Search

This paper introduces two main concepts, called a generalized Watson transform and a generalized skewWatson transform, which extend the notion of a Watson transform from its classical setting in one variable to higher dimensional and noncommutative situations. Several construction theorems are proved which provide necessary and sufficient conditions for an operator on a Hilbert space to be a generalized Watson transform or a generalized skew-Watson transform. Later papers in this series will treat applications of the theory to infinite-dimensional representation theory and integral operators on higher dimensional spaces.

In this paper, we prove that if S is an n-dimensional subspace of L(H), then S is ([i ] + 1)-reflexive, where [n ] denotes the greatest integer not n larger than '. By the result, we show that if 1?( ) = E Ai( )Bi is an 2=1 elementary operator on a C*-algebra A, then 'D is completely positive if and only if 'D is ([n1 ] + 1)-positive. In this paper, let H denote a complex Hilbert space. Let H(') denote the direct sum of n copies of H. For T E L(H), we write T(') for the operator on H(') which is the direct sum of n copies of T; the notation is extended to a subset of L(H) by S(n) = {T(n) E L(H(n)): T E S}. If S is a subspace of L(H), S is called n-reflexive if S(n) = ref (S(n)) =_ {T(n) E L(H(n)): T(n)X E [S(n)X], for all x E H (n)}, where [] denotes norm closed linear span. By the definiton, we have that if S is m-reflexive, then S is n-reflexive for n > m. A separating vector for a subspace S of L(H) is a vector x E H such that T 4 Tx, T E S, is an invective map. For x, y E H, let x 0 y denote the rank-one operator u | 4 (u, x)y. Let A denote a C*-algebra. Then A is called primitive, if A has a faithful irreducible representation on some Hilbert space. An elementary operator AP on A n is a linear mapping of the form AP: T F4 AiTBi, where {Ai},nL1 and {Bi},nU1 are i=l subsets of A. In this paper, we assume that all elementary operators are nonzero. A linear map 4J on A is positive (resp. hermitian-preserving) if 4)(T) is positive (resp. hermitian) for all positive (resp. hermitian) T in A. We define 4'n = 4{I 0 In: Mn(A) -4 Mn(A) by 4) 0 In((Tij)nxn) = (4)(Ti ))nxn. 4) is said to be n-positive if 4J 0 In is positive. If 4J is n-positive for all n, then 4J is said to be completely positive. In [4], Magajna states the following problem: For each positive integer r determine the smallest k = k(r) such that all rdimensional subspaces of L(H) are k-reflexive. In [4], Magajna proves k < r. In this paper, we prove that if S is an n-dimensional subspace of L(H), then S is ([n] + 1)-reflexive. Also by this result, we study complete positivity of elementary operators on a C*-algebra A. We prove that if n 4 ) = Ai( )Bi is an elementary operator on a C*-algebra A, then 4) is i=l1 Received by the editors July 8, 1996 and, in revised form, May 14, 1997. 1991 Mathematics Subject Classification. Primary 47B47, 47B49; Secondary 46L05.

