RIGHT SPECTRUM AND TRACE FORMULA OF SUBNORMAL TUPLE OF OPERATORS OF FINITE TYPE

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1 RIGHT SPECTRUM AND TRACE FORMULA OF SUBNORMAL TUPLE OF OPERATORS OF FINITE TYPE Daoxing Xia Abstract. This paper studies pure subnormal k-tuples of operators S = (S 1,, S k ) with finite rank of self-commutators. It determines the substantial part of the conjugate of the joint point spectrum of S = (S 1,, S k) which is the union of domains in Riemann surfaces in some algebraic varieties in C k. The concrete form of the principal current [CP1] related to S is also determined. Besides, some operator identities are found for S. Mathematics Subject Classification (2000). Primary 47B Introduction In this paper H is an infinite dimensional separable Hilbert space. A k-tuple of operators S = (S 1,, S k ) on H is said to be subnormal (or jointly subnormal) (cf. [Co2], [Cu2], [Pu], [EP]) if there is a k-tuple of commuting normal operators N = (N 1,, N k ) on a Hilbert space K, containing H as a subspace, satisfying S j = N j H, j = 1, 2,, k. Without loss of generality, we may only consider the m.n.e.(minimal normal extension), i. e. there is no improper reducing subspace of N in K H. S is said to be pure, if there is no improper reducing subspace of S in H. Yakubovich [Y1], [Y2] called a subnormal operator with finite rank self-commutator as a subnormal operator of finite type, cf. also [X5], [X6], [X8]. Let us call a subnormal k-tuple of operators S = (S 1,, S k ) is of finite type if rank[s j, S j] < +, j = 1, 2,, k. In this case, it is also that rank[s i, S j ] < +, i j, i, j = 1, 2,, k. In this paper we only study pure subnormal k-tuple of operators S of finite type on H.

2 2 Daoxing Xia The main mathematical tool of this study is the analytic model of S in [X1], [X2], [X3], [X5]. Let M = k j=1[s j, S j ]H. (1) Then dim M < +, since we only consider the operators of finite type. Let C ij def = [S i, S j ] M (2) as an operator on the finite dimensional space M, since M reduces [S i, S j]. Let Λ j def = (S j M ) L(M), (3) since M is invariant with respect to Sj, j = 1, 2,, k (cf.[x1]). Let and Then R ij (z) def = C ij (z Λ j ) 1 + Λ i, (4) Q ij (w, z) def = (w Λ i )(z Λ j ) C ij (5) P ij (z, w) def = det Q ij (z, w). (6) P ij (z, w) = P ji (w, z). The polynomial P ij (z, w) is with leading term z ν w ν where ν = dim M. For A = (A 1,, A k ), let sp jp (A ) be the conjugate of the joint point spectrum, i. e. the set of all (w 1,, w k ) C k satisfying the condition that there is a vector f H, f 0 such that A j f = w j f, j = 1, 2,, k. In 3, some relation between the right spectrum of a pure subnormal tuple S of operators of finite type and sp jp (S ) is given. In 5, we introduce a union of domains S in some Riemann surfaces which is in a domain in an algebraic variety S a. Those S and S a are determined basically by {P ij (, )} in (6). The aim of this paper is to determine a substantial part S of sp jp (S ) for pure subnormal k-tuple of operators S of finite type in Theorem 6.1. The principal current has been studied by Pincus and Carey [PC1], [PC2], [Pi]. In this paper, we will give the concrete form Theorem 7.1 of the principal current for the pure subnormal operators of finite type. Besides, in 7, we will give some operator identities for S, which are the generalization of the identity U +U + I = 0 for the unilateral shift U Preliminaries In order to make the proofs in this paper readable, we have to list the some basic facts in the theory of the analytic model of subnormal operators of finite type which is a special case in [X1], [X2], [X3].

