WHICH 2-HYPONORMAL 2-VARIABLE WEIGHTED SHIFTS ARE SUBNORMAL?
|
|
- Karen Morgan
- 5 years ago
- Views:
Transcription
1 WHICH -HYPONORMAL -VARIABLE WEIGHTED SHIFTS ARE SUBNORMAL? RAÚL E. CURTO, SANG HOON LEE, AND JASANG YOON Astract. It is well known that a -hyponormal unilateral weighted shift with two equal weights must e flat, and therefore sunormal. By contrast, a -hyponormal -variale weighted shift which is oth horizontally flat and vertically flat need not e sunormal. In this paper we identify a large class S of flat -variale weighted shifts for which -hyponormality is equivalent to sunormality. OnemeasureofthesizeofS is given y the fact that within S there are hyponormal shifts which are not sunormal. 1. Statement of the Main Results The Lifting Prolem for Commuting Sunormals (LPCS) asks for necessary and sufficient conditions for a pair of sunormal operators on Hilert space to admit commuting normal extensions. It is well known that the commutativity of the pair is necessary ut not sufficient ([1], [18], [19], [0]), and it has recently een shown that the joint hyponormality of the pair is necessary ut not sufficient [13]. Our previous work ([9], [10], [13], [14], [15], [4], [5]) has revealed that the nontrivial aspects of LPCS are est detected within the class H 0 of commuting pairs of sunormal operators; we thus focus our attention on this class. The class of sunormal pairs on Hilert space will e denoted y H, and for an integer k 1theclassofk-hyponormal pairs in H 0 will e denoted y H k. Clearly, H H k H H 1 H 0 ; the main results in [13] and [9] show that these inclusions are all proper. It is then natural to look for suclasses of H 0 on which sunormality and k-hyponormality agree, that is, classes on which sunormality can e detected with a matricial test. In this paper we identify a large class S H 0 on which -hyponormality and sunormality agree, that is, S H = S H. Concretely, S consists of all -variale weighted shifts T (T 1,T ) H 0 such that α (k1,0) = α (k1 +1,0) and β (0,k ) = β (0,k +1) for some k 1 1andk 1, where α and β denote the weight sequences of T 1 and T, respectively. One measure of the size of S is given y the fact that hyponormality and sunormality do not agree on S, thatiss H S H 1. Thus, S consists of nontrivial shifts for which -hyponormality and sunormality are equivalent, ut for which hyponormality and -hyponormality are different; that is, S is small enough to ensure that - hyponormality implies sunormality, ut large enough to separate hyponormality from sunormality. Each hyponormal shift in S is flat, thatis,α (k1,k ) = α (1,1) and β (k1,k ) = β (1,1) for all k 1,k 1. As a result, each hyponormal shift in S elongs to the class TCof shifts whose core is of tensor form (cf. Definition.8); TC is a class that we have studied in detail in [11]. Previously, in [10] we had proved that there exist -variale weighted shifts in TC which are hyponormal ut not sunormal. By contrast, the -hyponormal shifts in S are automatically sunormal. We prove our main results y comining the 15-point Test for -hyponormality [9] with the Sunormal Backward Extension Criterion [13] and Smul jan s Test for positivity of operator matrices 000 Mathematics Suject Classification. Primary 47B0, 47B37, 47A13; Secondary 44A60, 47-04, 47A0, 8A50. Key words and phrases. Jointly hyponormal pairs, -hyponormal pairs, sunormal pairs, -variale weighted shifts, flatness. Research partially supported y NSF Grants DMS and DMS
2 [1]. As a first step, we prove a propagation result for -hyponormal -variale weighted shifts (Theorem.10). We then seek conditions to guarantee the sunormality of T (T 1,T ) H. Our main results follow; first, we need a definition. Definition 1.1. S := {T (T 1,T ) H 0 : α (k1,0) = α (k1 +1,0) and β (0,k ) = β (0,k +1) for some k 1 1 and k 1}. (Here α and β denote the weight sequences of T 1 and T, respectively; cf. Section ). Theorem 1.. S H = S H. The generic form of the weight diagram of a hyponormal -variale weighted shift in S is given in Figure (ii). Using that notation, we can sharpen Theorem 1. as follows. Theorem 1.3. Let T (T 1,T ) S, with weight diagram given y Figure (ii), and assume that T is -hyponormal. Then T is sunormal, with Berger measure given as Theorem 1.4. S H S H 1. µ = 1 {[ (1 x ) y (1 a )]δ (0,0) +y (1 a )δ (0, ) +( x a y )δ (1,0) + a y δ (1, )}. Remark 1.5. Hyponormality alone does not imply flatness. While it is true that in the presence of hyponormality the Six-point Test creates L-shaped propagation (i.e., α k+ε1 = α k = α k = α k+ε and β k = β k+ε1 ; cf [15, Proof of Theorem 3.3]), without horizontal propagation (as guaranteed y the quadratic hyponormality of T 1 )thisl-propagation does not result in vertical propagation, needed to eventually lead to flatness. The same phenomenon arises in one variale, where hyponormality is a very soft condition (α k α k+1 for all k 0), while -hyponormality is quite rigid. The work in [6] (extending the ideas in []) revealed that, for unilateral weighted shifts with two equal weights, -hyponormality and sunormality are identical notions. In two variales, however, the analogous result does not hold, as the present work shows. Acknowledgment. Some of the results in this paper were motivated y calculations done with the software tool Mathematica [3].. Notation and Preliminaries For α {α n } n=0 a ounded sequence of positive real numers (called weights), let W α shift(α 0,α 1, ):l (Z + ) l (Z + ) e the associated unilateral weighted shift, defined y W α e n := α n e n+1 (all n 0), where {e n } n=0 is the canonical orthonormal asis in l (Z + ). Two special cases of significant interest are U + := shift(1, 1, ) (the (unweighted) unilateral shift) and S a := shift(a, 1, 1, )(0<a<1). The moments of α are given as { 1 if k =0 γ k γ k (α) := α 0 α k 1 if k>0 It is easy to see that W α is never normal, and that it is hyponormal if and only if α 0 α 1. Similarly, consider doule-indexed positive ounded sequences α k,β k l (Z +), k (k 1,k ) Z + := Z + Z + and let l (Z + ) e the Hilert space of square-summale complex sequences indexed y Z +. We define the -variale weighted shift T (T 1,T )y T 1 e k := α k e k+ε1 T e k := β k e k+ε, }.