We present a new proof for Morita equivalence of twisted crossed products by coactions within the abstract context of crossed products of Hilbert C∗-modules. In this context we are free from representing all C∗-algebras and Hilbert C∗-modules on Hilbert spaces. The notion of Morita equivalence of twisted coactions was introduced in [B]. In [B, Theorem 3.3] we established conditions on twisted coactions which are sufficient to ensure Morita equivalence of the corresponding crossed product C∗-algebras. Later [ER] gave a shorter proof for this result using their results on multipliers of imprimitivity bimodules. However in the proofs of both [B] and [ER], all C∗algebras and Hilbert C∗-modules need to be represented on Hilbert spaces. In this paper we present a new proof for [B, Theorem 3.3] based on the notion of crossed products of Hilbert C∗-modules introduced in [B2]. Crossed products of Hilbert C∗-modules in [B2] were defined as subspaces of adjointable operators between Hilbert C∗-modules. In this abstract context, we are free from representing all C∗-algebras and Hilbert C∗-modules on Hilbert spaces as in [B] and [ER]. As a consequence, the proof here is shorter and more elegant than that of [B]. Our approach is close to the spirit of [BS], and different from [ER]. Throughout this paper G is a locally compact group and N is a closed normal amenable subgroup of G. Recall from [M, Lemma 3] that there is a surjective homomorphism Ψ from C∗ r (G) into C ∗ r (G/N) such that Ψ(λ (r)) = λ (qN (r)), where q N : G → G/N is the quotient map, λ and λ are the left regular representations ofG andG/N . We denote by WG the unitary operator on L 2(G×G) defined by [WGξ](r, s) = ξ(r, r −1s). If f is an element of the Fourier algebra A(G), then Sf (WG) = Mf . Here Sf denotes the slice map, see [LPRS, §1]. To apply [B2, Theorem 1.6] to this paper, we need to show that WG is a regular multiplicative unitary. For any ξ, η ∈ L(G), we define ωη,ξ = 〈Tξ|η〉, ∀T ∈ B(L(G)). Then for any ω = ωη,ξ, we have 〈(id⊗ ω)(WG)ξ′|η′〉 = 〈Mω◦λGξ′|η′〉, ∀ξ′, η′ ∈ L(G). It then follows that ŜWG = C0(G), and the crossed product of [B2, Proposition 1.5] is just the crossed product of [LPRS, Definition 2.4]. The unitary operator Received by the editors October 23, 1995 and, in revised form, February 6, 1996. 1991 Mathematics Subject Classification. Primary 46L05, 22D25. c ©1997 American Mathematical Society 2109 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

In this paper we present a theorem which extends the results of an inequality originally due to Franz Rellich [4]. The theorem by Rellich establishes an inequality widely used in the spectral theory of partial differential operators. Our theorem allows for a broader range of application by extending the class of functions to which the theorem is applicable. Many authors call upon inequalities similar to the one established in our theorem in dealing with problems concerning essential self-adjointness of Schrodinger operators and other problems arising in oscillation theory of elliptic operators. In the first part of the paper we present Rellich's inequality and discuss some problems dealing with symmetric operators on Hilbert spaces where Rellich's inequality is a useful tool. We shall also discuss some important extensions of Rellich's work which were established by other mathematicians. One such extension was proved by W. Allegretto [1] in dealing with elliptic equations of order 2n. Another extension was established by U. W. Schmincke [5] in considering essential selfadjointness criteria of Schrodinger operators. Schmincke's extension is of particular interest to us due to his elegant proof. We follow Schmincke's method of proof. We then state and prove our generalization of Rellich's inequality along with a useful corollary. The paper concludes with a few brief comments on our result and other work which could be done with Rellich's inequality. RELLICH'S INEQUALITY Define the operator T by Tu = u cjxj4U on L 2(R n) with domain 92J(T) = CO(Rn\{O}). Then T is symmetric on 9(T). As Rellich mentions in [4] his inequality can be viewed as a means of finding the largest positive real number c such that the eigenvalues for the Friedrich's extension of T are nonnegative. Also, this could be viewed as finding the least point in the spectrum of Mu, u e CO (Rn\{O}), in the weighted Received by the editors August 5, 1987 and, in revised form, August 18, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 47F05; Secondary 26D10, 35P15, 47B25, 35J10.

This paper extends the following result of R. F. Olin and J. E. Thomson: A compact subset K K of the plane is the spectrum of an irreducible subnormal operator if and only if R ( K ) \mathcal {R}(K) has exactly one nontrivial Gleason part G G such that K K is the closure of G G . The main result of this paper is that the only additional requirement needed for the pair { K , K e } \left \{ {K,{K_e}} \right \} to be the spectrum and essential spectrum, respectively, is that K e {K_e} be a compact subset of K K which contains the boundary of K K . Additionally, results are obtained on the question of which index values can be specified on the various components of the complement of K e {K_e} .

Suppose that H is a finite dimensional discrete quantum group and K is a Hilbert space. This paper shows that if there exists an action γ of H on L(K) so that L(K) is a modular algebra and the inner product on K is H-invariant, then there is a unique C*-representation 9 of H on K supplemented by the γ. The commutant of θ(H) in L(K) is exactly the H-invariant subalgebra of L(K). As an application, a new proof of the classical Schur-Weyl duality theory of type A is given.