3 Subnormal tuple of operators 3 Let S = (S 1,, S k ) be a pure subnormal k-tuple of operators of finite type on H with m.n.e. N. Let E( ) be the spectral measure of N on sp(n). Let P M be the projection from H to M, where M is defined in (1). Let e( ) def = P M E( ) M (7) be the L(M)-valued positive measure on sp(n). Let Ĥ be the Hilbert space completion of the span of all functions k j=1 (λ j u j ) 1 a for a M and λ j ρ(s j ), j = 1, 2,, k, with respect to (f, h) = (f(u), h(u)). sp(n) Then there is a unitary operator U from H onto Ĥ such that and (US j U 1 f)(u) = u j f(u), u = (u 1,, u k ) sp(n) (US j U 1 f)(u) = u j (f(u) f(λ)) + Λ j f(λ) (8) where f(λ) = sp(n) f(u). This (US 1U 1,, US k U 1 ) is the analytic model of S. Without loss of generality, we may assume that H = Ĥ and U = I, i. e. we only have to consider the analytic model of S. For u σ(n), Q ij (u, u) = 0. (9) Let P (S) def = {u C k : P ij (u, u) = 0, i, j = 1, 2,, k} Then from (8) (cf. also [X3], [PX]), Let sp(n) P (S). µ j (z) def u j Λ j = sp(n) u j z, z ρ(n j). Then µ j ( ) is analytic on ρ(n j ) and µ j (z) = 0 for z ρ(s j ). Let T = (T 1,, T k ) be a k-tuple of commuting operators on H. If there is a finite dimensional subspace K H such that [T i, T j ]H K, T i K K for i, j = 1, 2,, k. Then T is said to be of finite type. In this case let C ij def = [T i, T j ] K, Λ i def = (T i K ) for i, j = 1, 2,, k. Define R ij (z) = C ij (z Λ j ) 1 + Λ i and Q ij (w, z) = (w Λ i )(z Λ j) C ij as in 1. The following lemma is useful for 4 and the future study.

4 4 Daoxing Xia Lemma 2.1 Let T be a k-tuple of operators of finite type. Then for z ρ(λ j ). [R mj (z), R nj (z)] = 0, m, n, j = 1, 2,, k, (10) Proof Without loss of generality, we may assume that m = j = 1, n = 2. Define A i = w i Ti, i = 1, 2, B 1 = z T i for z ρ(t 1 ), w i ρ(t i ). Then [A i, B 1 ] = [T i, T 1 ] = C i1 P K, where P K is the projection from H to K. Thus [A 1 i, B1 1 ] = A 1 i B1 1 C i1p K B1 1 A 1 i. Therefore P K A 1 1 A 1 2 B 1 1 K = P K A 1 1 [A 1 2, B 1 1 ] K + P K A 1 1 B 1 1 A 1 2 K. (11) The right-hand side of (11) equals P K A 1 1 A 1 2 B 1 1 C 21P K B 1 1 A 1 2 K + P K [A 1 1, B 1 1 ]A 1 2 K + P K B 1 1 A 1 1 A 1 2 K = P K A 1 1 A 1 2 B 1 1 C 21(z 1 Λ 1 ) 1 (w 2 Λ 2) 1 + P K A 1 1 B 1 1 C 11P K B 1 1 A 1 1 A 1 2 K +(z 1 Λ 1 ) 1 (w 1 Λ 1) 1 (w 2 Λ 2) 1. (12) since A 1 i K = (w i Λ i ) 1 and P K B1 1 K = (z Λ 1 ) 1. On the other hand Thus P K A 1 1 B 1 1 K = P K A 1 1 B 1 1 C 11P K B 1 1 A 1 1 K + P K B 1 1 A 1 1 K. P K A 1 1 B 1 1 K(I C 11 (z Λ 1 ) 1 (w 1 Λ 1) 1 ) = (z Λ 1 ) 1 (w 1 Λ 1) 1. Therefore From (11), (12) and (13), we have Thus Hence P K A 1 1 B 1 1 K = Q 11 (w 1, z) 1. (13) P K (A 1 A 1 2 B 1 1 ) KQ 21 (w 2, z)(z Λ 1 ) 1 (w 2 Λ 2) 1 = (Q 11 (w 1, z) 1 C 11 + I)(z Λ 1 ) 1 (w 1 Λ 1) 1 (w 2 Λ 2) 1 = Q 11 (w 1, z) 1 (w 2 Λ 2) 1 P K (A 1 1 A 1 2 B 1 1 ) KQ 21 (w 2, z)(z Λ 1 ) 1 = Q 11 (w 1, z) 1. P K (A 1 1 A 1 2 B 1 1 ) K = (z Λ 1 ) 1 (w 1 R 11 (z)) 1 (w 2 R 21 (z)) 1. From [A 1, A 2 ] = 0, we have [(w 1 R 11 (z)) 1, (w 2 R 21 (z)) 1 ] = 0 which proves the lemma.

