]. In the course of Arveson s studies [2 4], he considered the essential normality of quotients of standard Hilbert modules. For a dshift Hilbert module with finitemultiplicity, he established a pessential normality in case the submodule is generated by monomials [2]. That result on monomial submodules was generalized by Douglas to the case in which the dshift is replaced by more general weighted shifts [5]. In the dimension d = 2, Guo [7] obtained trace estimates, which implies that the homogeneous quotient modules of 2shift Hilbert module are essentially normal. In the recent work of Guo and Wang [8], these results were generalized to cases in which the 2shift is replaced by Uinvariant Hilbert modules with finitemultiplicity over 2dimensional unit ball. In [9], the essential normality and the a Khomology of quasihomogeneous quotient modules over a 2dimensional unit ball were characterized by a hard analysis, and the method Received July 3, 2006; accepted November 21, 2006 DOI: /s Corresponding author This work is partially supported by the National Natural Science Foundation of China (Grant No ), the Young Teacher Fund, the National Key Basic Research Project of China (Grant No. 2006CB805905) and the Specialized Research for the Doctoral Program
2 388 Kunyu GUO & Penghui WANG is different from the case of homogeneous quotient modules. The paper will be devoted to studying the essential normality of quotient modules of the Hardy module over the bidisk. At this point it is completely different from the Hilbert modules on the unit ball. It is well known that neither the Hardy module H 2 (D 2 ) nor its nonzero submodules are essentially normal. Let H 2 (D 2 ) be the Hardy module on the bidisk with the module action defined by multiplication of coordinate functions. The structure of submodules of H 2 (D 2 ) is much more complex than that of submodules of H 2 (D), and for more work on this subject, see [1, 6, 12 19] and references therein. Let M be a submodule of H 2 (D 2 ), set N = H 2 (D 2 ) M. Then N can be endowed with a C[z, w]module structure by p f = p(s z, S w )f, p C[z, w], f N, where S z = P N M z N and S w = P N M w N. Some efforts have been done to describe the structure of some certain quotient modules over the bidisk, see [20 23]. For any nonzero submodule M of H 2 (D 2 ), using the fact that the coordinate functions define a pair of isometries, both of infinite multiplicity, M is not essentially normal. However, in [24], Douglas and Misra showed that some quotient modules of H 2 (D 2 ) are essentially normal and some are not. Some other related work has been done in [11, 25]. If a quotient module N of H 2 (D 2 ) is essentially normal, then the C algebra C (N) generated by {S z, S w } is essentially commutative. A result of Yang [18,Theorem 4.4] implies that for any quotient module N of H 2 (D 2 ), the C algebra C (N) is irreducible. Hence, we have a C  extension 0 K C (N) C(σ e (S z, S w )) 0. This C extension gives an element of K 1 (σ e (S z, S w )), which is an invariant for the Hilbert modules [1,26,27]. In this paper, we mainly concern the homogenous quotient modules. Let I be an ideal of C[z, w], and [I] be the submodule of H 2 (D 2 ) generated by I. If I is homogeneous, then the submodule [I] also is homogeneous, and in this case, the quotient module H 2 (D 2 )/[I] is homogenous. Since the polynomial ring is Noetherian [28], the ideal I is generated by finitely many polynomials. This implies that I has a greatest common divisor p, and so, I can be uniquely written as I = pl, which is called the Beurling form of I (cf. [29]). In dimension d = 2, by a lemma of Yang [15], dim C[z, w]/l < and hence [p] [I] is of finite dimension, where [p] is the submodule of H 2 (D 2 ) generated by p. This means that H 2 (D 2 )/[p] and H 2 (D 2 )/[I] have the same essential normality. Notice if I is homogeneous, then both p and L are homogeneous. This paper is organized as follows. In sec. 2, we will consider the compactness of the commutators [P M, M z ] and [P M, M w ] for homogenous submodules M. For a homogenous submodule M = [p], both [P M, M z ] and [P M, M w ] are compact if and only if Z(p) D 2 T 2. In sec. 3, we will develop some properties of the asymptotic orthogonality. It plays an important role in the paper. Sec. 4 and sec. 5 are devoted to characterizing the essential normality of homogenous quotient modules of H 2 (D 2 ). Let p be a nonzero homogenous polynomial in C[z, w], then p
3 Homogenous quotient modules of Hardy modules on the bidisk 389 can be factorized as p(z, w) = (α i z β i w) ni. Set p 1 (z, w) = (α i z β i w) ni, p 2 (z, w) = (α i z β i w) ni. α i = β i α i β i Then p = p 1 p 2, (1.1) with Z(p 1 ) D 2 T 2 and Z(p 2 ) D 2 (D T) (T D). We state our main theorem as follows: Theorem 1.1. Let p be a nonzero homogenous polynomial in C[z, w], and p = p 1 p 2 as in (1.1), then the quotient module H 2 (D 2 )/[p] is essentially normal if and only if p 2 has one of the following forms : (1) p 2 = c, with c 0; (2) p 2 = αz + βw, with α β ; (3) p 2 = c(z αw)(w βz), with α < 1, β < 1 and c 0. In sec. 6, we will describe the essential spectrum of quotient modules. For any quotient module N of H 2 (D 2 ), the essential spectrum σ e (N) is defined by σ e (S z, S w ). It is shown that for a homogenous polynomial p, σ e (H 2 (D 2 )/[p]) = Z(p) D 2. Therefore, if a homogenous quotient module H 2 (D 2 )/[p] is essentially normal, then we have a C extension 0 K C (S z, S w ) C(Z(p) D 2 ) 0. It is shown that this extension yields a nontrivial Khomology element in K 1 (Z(p) D 2 ). Moreover, we also describe the essential spectrum of a general quotient module. Using the result in this paper, one can construct a quotient module N of H 2 (D 2 ), such that σ e (N) = D 2. 2 Compactness of [P M, M z ] and [P M, M w ] Let M be a submodule of H 2 (D 2 ), and N = H 2 (D 2 ) M be the corresponding quotient module. The quotient module N is called essentially normal, if both [S z, Sz ] and [S w, Sw ] are compact, where for two operators A and B, the operator [A, B] = AB BA is the commutator of A and B. If both S z and S w are essentially normal, then by FugledePutnam Theorem, the commutator [S z, S w] is compact. Since N is an invariant subspace of M z and M w, the quotient module N is essentially normal if and only if both M z N and M w N are essentially normal. In [24], Douglas and Misra established by a direct calculation that the quotient module H 2 (D 2 ) [(z w) 2 ] is essentially normal and H 2 (D 2 ) [z 2 ] is not. They also showed that H 2 (D 2 ) [z n w m ] is essentially normal, and this result was generalized by Clark in [25]. Recently, Izuchi and Yang [11] showed that if ϕ(w) H 2 (D) is an inner function, then H 2 (D 2 ) [z ϕ(w)] is essentially normal if and only if ϕ is a finite Blaschke product. In this section, the homogenous quotient modules H 2 (D 2 )/[p] with will be concerned. Z(p) D 2 T 2 Let p be a homogenous polynomial, and M = [p] be the submodule of H 2 (D 2 ) generated by p. We will prove that both [P M, M z ] and [P M, M w ] are compact if and only if Z(p) D 2 T 2, where P M denotes the orthogonal projection from H 2 (D 2 ) onto M.
4 390 Kunyu GUO & Penghui WANG As a corollary, we will prove that if Z(p) D 2 T 2, then the quotient module H 2 (D 2 )/[p] is essentially normal. At first, we need some results on the evaluation operator and the difference quotient operator, which are studied by Yang [16,17,19]. The evaluation operators L(0) and R(0) at z = 0 and w = 0 respectively are defined by L(0)f(z, w) = f(0, w), R(0)f(z, w) = f(z, 0), f H 2 (D 2 ). In fact, L(0) = 1 M z M z and R(0) = 1 M w M w are projections. Let M be a submodule of H 2 (D 2 ), and N = H 2 (D 2 ) M be the corresponding quotient module. Write then we can rewrite D z = P M M z N, D w = P M M w N, D z = P M M z (1 P M ) = [P M, M z ], D w = P M M w (1 P M ) = [P M, M w ]. The operators D z and D w are given by the difference quotient operators in [19], D zf(z, w) = f(z, w) f(0, w), D z wf(z, w) = f(z, w) f(z, 0). w The following proposition comes from [16], which gives the relationship between the evaluation operators and the difference quotient operators. Proposition 2.1 quotient module, then If M is a submodule of H 2 (D 2 ), let N = H 2 (D 2 ) M be the corresponding (1) S z S z + D zd z = I N, S ws w + D wd w = I N ; (2) S z S z + P NL(0)P N N = I N, S w S w + P NR(0)P N N = I N. Let M be a submodule of H 2 (D 2 ), and N = H 2 (D 2 ) M. Proposition 2.1 implies that, if both [P M, M z ] and L(0)P N are compact, then S z is essentially normal. Similarly, if both [P M, M w ] and R(0)P N are compact, then S w is essentially normal. We have the following Lemma 2.2 Let p be a homogeneous polynomial with the factorization n p(z, w) = λ (w α i z), with α i 0. Set M = [p] and N = H 2 (D 2 ) M, then [P M, M z ] is compact if and only if L(0)P N is compact. Similarly, the commutator [P M, M w ] is compact if and only if R(0)P N is compact. Proof. p as Notice that Firstly, we will prove that both ker S z and ker S z p(z, w) = λ n (w α i z) = λw n + zp 1 (z, w). ker S z = ker M z [p] = H 2 w [p], are of finite dimension. We rewrite where H 2 w = span{wn, n = 0, 1, 2,...}. If f ker S z, then f = k=0 a kw k and f [p]. For any m 0, λa m+n = λa m+n + M z f, w m p 1 = f, λw m+n + f, zw m p 1 = f, w m p = 0.
5 Homogenous quotient modules of Hardy modules on the bidisk 391 It follows that dim(ker Sz ) n + 1. Now we will show that ker S z = {0}. Notice that ker S z = {f f [p], zf [p]}. (2.1) For any f ker S z, let f = i=0 F i(z, w) be the homogeneous expansion of f. Since p is homogeneous and zf [p], zf i [p]. By the fact that F i is a polynomial, it follows that there is a polynomial q such that zf i = pq. Noting that GCD(z, p) = 1, this means that q/z is a polynomial, and hence F i = pq/z [p]. So f [p]. By (2.1), f = 0. Therefore ker S z = 0. By Proposition 2.1, if one of [P M, M z ] and L(0)P N is compact, then S z is Fredholm, and hence Sz is the essential inverse of S z. This insures that the other of [P M, M z ] and L(0)P N is compact. The same reason gives the case of [P M, M w ] and R(0)P N. We state the main theorem in this section as follows. Theorem 2.3. Let p be a homogenous polynomial in C[z, w]. Set M = [p], then both [P M, M z ] and [P M, M w ] are compact if and only if Z(p) D 2 T 2. Combining Theorem 2.3, Lemma 2.2 with Proposition 2.1, we have the next corollary. Corollary 2.4. Let p be a homogenous polynomial in C[z, w] satisfying Z(p) D 2 T 2, then the quotient module H 2 (D 2 ) [p] is essentially normal. Proof of the necessity of Theorem 2.3. one of [P M, M z ] and [P M, M w ] is not compact. We will prove that if Z(p) D 2 T 2, then at least As mentioned in the introduction, for any homogenous polynomial p, p can be factorized as p = λz k w l n (w α i z), with α i 0. Below, the necessity of Theorem 2.3 will be proved in two cases. Case 1. Both k and l are zero. Since Z(p) D 2 T 2, there is an i 0 such that α i0 1. Set β = α i0. Suppose now that 0 < β < 1. We rewrite p as p = (z βw)p, where p is a homogenous polynomial. Set N 1 = [z βw] and N 2 = [z βw] [p], then M = N 1 N 2. Since {1} {e j = ( βz) j +( βz) j 1 w+ +w j β 2j + β 2j } j=1 is an orthonormal basis of N 1, L(0)P N1 e j = 1 β 2j + β 2j wj. Then lim j L(0)P N1 e j = 1 β 2 0. It follows that L(0)P N1 is not compact, and hence L(0)P M is not compact. Lemma 2.2 implies that [P M, M z ] is compact if and only if L(0)P M is compact. Thus [P M, M z ] is not compact. The same reason implies that if β > 1, then [P M, M w ] is not compact. Case 2. At least one of k and l is not zero. We suppose now k 0, then p = zp, where p is a homogenous polynomial. It follows that M = [z] ([z] M), and hence {w j } j=1 M. Let m = deg p, then p = m i=0 a iz i w m i. Now taking a i0 0, we claim that P M M z i 0 M is not compact.