3 where ε 1 := (1, 0) and ε := (0, 1). Clearly, T 1 T = T T 1 β k+ε1 α k = α k+ε β k (all k Z + ). (.1) Trivially, a pair of unilateral weighted shifts W α and W β gives rise to a -variale weighted shift T (T 1,T ), if we let α (k1,k ) := α k1 and β (k1,k ) := β k (all k 1,k Z + ). In this case, T is sunormal (resp. hyponormal) if and only if so are T 1 and T ; in fact, under the canonical identification of l (Z + )withl (Z + ) l (Z + ), we have T 1 = I Wα and T = Wβ I,soTis also douly commuting. We now recall a well known characterization of sunormality for single variale weighted shifts, due to C. Berger (cf. [4, III.8.16]), and independently estalished y Gellar and Wallen [16]): W α is sunormal if and only if there exists a proaility measure ξ supported in [0, W α ], with W α suppξ, andsuchthatγ k (α) :=α 0 α k 1 = s k dξ(s) (k 1). We also recall the notion of moment of order k for a pair (α, β) satisfying (.1). Given k Z +, the moment of (α, β) oforderk is 1 if k =0 α (0,0) γ k γ k (α, β) :=... α (k 1 1,0) if k 1 1andk =0 β(0,0)... β (0,k 1) if k 1 =0andk 1. (.) α (0,0)... α (k 1 1,0) β (k 1,0)... β (k 1,k 1) if k 1 1andk 1 We remark that, due to the commutativity condition (.1), γ k can e computed using any nondecreasing path from (0, 0) to (k 1,k ). Moreover, T is sunormal if and only if there is a regular Borel proaility measure µ defined on the -dimensional rectangle R =[0,a 1 ] [0,a ](a i := T i )such that γ k = s k 1 t k dµ(s, t) (allk Z + )[17]. (.3) R Let H e a complex Hilert space and let B(H) denote the algera of ounded linear operators on H. For S, T B(H) let[s, T ] := ST TS. We say that an n-tuple T =(T 1,,T n )of operators on H is (jointly) hyponormal if the operator matrix [T, T] :=([Tj,T i]) n i,j=1 is positive on the direct sum of n copies of H (cf. [], [1]). The n-tuple T is said to e normal if T is commuting and each T i is normal, and T is sunormal if T is the restriction of a normal n-tuple to a common invariant suspace. Clearly, normal = sunormal = hyponormal. Moreover, the restriction of a hyponormal n-tuple to an invariant suspace is again hyponormal. The Bram-Halmos criterion states that an operator T B(H) is sunormal if and only if the k -tuple (T,T,,T k )is hyponormal for all k 1; we say that T is k-hyponormal when the latter condition holds. On the other hand, for a commuting pair T (T 1,T ) of operators on Hilert space we have Definition.1. (cf. [9]) A commuting pair T (T 1,T ) is called k-hyponormal if T(k) := (T 1,T,T1,T T 1,T,,Tk 1,T T1 k 1,,T k ) is hyponormal, or equivalently ([(T q T p 1 ),T n T1 m ])1 m+n k 0. 1 p+q k Clearly, sunormal = (k +1)-hyponormal = k-hyponormal for every k 1, and of course 1- hyponormality agrees with the usual definition of joint hyponormality (as aove). In [9] we otained the following multivariale version of the Bram-Halmos criterion for sunormality, which provided an astract answer to the LPCS, y showing that no matter how k-hyponormal the pair T might e, it may still fail to e sunormal. Theorem.. ([9, Theorem.3]) Let T (T 1,T ) e a commuting pair of sunormal operators on ahilertspace H. The following statements are equivalent. 3
4 (i) T is sunormal. (ii) T is k-hyponormal for all k Z +. In the single variale case, there are useful criteria for k-hyponormality ([6], [8]); for -variale weighted shifts, a simple criterion for joint hyponormality was given in([5]). The following characterization of k-hyponormality for -variale weighted shifts was given in [9, Theorem.4]. Theorem.3. Let T (T 1,T ) e a -variale weighted shift with weight sequences α {α k } and β {β k }. The following statements are equivalent. (a) T is k-hyponormal. () M u (k) :=(γ u+(m,n)+(p,q) )0 m+n k 0 for all u Z +. (For a sunormal pair T, thematrix 0 p+q k M u (k) is the truncation of the moment matrix associated to the Berger measure of T.) The following special cases of Theorem.3 will e essential for our work. Lemma.4. ([5])(Six-point Test; cf. Figure 1(i)) with weight sequences α and β. Then Let T (T 1,T ) e a -variale weighted shift, [T, T] 0 M k (1) 0(all k Z +) ( α k+ε 1 α k α k+ε β k+ε1 α k β k α k+ε β k+ε1 α k β k βk+ε βk ) 0 (all k Z +). (k 1,k +4) β k1,k +3 (k 1,k +3) α k1,k +3 β k1,k + β k1 +1,k + (k 1,k +) (k 1,k +) α k1,k + α k1 +1,k + β k1,k +1 β k1,k +1 β k1 +1,k +1 β k1 +,k +1 (k 1,k +1) α k1,k +1 (k 1 +1,k +1) α k1,k +1 α k1 +1,k +1 α k1 +,k +1 β k1,k β k1 +1,k β k1,k β k1 +1,k β k1 +,k β k1 +3,k α k1,k α k1 +1,k (k 1,k ) (k 1 +1,k ) (k 1 +,k ) (i) α k1,k α k1 +1,k α k1 +,k α k1 +3,k (k 1,k ) (k 1 +1,k ) (k 1 +,k ) (k 1 +3,k ) (k 1 +4,k ) (ii) Figure 1. Weight diagrams used in the Six-point Test and 15-point Test, respectively. 4
5 Lemma.5. ([9]) (15-point Test; cf. Figure 1(ii)) If T (T 1,T ) is -variale weighted shift with weight sequence α {α k } and β {β k }, then T is -hyponormal if and only if γ k1,k γ k1 +1,k γ k1,k +1 γ k1 +,k γ k1 +1,k +1 γ k1,k + γ k1 +1,k γ k1 +,k γ k1 +1,k +1 γ k1 +3,k γ k1 +,k +1 γ k1 +1,k + M (k1,k )() γ k1,k +1 γ k1 +1,k +1 γ k1,k + γ k1 +,k +1 γ k1 +1,k + γ k1,k +3 γ k1 +,k γ k1 +3,k γ k1 +,k +1 γ k1 +4,k γ k1 +3,k +1 γ k1 +,k 0, + γ k1 +1,k +1 γ k1 +,k +1 γ k1 +1,k + γ k1 +3,k +1 γ k1 +,k + γ k1 +1,k +3 γ k1,k + γ k1 +1,k + γ k1,k +3 γ k1 +,k + γ k1 +1,k +3 γ k1,k +4 for all k Z +. We now recall the following notation and terminology from [13]: (i) given a proaility measure µ on X Y R + R +,with 1 t L1 (µ), the extremal measure µ ext 1 (which is also a proaility measure) on X Y is given y dµ ext (s, t) :=(1 δ 0 (t)) dµ(s, t); t 1 t L 1 (µ) and (ii) given a measure µ on X Y, the marginal measure µ X is given y µ X := µ π 1 X, where π X : X Y X is the canonical projection onto X. Thus, µ X (E) =µ(e Y ), for every E X. We now list some results which are needed in the proof of Theorem 3.1. ( ) A B Lemma.6. (cf. [1], [7, Proposition.]) Let M e a operator matrix, where B C A and C are square matrices and B is a rectangular matrix. Then C 0 M 0 there exists W such that B = CW A W CW. For Lemma.7, Definition.8 and Theorem 3.1 the following two suspaces of l (Z + ) will e needed: M := {e k : k 1} and N := {e k : k 1 1}. Lemma.7. ([13]) (Sunormal ackward extension of a -variale weighted shift) Consider the -variale weighted shift whose weight diagram is given in Figure (i), and let T M denote the restriction of T to M. Assume that T M is sunormal