5 3. Right spectrum Subnormal tuple of operators 5 For a k-tuple of operators A = (A 1,, A k ) on Hilbert space H, let k sp r (A) = {(λ 1,, λ k ) C k : (A j λ j )(A j λ j ) is not invertible } j=1 be the right spectrum of A. It is obvious that sp r (A) sp jp (A ). Proposition 3.1 Let S = (S 1,, S k ) be a subnormal k-tuple of operators of finite type on H, with m.n.e. N = (N 1,, N k ). If λ = (λ 1,, λ k ) satisfies λ j σ(s j ) ρ(n j ), j = 1, 2,, k and λ sp r (S), then λ sp jp (S ). Proof If (λ 1,, λ k ) sp r (S), then there is a sequence {f m } H satisfying f m = 1, such that k lim (S j λ j )(S j λ j ) f m = 0, m j=1 since k j=1 (S j λ j )(S j λ j ) is self-adjoint. Thus lim (S j λ j ) f m = 0, j = 1, 2,, k. m since (S j λ j ) f m 2 ( j (S j λ j )(S j λ j ) f m, f m ). On the other hand from (8) (cf. [X1], [X3]) where (S j λ j )f m (u) = (u j λ j )f m (u) (u j Λ j )f m (Λ), f m (Λ) = sp(n) f m (u). We may choose a subsequence {f mn } of {f m } such that since f m (Λ) 1. Therefore, from and f mn (Λ) a M, as m n, f mn (u) = u j Λ j 1 f mn (Λ) + (Sj λ j )f mn (14) u j λ j u j λ j u j λ j dist(λ j, σ(n j )) > 0, for λ j ρ(n j ), it follows that as a sequence of vectors in H, {f mn } converges to g(u) = u j Λ j u j λ j a, j = 1, 2,, k. On the other hand, we have f mn (Λ) = µ j (λ j ) f mn (Λ) + u j λ j (S j λ j )f mn (u).

6 6 Daoxing Xia Thus a = µ j (λ j ) a, j = 1, 2,, k. But g = lim f mk = 1, therefore g is an eigenvector of S j corresponding to λ j, for j = 1, 2,, k, i. e. λ sp jp (S ). 4. Joint eigenvectors In order to study the joint eigenvectors for Sj and S k, we have to establish the following lemma. Lemma 4.1 Let S = (S 1,, S k ) be a pure subnormal k-tuple of operators on H with m.n.e. N = (N 1,, N k ). Let z l ρ(n l ), l = 1, 2,, n, n k and w ρ(n j ), then (u j Λ j ) n = (R n νl j(w) z l ) 1 µ j (w) + (u νl z l )(u j w) n m=1 (I + (z m m Λ νm )(R νmj(w) z m ) 1 ) (u νl z l ) n p=m+1 (R νpj(w) z p ) 1. (15) Proof For the simplicity of notation, we assume that ν l = l. Let us prove (15) by mathematical induction. First, let us prove it for n = 1. From (7) and (u i z)(u j w) = (I µ j(w) )Q ji (w, z) 1 Q ji (w, z) 1 µ i (z). (cf. [X1], [X3]). In [X1], it proves this identity in the case of i = j, but the method in the proof also applies to the case i j. Therefore, (w Λ j ) (u 1 z)(u j w) = (I µ j(w) )(z R 1j (w) ) 1 (z Λ 1 ) 1 (w R j1 (z)) 1 µ 1 (z)(w Λ j ), since (w Λ j )(z R ij (w) ) = Q ji (w, z), and (w R ji (z))(z Λ i ) = Q ji (w, z). But µ j (z)r ij (z) = R ij (z)µ j (z) (cf. [X1], [X3]). Again, in [X1] it only proves the case of i = j. But the proof may extend to the case of i j as well. We have (w Λ j ) From (u 1 z)(u j w) = (z R 1j (w) ) 1 (I µ j (w) ) (z Λ 1 ) 1 µ 1 (z)(z Λ 1 )(z R 1j (w) ) 1. (z Λ 1 ) 1 µ 1 (z) = (z Λ 1 ) 1 + u 1 z