6 392 Kunyu GUO & Penghui WANG Since p is homogenous, {f j = w j p/ p } j=1 is an orthonormal set of M. By Bessel s inequality, for sufficiently large j, P M M z i 0 M w j 2 = P M z i0 w j 2 ( ) z i0 w j 2, f i f i = z i0 w j, f j m+i0 2 = a i0 2 / p 2 0. Since w j converges to zero weakly, [P M, M z i 0 ] = P M M z i 0 M is not compact. Since {P M } e = {A B(H 2 (D 2 )) [P M, A] is compact} is a C algebra, the compactness of [P M, M z ] implies that [P M, M z i 0 ] is compact. And hence [P M, M z ] is not compact. The same reason implies that if l 0, then [P M, M w ] is not compact, thus completing the proof. The proof of the sufficiency of Theorem 2.3 is long. The remaining part of this section will be devoted to the proof of the sufficiency of Theorem 2.3. Let H n = {f n C[z, w] f n is homogenous with degree n}, then H 2 (D 2 ) can be decomposed as H 2 (D 2 ) = n=0 H n. It is easy to see that dim H n = n + 1. Given a homogenous polynomial p of degree m, then for sufficiently large n, and hence dim([p] H n ) = m. dim([p] H n ) = n m + 1, To prove the sufficiency of Theorem 2.3, we need the next lemma. In the case n = 1, the next lemma is considered in [16]. Lemma 2.5. For n 1, let M 0 = [(z w) n ], then both [P M0, M z ] and [P M0, M w ] are compact. Proof. then Let N k i Set N 0 = H 2 (D 2 ) M 0. For i = 1, 2,..., n, set N i = [(z w) i 1 ] [(z w) i ], N 0 = N 1 N n. = N i H k, k = 0, 1, 2,..., then N i has a homogeneous decomposition N i = Ni k. k=0 It is easy to see that for sufficiently large k, N k i is of dimension 1. Let e k i (z, w) be in N k i with e k i = 1. Define a unitary operator V : H 2 (D 2 ) H 2 (D 2 ) by V f(z, w) = f(w, z), f H 2 (D 2 ). Obviously, every N i is a reduced subspace of V, and hence every Ni k is a reduced subspace of V. It follows that there are complex numbers λ (k) i with λ (k) i = 1, such that e k i (w, z) = V e k i (z, w) = λ (k) i e k i (z, w).
7 Homogenous quotient modules of Hardy modules on the bidisk 393 This implies that z k, e k i = V z k, V e k i = w k, λ (k) i e k i = w k, e k i. (2.2) Below, we will prove that L(0)P Ni L(0) is compact for i = 1,..., n, by induction. {1, zk +z k 1 w+ +w k k+1, k = 1, 2,...} is an orthonormal basis of N 1, Since L(0)P N1 L(0)w k = wk k + 1, and this insures that L(0)P N1 L(0) is compact. Suppose now that L(0)P Ni L(0) is compact for all i < m. Since for sufficiently large k, L(0)P Ni L(0)w k = L(0)P N k i L(0)w k = w k, e k i ek i, wk w k, i = 1,..., m 1, we have lim k w k, e k i 2 = 0 for i = 1,..., m 1, and hence by (2.2) lim k m 1 z k, e k i e k i, w k m 1 lim k w k, e k i 2 = 0. (2.3) By [16, Lemma 4.1.2], for any quotient module N, the operator L(0)P N R(0) is HilbertSchmidt. Now take N = H 2 (D 2 ) [(z w) m ], then N = m N i. Since P Ni z k = P N k i z k = z k, e k i ek i, m m L(0)P N R(0)z k = L(0)P Ni R(0)z k = z k, e k i e k i, w k w k. Let L(0)P N R(0) H.S denote the HilbertSchmidt norm of L(0)P N R(0), then m 2 z k, e k i e k i, w k = L(0)P N R(0)z k 2 L(0)P N R(0) 2 H.s < +. Thus k=m By (2.3) and (2.4), we have It follows that k=m lim k m z k, e k i ek i, wk = 0. (2.4) lim k zk, e k m ek m, wk = 0. L(0)P Nm L(0)w k = w k, e k m ek m, wk = z k, e k m ek m, wk 0. So L(0)P Nm L(0) is compact, and hence for 1 i n, L(0)P Ni L(0) is compact. This insures that ( n L(0)P N0 L(0) = L(0) P Ni )L(0) is compact, and hence L(0)P N0 is compact. By Lemma 2.2, [P M0, M z ] is compact. The same reason implies that [P M0, M w ] is compact. Let p i C[z, w], i = 1, 2, setting r = GCD(p 1, p 2 ), the greatest common divisor of p 1 and p 2, then p 1 = r p 1 and p 2 = r p 2. If Z(r) D 2 =, then by [12, Proposition ], [r] = H 2 (D 2 ), and hence [p i ] = [ p i ]. To prove the sufficiency of Theorem 2.3, we still need several lemmas.
8 394 Kunyu GUO & Penghui WANG Lemma 2.6. Given p 1, p 2 C[z, w], if Z(p 1 ) Z(p 2 ) D 2 =, then [p 1 ] + [p 2 ] = [p 1 p 2 ]. Proof. At first, we claim that [p 1 ] + [p 2 ] is closed. Since Z(p 1 ) Z(p 2 ) D 2 =, as mentioned above, we may assume GCD(p 1, p 2 ) = 1. By [15, Lemma 6.1], it follows that [p 1 ] [p 2 ] = ([p 1 ] + [p 2 ]) is of finite dimension. Let M = [p 1 ] ([p 1 ] [p 2 ] ) and N = [p 2 ] ([p 1 ] [p 2 ] ), then [p 1 ] + [p 2 ] is closed if and only if M + N is closed. Now suppose that M + N is not closed, then we can take {f n f n = 1} n=1 M and {g n g n = 1} n=1 N, such that both f n and g n converge weakly to 0, and f n + g n tends to 0 as n. Since both Mp 1 f n = 0 and M p 2 g n = 0, we have lim ( (M p 1 Mp 1 + M p2 Mp 2 )f n, f n + (M p1 Mp 1 + M p2 Mp 2 )g n, g n ) = lim ( M p 2 M p 2 f n, f n + M p1 Mp 1 g n, g n + M p1 Mp 1 f n, f n + g n + M p1 Mp 1 g n, f n + M p2 Mp 2 g n, f n + g n + M p2 Mp 2 f n, g n ) = lim (M p 1 M p 1 + M p2 M p 2 )(f n + g n ), f n + g n lim M p 1 M p 1 + M p2 M p 2 f n + g n 2 = 0. Since Z(p 1 ) Z(p 2 ) D 2 =, ker(m p1 M p 1 + M p2 M 2 ) = [p 1] [p 2 ] is of finite dimension. By [30, Theorem 4.3] and [30, Lemma 4.5], the operator M p1 Mp 1 + M p2 Mp 2 has a closed range, and hence it is a positive Fredholm operator. This implies that there exist a positive invertible operator A and a compact operator K, such that M p1 M p 1 + M p2 M p 2 = A + K. Since both f n and g n converge weakly to zero as n, we have lim Kf n = 0 and lim Kg n = 0. Since A is positive and invertible, there are some c i > 0, i = 1, 2, such that and lim (M p 1 M p 1 + M p2 Mp 2 )f n, f n = lim (A + K)f n, f n lim c 1 f n 2 = c 1, lim (M p 1 M p 1 + M p2 M p 2 )g n, g n = lim (A + K)g n, g n lim c 2 g n 2 = c 2.
9 Homogenous quotient modules of Hardy modules on the bidisk 395 This contradiction implies that M + N is closed, and hence [p 1 ] + [p 2 ] is closed, the claim is proved. Using the characteristic space theory for polynomials in [12, Chapter 2], we have [p 1 ] [p 2 ] = [p 1 p 2 ], and hence [p 1 ] + [p 2 ] = ([p 1 ] [p 2 ]) = [p 1 p 2 ], thus completing the proof. Lemma 2.7. Let p i C[z, w], i = 1, 2, such that Z(p 1 ) Z(p 2 ) D 2 =. Write P i = P [pi] and set M = [p 1 p 2 ]. If both [P i, M z ] and [P i, M w ] are compact for i = 1, 2, then so are [P M, M z ] and [P M, M w ]. Proof. Since both [P i, M z ] and [P i, M w ] are compact for i = 1, 2, a simple reason shows that for any polynomial p, the commutator [P i, M p ] are compact for i = 1, 2. Since Z(p 1 ) Z(p 2 ) D 2 =, by Lemma 2.6, M = [p 1 ] + [p 2 ]. It is easy to check that Mp 1 M [p 2 ], and hence Mp 1 P M = P2 Mp 1 P M, where P2 = I P 2. Moreover, notice that M = [p 1 p 2 ] [p 2 ], then for any polynomial p, we have P M M p M p1 Mp 1 P M = P M (P 2 M p M p1 P2 )M p 1 P M = P M [P 2, M pp1 ]Mp 1 P M. This implies that the operator P M M p M p1 Mp 1 P M is compact. Similarly, the operator P M M p M p2 M p 2 P M is compact. Therefore, the operator is compact. P M M p (M p1 M p 1 + M p2 M p 2 )P M Since Z(p 1 ) Z(p 2 ) D 2 =, the same argument as in the proof of Lemma 2.6 implies that the operator M p1 Mp 1 + M p2 Mp 2 is a positive Fredholm operator, and hence there exist a finite rank operator F and a positive invertible operator X, such that M p1 Mp 1 + M p2 Mp 2 = X + F. Hence P M M p XP M = P M M p (M p1 M p 1 + M p2 M p 2 )P M P M M p F P M (2.5) is compact. Considering the matrix representation of X, X = X 11 X 12 X12 X 22 M M, and taking p = 1 in the left side of (2.5), we have that X 12 = P M XP M X 22 is Fredholm. is compact. Hence Noting that M is an invariant subspace, the operator M p has a matrix representation ( ) Sp D p M, and hence 0 R p M M p X = S px 11 + D p X12 S p X 12 + D p X 22 R p X12 R p X 22 M M.
