On completely hyperexpansive operators

Abstract

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-

Taylor spectrum approach to Brownian-type operators with quasinormal entry

“…For this, note that (T 1 , T 2 ) ⊆ ℝ × {0} ∪ {0} × ℝ if and only if p( (T 1 , T 2 )) = 0, where p(s, t) = s ⋅ t. Hence, applying Theorem 2.1(ii) gives the former equivalence in (11). The latter is a matter of routine verification.…”

“…Following [44], we call T ′ the Cauchy dual operator of T. Recall that if T is left-invertible, then so is T ′ and T = (T � ) � . Athavale noticed that the Cauchy dual operator of a completely hyperexpansive injective unilateral weighted shift is a subnormal contraction (see [11,Proposition 6] with t = 1 ), but not conversely (see [11,Remark 4]). The Cauchy dual subnormality problem asks whether the Cauchy dual operator of a completely hyperexpansive operator (see Sect.…”

“…This means that Brownian isometries cannot be characterized by the Taylor spectrum (|Q|, |E|). The above-mentioned concepts can be attributed to many authors, such as Agler [1] (m-contractivity), Richter [42] (2-expansivity), Aleman [6] (complete hyperexpansivity for special operators), Agler [2] (m-isometricity) and Athavale [11] (m-expansivity and complete hyperexpansivity). It is well known that a 2-isometry is m-isometric for every integer m ⩾ 2 (see [3][4][5]Paper I,§1]…”

Hyperexpansive composition operators

“…It is well known that subnormal operators are closely related to the theory of positive definite functions on the abelian semigroup (N, +, n * = n), while completely hyperexpansive operators to negative definite functions on (N, +, n * = n) (cf. [2,23]). More precisely, a bounded operator T is a subnormal contraction (resp.…”

“…[9,12,21,24]) we want to add characterizations of k-isometric, completely hyperexpansive and khyperexpansive composition operators. Hyperexpansive operators have already been studied in [1,2,10,11,17,20]. Note also that the class of 2-hyperexpansive operators has "small" intersections with some other known classes of operators.…”