7 Subnormal tuple of operators 7 we have (15) for n = 1. Suppose that (15) holds for n. Then from Lemma 2.1, (u j Λ j ) = + (w Λ n+1 n+1 n+1 j ) (u l z l )(u j w) (u l z l ) (u l z l )(u j w) and Qj(n+1) (w, z n+1 ) n+1 (u l z l )(u j w) ( (uj Λ j )(u n+1 z n+1 ) (u j w)(z n+1 Λ n+1 )) = n+1 (u l z l )(u j w) = it follows that = (u j Λ j ) (zn+1 Λ n+1 ), n n+1 (u l z l )(u j w) (u l z l ) (u j Λ j ) n+1 (u l z l )(u j w) (I (z n+1 n+1 Λ n+1 )(z n+1 R (n+1)j (w) ) 1 ) (u l z l ) (u j Λ j ) (z n n+1 R (n+1)j (w) ) 1. (u l z l )(u j w) which proves the lemma. Lemma 4.2 Under condition of Lemma 4.1, let w j σ(s j ) ρ(n j ), j = 1, 2. Suppose there are c ν C, ν = 1,, k and a vector a M satisfying and Let µ j (w j ) a = a, j = 1, 2 (16) R νj (w j ) a = c ν a, j = 1, 2; ν = 1, 2,, n. (17) f j (u) def = u j Λ j u j w j a, j = 1, 2

8 8 Daoxing Xia Then f j ( ) = f 2 ( ) as vectors in H and S j f j = w j f j, j = 1, 2. (18) Proof From (15), (16), (17) and Lemma 4.1, it is easy to calculate that f 1 (u) = f 2 (u), z l ρ(n l ), l = 1,, k. k k (u l z l ) (u l z l ) Let g(u) def = on the continuous part du ˆL of σ(n). Then by the Plemelj formula 1 g(u)f1 (u) = g(u)f 2 (u) n n (u l z l ) (u l z l ) l=2 for almost all points u 1 in L 1 = σ(s 1 ), where is ranging over a finite set of all points u in σ(n) with the same u 1. However z l is an arbitrary point in ρ(n l ), l = 2,, n. Therefore l=2 g(u)f 1 (u) = g(u)f 2 (u), for almost all u 1 L 1. Thus f 1 (u) = f 2 (u) for almost u ˆL. Similarly e({u})f 1 (u)= e({u})f 2 (u) for every point spectrum u in sp(n). Thus f 1 ( ) = f 2 ( ) as vectors in H. The formula (18) follows from (8) and (16). 5. Riemann surfaces and algebraic varieties associated with subnormal tuple of operators Let S = (S 1,, S k ) be a pure subnormal k-tuple of operators of finite type with m.n.e. N = (N 1,, N k ). Let O j σ(s j ) ρ(n j ), j = 1, 2,, k be non-empty open sets. Let Ŝj( ) be a univalent analytic function on O j, j = 1, 2,, k. Let O(O 1, Ŝ1,, O k, Ŝk) be defined as {z = (z 1,, z k ) C k : P mn (Ŝm(z m ), z n ) = 0 for z j O j, j = 1,, k}. (19) Let S 0 be the union of these neighborhoods {O(O 1, Ŝ1,, O k, Ŝk)}. On each O(O 1, Ŝ1,, O k, Ŝk), every function f(z j ), considered as a function on z = (z 1,, z j,, z k ) is a local coordinate, where f is an analytic univalent function on O j. Then S 0 is a union of domains in some Riemann surfaces. Proposition 5.1 The continuum part of sp(n) S 0, the boundary of S 0. Proof Let γ be a small arc in σ(n). Let γ j = {z j : (z 1,, z j,, z k ) γ} σ(n j ). Then there are domains O j σ(s j ) ρ(n j ) and D j ρ(s j ) such that D j γ j O j is a domain and γ j O j D j. From (9), P ij (z i, z j ) = 0, for (z 1,, z k ) γ. There are analytic functions Ŝj( ) on D j γ j O j satisfying S j (z) = z, z γ j. Actually these Ŝj( ) are branches of multivalued Schwarz functions associated with