Let T1,...,T, C B(H) be bounded operators on a Hilbert space NH such that TiT1 + . + TnTnT < I-H. Given a symmetry j on NH, i.e., j2 = j*j = I-, we define the j-symmetric commutant of {Ti,...,Tn to be the operator space {A C B(N): TiA = jATi, i = 1, ..., n}. In this paper we obtain lifting theorems for symmetric commutants. The result extends the Sz.-Nagy-Foias commutant lifting theorem (n = 1, j = I-H), the anticommutant lifting theorem of Sebestyen (n 1, j = -Ia), and the noncommutative commutant lifting theorem (j = Ia). Sarason's interpolation theorem for H' is extended to symmetric commutants on Fock spaces. 1. LIFTING THEOREM FOR SYMMETRIC COMMUTANTS Let F+ be the unital free semigroup on n generators s,... ., sr, and let e be its neutral element. For any o... sik E F+, we define its length 11 := k, and lel = 0. On the other hand, if Ti E B(H), i = 1,.. ., n, we denote T, :Til ... and Te := IH. Let us recall from [Pol], [Po2], and [Po4] a few results concerning the noncommutative dilation theory for n-tuples of operators. A sequence of operators T := [Ti,... , T], Ti E B(H), i = 1, ... , n, is called contractive (or row contraction) if TIT1* + *.. + TnTn < I-. We say that a sequence of isometries V := [VI, ... , 1V] on a Hilbert space IC D H is a minimal isometric dilation of T if the following properties are satisfied: (i VI VI* + + Vn Vn* < IK; (ii Vi* I = Ti* I i = 1, . . n; (iii) IC = VaEF+ VaH. Consider the full Fock space on n generators F2(Hn) := ClE Q Hm Secondary 30E05. The author was partially supported by NSF Grant DMS-9531954. ?2000 American Mathematical Society 1705 This content downloaded from 207.46.13.103 on Thu, 20 Oct 2016 04:32:19 UTC All use subject to http://about.jstor.org/terms

This paper deals with locatedness of convex subsets in inner product and Hilbert spaces which plays a crucial role in the constructive validity of many important theorems of analysis.

Let H H be a complex Hilbert space and let B ( H ) B(H) be the algebra of all bounded linear operators on H H . For c = ( c 1 , … , c k ) c=(c_{1},\dots ,c_{k}) , where c 1 ≥ ⋯ ≥ c k > 0 c_{1}\ge \cdots \ge c_{k}>0 and p ≥ 1 p\ge 1 , define the ( c , p ) (c,p) -norm of A ∈ B ( H ) A\in B(H) by \[ ‖ A ‖ c , p = ( ∑ i = 1 k c i s i ( A ) p ) 1 p , \|A\|_{c,p}=\left (\sum _{i=1}^{k} c_{i} s_{i}(A)^{p}\right )^{\frac {1}{p}} , \] where s i ( A ) s_{i}(A) denotes the i i th s s -numbers of A A . In this paper we study some basic properties of this norm and give a characterization of the extreme points of its closed unit ball. Using these results, we obtain a description of the corresponding isometric isomorphisms on B ( H ) B(H) .

Two of the most useful inequality formulas for bounded linear operators on a Hilbert space are the Ldwner-Heinz and Reid's inequalities. The first inequality was generalized by Furuta (so called the Furuta inequality in the literature). We shall generalize the second one and obtain its related results. It is shown that these two generalized fundamental inequalities are all equivalent to one another. 1. NOTATIONS AND INTRODUCTION Throughout the paper we use capital letters to denote bounded linear operators acting on a Hilbert space H. T is positive (written T > 0 ) in case (Tx, x) > 0 for all x EH. If S and T are Hermitian, we write T > S in case T-S > O. T = U I T I is the polar decomposition of T with U the partial isometry, whereas I T I is the positive square root of the operator T*T, and U*U is the initial projection. Let I denote the identity operator. Under the polar decomposition of T we recall a well-known relation: I T* Iq= U I T Iq U* for q > 0 [1, p. 752]. The aim of this article is to define and characterize a generalized Reid inequality. Consequently, related inequalities and improved inequalities are given. Recall that if S > 0 and SK is Hermitian, then for x EH, I (SKx, x) I? 11 K 11 (Sx, x) is Reid's inequality [8], and the inequality was sharpened by Halmos [4, pp. 51, 244], where II K II is replaced by the spectral radius of K. Let a, ,3 E [0, 1] with a + ,3 > 1. For x,y EH let us call I (SK I SK Ic+,-1 x,y) I'll K IIc+/31 Sax SI y I an extended Reid's inequality. We shall call, for an obvious reason, the inequality (2) in Theorem 1 below a generalized Reid inequality, whereas (1) is the well-known altered Furuta inequality [2]. Our main result is characterizations of this inequality.