10 396 Kunyu GUO & Penghui WANG By (2.5) again, S p X 12 + D p X 22 is compact. Since X 12 is compact, the operator D p X 22 is compact. The Fredholmness of X 22 implies that D p is compact. Now taking p = z and p = w, we have that both D z and D w are compact, that is, both [P M, M z ] and [P M, M w ] are compact, as desired. Proof of the sufficiency of Theorem 2.3. The hypothesis Z(p) D 2 T 2 implies that p can be factorized as m p(z, w) = λ (z α i w) ni, with α i = 1 and λ 0. Without loss of generality, we may assume λ = 1. Below, we will prove the theorem by induction. When m = 1 and α = 1, define a unitary operator U : H 2 (D 2 ) H 2 (D 2 ) by then Uf(z, w) = f(z, αw), for f H 2 (D 2 ), M z U = UM z, αm w U = UM w. Let M 0 = [(z w) n ] and M = [(z αw) n ], then P M0 = U P M U, P M 0 = U P M U. Notice that P M M z P M = UP M0 U M z UP M 0 U = UP M0 M z P M 0 U = U[P M0, M z ]U. Lemma 2.5 implies that [P M0, M z ] is compact, and hence [P M, M z ] = P M M z P M Similarly, [P M, M w ] is compact. Now we suppose that the theorem holds for m = k, that is, for is compact. k p k (z, w) = (z α i w) ni, with α i = 1 both [P k, M z ] and [P k, M w ] are compact, where P k = P [pk ]. For any β = 1 and β α i, i = 1,..., k, setting p k+1 = (z βw) n k+1 p k, write P k+1 = P [pk+1 ]. It suffices to show that both [P k+1, M z ] and [P k+1, M w ] are compact. Since β α i, Z(z βw) Z(p k ) = {0}, and hence by Lemma 2.7, both [P k+1, M z ] and [P k+1, M w ] are compact. This completes the proof. 3 Asymptotic orthogonality In this section, we will develop some properties of the asymptotic orthogonality for closed subspaces. It plays an important role in this paper.
11 Homogenous quotient modules of Hardy modules on the bidisk 397 Let H be a Hilbert space, N 1 and N 2 be two closed subspaces of H. The subspaces N 1 and N 2 are called asymptotically orthogonal, denoted by N 1 N 2 if P N1 P N2 is compact. Proposition 3.1. N 1 N 2, then N 1 + N 2 is closed. Let H be a Hilbert space, N 1 and N 2 be two closed subspaces of H. If Proof. Suppose N 1 + N 2 is not closed. Then there are {f n f n = 1} n=1 N 1 and {g n g n = 1} n=1 N 2 satisfying f n w 0 and gn w 0, such that fn + g n tends to zero as n. Since P N2 P N1 is compact, and since f n converges weakly to zero, lim P N 2 f n = lim P N 2 P N1 f n = 0. Similarly, lim P N 1 g n = 0. Now, on the one hand, lim (P N 2 + P N1 )(f n + g n ), (f n + g n ) lim P N 2 + P N1 f n + g n 2 = 0, on the other hand, lim (P N 2 + P N1 )(f n + g n ), (f n + g n ) = lim [ P N 1 f n, f n + P N2 g n, g n + P N1 g n, f n + g n + P N2 f n, f n + g n + P N1 f n, g n + P N2 g n, f n ] = lim ( f n 2 + g n 2 ) = 2. This contradiction shows that N 1 + N 2 is closed. The closeness of N 1 + N 2 is not enough to insure N 1 N 2. The following proposition helps understanding the asymptotic orthogonality. Proposition 3.2. Let H be a Hilbert space, N 1 and N 2 be two closed subspaces of H. Then N 1 and N 2 are asymptotically orthogonal if and only if N 1 + N 2 is closed and P N1+N 2 (P N1 + P N2 ) is compact. Proof. Suppose first that N 1 + N 2 is closed and P N1+N 2 (P N1 + P N2 ) is compact. Since P N1 P N2 + P N2 P N1 = (P N1 + P N2 ) 2 (P N1 + P N2 ) = (P N1 + P N2 ) 2 P N1+N 2 + P N1+N 2 (P N1 + P N2 ) = [(P N1 + P N2 P N1+N 2 )(P N1 + P N2 + P N1+N 2 )] +[P N1+N 2 (P N1 + P N2 )], we have that P N1 P N2 + P N2 P N1 is compact. It follows that P N1 P N2 P N1 P N2 = 1 2 P N 1 (P N1 P N2 + P N2 P N1 )P N1 P N2 is compact. Since (P N2 P N1 P N2 ) (P N2 P N1 P N2 ) = P N2 P N1 P N2 P N1 P N2, and hence P N2 P N1 P N2 is compact. From the equality P N2 P N1 P N2 = (P N1 P N2 ) (P N1 P N2 ),
12 398 Kunyu GUO & Penghui WANG we see that P N1 P N2 is compact. Suppose now that P N1 P N2 is compact. By Proposition 3.1, N 1 + N 2 is closed. Notice that the compactness of P N1 P N2 implies N 1 N 2 being of finite dimension, and hence we can assume without loss of generality that N 1 N 2 = {0}. Now, we claim that Q = P N1 + P N2, viewed as an operator on N 1 + N 2, is Fredholm. At first, for f 1 N 1, f 2 N 2, if Q(f 1 + f 2 ) = 0, then by N 1 N 2 = {0}, f 1 + P N1 f 2 = 0 and f 2 + P N2 f 1 = 0, and hence f 2 = P N2 P N1 f 2. Since N 1 N 2 = {0}, we have f 2 = 0. Thus f 1 = 0. Therefore ker Q = {0}. To prove that Q is Fredholm, it suffices to show that Q has a closed range. Let N 1 N 2 = {(f 1, f 2 ) f i N i, (f 1, f 2 ) = f 1 + f 2 }, then N 1 N 2 is a Banach space. Define an operator T : N 1 N 2 N 1 +N 2 by T (f 1, f 2 ) = f 1 +f 2, then T is bounded and invertible. It follows that there are c 0 > 0 and c 1 > 0 such that c 0 f 1 + f 2 f 1 + f 2 c 1 f 1 + f 2, for f 1 N 1, f 2 N 2. Notice that P N1 P N2 P N1 can be regarded as an operator on N 1, and it is compact. By the Fredholm Alternative Theorem [31,Theorem 5.22], for any ε > 0, there is a finitely codimensional subspace N1 ε of N 1 such that, for any f N1 ε, P N2 f = (P N1 P N2 P N1 ) 1 2 f < ε f. Similarly, there is a finitely codimensional subspace N2 ε of N 2, such that for any f N2 ε, P N1 f = (P N2 P N1 P N2 ) 1 2 f < ε f. Hence for f 1 N1 ε and f 2 N2 ε, (P N1 + P N2 )(f 1 + f 2 ) f 1 + f 2 ( P N1 f 2 + P N2 f 1 ) = f 1 + f 2 ( (P N2 P N1 P N2 ) 1 2 f2 + (P N1 P N2 P N1 ) 1 2 f1 ) f 1 + f 2 ε( f 1 + f 2 ) c 0 f 1 + f 2 εc 1 f 1 + f 2 = (c 0 εc 1 ) f 1 + f 2. We can take ε to be enough small, such that c 0 εc 1 > 0. Now, fix ε, since Ni ε is a finitely codimensional subspace of N i, the space N ε = N1 ε + N2 ε is a finitely codimensional subspace of N 1 + N 2. The above reason implies that Q N ε is bounded below, and hence Q has a closed range. Therefore, the operator Q is Fredholm, and this means that 0 / σ e (Q). Now, since P N1 P N2 is compact, Q 2 Q = (P N1 + P N2 ) 2 (P N1 + P N2 ) = P N1 P N2 + P N2 P N1 is compact, and hence σ e (Q) {0, 1}. It follows that σ e (Q) = {1}. Since Q is selfadjoint, there is an exact sequence, 0 K C (Q) + K π C 0,
13 Homogenous quotient modules of Hardy modules on the bidisk 399 where K is the ideal of all compact operators on N 1 +N 2 and C (Q) is the C algebra generated by Q and the identity operator I on N 1 + N 2. Since Q is Fredholm, π(q) 0. Notice that π(q) 2 = π(q), and we have π(q) = 1. It follows that I Q is compact. This insures that P N1+N 2 (P N1 + P N2 ) is compact, as desired. Let H be an infinitely dimensional Hilbert space, T B(H). Suppose N 1 and N 2 are invariant subspaces of T satisfying N 1 N 2, by Proposition 3.1, N = N 1 + N 2 is closed. It is easy to see that N is an invariant subspace of T. Let T 1 = T N1 and T 2 = T N2 be the restrictions of T to N 1 and N 2 respectively. The following theorem maybe is familiar to some readers, we sketch it here for convenience. Theorem 3.3 Under the above assumption, if both T 1 and T 2 are essentially normal, then T N, the restriction of T to N, is essentially normal. Proof. At first, we consider the matrix of T N relative to the decomposition N = N 1 (N N 1 ) to obtain T N = P N 1 T P N1 P N1 T P N N1 P N N1 T P N1 P N N1 T P N N1 Since N 1 is an invariant subspace of T, we have T P N1 = P N1 T P N1, and hence P N N1 T P N1 = P N N1 P N1 T P N1 = 0. Since N 2 is also an invariant subspace of T, we have T P N2 = P N2 T P N2, and hence. P N1 T P N N1 = P N1 T (P N N1 P N2 ) + P N1 T P N2 = P N1 T (P N N1 P N2 ) + P N1 P N2 T P N2. Since N 1 N 2, by Proposition 3.2, both P N N1 P N2 and P N1 P N2 are compact, and hence P N1 T P N N1 is compact. A simple reason implies that T N is essentially normal if and only if both P N1 T P N1 and P N N1 T P N N1 are essentially normal. Since T 1 is essentially normal, P N1 T P N1 = T 1 is essentially normal. Now noticing that P N N1 P N2 is compact, we have P N N1 T P N N1 = P N2 T P N2 + a compact operator. Since P N2 T P N2 = T 2 is essentially normal, this implies that P N N1 T P N N1 is essentially normal, and hence T N is essentially normal, thus completing the proof. Now, let us turn to quotient modules of H 2 (D 2 ). Let M 1, M 2 be two submodules of H 2 (D 2 ), and N 1 = H 2 (D 2 ) M 1, N 2 = H 2 (D 2 ) M 2 be the corresponding quotient modules. If N 1 N 2, then by Proposition 3.1, N = N 1 + N 2 is closed, and hence N = H 2 (D 2 ) (M 1 M 2 ). As an immediate application of Theorem 3.3, we have Corollary 3.4. Under the above assumption, if both N 1 and N 2 are essentially normal, then N is essentially normal. 4 The sufficiency of Theorem 1.1 In this section, we use some results of the asymptotic orthogonality to prove the sufficiency of Theorem 1.1. To this end, we still need several facts. Proposition 4.1. Let p 1 and p 2 be two homogenous polynomials in C[z, w], setting N 1 = H 2 (D 2 ) [p 1 ] and N 2 = H 2 (D 2 ) [p 2 ], then the followings are equivalent :