9 Subnormal tuple of operators 9 those subnormal operators S j (cf. [AS], [X5], [X6], [X7]), j = 1, 2,, k. Thus P ji (Ŝj(z j ), z i ) = 0, (z 1,, z k ) γ. Therefore γ O(O 1, Ŝ1,, O k, Ŝk). Suppose P (w) = k j=0 p k jw j and Q = k j=0 q k jw j are polynomials with constant coefficients p j and q i. Assume that p 0 0, q 0 0. Let be the resolvent of P and Q, where R(P, Q) def = det(r ij ) i,j=1,2,,2k r ij = p j i for 1 i j k, r ij = q i k j for 1 i k j, k + 1 i 2k, and r ij = 0 for other pairs of i and j. It is well-known that if there is a common solution of P ( ) = 0 and Q( ) = 0, then R(P, Q) = 0. Consider the algebraic variety S a def = i j {(u 1,, u k ) C k : R(P ii (, u i ), P ij (, u j )) = 0}. It is easy to see the following. Lemma 5.1 S 0 S a. Let γ be a small arc in the sp(n) satisfying the condition that for each j the mapping γ γ j = {u j C : (u 1,, u j,, u k ) γ} is one to one and γ j is a simple arc (there is no node in γ j ). For each γ j there is a simply connected domain O j σ(s j ) ρ(n j ) such that γ j is in the boundary of O j and there is an analytic function Ŝj( ) on O j such that P jj (Ŝj(u), u) = 0, u O j and the boundary value of Ŝj( ) satisfying since P jj (z, z) = 0, z γ j. Let Ŝ j (z) = z, z γ j, O γ = O(O 1, Ŝ1,, O k, Ŝk) (20) defined in (19) by these {O j, Ŝj( )}. Let S be the union of those component in S 0 which contains some O γ as a subset. This Ŝj( ) in (20) is a branch of the Schwarz function related to the subnormal operator S j (cf. [X5], [X6], [X7], [Y1] and [Y2]). Thus the boundary of S must be also in sp(n). Therefore S = sp(n). Theorem 5.1 Let S be a pure subnormal k-tuple of operators of finite type, then S S a. Let f be an analytic function on a neighborhood of σ(s 1 ) σ(s k ). Define u j Λ j µ j (z; f) = u j z f(u), z ρ(n j) σ(n) as in [X2] and [X3]. Then µ j ( ; f) is an analytic function on ρ(n j ) and µ j (z; f) = 0, z ρ(s j ).

10 10 Daoxing Xia By Plemelj formula, the boundary value of µ j ( ; f) at the continuum part L j σ(n j ) is (u j Λ j ) f(u), du j where is ranging over a finite set of points of u sp(n) with the same j-th coordinate u j. Therefore, there are L(M)-valued analytic functions ˆµ j (u) on S 0 such that on the continuum part L of sp(n) It is easy to see that ˆµ j (u) = (u j Λ j ) du j. (u j Λ j ) 1 ˆµ j (u) = (u k Λ k ) 1 ˆµ k (u) du j du k, for u S. (21) Remark Let π be a linear function a 1 u 1 + +a k u k and S π = a 1 S 1 +, +a k S k (cf. [PZ]). We may choose k linear functions π 1, π 2,, π k which are linearly independent on C k such that if we replace S 1,, S k by S π1,, S πk respectively then ˆµ j (u) = µ j (u j ) for u S S and sp jp (S ) Let M u be the range of ˆµ j (u) for u S, which is well-defined by (21) and M u {0} except a finite set F S. Theorem 6.1 Let S = (S 1,, S k ) be a pure subnormal k-tuple of operators of finite type. Then there is a finite set F such that For w S \ F, and a M w \ {0}, the vector satisfies f 0 and Proof S \ F sp jp (S ). (22) f(u) = u j Λ j u j w j a, j = 1, 2,, k (23) S j f = w j f, j = 1, 2,, k. (24) For u in the continuum part of sp(n), µ j (u) = du j (u j Λ j ). (25) By (7) and (25), we may prove that ˆµ j (u) M µ j (u) M for a.e. u σ(n). Thus (I µ j (u j ) )ˆµ j (u) = 0 for a. e. u sp(n). Therefore (I µ j (u j ) )ˆµ j (u) = 0 for u S, since µ j ( ) and ˆµ j ( ) are analytic functions. Hence for a M u, w S µ j (w j ) a = a, for j = 1,, k.