with associated measure µ M, and that W 0 := shift(α 00,α 10, ) is sunormal with associated measure ν. Then T is sunormal if and only if (i) 1 t L1 (µ M ); (ii) β00 ( 1 t L 1 (µ M ) ) 1 ; (iii) β00 1 t L 1 (µ M ) (µ M) X ext ν. Moreover, if β00 1 t L 1 (µ M ) =1, then (µ M) X ext = ν. In the case when T is sunormal, Berger measure µ of T is given y dµ(s, t) =β00 1 t d(µ M ) ext (s, t)+(dν(s) β 1 00 L 1 (µ M ) t d(µ M ) X ext (s))dδ 0(t). L 1 (µ M ) J. Stampfli showed in [] that for sunormal weighted shifts W α a propagation phenomenon occurs that forces the flatness of W α whenever two equal weights are present. Later, R.E. Curto proved in [6] that a hyponormal weighted shift with three equal weights cannot e quadratically hyponormal without eing flat: If W α is quadratically hyponormal and α n = α n+1 = α n+ for some n 0, then α 1 = α = α 3 =. Y. Choi [3] improved this result, that is, if W α e quadratically hyponormal and α n = α n+1 for some n 1, then W α is flat, thatis,w α shift(α 0,α 1,α 1, ). In Theorem.10 we show that similar propagation phenomena occur for -hyponormal -variale weighted shifts T. In two variales, the flatness of T is captured y the so-called core of T, c(t) (cf. Definition.8 elow), as follows: T is flat when c(t) is a -variale weighted shift of tensor form. 5
6 Definition.8. (i) The core of a -variale weighted shift T is the restriction of T to M N,in symols, c(t) :=T M N. (ii) A -variale weighted shift T is said to e of tensor form if T = (I W α,w β I). When T is sunormal, this is equivalent to requiring that the Berger measure e a Cartesian product ξ η. (iii) The class of all -variale weighted shift T H 0 whose core is of tensor form will e denoted y TC, that is, TC:= {T H 0 : c(t) is of tensor form}. We now recall that a -variale weighted shift T is said to e horizontally flat when α (k1,k ) = α (1,1) for all k 1,k 1; and we call T vertically flat when β (k1,k ) = β (1,1) for all k 1,k 1. We say T is flat if T is horizontally and vertically flat, and that T is symmetrically flat if T is flat and α (1,1) = β (1,1). The next result shows the extent to which propagation holds in the presence of (joint) hyponormality. Proposition.9. ([15]) LetT e a commuting hyponormal -variale weighted shift whose weight diagram is given in Figure (i). Then (i) If α k+ε1 = α k for some k, thenα k+ε = α k and β k+ε1 = β k ; (ii) If T 1 is quadratically hyponormal, if T is sunormal, and if α (k1,k )+ε 1 = α (k1,k ) for some k 1,k 0, thent is horizontally flat. On the other hand, under the assumption of -hyponormality (and without assuming the sunormality of either T 1 or T ), we can prove that T is flat whenever α k+ε1 = α k and β m+ε = β m for some k and m. We do this using the 15-point Test (Lemma.5). Theorem.10. Let T e a -hyponormal -variale weighted shift. If α (k1,k )+ε 1 = α (k1,k ) for some k 1,k 1, thent is horizontally flat. If instead, β (k1,k )+ε = β (k1,k ) for some k 1,k 1, then T is vertically flat. Proof. Without loss of generality, assume α (k1,k )+ε 1 = α (k1,k ) = 1. Then y the aove mentioned result of Y. Choi and Proposition.9(i), we can see that T 1 is of tensor form when restricted to the suspace {e m : m 1 1andm k }. If k = 1 we are done, so assume k and consider the matrix γ k1,k γ k1 +1,k γ k1,k 1 γ k1 +,k γ k1 +1,k 1 γ k1,k γ k1 +1,k γ k1 +,k γ k1 +1,k 1 γ k1 +3,k γ k1 +,k 1 γ k1 +1,k M (k1,k )() = γ k1,k 1 γ k1 +1,k 1 γ k1,k γ k1 +,k 1 γ k1 +1,k γ k1,k +1 γ k1 +,k γ k1 +3,k γ k1 +,k 1 γ k1 +4,k γ k1 +3,k 1 γ k1 +,k, γ k1 +1,k 1 γ k1 +,k 1 γ k1 +1,k γ k1 +3,k 1 γ k1 +,k γ k1 +1,k +1 γ k1,k γ k1 +1,k γ k1,k +1 γ k1 +,k γ k1 +1,k +1 γ k1,k + which we know is positive semidefinite since T is -hyponormal (Lemma.5). It suffices to prove that α k ε = 1. Let us focus on the principal sumatrix M 0 determined y rows and columns 1, 3 and 5. From (.) we have 1 βk ε βk ε α k ε M = βk ε βk ε βk ε βk ε βk ε 0. (.4) βk ε α k ε βk ε βk ε βk ε βk ε For notational convenience, let a := βk ε, := βk ε and c := α k ε. Thus, M 1 a ac ( a a a a a a a 0 N := ) c a a ac a a c a a c 0. 6
7 If a = then we must necessarily have c =1,thatis,α k ε = 1, as desired. Assume, therefore, that a<. It follows that N 0 det N 0 (a a )(a a c ) (a a c) 0 a 3 (c 1) 0 c =1, that is, α k ε = 1, as desired. 3. Proofs of the Main Results We are now ready to prove our main results, which we restate for the reader s convenience. Theorem 3.1. S H = S H. Proof. By Proposition.9 and Theorem.10, we can assume, without loss of generality, that α (k1,k ) = α (1,0) =1(allk 1 1,k 0) and β (k1,k ) = β (0,1) 1(allk 1 0,k 1). For notational convenience, we let a := α (0,1) < 1, := β (0,1) 1, x := α (0,0) < 1, and y := β (0,0) < 1. We summarize this information in Figure (ii). (0, 3)... (0, 3)... β 0 γ1,3 γ 1, γ,3 γ, T (0, ) α 0 α 1 α T (0, ) a 1 1 β 01 γ1, γ 1,1 γ, γ,1 (0, 1) α 01 α 11 α 1 (0, 1) a 1 1 β 00 γ1,1 γ 1,0 γ,1 γ,0 y ay x ay x α 00 α 10 α 0 (0, 0) (1, 0) (, 0) (3, 0) x 1 1 (0, 0) (1, 0) (, 0) (3, 0) T 1 (i) T 1 (ii) Figure. Weight diagrams of the -variale weighted shifts in Lemma.7 and Theorem 3.1, respectively. Let C k := {(a,, x, y) :T H k } (k =1,, ). Clearly, C C C 1. We first descrie concretely the set C in terms of necessary and sufficient conditions on the four parameters a,, x and y that guarantee the -hyponormality of the pair T. We then estalish that C C. Finally, in Theorem 3.3 we will show that C C 1. It is straightforward to verify that T M = (I shift(a, 1, 1, ),U+ I) andt N = (I U +,shift( ay x,,, ) I); thus, T M and T N are sunormal. As a consequence, to descrie C 7
8 we only need to apply the 15-point Test (Lemma.5) at k =(0, 0), that is, we need to guarantee that M (0,0) () 0. We thus have: T H M (0,0) () 0. Now, since the moments γ k (k Z +) associated with T are 1 if k 1 =0andk =0 x if k 1 1andk =0 γ k = y (k 1) if k 1 =0andk 1 a y (k 1) if k 1 1andk 1, it follows that 1 x y x a y y x x a y x a y a y M (0,0) () y a y y a y a y 4 y x x a y x a y a y a y a y a y a y a y a 4 y y a y 4 y a y a 4 y 6 y M := 1 x y a y x x a y a y y a y y a y a y a y a y a y 0 0 (since the sixth row is a multiple of the third row, and the second and fourth rows are identical). If we now interchange the second and third rows and columns, we see that the positivity of M is determined y the positivity of ( ) ( 1 y x a y ) y y a y a y ( x a y a y a y ) ( x a y a y a y ) ( A := Thus, from Lemma.6 (with C = B and W = I), we have M 0 A B 0. Now oserve that B 0 ay x and ( 1 x y A B = (1 a ) y (1 a ) y (1 a ) B ). ) B 0. B Since the (1, 1)-entry of A B is always positive, the positivity of A B is completely determined y its determinant; that is, A B 0 det(a B) 0 y (1 a ) (1 x ). It follows that T H ay x and y (1 a ) (1 x ). (3.1) We thus see that C = {(a,, x, y) :0<x<1, 0 <y<1, ay x, andy (1 a ) (1 x )}. (3.) 8