14 400 Kunyu GUO & Penghui WANG (i) N 1 N 2, (ii) for any f n N 1 H n, g n N 2 H n satisfying f n = 1 and g n = 1, lim f n, g n = 0, (iii) for f n N 1 H n with f n = 1, there exist q n [p 2 ] H n such that lim f n q n = 0. Proof. (i)= (iii): Since H 2 (D 2 ) = N 2 [p 2 ], there are f n (1) N 2 H n and f n (2) [p 2 ] H n such that f n = f n (1) + f n (2). Since P N2 P N1 is compact and f n converges weakly to zero, lim f n f (2) n = lim (1) f n = lim P N 2 P N1 f n = 0. (iii)= (ii): By (iii), for any f n N 1 H n with f n = 1, there is q n [p 2 ] H n such that lim f n q n = 0. It follows that for any g n N 2 H n with g n = 1, we have lim f n, g n lim ( f n q n, g n + q n, g n ) lim f n q n g n = 0. (ii)= (i): Set k 1 = deg p 1 and k 2 = deg p 2. Then, as mentioned in sec. 2, there is a positive integer N, such that for any n N, dim(n 1 H n ) = k 1 and dim(n 2 H n ) = k 2. Let {e i n }k1 and {f j n }k2 j=1 be orthonormal bases of N 1 H n and N 2 H n respectively, then there is a finite rank operator F, so that ( P N1 P N2 = F + = F + k 1 n=n ( k1 n=n e i n e i n )( k 2 m=n j=1 k 2 ) fn, j e i n e i n fn j. j=1 f j m f j m Since k 1 k2 j=1 f n j, ei n ei n f n j can be viewed as an operator from N 2 H n to N 1 H n, we have ( k1 k 2 ) ( k1 k 2 ) fn, j e i n e i n fn j = fn, j e i n e i n fn j. n=n j=1 Since for any i, j, lim f j n, e i n = 0, this gives lim n=n j=1 k 1 k 2 fn, j e i n e i n fn j = 0. j=1 Therefore, P N1 P N2 is compact. From Proposition 4.1, the next corollary follows easily. Corollary 4.2. Let α 1 < 1 and α 2 < 1, then [z α 1 w] [w α 2 z]. )
15 Homogenous quotient modules of Hardy modules on the bidisk 401 Proof. and Let e k = (ᾱ 1z) k + (ᾱ 1 z) k 1 w + w k α1 2k + α 1 2k , f k = (ᾱ 2w) k + (ᾱ 2 w) k 1 z + z k α2 2k + α 2 2k , then e k [z α 1 w] H k and f k [w α 2 z] H k, with e k = 1 and f k = 1. Note that dim([z α 1 w] H k ) = 1 and dim([w α 2 z] H k ) = 1. It is easy to verify lim e k, f k = 0, by Proposition 4.1, Proposition 4.3. [z α 1 w] [w α 2 z]. Let p i, i = 1, 2, 3, be the homogenous polynomials in C[z, w] satisfying GCD(p 2, p 3 ) = 1. If both [p 1 ] [p 2 ] and [p 1 ] [p 3 ], then [p 1 ] [p 2 p 3 ]. Proof. Since p 2 and p 3 are homogenous and GCD(p 2, p 3 ) = 1, Z(p 2 ) Z(p 3 ) = {0}. By Lemma 2.6, [p 2 p 3 ] = [p 2 ] + [p 3 ]. Below, we will prove [p 1 ] ([p 2 ] + [p 3 ] ). By [15, Lemma 6.1], [p 2 ] [p 3 ] is of finite dimension, and hence for sufficiently large n, ([p 2 ] H n ) ([p 3 ] H n ) = {0}. Using the argument in the proof of Lemma 2.6, there are constants c i > 0, i = 1, 2, such that for sufficiently large n and f n [p 2 ] H n, g n [p 3 ] H n, f n c 1 f n + g n and g n c 2 f n + g n. Suppose now f n +g n = 1, then f n c 1 and g n c 2. Since [p 1 ] [p 2 ] and [p 1 ] [p 3 ], by Proposition 4.1, for any h n [p 1 ] H n satisfying h n = 1, lim (f n + g n ), h n = lim f n, h n + lim g n, h n = 0. By Proposition 4.1 again, [p 1 ] ([p 2 ] + [p 3 ] ), thus completing the proof. Lemma 4.4. For α 1 and β = 1, the quotient modules [z αw] [(z βw) n ]. Proof. Without loss of generality, assume α < 1. Setting α = α β, we claim that [z α w] [(z w) n ]. Since α < 1, it follows from Corollary 4.2 that [z] [w ᾱ z]. Now, let N i = [(z w) i 1 ] [(z w) i ], and e k i N i H n with e k i = 1. The argument in the proof of Lemma 2.5 implies that for i = 1, 2,..., n, lim k w k, e k i = 0. By Proposition 4.1, P [z] P N is i k compact, and hence P [z] P [(z w) n ] is compact, which means [z] [(z w) n ]. Since α < 1, GCD((ᾱ z w), (z w) n ) = 1, and hence by Proposition 4.3, the quotient modules [z]
16 402 Kunyu GUO & Penghui WANG [(ᾱ z w)(z w) n ]. By Proposition 4.1, it follows that there are polynomials p k H k n, such that lim k wk+1 (ᾱ z w)(z w) n p k = 0. Now, setting f k 1 = k 1 i=0 (ᾱ z) i w k 1 i, we have (ᾱ z) k w k = (ᾱ z w)f k 1, and hence 0 lim (ᾱ z w)f k (ᾱ z w)(z w) n p k lim (ᾱ z) k+1 + lim wk+1 (ᾱ z w)(z w) n p k = 0. Since for any f H 2 (D 2 ), (ᾱ z w)f(z, w) 2 = (ᾱ z w)f(z, w) 2 dm 2 (1 α ) 2 f 2, T 2 we have 0 lim k f k (z w) n p k 1 1 α Noticing that lim k f k = (1 α 2 ) 1 2, we have lim k (ᾱ z w)f k (ᾱ z w)(z w) n p k = 0. lim f k (z w) n p k / f k = 0. k Taking q k = p k f k, then q k [(z w) n ]. Since f k f k [z α w] H k, by Proposition 4.1, the quotient modules [z α w] [(z w) n ]. The claim is proved. For β = 1, define a unitary operator U : H 2 (D 2 ) H 2 (D 2 ) by Then it is easy to see that Uf(z, w) = f(z, βw) for f H 2 (D 2 ). [z αw] = U[z α w], [(z βw) n ] = U[(z w) n ]. This implies that [z αw] [(z βw) n ], thus completing the proof. For a homogenous polynomial p satisfying Z(p) D 2 T 2, p can be factorized as p(z, w) = λ m (z α i w) ni, with α i = 1 and λ 0. Combining Lemma 4.4 with Proposition 4.3, we have the following Lemma 4.5. Let p be a homogenous polynomial in C[z, w] satisfying then for α 1, [z αw] [p]. Z(p) D 2 T 2, With the above preparations, we are able to prove the sufficiency of Theorem 1.1. As mentioned in the introduction, let p be a nonzero homogenous polynomial in C[z, w], then p can be factorized as p = p 1 p 2,
17 Homogenous quotient modules of Hardy modules on the bidisk 403 with Z(p 1 ) D 2 T 2 and Z(p 2 ) D 2 (D T) (T D). We restate the sufficiency of Theorem 1.1 as follows. Theorem 4.6. Let p be a nonzero homogenous polynomial in C[z, w], and p = p 1 p 2 as in (1.1), if p 2 has one of the following forms: (1) p 2 = c, with c 0; (2) p 2 = αz + βw, with α β ; (3) p 2 = c(z αw)(w βz), with α < 1, β < 1 and c 0, then the quotient module H 2 (D 2 )/[p] is essentially normal. Proof. (1) Suppose p 2 = c, then [p] = [p 1 ]. Corollary 2.4 implies that the quotient module H 2 (D 2 )/[p] is essentially normal. (2) Suppose p 2 = αz + βw, with α β. Without loss of generality, we may assume that α 0. Notice that Z(p 2 ) Z(p 1 ) D 2 =. By Lemma 2.6, [p] = [p 1 ] + [p 2 ]. Since α 0 and α β, [p 2 ] = [ z β α w ], β with 1. α Notice that Z(p 1 ) D 2 T 2. Lemma 4.5 insures [p 2 ] [p 1 ]. To show that [p] is essentially normal, by Corollary 3.4, it suffices to show that both [p 1 ] and [p 2 ] are essentially normal. Since Z(p 1 ) D 2 T 2, the essential normality of [p 1 ] comes from Corollary 2.4. Now, set α = β α, then [p 2] = [z α w]. It follows that e k = (ᾱ z) k + (ᾱ z) k 1 w + + w k α 2k + α 2k is an orthonormal basis of [p 2 ]. So S z e k = ze k, e k+1 e k+1, and hence it is easy to verify that S z is essentially normal. Similarly, S w is essentially normal. This implies that [p 2 ] is essentially normal. (3) Suppose p 2 = c(z αw)(w βz), with α < 1, β < 1 and c 0, then [p] = [(z αw)(w βz)p 1 ]. Since by Lemma 2.6, we have Z(z αw) Z((w βz)p 1 ) = {0}, [p] = [z αw] + [(w βz)p 1 ]. As done in proof (2), the same reason shows that both [z αw] and [(w βz)p 1 ] are essentially normal. To show that [p] is essentially normal, by Corollary 3.4, it suffices to show [z αw] [(w βz)p 1 ]. Since α < 1, β < 1, Corollary 4.2 insures [z αw] [w βz]. Since α < 1 and Z(p 1 ) D 2 T 2, Lemma 4.5 implies [z αw] [p 1 ].
18 404 Kunyu GUO & Penghui WANG Notice that Z(w βz) Z(p 1 ) D 2 =, then by Proposition 4.3, [z αw] [(w βz)p 1 ], as desired, thus completing the proof. 5 The necessity of Theorem 1.1 In this section, we will prove the necessity of Theorem 1.1. In fact, we will prove the following result, from which the necessity of Theorem 1.1 easily follows. Theorem 5.1. Let M be a submodule of H 2 (D 2 ), and α i < 1, i = 1, 2, then the quotient module [(z α 1 w)(z α 2 w)m ] is not essentially normal. Similarly, the quotient module [(w α 1 z)(w α 2 z)m ] is not essentially normal. Let p be a homogenous polynomial. Factorizing p = p 1 p 2 as in Theorem 1.1, if p 2 has not one of the forms in Theorem 1.1, then p can be factorized as p = (z α 1 w)(z α 2 w)p, or p = (w α 1 z)(w α 2 z)p, with α i < 1, i = 1, 2. Now taking M = [p ] in Theorem 5.1, the necessity of Theorem 1.1 immediately comes from Theorem 5.1. Let H be an infinitely dimensional Hilbert space, and T be a bounded linear operator on H. For an invariant space M of T, T has a matrix representation, T = P M T M P M T M P M T M M M. To prove Theorem 5.1, the following lemma is needed, whose proof is similar to the proofs of [3, Theorem 1] and [5, Theorem 4.3]. Lemma 5.2. Under the above assumption, if P M T M is essentially normal, then the followings are equivalent: (1) T is essentially normal, (2) P M T M is compact, and P M T M is essentially normal. To continue, the following lemma is needed. Lemma 5.3. Given α i < 1, i = 1, 2, write M = [(z α 1 w)(z α 2 w)] and N = H 2 (D 2 ) M. Then the quotient module N is not essentially normal. Proof. We will show that S z is not essentially normal. Let V 1 = [z α 1 w] and V 2 = [z α 1 w] M, then N = V 1 V 2. For any f V 1 and g V 2, it is easy to see Sz f, g = f, zg = 0, that is, V 1 is an invariant subspace of Sz. It follows that S z has the matrix representation, S z = A z B z C z V 1 V 2.