11 Subnormal tuple of operators 11 On the other hand, from (9), we have (S ν (w ν ) R νj (w j ) )ˆµ m (w) = 0, w γ (26) for w γ and j, ν, m = 1, 2,, k where {S j ( )} are those functions in the proof of Proposition 5.1, since S ν (w ν ) = w ν for w γ. Thus (26) holds good for w O γ. By analytic continuation, (26) holds good for w S. Therefore (16) and (17) are satisfied if a M w and c ν = S ν (w ν ). From Lemma 4.2, it follows (23) and (24) which proves (22) and hence Theorem 6.1. Conjecture : sp jp (S ) \ S is a finite set. 7. Principal current. The short introduction of the principal current related to the present work can be seen in the introduction of [PX] or [PZ] and the papers [CP1], [CP2], [Pi]. Let us first list the following two lemmas. For a compact set σ C n, let A(σ) be the algebra generated by analytic functions f on a neighborhood of σ and its conjugate. Lemma 7.1 [PX] Let S be a pure subnormal k-tuple of operators of finite type with m.n.e. N. Then trace[f(s), h(s)] = 1 mfdh 2πi L for f, h A(sp(N)), where L is the union of a finite collection of closed curves and is also the union of a finite collection of algebraic arcs such that sp(n) is the union of L and a finite set. Furthermore, m(u) is an integer valued multiplicity function which is constant on the irreducible pieces (simple closed curves) of L. In this lemma, m(u) = 2πi trace((u j Λ j ) du j ), for u L. Thus we have the following Lemma 7.2 The function m( ) defined in Lemma 7.1 is the boundary value of the function m(u) def = dim M u = rank ˆµ j (u), u S. Lemma 7.3 [PZ] Suppose that S is a pure subnormal k-tuple of finite tuple with m.n.e. N. The principal current of S can be represented as l(fdh) = i trace[f(s), h(s)] l(fdh) = 1 2π l C l m l fdh where {C l } is the collection of cycles sp ess (S) and the weights m l are spectral multiplicity of N at any regular point ζ of C l.

12 12 Daoxing Xia In this lemma, m l is the value of function m(u) for u C l. By the fact that sp(n) is the boundary of S, Cartan s formula dω = ω, σ Lemma and the fact that the non-negative integer valued function m( ) is piece-wise constant. We have the following Theorem 7.1 Let S = (S 1,, S k ) be a pure subnormal k-tuple of operators of finite type. Then for f, h A(σ(S 1 ) σ(s k )), i trace[f(s), h(s)] = 1 mdf dh, 2π where m( ) = rank ˆµ j ( ). Thus m( ) can be considered as an extension of the concept of Pincus principal function from subnormal operators to subnormal k-tuple of operators. σ S 8. Operator identities. It is well-known that the unilateral shift U + satisfies the identity U +U + I = 0. (27) This identity also characterizes the unilateral shift. In this section we will introduce some operator identities for the subnormal tuple of operators of finite type which are the generalization of (27). Let p(w, z) = p mn w m z n be a polynomial and A and B be operators. We adopt the Weyl ordering p(a, B) def = p mn A m B n. Theorem 8.1 Let S = (S 1,, S k ) be a subnormal k-tuple of operators of finite type. Let P ij (u i, u j ) be the polynomial defined in (6) and R(P ii (, u i ), P ij (, u j )) be the polynomial of u i, u j defined in 5. Then for i, j = 1, 2,, k, P ij (S i, S j ) = 0, (28) and Proof Suppose R(P ii (, S i ), P ij (, S j )) = 0. (29) P ij (w, z) = m,n p mn w m z n.