9 We will now prove that C C. Let (a,, x, y) C. Let z := ay x. Case 1. If z =, thent N (I U +,U + I), and a straightforward application of Lemma.7 in the s direction shows that T is sunormal if and only if x y if and only if a 1(since x = ay), which is true. Thus, (a,, x, y) C. Case. Assume now that z<1. Since T N (I U +,S z/ I) is sunormal with Berger measure µ N δ 1 [(1 z )δ 0 + z δ ], we can think of T as a ackward extension of T N (in the s direction) and apply Lemma.7. Note that 1 s L 1 (µ N ) =1,sod(µ N ) ext (s, t) (1 δ 0 (s)) 1 s dµ N (s, t) = dδ 1 (s)[(1 z )dδ 0 (t)+ z dδ (t)] and α 1 00 s L 1 (µ N ) (µ N ) Y ext = x [(1 z )δ 0 + z δ ]. Thus, y Lemma.7, T is sunormal α 1 00 s L 1 (µ N ) (µ N ) Y ext η 0 (where η 0 denotes the Berger measure of shift(y,,, )) x [(1 z )δ 0 + z δ ] (1 y )δ 0 + y δ (3.3) x (1 z ) (1 y )and x z y y (1 a ) (1 x ), as in the last condition in (3.). Therefore, T is sunormal, and the proof is complete. Theorem 3.. Let T (T 1,T ) S, with weight diagram given y Figure (ii), and assume that T is -hyponormal. Then T is sunormal, with Berger measure given as µ = 1 {[ (1 x ) y (1 a )]δ (0,0) (3.4) +y (1 a )δ (0, ) +( x a y )δ (1,0) + a y δ (1, )}. Proof. We apply the main result in [11], in the special form needed here; cf. [11, Proposition 3.1]. For a -variale sunormal weighted shift with weight diagram given y Figure (ii), the Berger measure is µ = ϕ (δ 0 δ )+y 1 t δ 0 ( ψ δ )+ξ x δ, (3.5) L 1 (ψ) where ψ =(1 a )δ, ϕ = ξ x y 1 a δ 0 a y δ 1, d ψ(t) 1 := dψ(t) andξ t 1 t x is the Berger L 1 (ψ) measure of S x. A straightforward calculation shows that ψ = δ, so that (3.5) ecomes µ = {[ (1 x ) y (1 a ) ]δ 0 + x a y δ 1 } (δ 0 δ ) +[(1 x )δ 0 + x δ 1 ] δ, which easily leads to the desired formula (3.4). We conclude this section with a proof of Theorem 1.4, which we reformulate in terms of C 1 and C. Toward this end, we need a description of C 1 := {(a,, x, y) :T H 1 }: y the Six-point Test (Lemma.4), T is hyponormal if and only if M (0,0) (1) 1 x y x x a y 0. y a y y Theorem 3.3. C C 1. 9
10 Proof. Since x<1, M (0,0) (1) 0 if and only if det M (0,0) (1) 0, that is, if and only if P 1 := y ( x a 4 y x 4 +a x y x y ) 0. Let x> and let a := x 1. It follows that 1 a =(1 x ), so that P := (1 x ) y (1 a )= (1 x ) y (1 x )=( y )(1 x ), and it suffices to choose y such that <y<to make P < 0, and thus reak -hyponormality (cf. (3.1)). On the other hand, P 1 P x y + y 4 a (x a ) = y ( x x 4 y + x y ) = ( x y )(1 x ), which can e made nonnegative y taking x close to 1. This shows that C C 1. References [1] M. Arahamse, Commuting sunormal operators, Ill. Math. J. (1978), [] A. Athavale, On joint hyponormality of operators, Proc.Amer.Math.Soc. 103(1988), [3] Y. Choi, A propagation of the quadratically hyponormal weighted shifts, Bull. Korean Math. Soc. 37(000), [4] J. Conway, The Theory of Sunormal Operators, Mathematical Surveys and Monographs, vol. 36, Amer. Math. Soc., Providence, [5] R. Curto, Joint hyponormality: A ridge etween hyponormality and sunormality, Proc. Symposia Pure Math. 51(1990), [6] R. Curto, Quadratically hyponormal weighted shifts, Integral Equations Operator Theory 13(1990), [7] R. Curto and L. Fialkow, Solution of the truncated complex moment prolem with flat data, Memoirs Amer. Math. Soc., no. 568, Amer. Math. Soc., Providence, [8] R. Curto, S.H. Lee and W.Y. Lee, A new criterion for k-hyponormality via weak sunormality, Proc. Amer. Math. Soc. 133(005), [9] R. Curto, S.H. Lee and J. Yoon, k-hyponormality of multivariale weighted shifts, J. Funct. Anal. 9(005), [10] R. Curto, S.H. Lee and J. Yoon, Hyponormality and sunormality for powers of commuting pairs of sunormal operators, J. Funct. Anal. 45(007), [11] R. Curto, S.H. Lee and J. Yoon, Reconstruction of the Berger measure when the core is of tensor form, Oper. Theory Adv. Appl., to appear. [1] R. Curto, P. Muhly and J. Xia, Hyponormal pairs of commuting operators, Oper. Theory Adv. Appl. 35(1988), 1-. [13] R. Curto, J. Yoon, Jointly hyponormal pairs of sunormal operators need not e jointly sunormal, Trans. Amer. Math. Soc. 358(006), [14] R. Curto and J. Yoon, Disintegration-of-measure techniques for multivariale weighted shifts, Proc. London Math. Soc. 9(006), [15] R. Curto and J. Yoon, Propagation phenomena for hyponormal -variale weighted shifts, J. Operator Theory, to appear. [16] R. Gellar and L.J. Wallen, Sunormal weighted shifts and the Halmos-Bram criterion, Proc. Japan Acad. 46(1970), [17] N.P. Jewell and A.R. Luin, Commuting weighted shifts and analytic function theory in several variales, J. Operator Theory 1(1979), [18] A. Luin, Weighted shifts and product of sunormal operators, Indiana Univ. Math. J. 6(1977), [19] A. Luin, Extensions of commuting sunormal operators, Lecture Notes in Math. 693(1978), [0] A. Luin, A sunormal semigroup without normal extension, Proc. Amer. Math. Soc. 68(1978), [1] J.L. Smul jan, An operator Hellinger integral (Russian), Mat. S. 91(1959), [] J. Stampfli, Which weighted shifts are sunormal?, Pacific J. Math. 17(1966), [3] Wolfram Research, Inc. Mathematica, Version 4., Wolfram Research Inc., Champaign, IL,
11 [4] J. Yoon, Disintegration of measures and contractive -variale weighted shifts, Integral Equations Operator Theory, to appear. [5] J. Yoon, Schur product techniques for commuting multivariale weighted shifts, J. Math. Anal. Appl. 333(007), Department of Mathematics, The University of Iowa, Iowa City, Iowa 54 address: URL: Department of Mathematics, Chungnam National University, Daejeon, , Korea address: Department of Mathematics, The University of Texas - Pan American, Edinurg, Texas address: jasangyoon@hotmail.com 11
Toral and Spherical Aluthge Transforms
Toral and Spherical Aluthge Transforms (joint work with Jasang Yoon) Raúl E. Curto University of Iowa INFAS, Des Moines, IA November 11, 2017 (our first report on this research appeared in Comptes Rendus