19 Homogenous quotient modules of Hardy modules on the bidisk 405 Since V 1 = [z αw], Theorem 4.6 implies that A z = P V1 M z V1 is essentially normal. To show that S z is not essentially normal, by Lemma 5.2, it suffices to show that B z is not compact. Notice that for n 2, dim V i H n = 1, i = 1, 2. Let e n = (ᾱ 1z) n + (ᾱ 1 z) n 1 w + + w n α1 2n + α 1 2(n 1) + + 1, then e n V 1 H n and e n = 1. For g n V 2 H n satisfying g n = 1, we have B z e n = ze n, g n+1 g n+1. Hence B z is compact if and only if ze n, g n+1 0 as n. We will show that B z is not compact in the following three cases. Case 1. α 1 α 2. Since Z(z α 1 w) Z(z α 2 w) = {0}, by Lemma 2.4, N = M = [z α 1 w] + [z α 2 w]. Set and e n = (ᾱ 1z) n + (ᾱ 1 z) n 1 w + + w n α1 2n + α 1 2(n 1) + + 1, f n = (ᾱ 2z) n + (ᾱ 2 z) n 1 w + + w n α2 2n + α 2 2(n 1) Since α 1 α 2, by the Cauchy inequality, e n, f n n α 1 α 2 j/( n n ) 1 α 1 2j α 2 2j 2 < 1. j=0 It follows that for n 2, {e n, f n } is a linear basis of N H n. orthogonalization, j=0 j=0 g n = ( f n f n, e n e n )/( 1 en, f n 2) 1 2. Since α i < 1, i = 1, 2, by direct calculations, we have Using the GramSchmidt and lim e n+1, f n+1 = lim ze n, f n+1 = α 2 (1 α1 2 )(1 α 2 2 ), 1 ᾱ 1 α 2 (1 α1 2 )(1 α 2 2 ), 1 ᾱ 1 α 2 lim ze n, e n+1 = α 1. Set c 0 = 1 ᾱ1α2 (1 α 1 2 )(1 α 2 2 ) (1 ᾱ 1α 2) α 1 α 2, then we have lim ze ze n, f n+1 e n+1, f n+1 ze n, e n+1 n, g n+1 = lim (1 e n, f n 2 ) 1 2 = c 0 (α 2 α 1 ).
20 406 Kunyu GUO & Penghui WANG In the case α 1 α 2, we have c 0 0, and hence lim ze n, g n+1 0. This implies that in this case, B z is not compact. Case 2. α 1 = α 2 = α 0. Let e n = (ᾱz)n + (ᾱz) n 1 w + + w n α 2n + α 2n Now it is easy to verify that for any nonnegative integers j 1, j 2, (z αw) 2 z j1 w j2, (n + 1)(ᾱz) n + n(ᾱz) n 1 w + + w n = 0, for n 2. It follows that (n + 1)(ᾱz) n + n(ᾱz) n 1 w + + w n N H n. Write h n = (n + 1)(ᾱz)n + n(ᾱz) n 1 w + + w n (n + 1)2 α 2n + n 2 α 2n , then h n = 1. For n 2, by the Cauchy inequality, ( n )/( n n ) 1 e n, h n = (j + 1) α 2j (j + 1) 2 α 2j α 2j 2 < 1. j=0 Since dim N H n = 2 for n 2, {e n, h n } is a linear basis of N H n. Using the GramSchmidt orthogonalization again, j=0 j=0 g n = (h n h n, e n e n ) / (1 e n, h n 2 ) 1 2 V2 H n, and g n = 1. Now, we will show that Since e n+1, g n+1 = 0 and α 0, (5.1) holds if and only if lim ze n, g n+1 0. (5.1) lim ᾱze n be n+1, g n+1 0 for any b R. Taking we have b = ( n+1 / n ) 1 α 2j α 2j 2, j=1 j=1 ᾱze n be n+1 = w n+1 /( n j=1 α 2j ) 1 2. Therefore, it suffices to show that 1 α 2 lim wn+1, g n+1 = lim Since 0 < α < 1, by direct calculations, we have w n+1 /( n j=1 α 2j ) 1 2, gn+1 0. lim wn, e n = 1 α 2,
21 Homogenous quotient modules of Hardy modules on the bidisk 407 and / lim wn, h n = (1 α 2 ) 3 2 (1 + α 2 ) 1 2, lim e 1 n, h n =. 1 + α 2 Therefore, lim wn, g n = ( lim w n, h n e n, h n w n, e n ) = 1 lim 1 en, h n 2 α Therefore, in this case, B z is not compact. Case 3. ( ) (1 α 2 ) 3 2 (1 α 2 ) α 1 = α 2 = 0. This case is considered in [24], we sketch the proof here for convenience. M = N 1 N 2, where N 1 = H 2 (D 2 ) zh 2 (D 2 ) and N 2 = zh 2 (D 2 ) z 2 H 2 (D 2 ). It is easy to see that N 1 = span{w n n = 0, 1,...}, N 2 = span{zw n n = 0, 1,...}. Since S z can be decomposed as S z w n = zw n and S z zw n = 0, S z = 0 0 U 0 N 1 N 2, where Uw n = zw n. That is, B z = U is not compact. The above reason shows that S z is not essentially normal, thus completing the proof. The proof of Theorem 5.1. submodule M of H 2 (D 2 ), the quotient module is not essentially normal. Let We will only prove that, if α i < 1 for i = 1, 2, then for any N = H 2 (D 2 ) [(z α 1 w)(z α 2 w)m ] M 1 = [z α 1 w], M 2 = [(z α 1 w)(z α 2 w)] and M 3 = [(z α 1 w)(z α 2 w)m ], then N can be decomposed as N = M 1 (M 1 M 2 ) (M 2 M 3 ). Relative to this decomposition, S z has the matrix representation, S z = A z B z C z E z F z G z M 1 M 1 M 2. M 2 M 3
22 408 Kunyu GUO & Penghui WANG Since M 1 = [z α 1 w], Theorem 4.6 implies that A z = P M 1 M z M 1 is essentially normal. The ( ) argument in the proof of Lemma 5.3 implies that B z is not compact, and hence Bz is not E z compact. By Lemma 5.2, S z is not essentially normal, and hence Theorem 5.1 is proved. 6 Essential spectrum of the quotient modules and Khomology In this section, we will describe the essential spectrum of quotient modules of H 2 (D 2 ). Some similar techniques can be seen in [10]. The Khomology will also been considered. Let M be submodule of H 2 (D 2 ), Z(M) = {λ D 2 f(λ) = 0, f M}, and Z (M) is defined by { } Z (M) = λ D 2 there are λ n Z(M), such that lim λ n = λ. Theorem 6.1. σ e (N). For any submodule M of H 2 (D 2 ), set N = H 2 (D 2 ) M. Then Z (M) Proof. The proof is routine. Some similar techniques appear in [10]. For the reader s convenience, we give the detail of the proof. Given λ Z (M), by the definition of Z (M), there is µ n Z(M) such that lim µ n = λ. If the pair (λ 1 S z, λ 2 S w ) is Fredholm, by a result of Curto [32, Corollary 3.11], the operator A = λ 1 S z λ 2 S w (λ 2 S w ) (λ 1 S z ) : N N N N is Fredholm. Set T 1 = λ 1 S z and T 2 = λ 2 S w. It follows that AA = T 1T1 + T 2 T2 0 0 T1 T 1 + T2 T 2 is Fredholm. Hence (λ 1 S z )(λ 1 S z ) + (λ 2 S w )(λ 2 S w ) = T 1 T1 + T 2T2 is Fredholm. This implies that there exist a positive invertible operator B and a compact operator K, such that (λ 1 S z )(λ 1 S z ) + (λ 2 S w )(λ 2 S w ) = B + K. Now, let k µn be the normalized reproducing kernel of H 2 (D 2 ) at µ n. Since k µn converges weakly to zero, and since B is positive and invertible, there is a positive constant c such that lim [(λ 1 S z )(λ 1 S z ) + (λ 2 S w )(λ 2 S w ) ]k µn, k µn = lim (B + K)k µ n, k µn = lim Bk µ n, k µn c. However, since µ n = (µ (1) n, µ (2) n ) Z(M), for any f M, f, k µn = 0, and then k µn N. It follows that lim [(λ 1 S z )(λ 1 S z ) + (λ 2 S w )(λ 2 S w ) ]k µn, k µn = lim ( λ 1 µ (1) n 2 + λ 2 µ (2) n 2 ) = 0.
23 Homogenous quotient modules of Hardy modules on the bidisk 409 This contradiction implies Z (M) σ e (N), as desired. Example. Let ᾱ n α n z φ(z) = α n 1 ᾱ n z, ϕ(w) = β n β n w β n 1 β n w n=1 be two infinite Blaschke products. It is wellknown that there exist infinite Blaschke products φ and ϕ, such that T Z(φ) and T Z(ϕ). For detailed information of this kind of inner functions, one can see [33, sec. 3]. Now, let M = [φϕ], then Z (M) = D 2. By Theorem 6.1, D 2 σ e (S z, S w ). By [19, Theorem 4.3], σ e (S z, S w ) = D 2. Theorem 6.2. Given a homogenous polynomial p. Set N = H 2 (D 2 ) [p], then σ e (N) = Z(p) D 2. Proof. On the one hand, since p is a homogenous polynomial, Z ([p]) = Z([p]) D 2 = Z(p) D 2. By Theorem 6.1, Z(p) D 2 σ e (N). On the other hand, by [19, Theorem 4.3], σ e (N) D 2. Since p(s z, S w ) = 0, by the Spectral Mapping Theorem [34,Theorem 4.8], σ e (N) Z(p) D 2. This completes the proof. Let p be a homogenous polynomial. If the quotient module [p] is essentially normal, then we get an extension 0 K C ([p] ) C(Z(p) D 2 ) 0. This extension yields a Khomology element in K 1 (Z(p) D 2 ), which is denoted by e p. By Theorem 1.1, the polynomial p can be factorized as p = (w α 1 z) n1 (z α 2 w) n2 m i=3 n=1 (z α i w) ni, with α i < 1, n i 1 for i = 1, 2, and α i = 1 for i 3. Set p 1 = (w α 1 z) n1, and p i = (z α i w) ni for i 2. Without loss of generality, assume that Z(p i ) Z(p j ) = {0} for i j. The same argument as [8, Proposition 4.2] implies that e p = e p1 e p2 e pm. (6.1) Below, we will show that if deg p i 0, then e pi are nontrivial Khomology elements in K 1 (Z(p i ) D 2 ) for i = 1,..., m. In fact, we will show that the corresponding extension is not split. Proposition 6.3. Let α be a complex number with α < 1, then the extension 0 K C ([z αw] ) C(Z(z αw) D 2 ) 0 is not split.