Let 0 < p, q, r E R be real numbers with p + 2r < (1 + 2r)q and 1 < q. Furuta (1987) proved that if bounded linear operators A, B E B(H) P+2r 1 on a Hilbert space H satisfy 0 < B < A, then B q < (BrAPBr) q. This inequality is called the Furuta inequality and has many applications. In this paper, we prove that the Furuta inequality holds in a unital hermitian Banach *-algebra with continuous involution. Let A, B be bounded linear operators on a Hilbert space H. The celebrated Lbwner-Heinz inequality states the following; Theorem A (L6wner-Heinz inequality [4], [5]). Let A, B E B(H) satisfy 0 < B < A. If 0 < p < 1, then BP < AP. For an extension of Theorem A, Furuta obtained the following interesting inequality in [1] and one page elementary proof in [2]. Theorem B (Furuta inequality [1], [2]). Let 0 < p, q, r G R and A, B E B(H) satisfy 0 < B < A. If p + 2r < (1 + 2r)q and 1 < q as shown in the figure, then B q < (BrAPBr) q.

In this paper, we show that a pseudo-differential operator associated to a symbol a ∈ L ∞ ( R × R , L ( H ) ) a\in L^{\infty }(\mathbb {R}\times \mathbb {R},\mathcal {L}(H)) ( H H being a Hilbert space) which admits a holomorphic extension to a suitable sector of C \mathbb {C} acts as a bounded operator on L 2 ( R , H ) L^{2}(\mathbb {R},H) . By showing that maximal L p L^{p} -regularity for the non-autonomous parabolic equation u ′ ( t ) + A ( t ) u ( t ) = f ( t ) , u ( 0 ) = 0 u’(t) + A(t)u(t) = f(t), u(0)=0 is independent of p ∈ ( 1 , ∞ ) p\in (1,\infty ) , we obtain as a consequence a maximal L p ( [ 0 , T ] , H ) L^{p}([0,T],H) -regularity result for solutions of the above equation.

Let A , B ∈ B ( H ) A, B \in B(H) be bounded linear operators on a Hilbert space H H satisfying O ≤ B ≤ A O\leq B\leq A . Furuta showed the operator inequality ( A r B p A r ) 1 q ≤ (A^{r}B^{p}A^{r})^{\frac {1}{q}}\leq A p + 2 r q A^{\frac {p+2r}{q}} as long as positive real numbers p , q , r p , q, r satisfy p + 2 r ≤ ( 1 + 2 r ) q p+2r\leq (1+2r)q and 1 ≤ q 1\leq q . In this paper, we show this inequality is valid if negative real numbers p , q , r p, q, r satisfy a certain condition. Also, we investigate the optimality of that condition.

In this paper we solve a mapping problem for a particular class of Hilbert modules over an algebra multipliers of a diagonal Nevanlinna-Pick (NP) kernel. In this case, the regular representation provides a multiplier norm which induces the topology on the algebra. In particular, we show that, in an appropriate category, a certain class of Hilbert modules are projective. In addition, we establish a commutant lifting theorem for diagonal NP kernels.

The purpose of this paper is to prove that a completely positive projection on a Hilbert space associated with a standard form of a von Neumann algebra induces the existence of a conditional expectation of the von Neumann algebra with respect to a normal state, and we consider the application to a standard form of an injective von Neumann algebra.