More informationEXISTENCE OF NON-SUBNORMAL POLYNOMIALLY HYPONORMAL OPERATORS
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 25, Number 2, October 1991 EXISTENCE OF NON-SUBNORMAL POLYNOMIALLY HYPONORMAL OPERATORS RAUL E. CURTO AND MIHAI PUTINAR INTRODUCTION In
More informationMINIMAL REPRESENTING MEASURES ARISING FROM RANK-INCREASING MOMENT MATRIX EXTENSIONS
J. OPERATOR THEORY 42(1999), 425 436 c Copyright by Theta, 1999 MINIMAL REPRESENTING MEASURES ARISING FROM RANK-INCREASING MOMENT MATRIX EXTENSIONS LAWRENCE A. FIALKOW Communicated by Florian-Horia Vasilescu
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 so-called quasitriangular property dim
More informationPolynomially hyponormal operators
Polynomially hyponormal operators Raúl Curto and Mihai Putinar To the memory of Paul R. Halmos Abstract. A survey of the theory of k-hyponormal operators starts with the construction of a polynomially
More informationMoment Infinitely Divisible Weighted Shifts
Moment Infinitely Divisible Weighted Shifts (joint work with Chafiq Benhida and George Exner) Raúl E. Curto GPOTS 2016 Raúl E. Curto (Urbana, IL, 5/26/2016) Infinitely Divisible Shifts 1 / 47 Some Key
More informationAnn. Acad. Rom. Sci. Ser. Math. Appl. Vol. 8, No. 1/2016. The space l (X) Namita Das
ISSN 2066-6594 Ann. Acad. Rom. Sci. Ser. Math. Appl. Vol. 8, No. 1/2016 The space l (X) Namita Das Astract In this paper we have shown that the space c n n 0 cannot e complemented in l n n and c 0 (H)
More informationPOSITIVE MAP AS DIFFERENCE OF TWO COMPLETELY POSITIVE OR SUPER-POSITIVE MAPS
Adv. Oper. Theory 3 (2018), no. 1, 53 60 http://doi.org/10.22034/aot.1702-1129 ISSN: 2538-225X (electronic) http://aot-math.org POSITIVE MAP AS DIFFERENCE OF TWO COMPLETELY POSITIVE OR SUPER-POSITIVE MAPS
More informationA New Necessary Condition for the Hyponormality of Toeplitz Operators on the Bergman Space
A New Necessary Condition for the Hyponormality of Toeplitz Operators on the Bergman Space Raúl Curto IWOTA 2016 Special Session on Multivariable Operator Theory St. Louis, MO, USA; July 19, 2016 (with
More informationdominant positive-normal. In view of (1), it is very natural to study the properties of positivenormal
Bull. Korean Math. Soc. 39 (2002), No. 1, pp. 33 41 ON POSITIVE-NORMAL OPERATORS In Ho Jeon, Se Hee Kim, Eungil Ko, and Ji Eun Park Abstract. In this paper we study the properties of positive-normal operators
More informationEVALUATIONS OF EXPECTED GENERALIZED ORDER STATISTICS IN VARIOUS SCALE UNITS
APPLICATIONES MATHEMATICAE 9,3 (), pp. 85 95 Erhard Cramer (Oldenurg) Udo Kamps (Oldenurg) Tomasz Rychlik (Toruń) EVALUATIONS OF EXPECTED GENERALIZED ORDER STATISTICS IN VARIOUS SCALE UNITS Astract. We
More informationSubclasses of Analytic Functions. Involving the Hurwitz-Lerch Zeta Function
International Mathematical Forum, Vol. 6, 211, no. 52, 2573-2586 Suclasses of Analytic Functions Involving the Hurwitz-Lerch Zeta Function Shigeyoshi Owa Department of Mathematics Kinki University Higashi-Osaka,
More informationNEW CONSTRUCTION OF THE EAGON-NORTHCOTT COMPLEX. Oh-Jin Kang and Joohyung Kim
Korean J Math 20 (2012) No 2 pp 161 176 NEW CONSTRUCTION OF THE EAGON-NORTHCOTT COMPLEX Oh-Jin Kang and Joohyung Kim Abstract The authors [6 introduced the concept of a complete matrix of grade g > 3 to
More informationON THE SIMILARITY OF CENTERED OPERATORS TO CONTRACTIONS. Srdjan Petrović
ON THE SIMILARITY OF CENTERED OPERATORS TO CONTRACTIONS Srdjan Petrović Abstract. In this paper we show that every power bounded operator weighted shift with commuting normal weights is similar to a contraction.
More informationON KRONECKER PRODUCTS OF CHARACTERS OF THE SYMMETRIC GROUPS WITH FEW COMPONENTS
ON KRONECKER PRODUCTS OF CHARACTERS OF THE SYMMETRIC GROUPS WITH FEW COMPONENTS C. BESSENRODT AND S. VAN WILLIGENBURG Abstract. Confirming a conjecture made by Bessenrodt and Kleshchev in 1999, we classify
More informationSTRUCTURAL AND SPECTRAL PROPERTIES OF k-quasi- -PARANORMAL OPERATORS. Fei Zuo and Hongliang Zuo
Korean J Math (015), No, pp 49 57 http://dxdoiorg/1011568/kjm01549 STRUCTURAL AND SPECTRAL PROPERTIES OF k-quasi- -PARANORMAL OPERATORS Fei Zuo and Hongliang Zuo Abstract For a positive integer k, an operator
More informationBerger measures for transformations of subnormal weighted shifts
Berger measures for transformations of subnormal weighted shifts (joint work with George Exner) Raúl E. Curto 4th Small Workshop on Operator Theory University of Agriculture, Kraków, July 8, 2014 Raúl
More informationON M-HYPONORMAL WEIGHTED SHIFTS. Jung Sook Ham, Sang Hoon Lee and Woo Young Lee. 1. Introduction
ON M-HYPONORMAL WEIGHTED SHIFTS Jung Soo Ham, Sang Hoon Lee and Woo Young Lee Abstract Let α {α n} n0 be a weight sequence and let Wα denote the associated unilateral weighted shift on l Z + In this paper
More informationUpper Bounds for Stern s Diatomic Sequence and Related Sequences
Upper Bounds for Stern s Diatomic Sequence and Related Sequences Colin Defant Department of Mathematics University of Florida, U.S.A. cdefant@ufl.edu Sumitted: Jun 18, 01; Accepted: Oct, 016; Pulished:
More informationSpecial Precovered Categories of Gorenstein Categories
Special Precovered Categories of Gorenstein Categories Tiwei Zhao and Zhaoyong Huang Department of Mathematics, Nanjing University, Nanjing 9, Jiangsu Province, P. R. China Astract Let A e an aelian category
More informationON OPERATORS WITH AN ABSOLUTE VALUE CONDITION. In Ho Jeon and B. P. Duggal. 1. Introduction
J. Korean Math. Soc. 41 (2004), No. 4, pp. 617 627 ON OPERATORS WITH AN ABSOLUTE VALUE CONDITION In Ho Jeon and B. P. Duggal Abstract. Let A denote the class of bounded linear Hilbert space operators with
More informationRepresentation theory of SU(2), density operators, purification Michael Walter, University of Amsterdam
Symmetry and Quantum Information Feruary 6, 018 Representation theory of S(), density operators, purification Lecture 7 Michael Walter, niversity of Amsterdam Last week, we learned the asic concepts of
More informationON A CLASS OF OPERATORS RELATED TO PARANORMAL OPERATORS
J. Korean Math. Soc. 44 (2007), No. 1, pp. 25 34 ON A CLASS OF OPERATORS RELATED TO PARANORMAL OPERATORS Mi Young Lee and Sang Hun Lee Reprinted from the Journal of the Korean Mathematical Society Vol.