24 410 Kunyu GUO & Penghui WANG Proof. By Theorem 6.2, σ e ([z αw] ) = Z(z αw) D 2 = {(αw, w) w = 1}. By the Spectral Mapping Theorem [34,Theorem 4.8], σ e (S w ) = T. This implies that S w is Fredholm. It is easy to see that the Fredholm index Ind(S w ) = 1. By [8, Proposition 4.6], the extension 0 K C ([z αw] ) C(Z(z αw) D 2 ) 0 is not split, as desired. Proposition 6.4. Let α be a complex number with α = 1, then the extension 0 K C ([(z αw) n ] ) C(Z(z αw) D 2 ) 0 is not split. Proof. The same reason as in the proof of Proposition 6.3 shows that S z is Fredholm. The argument in the proof of Lemma 2.2 implies that ker S z = {0}. Now since 1 ker Sz, the Fredholm index Ind(S z ) 0. By [8, Proposition 4.6] again, the extension 0 K C ([(z αw) n ] ) C(Z(z αw) D 2 ) 0 is not split, thus completing the proof. Combining Proposition 6.3, Proposition 6.4 with (6.1), we have the following theorem. Theorem 6.5. Let p be a homogenous polynomial. If [p] is essentially normal, then the short exact sequence 0 K C ([p] ) C(Z(p) D 2 ) 0 is not split. References [1] Douglas R, Paulsen V. Hilbert modules over function algebras. Pitman Research Notes in Mathematics Series, 217, 1989 [2] Arveson W. psummable commutators in dimension d. J Oper Theory, 54: (2005) [3] Arveson W. Quotients of standard Hilbert modules. Trans AMS, to appear [4] Arveson W. The dirac operator of a commuting dtuple. J Funct Anal, 189: (2002) [5] Douglas R. Essentially reductive Hilbert modules. J Oper Theory, 55: (2006) [6] Douglas R. Invariants for Hilbert Modules. Proceedings of Symposia in Pure Mathematics, Vol. 51, Part 1, 1990, [7] Guo K. Defect operator for submodules of Hd 2. J Reine Angew Math, 573: (2004) [8] Guo K, Wang K. Essentially normal Hilbert modules and Khomology. Preprint [9] Guo K, Wang K. Essentially normal Hilbert modules and Khomology II: Quasihomogeneous Hilbert modules over two dimensional unit ball. Preprint [10] Guo K, Duan Y. Spectrum property of the submodule of the Hardy space over B d. Studia Math, to appear [11] Izuchi K, Yang R. N ϕtype quotient modules on the torus. Preprint [12] Chen X, Guo K. Analytic Hilbert modules. πchapman & Hall/CRC Research Notes in Math, 433, 2003 [13] Curto R, Muhly P, Yan K. The C algebra of an homogeneous ideal in two variables is type I. Current Topics in Operator Algebras (Nara, 1990). River Edge, NJ: World Sci. Publishing,
25 Homogenous quotient modules of Hardy modules on the bidisk 411 [14] Guo K, Yang R. The core function of Hardy submodules over the bidisk. Indiana Univ Math J, 53: (2004) [15] Yang R. The BergerShaw theorem in the Hardy module over the bidisk. J Oper Theory, 42: (1999) [16] Yang R. Operator theory in the Hardy space over the bidisk, II. Int Equ Oper Theory, 42: (2002) [17] Yang R. Operator theory in the Hardy space over the bidisk, III. J Funct Anal, 186: (2001) [18] Yang R. The core operators and Congruent submodules. J Funct Anal, 228: (2005) [19] Yang R. On twovariable Jordan block (II). Int Equ Oper Theory, to appear [20] Douglas R, Misra G, Varughese C. Some geometric invariants from resolutions of Hilbert modules. Systems, approximation, singular integral operators, and related topics (Bordeaus, 2000). Operator Theory: Advances and Applications, Vol. 129, Basel: GBirkhauser, [21] Douglas R, Misra G, Varughese C. On quotient modules, the case of arbitrary multiplicity, J Funct Anal, 210, No. 1: (2000) [22] Ferguson S, Rochberg R. Higerorder HilbertSchmidt Hankel Forms and tensors of analytic kernels. Math Scand, to appear [23] Ferguson S, Rochberg R. Description of certain quotient Hilbert modules. Preprint [24] Douglas R, Misra G. Some Calculations for Hilbert modules, J Orissa Math Soc, 12 15: 75 85, ( ) [25] Clark D. Restrictions of H p functions in the polydisk, Amer J Math, 110: (1988) [26] Brown L, Douglas R, Fillmore P. Extension of C algebras and Khomology. Ann of Math, 105: (1977) [27] Brown L, Douglas R, Fillmore P. Unitary equivalence modulo the compact operators and extensions of C algebra. Lecture notes in Math, 345, 1973 [28] Zariski O, Samuel P. Commutative algebra, Vol. (I),(II). Princeton: Van Nostrand, 1958/1960 [29] Guo K. Equivalence of Hardy submodules generated by polynomials. J Funct Anal, 178: (2000) [30] Guo K, Wang P. Defect operators and Fredholmness for Toeplitz pairs with inner symbols. J Oper Theory, to appear [31] Douglas R. Banach algebra Techniques in Operator Theory. New York: SpringerVerlag, 1997 [32] Curto R. Fredholm and invertible ntuples of operators. The Deformation problem. Trans AMS, 266: (1981) [33] Arveson W. Subalgebras of C algebras. Acta Math, 123: (1969) [34] Taylor J. The analytic functional calculus for several commuting operators. Acta Math, 125: 1 38 (1970)
TRANSLATION INVARIANCE OF FOCK SPACES
TRANSLATION INVARIANCE OF FOCK SPACES KEHE ZHU ABSTRACT. We show that there is only one Hilbert space of entire functions that is invariant under the action of naturally defined weighted translations.
More informationON THE INDEX OF INVARIANT SUBSPACES IN SPACES OF ANALYTIC FUNCTIONS OF SEVERAL COMPLEX VARIABLES
ON THE INDEX OF INVARIANT SUBSPACES IN SPACES OF ANALYTIC FUNCTIONS OF SEVERAL COMPLEX VARIABLES JIM GLEASON, STEFAN RICHTER, AND CARL SUNDBERG Abstract. Let B d be the open unit ball in C d,d 1, and Hd
More informationWEYL S THEOREM FOR PAIRS OF COMMUTING HYPONORMAL OPERATORS
WEYL S THEOREM FOR PAIRS OF COMMUTING HYPONORMAL OPERATORS SAMEER CHAVAN AND RAÚL CURTO Abstract. Let T be a pair of commuting hyponormal operators satisfying the socalled quasitriangular property dim
More informationLINEAR FRACTIONAL COMPOSITION OPERATORS ON H 2
J Integral Equations and Operator Theory (988, 5 60 LINEAR FRACTIONAL COMPOSITION OPERATORS ON H 2 CARL C COWEN Abstract If ϕ is an analytic function mapping the unit disk D into itself, the composition
More informationESSENTIALLY COMMUTING HANKEL AND TOEPLITZ OPERATORS
ESSENTIALLY COMMUTING HANKEL AND TOEPLITZ OPERATORS KUNYU GUO AND DECHAO ZHENG Abstract. We characterize when a Hankel operator a Toeplitz operator have a compact commutator. Let dσ(w) be the normalized
More informationTOEPLITZ OPERATORS. Toeplitz studied infinite matrices with NWSE diagonals constant. f e C :
TOEPLITZ OPERATORS EFTON PARK 1. Introduction to Toeplitz Operators Otto Toeplitz lived from 18811940 in Goettingen, and it was pretty rough there, so he eventually went to Palestine and eventually contracted
More informationFinitedimensional spaces. C n is the space of ntuples x = (x 1,..., x n ) of complex numbers. It is a Hilbert space with the inner product
Chapter 4 Hilbert Spaces 4.1 Inner Product Spaces Inner Product Space. A complex vector space E is called an inner product space (or a prehilbert space, or a unitary space) if there is a mapping (, )
More information1 Compact and Precompact Subsets of H
Compact Sets and Compact Operators by Francis J. Narcowich November, 2014 Throughout these notes, H denotes a separable Hilbert space. We will use the notation B(H) to denote the set of bounded linear
More informationRIGHT SPECTRUM AND TRACE FORMULA OF SUBNORMAL TUPLE OF OPERATORS OF FINITE TYPE
RIGHT SPECTRUM AND TRACE FORMULA OF SUBNORMAL TUPLE OF OPERATORS OF FINITE TYPE Daoxing Xia Abstract. This paper studies pure subnormal ktuples of operators S = (S 1,, S k ) with finite rank of selfcommutators.
More informationEXISTENCE OF NONSUBNORMAL POLYNOMIALLY HYPONORMAL OPERATORS
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 25, Number 2, October 1991 EXISTENCE OF NONSUBNORMAL POLYNOMIALLY HYPONORMAL OPERATORS RAUL E. CURTO AND MIHAI PUTINAR INTRODUCTION In
More informationDIAGONAL TOEPLITZ OPERATORS ON WEIGHTED BERGMAN SPACES
DIAGONAL TOEPLITZ OPERATORS ON WEIGHTED BERGMAN SPACES TRIEU LE Abstract. In this paper we discuss some algebraic properties of diagonal Toeplitz operators on weighted Bergman spaces of the unit ball in
More informationTHE BERGMAN KERNEL FUNCTION. 1. Introduction
THE BERGMAN KERNEL FUNCTION GAAHAR MISRA Abstract. In this note, we point out that a large family of n n matrix valued kernel functions defined on the unit disc C, which were constructed recently in [9],
More informationPatryk Pagacz. Characterization of strong stability of powerbounded operators. Uniwersytet Jagielloński
Patryk Pagacz Uniwersytet Jagielloński Characterization of strong stability of powerbounded operators Praca semestralna nr 3 (semestr zimowy 2011/12) Opiekun pracy: Jaroslav Zemanek CHARACTERIZATION OF
More informationNormality of adjointable module maps
MATHEMATICAL COMMUNICATIONS 187 Math. Commun. 17(2012), 187 193 Normality of adjointable module maps Kamran Sharifi 1, 1 Department of Mathematics, Shahrood University of Technology, P. O. Box 3619995161316,
More informationADJOINTS, ABSOLUTE VALUES AND POLAR DECOMPOSITIONS
J. OPERATOR THEORY 44(2000), 243 254 c Copyright by Theta, 2000 ADJOINTS, ABSOLUTE VALUES AND POLAR DECOMPOSITIONS DOUGLAS BRIDGES, FRED RICHMAN and PETER SCHUSTER Communicated by William B. Arveson Abstract.
More information1. Partial Fraction Expansion All the polynomials in this note are assumed to be complex polynomials.