The purpose of this paper is to show the following: Let 0 > p > 1 2 0 > p > \frac {1}{2} . If T is a p-hyponormal operator on a Hilbert space, then \[ ‖ ( T ∗ T ) p − ( T T ∗ ) p ‖ ≤ p π ∬ σ ( T ) ρ 2 p − 1 d ρ d θ . \left \| {{{({T^ \ast }T)}^p} - {{(T{T^ \ast })}^p}} \right \| \leq \frac {p}{\pi }\iint _{\sigma (T)} {{\rho ^{2p - 1}}d\rho d\theta }. \] That is, Putnam’s inequality holds for p-hyponormal operators.

In this paper we establish a surprising new identity for Parseval frames in a Hilbert space. Several variations of this result are given, including an extension to general frames. Finally, we discuss the derived results.

This paper focuses on a multiplicative property of the Berezin symbol A, of a given linear map A: Z P R, where Z is a functional Hilbert space of analytic functions. We show AB = AB for all B in R'(R) if and only if A is a multiplication operator Mp , where (p is a multiplier. We also present a version of this result for vector-valued functional Hilbert spaces.

The spectral representation for a group of unitary operators acting on a Hilbert space where the parameter set is a separable real Hilbert space is obtained. The usual spectral representation of such a group of unitary operators is when the parameter set is a locally compact abelian group (Stone's theorem). The main result used in the proof is the Bochner theorem on the representation of positive definite functions on a real Hilbert space. Introduction. This paper discusses a spectral representation theorem for a group of unitary operators acting on a Hilbert space H, where the parameter set is a real Hilbert space T. The usual spectral representation of such a group of unitary operators (the well-known Stone theorem) is when the set T is a locally compact abelian group [3, p. 392]. The main result used in the course of our work is the Bochner theorem on the representation of positive definite functions on a real Hilbert space T. This theorem is included in the interesting work of L. Gross [2] in Harmonic analysis on Hilbert space. 1. Preliminaries. We begin this section by introducing some notation which will be used later. T will be a real Hilbert space. We will use s, t, etc. for points in T. By a Borel set in T we shall mean a set in the ?-algebra determined by the collection of open sets in T. In T the inner product and norm will be denoted by ( , .)T and I-IT. A complex-valued function T on T will be called positive definite if for every finite set tl, * * *, tn of points in T the matrix (ais) defined by aij = T (ti tj) is nonnegative definite. The following definition is due to L. Gross [2, p. 5]. 1.1 DEFINITION (TOPOLOGY iFOR T). The topology of iis defined as the weakest topology for which all Hilbert-Schmidt operators on Tinto T are continuous from zto the strong topology of T. Received by the editors April 26, 1971. AMS 1970 subject classifications. Primary 43A35, 47D10; Secondary 60G25.

The double commutant of an algebraic operator on a complex Hilbert space is equal to the algebra (with identity) generated by that operator. Introduction. Let fl be an algebra over some field, and let S? be a subset of t. The commutant of 5 in W is the set S?' of all elements of Qi which commute with every element of Yf. The double commutant of Y? in W is the commutant of Y?' in W. If 9Y consists of a single element s, we shall speak of the commutant and double commutant of s rather than of {s}. Let A be a linear transformation on a vector space. We say that A is algebraic if there exists a polynomial p such that p(A) = 0. In this paper we shall prove that if X' is a complex Hilbert space, and A belonging to B(Q#) (the algebra of all bounded linear' transformations on JX) is algebraic, then the double commutant of A in B(,#') is equal to the algebra generated by A. This generalizes the well-known analogous result for finite-dimensional spaces (see [2, p. 113]), since any linear transformation on a finitedimensional vector space is algebraic. It is already known (see [3 p. 72]) that if T is an algebraic linear transformation on a vector space I then the double commutant of T in L('f) (the algebra of all linear transformations on f) is equal to the algebra generated by T. This result does not imply our result for B(JV') however, since in general L(X#) is far larger than B(Q#). In the forthcoming the word operator will mean bounded linear transformation, and Hilbert space will mean complex Hilbert space. 1. Notation. If A is an algebraic operator on a Hilbert space JX', we denote by WA the algebra (with identity) generated by A in B(J). REMARK. In discussing operators on Hilbert space one often speaks of the weakly closed algebras which they generate. Since the algebra generated by an algebraic operator is finite dimensional it is closed in all of the usual Received by the editors September 29, 1971. AMS 1970 subject classifications. Primary 46L15, 47B99.