More informationCauchy Duals of 2-hyperexpansive Operators. The Multivariable Case
Operators Cauchy Dual to 2-hyperexpansive Operators: The Multivariable Case Raúl E. Curto Southeastern Analysis Meeting XXVII, Gainesville (joint work with Sameer Chavan) raul-curto@uiowa.edu http://www.math.uiowa.edu/
More informationSUMS OF UNITS IN SELF-INJECTIVE RINGS
SUMS OF UNITS IN SELF-INJECTIVE RINGS ANJANA KHURANA, DINESH KHURANA, AND PACE P. NIELSEN Abstract. We prove that if no field of order less than n + 2 is a homomorphic image of a right self-injective ring
More informationPolynomials Characterizing Hyper b-ary Representations
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 21 (2018), Article 18.4.3 Polynomials Characterizing Hyper -ary Representations Karl Dilcher 1 Department of Mathematics and Statistics Dalhousie University
More informationLie Weak Amenability of Triangular Banach Algebra
Journal of Mathematical Research with Applications Sept 217 Vol 37 No 5 pp 63 612 DOI:1377/jissn:295-265121751 Http://jmredluteducn ie Weak Amenaility of Triangular anach Algera in CHEN 1 Fangyan U 2 1
More information#A50 INTEGERS 14 (2014) ON RATS SEQUENCES IN GENERAL BASES
#A50 INTEGERS 14 (014) ON RATS SEQUENCES IN GENERAL BASES Johann Thiel Dept. of Mathematics, New York City College of Technology, Brooklyn, New York jthiel@citytech.cuny.edu Received: 6/11/13, Revised:
More informationA New Necessary Condition for the Hyponormality of Toeplitz Operators on the Bergman Space
A New Necessary Condition for the Hyponormality of Toeplitz Operators on the Bergman Space Željko Čučković Department of Mathematics, University of Toledo, Toledo, Ohio 43606 Raúl E Curto Department of
More informationMeans of unitaries, conjugations, and the Friedrichs operator
J. Math. Anal. Appl. 335 (2007) 941 947 www.elsevier.com/locate/jmaa Means of unitaries, conjugations, and the Friedrichs operator Stephan Ramon Garcia Department of Mathematics, Pomona College, Claremont,
More informationCyclic Direct Sum of Tuples
Int. J. Contemp. Math. Sciences, Vol. 7, 2012, no. 8, 377-382 Cyclic Direct Sum of Tuples Bahmann Yousefi Fariba Ershad Department of Mathematics, Payame Noor University P.O. Box: 71955-1368, Shiraz, Iran
More informationTHE FULL C* -ALGEBRA OF THE FREE GROUP ON TWO GENERATORS
PACIFIC JOURNAL OF MATHEMATICS Vol. 87, No. 1, 1980 THE FULL C* -ALGEBRA OF THE FREE GROUP ON TWO GENERATORS MAN-DUEN CHOI C*(F 2 ) is a primitive C*-algebra with no nontrivial projection. C*(F 2 ) has
More informationSOLUTION OF THE TRUNCATED PARABOLIC MOMENT PROBLEM
SOLUTION OF THE TRUNCATED PARABOLIC MOMENT PROBLEM RAÚL E. CURTO AND LAWRENCE A. FIALKOW Abstract. Given real numbers β β (2n) {β ij} i,j 0,i+j 2n, with γ 00 > 0, the truncated parabolic moment problem
More informationDeterminants of generalized binary band matrices
Determinants of generalized inary and matrices Dmitry Efimov arxiv:17005655v1 [mathra] 18 Fe 017 Department of Mathematics, Komi Science Centre UrD RAS, Syktyvkar, Russia Astract Under inary matrices we
More informationMOMENT SEQUENCES AND BACKWARD EXTENSIONS OF SUBNORMAL WEIGHTED SHIFTS
J. Austral. Math. Soc. 73 (2002), 27-36 MOMENT SEQUENCES AND BACKWARD EXTENSIONS OF SUBNORMAL WEIGHTED SHIFTS THOMAS HOOVER, IL BONG JUNG and ALAN LAMBERT (Received 15 February 2000; revised 10 July 2001)
More informationMOMENT INFINITELY DIVISIBLE WEIGHTED SHIFTS
MOMENT INFINITELY DIVISIBLE WEIGHTED SHIFTS CHAFIQ BENHIDA, RAÚL E. CURTO, AND GEORGE R. EXNER Abstract. We say that a weighted shift W α with (positive) weight sequence α : α 0,α 1,... is moment infinitely
More information10 Lorentz Group and Special Relativity
Physics 129 Lecture 16 Caltech, 02/27/18 Reference: Jones, Groups, Representations, and Physics, Chapter 10. 10 Lorentz Group and Special Relativity Special relativity says, physics laws should look the
More informationWEYL-HEISENBERG FRAMES FOR SUBSPACES OF L 2 (R)
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Numer 1, Pages 145 154 S 0002-9939(00)05731-2 Article electronically pulished on July 27, 2000 WEYL-HEISENBERG FRAMES FOR SUBSPACES OF L 2 (R)
More informationTHREE LECTURES ON QUASIDETERMINANTS
THREE LECTURES ON QUASIDETERMINANTS Robert Lee Wilson Department of Mathematics Rutgers University The determinant of a matrix with entries in a commutative ring is a main organizing tool in commutative
More informationContinuity properties for linear commutators of Calderón-Zygmund operators
Collect. Math. 49, 1 (1998), 17 31 c 1998 Universitat de Barcelona Continuity properties for linear commutators of Calderón-Zygmund operators Josefina Alvarez Department of Mathematical Sciences, New Mexico
More informationOn Systems of Diagonal Forms II
On Systems of Diagonal Forms II Michael P Knapp 1 Introduction In a recent paper [8], we considered the system F of homogeneous additive forms F 1 (x) = a 11 x k 1 1 + + a 1s x k 1 s F R (x) = a R1 x k
More informationSung-Wook Park*, Hyuk Han**, and Se Won Park***
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 16, No. 1, June 2003 CONTINUITY OF LINEAR OPERATOR INTERTWINING WITH DECOMPOSABLE OPERATORS AND PURE HYPONORMAL OPERATORS Sung-Wook Park*, Hyuk Han**,
More informationZeroing the baseball indicator and the chirality of triples
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 7 (2004), Article 04.1.7 Zeroing the aseall indicator and the chirality of triples Christopher S. Simons and Marcus Wright Department of Mathematics
More informationChapter 2 Canonical Correlation Analysis
Chapter 2 Canonical Correlation Analysis Canonical correlation analysis CCA, which is a multivariate analysis method, tries to quantify the amount of linear relationships etween two sets of random variales,
More informationChapter 2: Matrix Algebra
Chapter 2: Matrix Algebra (Last Updated: October 12, 2016) These notes are derived primarily from Linear Algebra and its applications by David Lay (4ed). Write A = 1. Matrix operations [a 1 a n. Then entry
More informationCHARACTERISTIC POLYNOMIAL PATTERNS IN DIFFERENCE SETS OF MATRICES
CHARACTERISTIC POLYNOMIAL PATTERNS IN DIFFERENCE SETS OF MATRICES MICHAEL BJÖRKLUND AND ALEXANDER FISH Abstract. We show that for every subset E of positive density in the set of integer squarematrices
More informationTWO REMARKS ABOUT NILPOTENT OPERATORS OF ORDER TWO
TWO REMARKS ABOUT NILPOTENT OPERATORS OF ORDER TWO STEPHAN RAMON GARCIA, BOB LUTZ, AND DAN TIMOTIN Abstract. We present two novel results about Hilbert space operators which are nilpotent of order two.
More informationFACTORING A QUADRATIC OPERATOR AS A PRODUCT OF TWO POSITIVE CONTRACTIONS
FACTORING A QUADRATIC OPERATOR AS A PRODUCT OF TWO POSITIVE CONTRACTIONS CHI-KWONG LI AND MING-CHENG TSAI Abstract. Let T be a quadratic operator on a complex Hilbert space H. We show that T can be written
More informationExpansion formula using properties of dot product (analogous to FOIL in algebra): u v 2 u v u v u u 2u v v v u 2 2u v v 2
Least squares: Mathematical theory Below we provide the "vector space" formulation, and solution, of the least squares prolem. While not strictly necessary until we ring in the machinery of matrix algera,
More informationENTANGLED STATES ARISING FROM INDECOMPOSABLE POSITIVE LINEAR MAPS. 1. Introduction
Trends in Mathematics Information Center for Mathematical Sciences Volume 6, Number 2, December, 2003, Pages 83 91 ENTANGLED STATES ARISING FROM INDECOMPOSABLE POSITIVE LINEAR MAPS SEUNG-HYEOK KYE Abstract.