Partial Fraction Expansion All the polynomials in this note are assumed to be complex polynomials A rational function / is a quotient of two polynomials P, Q with 0 By Fundamental Theorem of algebra, =
More informationON OPERATORS WHICH ARE POWER SIMILAR TO HYPONORMAL OPERATORS
Jung, S., Ko, E. and Lee, M. Osaka J. Math. 52 (2015), 833 847 ON OPERATORS WHICH ARE POWER SIMILAR TO HYPONORMAL OPERATORS SUNGEUN JUNG, EUNGIL KO and MEEJUNG LEE (Received April 1, 2014) Abstract In
More informationFredholmness of Some Toeplitz Operators
Fredholmness of Some Toeplitz Operators Beyaz Başak Koca Istanbul University, Istanbul, Turkey IWOTA, 2017 Preliminaries Definition (Fredholm operator) Let H be a Hilbert space and let T B(H). T is said
More informationIntroduction to Index Theory. Elmar Schrohe Institut für Analysis
Introduction to Index Theory Elmar Schrohe Institut für Analysis Basics Background In analysis and pde, you want to solve equations. In good cases: Linearize, end up with Au = f, where A L(E, F ) is a
More informationCHAPTER VIII HILBERT SPACES
CHAPTER VIII HILBERT SPACES DEFINITION Let X and Y be two complex vector spaces. A map T : X Y is called a conjugatelinear transformation if it is a reallinear transformation from X into Y, and if T (λx)
More informationWEIGHTED COMPOSITION OPERATORS BETWEEN DIRICHLET SPACES
Acta Mathematica Scientia 20,3B(2):64 65 http://actams.wipm.ac.cn WEIGHTE COMPOSITION OPERATORS BETWEEN IRICHLET SPACES Wang Maofa ( ) School of Mathematics and Statistics, Wuhan University, Wuhan 430072,
More informationOn the vanishing of Tor of the absolute integral closure
On the vanishing of Tor of the absolute integral closure Hans Schoutens Department of Mathematics NYC College of Technology City University of New York NY, NY 11201 (USA) Abstract Let R be an excellent
More informationFormal power series rings, inverse limits, and Iadic completions of rings
Formal power series rings, inverse limits, and Iadic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely
More informationAlgebra Exam Syllabus
Algebra Exam Syllabus The Algebra comprehensive exam covers four broad areas of algebra: (1) Groups; (2) Rings; (3) Modules; and (4) Linear Algebra. These topics are all covered in the first semester graduate
More informationON THE GENERALIZED FUGLEDEPUTNAM THEOREM M. H. M. RASHID, M. S. M. NOORANI AND A. S. SAARI
TAMKANG JOURNAL OF MATHEMATICS Volume 39, Number 3, 239246, Autumn 2008 0pt0pt ON THE GENERALIZED FUGLEDEPUTNAM THEOREM M. H. M. RASHID, M. S. M. NOORANI AND A. S. SAARI Abstract. In this paper, we prove
More informationInvertible Composition Operators: The product of a composition operator with the adjoint of a composition operator.
Invertible Composition Operators: The product of a composition operator with the adjoint of a composition operator. John H. Clifford, Trieu Le and Alan Wiggins Abstract. In this paper, we study the product
More informationLemma 3. Suppose E, F X are closed subspaces with F finite dimensional.
Notes on Fredholm operators David Penneys Let X, Y be Banach spaces. Definition 1. A bounded linear map T B(X, Y ) is called Fredholm if dim(ker T ) < and codim(t X)
More informationTHE DIRAC OPERATOR OF A COMMUTING dtuple. William Arveson Department of Mathematics University of California Berkeley CA 94720, USA
THE DIRAC OPERATOR OF A COMMUTING dtuple William Arveson Department of Mathematics University of California Berkeley CA 94720, USA Abstract. Given a commuting dtuple T = (T 1,..., T d ) of otherwise
More informationOn z ideals in C(X) F. A z a r p a n a h, O. A. S. K a r a m z a d e h and A. R e z a i A l i a b a d (Ahvaz)
F U N D A M E N T A MATHEMATICAE 160 (1999) On z ideals in C(X) by F. A z a r p a n a h, O. A. S. K a r a m z a d e h and A. R e z a i A l i a b a d (Ahvaz) Abstract. An ideal I in a commutative ring
More informationUNCERTAINTY PRINCIPLES FOR THE FOCK SPACE
UNCERTAINTY PRINCIPLES FOR THE FOCK SPACE KEHE ZHU ABSTRACT. We prove several versions of the uncertainty principle for the Fock space F 2 in the complex plane. In particular, for any unit vector f in
More informationarxiv: v1 [math.fa] 9 Dec 2017
COWENDOUGLAS FUNCTION AND ITS APPLICATION ON CHAOS LVLIN LUO arxiv:1712.03348v1 [math.fa] 9 Dec 2017 Abstract. In this paper we define CowenDouglas function introduced by CowenDouglas operator on Hardy
More informationThe Choquet boundary of an operator system
1 The Choquet boundary of an operator system Matthew Kennedy (joint work with Ken Davidson) Carleton University Nov. 9, 2013 Operator systems and completely positive maps Definition An operator system
More informationA COMMENT ON FREE GROUP FACTORS
A COMMENT ON FREE GROUP FACTORS NARUTAKA OZAWA Abstract. Let M be a finite von Neumann algebra acting on the standard Hilbert space L 2 (M). We look at the space of those bounded operators on L 2 (M) that
More informationTHE CLOSED RANGE PROPERTY FOR BANACH SPACE OPERATORS
THE CLOSED RANGE PROPERTY FOR BANACH SPACE OPERATORS THOMAS L. MILLER AND VLADIMIR MÜLLER Abstract. Let T be a bounded operator on a complex Banach space X. If V is an open subset of the complex plane,
More information1 Invariant subspaces
MATH 2040 Linear Algebra II Lecture Notes by Martin Li Lecture 8 Eigenvalues, eigenvectors and invariant subspaces 1 In previous lectures we have studied linear maps T : V W from a vector space V to another
More informationMath Linear Algebra II. 1. Inner Products and Norms
Math 342  Linear Algebra II Notes 1. Inner Products and Norms One knows from a basic introduction to vectors in R n Math 254 at OSU) that the length of a vector x = x 1 x 2... x n ) T R n, denoted x,
More informationarxiv: v1 [math.fa] 13 Dec 2016 CHARACTERIZATION OF TRUNCATED TOEPLITZ OPERATORS BY CONJUGATIONS
arxiv:1612.04406v1 [math.fa] 13 Dec 2016 CHARACTERIZATION OF TRUNCATED TOEPLITZ OPERATORS BY CONJUGATIONS KAMILA KLIŚGARLICKA, BARTOSZ ŁANUCHA AND MAREK PTAK ABSTRACT. Truncated Toeplitz operators are
More information7.3 Singular Homology Groups
184 CHAPTER 7. HOMOLOGY THEORY 7.3 Singular Homology Groups 7.3.1 Cycles, Boundaries and Homology Groups We can define the singular pchains with coefficients in a field K. Furthermore, we can define the
More informationSome Properties of Closed Range Operators
Some Properties of Closed Range Operators J. FarrokhiOstad 1,, M. H. Rezaei gol 2 1 Department of Basic Sciences, Birjand University of Technology, Birjand, Iran. Email: javadfarrokhi90@gmail.com, j.farrokhi@birjandut.ac.ir
More informationExercise Solutions to Functional Analysis
Exercise Solutions to Functional Analysis Note: References refer to M. Schechter, Principles of Functional Analysis Exersize that. Let φ,..., φ n be an orthonormal set in a Hilbert space H. Show n f n
More informationarxiv: v2 [math.oa] 21 Nov 2010
NORMALITY OF ADJOINTABLE MODULE MAPS arxiv:1011.1582v2 [math.oa] 21 Nov 2010 K. SHARIFI Abstract. Normality of bounded and unbounded adjointable operators are discussed. Suppose T is an adjointable operator
More information1 The Projection Theorem
Several Important Theorems by Francis J. Narcowich November, 14 1 The Projection Theorem Let H be a Hilbert space. When V is a finite dimensional subspace of H and f H, we can always find a unique p V
More informationRemarks on FugledePutnam Theorem for Normal Operators Modulo the HilbertSchmidt Class
International Mathematical Forum, Vol. 9, 2014, no. 29, 13891396 HIKARI Ltd, www.mhikari.com http://dx.doi.org/10.12988/imf.2014.47141 Remarks on FugledePutnam Theorem for Normal Operators Modulo the
More informationRolle s Theorem for Polynomials of Degree Four in a Hilbert Space 1
Journal of Mathematical Analysis and Applications 265, 322 33 (2002) doi:0.006/jmaa.200.7708, available online at http://www.idealibrary.com on Rolle s Theorem for Polynomials of Degree Four in a Hilbert
More informationALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA
ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND
More informationGlobal holomorphic functions in several noncommuting variables II
arxiv:706.09973v [math.fa] 29 Jun 207 Global holomorphic functions in several noncommuting variables II Jim Agler U.C. San Diego La Jolla, CA 92093 July 3, 207 John E. McCarthy Washington University St.
More informationTrace Class Operators and Lidskii s Theorem
Trace Class Operators and Lidskii s Theorem Tom Phelan Semester 2 2009 1 Introduction The purpose of this paper is to provide the reader with a selfcontained derivation of the celebrated Lidskii Trace
More informationBoolean InnerProduct Spaces and Boolean Matrices
Boolean InnerProduct Spaces and Boolean Matrices Stan Gudder Department of Mathematics, University of Denver, Denver CO 80208 Frédéric Latrémolière Department of Mathematics, University of Denver, Denver
More informationNoetherian property of infinite EI categories
Noetherian property of infinite EI categories Wee Liang Gan and Liping Li Abstract. It is known that finitely generated FImodules over a field of characteristic 0 are Noetherian. We generalize this result
More informationFunctional Analysis II held by Prof. Dr. Moritz Weber in summer 18
Functional Analysis II held by Prof. Dr. Moritz Weber in summer 18 General information on organisation Tutorials and admission for the final exam To take part in the final exam of this course, 50 % of
More informationTHE S 1 EQUIVARIANT COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES
Horiguchi, T. Osaka J. Math. 52 (2015), 1051 1062 THE S 1 EQUIVARIANT COHOMOLOGY RINGS OF (n k, k) SPRINGER VARIETIES TATSUYA HORIGUCHI (Received January 6, 2014, revised July 14, 2014) Abstract The main
More informationAn introduction to some aspects of functional analysis
An introduction to some aspects of functional analysis Stephen Semmes Rice University Abstract These informal notes deal with some very basic objects in functional analysis, including norms and seminorms
More informationOn the centralizer of a regular, semisimple, stable conjugacy class. Benedict H. Gross
On the centralizer of a regular, semisimple, stable conjugacy class Benedict H. Gross Let k be a field, and let G be a semisimple, simplyconnected algebraic group, which is quasisplit over k. The theory
More informationHonors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35
Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime
More informationTHE REGULAR ELEMENT PROPERTY
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 7, July 1998, Pages 2123 2129 S 00029939(98)042579 THE REGULAR ELEMENT PROPERTY FRED RICHMAN (Communicated by Wolmer V. Vasconcelos)
More informationEXTREMAL MULTIPLIERS OF THE DRURYARVESON SPACE
EXTREMAL MULTIPLIERS OF THE DRURYARVESON SPACE MT JURY AND RTW MARTIN Abstract We give a new characterization of the socalled quasiextreme multipliers of the DruryArveson space Hd 2, and show that
More informationThe Polynomial Numerical Index of L p (µ)
KYUNGPOOK Math. J. 53(2013), 117124 http://dx.doi.org/10.5666/kmj.2013.53.1.117 The Polynomial Numerical Index of L p (µ) Sung Guen Kim Department of Mathematics, Kyungpook National University, Daegu
More informationBinomial Exercises A = 1 1 and 1
Lecture I. Toric ideals. Exhibit a point configuration A whose affine semigroup NA does not consist of the intersection of the lattice ZA spanned by the columns of A with the real cone generated by A.