13 Subnormal tuple of operators 13 Then by (9), P ij (u i, u j ) = 0 for (u 1,, u k ) sp(n). Thus for f, g H (P ij (Si, S j )f, g) = p mn (Si m S n j f, g) = p mn (S n j f, S m i g) = p mn (u n j f(u), u m i g(u)) = sp(n) sp(n) (P ij (u i, u j )f(u), g(u)) = 0, which proves (28). Similarly, we may prove (29), since R(P ii (, u i ), P ij (, u j )) = 0 for (u 1,, u k ) sp(n). References [AS] D. Aharonov and H. S. Sapiro, Domains on which analytic functions satisfy quadrature identities, J. Anal. Math. 30 (1976), [Co1] J. B. Conway, Theory of Subnormal Operators, Math. Surv. Mon. 36 (1991). [Co2] J. B. Conway, Towards a functional calculus for subnormal tuples: the minimal normal extension, 138 (1977), [CP1] R. W. Carey and J. D. Pincus, Principal currents, Integr. Equ. Oper. Theory 8 (1985), [CP2] R. W. Carey and J. D. Pincus, Reciprocity for Fredholm operators, Integr. Equ. Oper. Theory 90 (1986), [Cu1] R. E. Curto, Connections between Harte and Taylor spectra, Rev. Roum. Math. Pure Appl. 31 (1986), [Cu2] R. E. Curto, Joint hyponormality: a bridge between hyponormality and subnormality, Pro. Symp. Pure Math. 5 (1990), [EP] J. Eschmeier and M. Putinar, Some remarks on spherical isometries, in vol. Systems, Approximation, Singular Integral Operators and Related Topics (A. A. Borichev and N. K. Nikolskii, eds), Birkhauser, Bassel et al., 2001, [Pi] J. D. Pincus, The principal index, Proc. Symp. Pure Math. 51 (1990), [Pu] M. Putinar, Spectral inclusion for subnormal n-tuples, proc. Amer. Math. Soc. 90 (1984),

14 14 Daoxing Xia [PX] J. D. Pincus, D. Xia, A trace formula for subnormal tuples, Integr. Equ. Oper. Theory. 14 (1991), [PZ] J. D. Pincus, D. Zheng, A remark on the spectral multiplicity of normal extensions of commuting subnormal operator tuples, Integr. Equ. Oper. Theory 16 (1993), [T1] J. L. Taylor, A joint spectrum for several commuting operators, Jour. Functional Analysis 6 (1970), [T2] J. L. Taylor, The analytic functional calculus for several commuting operators, Acta Mathematica 125 (1970), [X1] D. Xia, On the analytic model of a subnormal operator, Integr. Equ. Oper. Theory 10 (1987), [X2] D. Xia, Analytic theory of subnormal operators, Integr. Equ. Oper. Theory 10 (1987), [X3] D. Xia, Analytic theory of a subnormal n-tuple of operators, Proc. Symp. Pure Math. 51 (1990), [X4] D. Xia, Trace formulas for a class of subnormal tuple of operators, Integr. Equ. Oper. Theory 17 (1993), [X5] D. Xia, On pure subnormal operators with finite rank self-commutators and related operator tuples, Integr. Equ. Oper. Theory 24 (1996), [X6] D. Xia, On a class of operators with finite rank self-commutators, Integr. Equa. Oper. Theory 33 (1999), [X7] D. Xia, Trace formulas for some operators related to quadrature domains in Riemann surfaces, Integr. Equ. Oper. Theory 47 (2003), [X8] D. Xia, On a class of hyponormal operators of finite type, to appear in Integr. Equ. Oper. Theory. [Y1] D. V. Yakubovich, Subnormal operators of finite type I, Xia s model and real algebraic curves, Revista Matem. Iber. 14 (1998), [Y2] D. V. Yakubovich, Subnormal operators of finite type II, Structure theorems, Revista Matem. Iber. 14 (1998), Daoxing Xia Department of Mathematics Vanderbilt University Nashville, TN USA daoxingxia@netscape.net