More informationHOMOLOGICAL PROPERTIES OF MODULES OVER DING-CHEN RINGS
J. Korean Math. Soc. 49 (2012), No. 1, pp. 31 47 http://dx.doi.org/10.4134/jkms.2012.49.1.031 HOMOLOGICAL POPETIES OF MODULES OVE DING-CHEN INGS Gang Yang Abstract. The so-called Ding-Chen ring is an n-fc
More informationA 3 3 DILATION COUNTEREXAMPLE
A 3 3 DILATION COUNTEREXAMPLE MAN DUEN CHOI AND KENNETH R DAVIDSON Dedicated to the memory of William B Arveson Abstract We define four 3 3 commuting contractions which do not dilate to commuting isometries
More informationA rank formula for the self-commutators of rational Toeplitz tuples
wang et al Journal of Inequalities and Applications 2016) 2016:184 DOI 101186/s13660-016-1125-x R E S E A R C Open Access A rank formula for the self-commutators of rational Toeplitz tuples In Sung wang
More informationSTRONG NORMALITY AND GENERALIZED COPELAND ERDŐS NUMBERS
#A INTEGERS 6 (206) STRONG NORMALITY AND GENERALIZED COPELAND ERDŐS NUMBERS Elliot Catt School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, New South Wales, Australia
More informationPERMANENTLY WEAK AMENABILITY OF REES SEMIGROUP ALGEBRAS. Corresponding author: jabbari 1. Introduction
International Journal of Analysis and Applications Volume 16, Number 1 (2018), 117-124 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-16-2018-117 PERMANENTLY WEAK AMENABILITY OF REES SEMIGROUP
More informationSCALE INVARIANT FOURIER RESTRICTION TO A HYPERBOLIC SURFACE
SCALE INVARIANT FOURIER RESTRICTION TO A HYPERBOLIC SURFACE BETSY STOVALL Abstract. This result sharpens the bilinear to linear deduction of Lee and Vargas for extension estimates on the hyperbolic paraboloid
More informationGaps between classes of matrix monotone functions
arxiv:math/0204204v [math.oa] 6 Apr 2002 Gaps between classes of matrix monotone functions Frank Hansen, Guoxing Ji and Jun Tomiyama Introduction April 6, 2002 Almost seventy years have passed since K.
More informationJACOBSON S THEOREM FOR BILINEAR FORMS IN CHARACTERISTIC 2
JACOBSON S THEOREM FOR BILINEAR FORMS IN CHARACTERISTIC 2 AHMED LAGHRIBI Astract The aim of this note is to extend to ilinear forms in characteristic 2 a result of Jacoson which states that over any field,
More informationUltragraph C -algebras via topological quivers
STUDIA MATHEMATICA 187 (2) (2008) Ultragraph C -algebras via topological quivers by Takeshi Katsura (Yokohama), Paul S. Muhly (Iowa City, IA), Aidan Sims (Wollongong) and Mark Tomforde (Houston, TX) Abstract.
More informationCONTINUOUS DEPENDENCE ESTIMATES FOR VISCOSITY SOLUTIONS OF FULLY NONLINEAR DEGENERATE ELLIPTIC EQUATIONS
Electronic Journal of Differential Equations, Vol. 20022002), No. 39, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu login: ftp) CONTINUOUS DEPENDENCE
More informationNOTE. A Note on Fisher's Inequality
journal of cominatorial theory, Series A 77, 171176 (1997) article no. TA962729 NOTE A Note on Fisher's Inequality Douglas R. Woodall Department of Mathematics, University of Nottingham, Nottingham NG7
More informationREVERSALS ON SFT S. 1. Introduction and preliminaries
Trends in Mathematics Information Center for Mathematical Sciences Volume 7, Number 2, December, 2004, Pages 119 125 REVERSALS ON SFT S JUNGSEOB LEE Abstract. Reversals of topological dynamical systems
More informationLARGE DEVIATIONS OF TYPICAL LINEAR FUNCTIONALS ON A CONVEX BODY WITH UNCONDITIONAL BASIS. S. G. Bobkov and F. L. Nazarov. September 25, 2011
LARGE DEVIATIONS OF TYPICAL LINEAR FUNCTIONALS ON A CONVEX BODY WITH UNCONDITIONAL BASIS S. G. Bobkov and F. L. Nazarov September 25, 20 Abstract We study large deviations of linear functionals on an isotropic
More informationQuivers. Virginia, Lonardo, Tiago and Eloy UNICAMP. 28 July 2006
Quivers Virginia, Lonardo, Tiago and Eloy UNICAMP 28 July 2006 1 Introduction This is our project Throughout this paper, k will indicate an algeraically closed field of characteristic 0 2 Quivers 21 asic
More informationBoundedly complete weak-cauchy basic sequences in Banach spaces with the PCP
Journal of Functional Analysis 253 (2007) 772 781 www.elsevier.com/locate/jfa Note Boundedly complete weak-cauchy basic sequences in Banach spaces with the PCP Haskell Rosenthal Department of Mathematics,
More informationEXTENSIONS OF EXTENDED SYMMETRIC RINGS
Bull Korean Math Soc 44 2007, No 4, pp 777 788 EXTENSIONS OF EXTENDED SYMMETRIC RINGS Tai Keun Kwak Reprinted from the Bulletin of the Korean Mathematical Society Vol 44, No 4, November 2007 c 2007 The
More informationThe Airy function is a Fredholm determinant
Journal of Dynamics and Differential Equations manuscript No. (will e inserted y the editor) The Airy function is a Fredholm determinant Govind Menon Received: date / Accepted: date Astract Let G e the
More informationNOTES ON PRODUCT SYSTEMS
NOTES ON PRODUCT SYSTEMS WILLIAM ARVESON Abstract. We summarize the basic properties of continuous tensor product systems of Hilbert spaces and their role in non-commutative dynamics. These are lecture
More informationPAijpam.eu CLASS OF (A, n)-power QUASI-NORMAL OPERATORS IN SEMI-HILBERTIAN SPACES
International Journal of Pure and Applied Mathematics Volume 93 No. 1 2014, 61-83 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v93i1.6
More informationUpper triangular matrices and Billiard Arrays
Linear Algebra and its Applications 493 (2016) 508 536 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.com/locate/laa Upper triangular matrices and Billiard Arrays
More informationPACKING-DIMENSION PROFILES AND FRACTIONAL BROWNIAN MOTION
PACKING-DIMENSION PROFILES AND FRACTIONAL BROWNIAN MOTION DAVAR KHOSHNEVISAN AND YIMIN XIAO Abstract. In order to compute the packing dimension of orthogonal projections Falconer and Howroyd 997) introduced
More informationGenetic Algorithms applied to Problems of Forbidden Configurations
Genetic Algorithms applied to Prolems of Foridden Configurations R.P. Anstee Miguel Raggi Department of Mathematics University of British Columia Vancouver, B.C. Canada V6T Z2 anstee@math.uc.ca mraggi@gmail.com
More informationSVETLANA KATOK AND ILIE UGARCOVICI (Communicated by Jens Marklof)
JOURNAL OF MODERN DYNAMICS VOLUME 4, NO. 4, 010, 637 691 doi: 10.3934/jmd.010.4.637 STRUCTURE OF ATTRACTORS FOR (a, )-CONTINUED FRACTION TRANSFORMATIONS SVETLANA KATOK AND ILIE UGARCOVICI (Communicated
More informationOn the simplest expression of the perturbed Moore Penrose metric generalized inverse
Annals of the University of Bucharest (mathematical series) 4 (LXII) (2013), 433 446 On the simplest expression of the perturbed Moore Penrose metric generalized inverse Jianbing Cao and Yifeng Xue Communicated
More informationMi Ryeong Lee and Hye Yeong Yun
Commun. Korean Math. Soc. 28 (213, No. 4, pp. 741 75 http://dx.doi.org/1.4134/ckms.213.28.4.741 ON QUASI-A(n,k CLASS OPERATORS Mi Ryeong Lee and Hye Yeong Yun Abstract. To study the operator inequalities,
More informationSOME PROPERTIES OF N-SUPERCYCLIC OPERATORS
SOME PROPERTIES OF N-SUPERCYCLIC OPERATORS P S. BOURDON, N. S. FELDMAN, AND J. H. SHAPIRO Abstract. Let T be a continuous linear operator on a Hausdorff topological vector space X over the field C. We
More informationEXTENSION OF VECTOR-VALUED INTEGRAL POLYNOMIALS. Introduction
EXTENSION OF VECTOR-VALUED INTEGRAL POLYNOMIALS DANIEL CARANDO AND SILVIA LASSALLE Abstract. We study the extendibility of integral vector-valued polynomials on Banach spaces. We prove that an X-valued
More informationJeong-Sik Kim, Yeong-Moo Song and Mukut Mani Tripathi
Bull. Korean Math. Soc. 40 (003), No. 3, pp. 411 43 B.-Y. CHEN INEQUALITIES FOR SUBMANIFOLDS IN GENERALIZED COMPLEX SPACE FORMS Jeong-Sik Kim, Yeong-Moo Song and Mukut Mani Tripathi Abstract. Some B.-Y.