More informationMath 123 Homework Assignment #2 Due Monday, April 21, 2008
Math 123 Homework Assignment #2 Due Monday, April 21, 2008 Part I: 1. Suppose that A is a C algebra. (a) Suppose that e A satisfies xe = x for all x A. Show that e = e and that e = 1. Conclude that e
More informationMath 121 Homework 5: Notes on Selected Problems
Math 121 Homework 5: Notes on Selected Problems 12.1.2. Let M be a module over the integral domain R. (a) Assume that M has rank n and that x 1,..., x n is any maximal set of linearly independent elements
More informationSREGULARITY AND THE CORONA FACTORIZATION PROPERTY
MATH. SCAND. 99 (2006), 204 216 SREGULARITY AND THE CORONA FACTORIZATION PROPERTY D. KUCEROVSKY and P. W. NG Abstract Stability is an important and fundamental property of C algebras. Given a short exact
More informationA linear algebra proof of the fundamental theorem of algebra
A linear algebra proof of the fundamental theorem of algebra Andrés E. Caicedo May 18, 2010 Abstract We present a recent proof due to Harm Derksen, that any linear operator in a complex finite dimensional
More informationA short proof of Klyachko s theorem about rational algebraic tori
A short proof of Klyachko s theorem about rational algebraic tori Mathieu Florence Abstract In this paper, we give another proof of a theorem by Klyachko ([?]), which asserts that Zariski s conjecture
More informationTHEORY OF BERGMAN SPACES IN THE UNIT BALL OF C n. 1. INTRODUCTION Throughout the paper we fix a positive integer n and let
THEORY OF BERGMAN SPACES IN THE UNIT BALL OF C n RUHAN ZHAO AND KEHE ZHU ABSTRACT. There has been a great deal of work done in recent years on weighted Bergman spaces A p α on the unit ball of C n, where
More information6 Lecture 6: More constructions with Huber rings
6 Lecture 6: More constructions with Huber rings 6.1 Introduction Recall from Definition 5.2.4 that a Huber ring is a commutative topological ring A equipped with an open subring A 0, such that the subspace
More informationMODULES OVER A PID. induces an isomorphism
MODULES OVER A PID A module over a PID is an abelian group that also carries multiplication by a particularly convenient ring of scalars. Indeed, when the scalar ring is the integers, the module is precisely
More informationRudin orthogonality problem on the Bergman space
Journal of Functional Analysis 261 211) 51 68 www.elsevier.com/locate/jfa Rudin orthogonality problem on the Bergman space Kunyu Guo a, echao Zheng b,c, a School of Mathematical Sciences, Fudan University,
More informationSTRONG SINGULARITY OF SINGULAR MASAS IN II 1 FACTORS
Illinois Journal of Mathematics Volume 5, Number 4, Winter 2007, Pages 077 084 S 0092082 STRONG SINGULARITY OF SINGULAR MASAS IN II FACTORS ALLAN M. SINCLAIR, ROGER R. SMITH, STUART A. WHITE, AND ALAN
More informationC Algebra B H (I) Consisting of Bessel Sequences in a Hilbert Space
Journal of Mathematical Research with Applications Mar., 2015, Vol. 35, No. 2, pp. 191 199 DOI:10.3770/j.issn:20952651.2015.02.009 Http://jmre.dlut.edu.cn C Algebra B H (I) Consisting of Bessel Sequences
More informationThe Geometric Approach for Computing the Joint Spectral Radius
Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 2005 Seville, Spain, December 1215, 2005 TuB08.2 The Geometric Approach for Computing the Joint Spectral
More informationHyponormal and Subnormal Toeplitz Operators
Hyponormal and Subnormal Toeplitz Operators Carl C. Cowen This paper is my view of the past, present, and future of Problem 5 of Halmos s 1970 lectures Ten Problems in Hilbert Space [12] (see also [13]):
More informationON BOUNDEDNESS OF MAXIMAL FUNCTIONS IN SOBOLEV SPACES
Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 29, 2004, 167 176 ON BOUNDEDNESS OF MAXIMAL FUNCTIONS IN SOBOLEV SPACES Piotr Haj lasz and Jani Onninen Warsaw University, Institute of Mathematics
More informationChapter 8. Padic numbers. 8.1 Absolute values
Chapter 8 Padic numbers Literature: N. Koblitz, padic Numbers, padic Analysis, and ZetaFunctions, 2nd edition, Graduate Texts in Mathematics 58, Springer Verlag 1984, corrected 2nd printing 1996, Chap.
More informationLecture 2. (1) Every P L A (M) has a maximal element, (2) Every ascending chain of submodules stabilizes (ACC).
Lecture 2 1. Noetherian and Artinian rings and modules Let A be a commutative ring with identity, A M a module, and φ : M N an Alinear map. Then ker φ = {m M : φ(m) = 0} is a submodule of M and im φ is
More informationDefinitions, Theorems and Exercises. Abstract Algebra Math 332. Ethan D. Bloch
Definitions, Theorems and Exercises Abstract Algebra Math 332 Ethan D. Bloch December 26, 2013 ii Contents 1 Binary Operations 3 1.1 Binary Operations............................... 4 1.2 Isomorphic Binary
More informationRing Theory Problems. A σ
Ring Theory Problems 1. Given the commutative diagram α A σ B β A σ B show that α: ker σ ker σ and that β : coker σ coker σ. Here coker σ = B/σ(A). 2. Let K be a field, let V be an infinite dimensional
More informationOn Compact JustNonLie Groups
On Compact JustNonLie Groups FRANCESCO RUSSO Mathematics Department of Naples, Naples, Italy Email:francesco.russo@dma.unina.it Abstract. A compact group is called a compact JustNonLie group or a
More informationLinear Algebra and Dirac Notation, Pt. 2
Linear Algebra and Dirac Notation, Pt. 2 PHYS 500  Southern Illinois University February 1, 2017 PHYS 500  Southern Illinois University Linear Algebra and Dirac Notation, Pt. 2 February 1, 2017 1 / 14
More informationTRANSLATIONINVARIANT FUNCTION ALGEBRAS ON COMPACT GROUPS
PACIFIC JOURNAL OF MATHEMATICS Vol. 15, No. 3, 1965 TRANSLATIONINVARIANT FUNCTION ALGEBRAS ON COMPACT GROUPS JOSEPH A. WOLF Let X be a compact group. $(X) denotes the Banach algebra (point multiplication,
More informationHardy spaces of Dirichlet series and function theory on polydiscs
Hardy spaces of Dirichlet series and function theory on polydiscs Kristian Seip Norwegian University of Science and Technology (NTNU) Steinkjer, September 11 12, 2009 Summary Lecture One Theorem (H. Bohr)
More informationMath 25a Practice Final #1 Solutions
Math 25a Practice Final #1 Solutions Problem 1. Suppose U and W are subspaces of V such that V = U W. Suppose also that u 1,..., u m is a basis of U and w 1,..., w n is a basis of W. Prove that is a basis
More informationarxiv: v4 [math.rt] 14 Jun 2016
TWO HOMOLOGICAL PROOFS OF THE NOETHERIANITY OF FI G LIPING LI arxiv:163.4552v4 [math.rt] 14 Jun 216 Abstract. We give two homological proofs of the Noetherianity of the category F I, a fundamental result
More informationC.6 Adjoints for Operators on Hilbert Spaces
C.6 Adjoints for Operators on Hilbert Spaces 317 Additional Problems C.11. Let E R be measurable. Given 1 p and a measurable weight function w: E (0, ), the weighted L p space L p s (R) consists of all
More informationOrthogonal Pure States in Operator Theory
Orthogonal Pure States in Operator Theory arxiv:math/0211202v2 [math.oa] 5 Jun 2003 Jan Hamhalter Abstract: We summarize and deepen existing results on systems of orthogonal pure states in the context
More informationAnother proof of the global F regularity of Schubert varieties
Another proof of the global F regularity of Schubert varieties Mitsuyasu Hashimoto Abstract Recently, Lauritzen, RabenPedersen and Thomsen proved that Schubert varieties are globally F regular. We give
More informationHASSEMINKOWSKI THEOREM
HASSEMINKOWSKI THEOREM KIM, SUNGJIN 1. Introduction In rough terms, a localglobal principle is a statement that asserts that a certain property is true globally if and only if it is true everywhere locally.
More informationOn some properties of elementary derivations in dimension six
Journal of Pure and Applied Algebra 56 (200) 69 79 www.elsevier.com/locate/jpaa On some properties of elementary derivations in dimension six Joseph Khoury Department of Mathematics, University of Ottawa,
More informationDimension Theory. Chapter The Calculus of Finite Differences Lemma Lemma
Chapter 5 Dimension Theory The geometric notion of the dimension of an affine algebraic variety V is closely related to algebraic properties of the coordinate ring of the variety, that is, the ring of
More informationSome Inequalities for Commutators of Bounded Linear Operators in Hilbert Spaces
Some Inequalities for Commutators of Bounded Linear Operators in Hilbert Spaces S.S. Dragomir Abstract. Some new inequalities for commutators that complement and in some instances improve recent results
More informationSpectral Measures, the Spectral Theorem, and Ergodic Theory
Spectral Measures, the Spectral Theorem, and Ergodic Theory Sam Ziegler The spectral theorem for unitary operators The presentation given here largely follows [4]. will refer to the unit circle throughout.
More informationMultiplier Operator Algebras and Applications
Multiplier Operator Algebras and Applications David P. Blecher* Department of Mathematics, University of Houston, Houston, TX 77204. Vrej Zarikian Department of Mathematics, University of Texas, Austin,
More informationMath 210C. The representation ring
Math 210C. The representation ring 1. Introduction Let G be a nontrivial connected compact Lie group that is semisimple and simply connected (e.g., SU(n) for n 2, Sp(n) for n 1, or Spin(n) for n 3). Let
More informationSERRE FINITENESS AND SERRE VANISHING FOR NONCOMMUTATIVE P 1 BUNDLES ADAM NYMAN
SERRE FINITENESS AND SERRE VANISHING FOR NONCOMMUTATIVE P 1 BUNDLES ADAM NYMAN Abstract. Suppose X is a smooth projective scheme of finite type over a field K, E is a locally free O X bimodule of rank
More informationIr O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )
Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O
More informationarxiv: v1 [math.ca] 1 Jul 2013
Carleson measures on planar sets Zhijian Qiu Department of mathematics, Southwestern University of Finance and Economics arxiv:1307.0424v1 [math.ca] 1 Jul 2013 Abstract In this paper, we investigate what
More informationQuasinormalty and subscalarity of class pwa(s, t) operators
Functional Analysis, Approximation Computation 9 1 17, 61 68 Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://wwwpmfniacrs/faac Quasinormalty subscalarity of
More informationON THE LINEAR SPAN OF THE PROJECTIONS IN CERTAIN SIMPLE C ALGEBRAS
ON THE LINEAR SPAN OF THE PROJECTIONS IN CERTAIN SIMPLE C ALGEBRAS L.W. MARCOUX 1 Abstract. In this paper we show that if a C algebra A admits a certain 3 3 matrix decomposition, then every commutator
More informationCOMPOSITION OPERATORS INDUCED BY SYMBOLS DEFINED ON A POLYDISK
COMPOSITION OPERATORS INDUCED BY SYMBOLS DEFINED ON A POLYDISK MICHAEL STESSIN AND KEHE ZHU* ABSTRACT. Suppose ϕ is a holomorphic mapping from the polydisk D m into the polydisk D n, or from the polydisk
More informationTopics in linear algebra
Chapter 6 Topics in linear algebra 6.1 Change of basis I want to remind you of one of the basic ideas in linear algebra: change of basis. Let F be a field, V and W be finite dimensional vector spaces over
More informationTIGHT CLOSURE IN NON EQUIDIMENSIONAL RINGS ANURAG K. SINGH
TIGHT CLOSURE IN NON EQUIDIMENSIONAL RINGS ANURAG K. SINGH 1. Introduction Throughout our discussion, all rings are commutative, Noetherian and have an identity element. The notion of the tight closure
More information