ABSTRACT. A notion of joint hyponormality is introduced for a collection of bounded linear operators on a separable Hubert space.It is the purpose of this note to introduce a notion of joint hyponormality for a collection of bounded linear operators on a separable Hubert space ßtf in a way that will meet the following conditions.(a) The notion of joint hyponormality will be in some sense a natural generalization of the notion of hyponormality for a single operator.(b) The notion will be at least as strong as requiring that the linear span of a given collection of operators consists of hyponormal operators.(c) The notion will relate in a reasonable way to the questions pertaining to commuting normal extensions of commuting operators.In some ways, the present note continues the investigations started in [3], but in addition it paraphrases the statements of some known results and open problems in subnormal operator theory in terms of joint hyponormality.We begin by fixing some notation. The set of bounded linear operators on a Hubert space %? will be denoted by 3 §(%f), while J^'"1' will stand for the direct sum of ßf with itself m times. An mxm operator matrix sf = (Ay) will be the matrix with the operator Ay as the (ij)tb entry. The commutator AB -BA of two elements A and B in 3h'(ßf) will be denoted by [A, B]
Let D be a strictly pseudoconvex bounded domain in C m with C 2 boundary ∂D. If a subnormal m-tuple T of Hilbert space operators has the spectral measure of its minimal normal extension N supported on ∂D, then T is referred to as a ∂D-isometry. Using some non-trivial approximation theorems in the theory of several complex variables, we establish a commutant lifting theorem for those ∂D-isometries whose (joint) Taylor spectra are contained in a special superdomain Ω of D. Further, we provide a functiontheoretic characterization of those subnormal tuples whose Taylor spectra are contained in Ω and that are quasisimilar to a certain (fixed) ∂D-isometry T (of which the multiplication tuple on the Hardy space of the unit ball in C m is a rather special example). Mathematics Subject Classification (2000). Primary 47B20; Secondary 30E10, 32A30.
Abstract. We introduce and discuss a class of operators, to be referred to as the class of completely hyperexpansive operators, which is in some sense antithetical to the class of contractive subnormals. The new class is intimately related to the theory of negative definite functions on abelian semigroups. The known interplay between positive and negative definite functions from the theory of harmonic analysis on semigroups can be exploited to reveal some interesting connections between subnormals and completely hyperexpansive operators.If H is a complex infinite-dimensional separable Hilbert space, then B(H) will denote the set of bounded linear operators on H. Recall that S in B(H) is said to be subnormal if there exist a Hilbert space K containing H and a normal operator N in B(K) such that N H ⊂ H and N/H = S. For all of the elementary results pertaining to subnormals (and related classes of operators such as hyponormals) that are stated below without proof, the reader is referred to [Co]. If {e n } n≥0 is an orthonormal basis for H, then a weighted shift operator T on H with the weight sequence {α n } n≥0 is defined through the relations T e n = α n e n+1 (n ≥ 0). We will always assume that α n > 0 for all n. An excellent reference for the basic properties of weighted shifts is [S]. We will often use the notation T : {α n } to indicate a weighted shift. Of particular interest to us are the shifts U : {1} (the unilateral shift), B : { √ n + 1/ √ n + 2} (the Bergman shift) and D : { √ n + 2/ √ n + 1} (the Dirichlet shift). The shifts U and B are subnormal with U in fact being an isometry; while D is a 2-isometry (I − 2D * D + D * 2 D 2 = 0, I and 0 being the identity and zero operator respectively).It is the purpose of this note to introduce and discuss a class of operators that is in some sense antithetical to the class of contractive subnormals. The connections between the theory of positive definite functions on abelian semigroups and the theory of subnormals are well-known and will be touched upon briefly in the sequel. The new class of operators, to be referred to as the class of completely hyperexpansive operators, is intimately related to the theory of negative definite functions on abelian semigroups. The shift U and the shift D are rather special examples of the new class. The known connections between positive and negative definite functions will allow us to correlate U and D as well as B and D in meaningful ways. Using the idea of the Laplace Transform, we will be able to associate with every hyperex-
Necessary and sufficient conditions in terms of operator polynomials are obtained for an m-tuple 7" = (Tx,...,Tm) of commuting bounded linear operators on a separable Hubert space JÉ" to extend to an m-tuple S = (Sx,.. .,Sm) of operators on some Hubert space X, where each S, is realized as a *-representation of the adjoint of a multiplication operator on the tensor product of a special type of functional Hubert spaces. Also, necessary and sufficient conditions in terms of operator polynomials are obtained for T to have a commuting normal extension. 0. Introduction. In this paper, some results in [1 and 2] for a single bounded linear operator T on a separable Hilbert space 34" are generalized to m commuting operators on 34?. In [1], Agler introduces a special class of functional Hilbert spaces M and describes conditions under which an operator T on 34? extends to M*(oc), where M denotes the multiplication operator on M, and M*(oo) denotes the countable direct sum of M* with itself. Special cases of spaces Ji', of which the classical Hardy space is the prototype, are considered in [1,2]. The relevant conditions for the kind of extension of T referred to above are expressed in terms of the positivity of certain operator polynomials involving T and T*. These conditions are closely related to the reproducing kernel associated with the space Ji'. In [2], necessary and sufficient conditions are also given for a contraction T to be subnormal. These conditions are really the requirement that a certain sequence of polynomials in T and T* be positive. It is natural to seek generalizations of these results to m commuting operators Tx,...,Tm on 347. An appropriate model for this generalization is obtained by constructing a finite tensor product of spaces Ji and exploiting the well-known fact that the reproducing kernel of such a tensor product is the product of the reproducing kernels of the individual spaces. Suitable modifications of the reproducing kernels which render them holomorphic on the unit polydisc can be used to describe analogous extension results for T = (7\,..., Tm). The question of T having a commuting normal extension N = (Nx,..., Nm) turns out to have a direct link with the multi-dimensional Hausdorff Moment Problem from the theory of probability.
ABSTRACT. A notion of joint hyponormality is introduced for a collection of bounded linear operators on a separable Hubert space.It is the purpose of this note to introduce a notion of joint hyponormality for a collection of bounded linear operators on a separable Hubert space ßtf in a way that will meet the following conditions.(a) The notion of joint hyponormality will be in some sense a natural generalization of the notion of hyponormality for a single operator.(b) The notion will be at least as strong as requiring that the linear span of a given collection of operators consists of hyponormal operators.(c) The notion will relate in a reasonable way to the questions pertaining to commuting normal extensions of commuting operators.In some ways, the present note continues the investigations started in [3], but in addition it paraphrases the statements of some known results and open problems in subnormal operator theory in terms of joint hyponormality.We begin by fixing some notation. The set of bounded linear operators on a Hubert space %? will be denoted by 3 §(%f), while J^'"1' will stand for the direct sum of ßf with itself m times. An mxm operator matrix sf = (Ay) will be the matrix with the operator Ay as the (ij)tb entry. The commutator AB -BA of two elements A and B in 3h'(ßf) will be denoted by [A, B]
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