More informationLet N > 0, let A, B, and C be constants, and let f and g be any functions. Then: S2: separate summed terms. S7: sum of k2^(k-1)
Summation Formulas Let > 0, let A, B, and C e constants, and let f and g e any functions. Then: k Cf ( k) C k S: factor out constant f ( k) k ( f ( k) ± g( k)) k S: separate summed terms f ( k) ± k g(
More informationAbstract. In this article, several matrix norm inequalities are proved by making use of the Hiroshima 2003 result on majorization relations.
HIROSHIMA S THEOREM AND MATRIX NORM INEQUALITIES MINGHUA LIN AND HENRY WOLKOWICZ Abstract. In this article, several matrix norm inequalities are proved by making use of the Hiroshima 2003 result on majorization
More informationInvariant subspaces for operators whose spectra are Carathéodory regions
Invariant subspaces for operators whose spectra are Carathéodory regions Jaewoong Kim and Woo Young Lee Abstract. In this paper it is shown that if an operator T satisfies p(t ) p σ(t ) for every polynomial
More informationarxiv:math/ v4 [math.oa] 28 Dec 2005
arxiv:math/0506151v4 [math.oa] 28 Dec 2005 TENSOR ALGEBRAS OF C -CORRESPONDENCES AND THEIR C -ENVELOPES. ELIAS G. KATSOULIS AND DAVID W. KRIBS Abstract. We show that the C -envelope of the tensor algebra
More informationFORMAL TAYLOR SERIES AND COMPLEMENTARY INVARIANT SUBSPACES
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCITEY Volume 45, Number 1, July 1974 FORMAL TAYLOR SERIES AND COMPLEMENTARY INVARIANT SUBSPACES DOMINGO A. HERRERO X ABSTRACT. A class of operators (which includes
More informationSome Generalizations of Caristi s Fixed Point Theorem with Applications
Int. Journal of Math. Analysis, Vol. 7, 2013, no. 12, 557-564 Some Generalizations of Caristi s Fixed Point Theorem with Applications Seong-Hoon Cho Department of Mathematics Hanseo University, Seosan
More informationarxiv: v1 [quant-ph] 4 Jul 2013
GEOMETRY FOR SEPARABLE STATES AND CONSTRUCTION OF ENTANGLED STATES WITH POSITIVE PARTIAL TRANSPOSES KIL-CHAN HA AND SEUNG-HYEOK KYE arxiv:1307.1362v1 [quant-ph] 4 Jul 2013 Abstract. We construct faces
More information1 Systems of Differential Equations
March, 20 7- Systems of Differential Equations Let U e an open suset of R n, I e an open interval in R and : I R n R n e a function from I R n to R n The equation ẋ = ft, x is called a first order ordinary
More informationCHAPTER 5. Linear Operators, Span, Linear Independence, Basis Sets, and Dimension
A SERIES OF CLASS NOTES TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS LINEAR CLASS NOTES: A COLLECTION OF HANDOUTS FOR REVIEW AND PREVIEW OF LINEAR THEORY
More informationActa Mathematica Academiae Paedagogicae Nyíregyháziensis 24 (2008), ISSN
Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 24 (2008), 313 321 www.emis.de/journals ISSN 1786-0091 DUAL BANACH ALGEBRAS AND CONNES-AMENABILITY FARUK UYGUL Abstract. In this survey, we first
More informationThis paper was published in Connections in Discrete Mathematics, S. Butler, J. Cooper, and G. Hurlbert, editors, Cambridge University Press,
This paper was published in Connections in Discrete Mathematics, S Butler, J Cooper, and G Hurlbert, editors, Cambridge University Press, Cambridge, 2018, 200-213 To the best of my knowledge, this is the
More informationDisconjugate operators and related differential equations
Disconjugate operators and related differential equations Mariella Cecchi, Zuzana Došlá and Mauro Marini Dedicated to J. Vosmanský on occasion of his 65 th birthday Abstract: There are studied asymptotic
More informationWeighted Sums of Orthogonal Polynomials Related to Birth-Death Processes with Killing
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 8, Number 2, pp. 401 412 (2013) http://campus.mst.edu/adsa Weighted Sums of Orthogonal Polynomials Related to Birth-Death Processes
More informationOperator positivity and analytic models of commuting tuples of operators
STUDIA MATHEMATICA Online First version Operator positivity and analytic models of commuting tuples of operators by Monojit Bhattacharjee and Jaydeb Sarkar (Bangalore) Abstract. We study analytic models
More informationPatryk Pagacz. Characterization of strong stability of power-bounded operators. Uniwersytet Jagielloński
Patryk Pagacz Uniwersytet Jagielloński Characterization of strong stability of power-bounded operators Praca semestralna nr 3 (semestr zimowy 2011/12) Opiekun pracy: Jaroslav Zemanek CHARACTERIZATION OF
More informationWeyl s Theorem for Algebraically Paranormal Operators
Integr. equ. oper. theory 47 (2003) 307 314 0378-620X/030307-8, DOI 10.1007/s00020-002-1164-1 c 2003 Birkhäuser Verlag Basel/Switzerland Integral Equations and Operator Theory Weyl s Theorem for Algebraically
More informationCharacterization of invariant subspaces in the polydisc
Characterization of invariant subspaces in the polydisc Amit Maji IIIT Guwahati (Joint work with Aneesh M., Jaydeb Sarkar & Sankar T. R.) Recent Advances in Operator Theory and Operator Algebras Indian
More informationSome Restricted Plane partitions and Associated Lattice Paths S. Bedi Department of Mathematics, D.A.V College, Sector 10 Chandigarh , India
Some Restricted Plane partitions and Associated Lattice Paths S. Bedi Department of Mathematics, D.A.V College, Sector 10 Chandigarh - 160010, India Abstract. Anand and Agarwal, (Proc. Indian Acad. Sci.
More informationarxiv:math/ v1 [math.sp] 27 Mar 2006
SHARP BOUNDS FOR EIGENVALUES OF TRIANGLES arxiv:math/0603630v1 [math.sp] 27 Mar 2006 BART LOMIEJ SIUDEJA Astract. We prove that the first eigenvalue of the Dirichlet Laplacian for a triangle in the plane
More informationA converse Gaussian Poincare-type inequality for convex functions
Statistics & Proaility Letters 44 999 28 290 www.elsevier.nl/locate/stapro A converse Gaussian Poincare-type inequality for convex functions S.G. Bokov a;, C. Houdre ; ;2 a Department of Mathematics, Syktyvkar
More informationSang-baek Lee*, Jae-hyeong Bae**, and Won-gil Park***
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 6, No. 4, November 013 http://d.doi.org/10.14403/jcms.013.6.4.671 ON THE HYERS-ULAM STABILITY OF AN ADDITIVE FUNCTIONAL INEQUALITY Sang-baek Lee*,
More informationNon-separable AF-algebras
Non-separable AF-algebras Takeshi Katsura Department of Mathematics, Hokkaido University, Kita 1, Nishi 8, Kita-Ku, Sapporo, 6-81, JAPAN katsura@math.sci.hokudai.ac.jp Summary. We give two pathological
More information