Some Quadratically Hyponormal Weighted Shifts

In the study of the gaps between subnormality and hyponormality both quadratic hyponormality and the related property positive quadratic hyponormality have been considered, especially for weighted shift operators. In particular, these have been studied for shifts with the first two weights equal and with Bergman tail or recursively generated tail. In this article, we characterize the allowed first two equal weights for quadratic hyponormality with Bergman tail, and the allowed first two equal weights for positive quadratic hyponormality with recursively generated tail.      

Disintegration of Measures and Contractive 2-Variable Weighted Shifts

In this paper we give a new proof of the existence of disintegration measures using the Hausdorff Moment Problem on a Borel measurable space X × Y, where X ≡ Y is the unit interval. Using this new tool, we can give an abstract solution, moreover, and a concrete necessary condition for the Lifting Problem for contractive 2-variable weighted shifts. In addition, we have a new, computable, and sufficient condition for the Lifting Problem for 2-variable weighted shifts, and an improved version of the Curto-Muhly-Xia conjecture [8] for 2-variable weighted shifts.     

关于加权移位算子的n-亚正规性

对于较Bergman加权移位序列更一般的序列,利用无穷维矩阵的正定性得到了其n-亚正规性,推广了已有的一些结论.     

Quadratically Hyponormal Recursively Generated Weighted Shifts Need Not Be Positively Quadratically Hyponormal

We study a class of weighted shifts W _α defined by a recursively generated sequence α ≡ α₀, … , α _m−2, (α _m−1, α _m , α _m+1)^∧ and characterize the difference between quadratic hyponormality and positive quadratic hyponormality. We show that a shift in this class is positively quadratically hyponormal if and only if it is quadratically hyponormal and satisfies a finite number of conditions. Using this characterization, we give a new proof of [12, Theorem 4.6], that is, for m = 2, W _α is quadratically hyponormal if and only if it is positively quadratically hyponormal. Also, we give some new conditions for quadratic hyponormality of recursively generated weighted shift W _α (m ≥ 2). Finally, we give an example to show that for m ≥ 3, a quadratically hyponormal recursively generated weighted shift W _α need not be positively quadratically hyponormal.     

Quadratically hyponormal weighted shifts and their examples

We discuss some characterizations for the quadratical hyponormal unilateral weighted shiftW with a weight sequence , which give a distinction example for quadratical hyponormality and positively quadratical hyponormality. In addition, we consider a recursively quadratically hyponormal weighted shift with a recursive weight : {ie480-1} which is a back step extension of subnormal completion ofu,v, andw with0, and prove that the recursively weighted shiftW is quadratically hyponormal if and only if it is positively quadratically hyponormal.Research partially supported by KOSEF 971-0102-006-2 and the Basic Science Research Institute Program, Ministry of Education, 1997, BSRI-97-1401.     

On n-contractive and n-hypercontractive Operators

We consider in this paper the classes of n-hypercontractive Hilbert space operators, primarily weighted shifts, and obtain results for back step extensions of recursively generated subnormal weighted shifts and for perturbations in the first weight of the Bergman shift. We compare the results with those for the classes of k-hyponormal operators, and recapture, by an n-hypercontractive approach, a subnormality result originally proved in the k-hyponormal context.     

On n-Contractive and n-Hypercontractive Operators, II

This paper studies the n-contractive and n-hypercontractive Hilbert space operators (n = 1, 2, . . .), classes weaker than, but related to, the class of subnormal operators. The k-hyponormal operators are the more thoroughly explored examples of classes weaker than subnormal; we show that k-hyponormality implies 2k-contractivity. Turning to weighted shifts, it is shown that if a weighted shift is extremal in the sense that the general nonnegativity test for n-contractivity is satisfied with equality to zero, then the shift is necessarily the unweighted unilateral shift. Also considered are the n-contractivity of back step extensions and perturbations of subnormal weighted shifts and some connections with the Berger measure of a subnormal shift. The second author was supported by the Korean Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF-2006-312-C00027). The third author was supported by the Korean Research Foundation Grant funded by the Korean Government (MOEHRD) (KRF- 2007-359-C00005.     

An Isometric Bilateral Shift that is Weakly Supercyclic

Ansari and Bourdon showed that an isometry on an infinite dimensional Banach space cannot be norm supercyclic. However, in this paper, we show there does exist a weakly supercyclic isometry on the Banach space      

Weighted Hardy inequalities for decreasing sequences and functions

We obtain a complete characterization of the weights for which Hardy's inequality holds on the cone of non-increasing sequences. Our proofs translate immediately to the analogous inequality for non-increasing functions, thereby also completing the investigation in that direction. As an application of our results we characterize the boundedness of the Hardy-Littlewood maximal operator on Lorentz sequence spaces.     

J-Class Weighted Shifts on the Space of Bounded Sequences of Complex Numbers

We provide a characterization of J-class and J ^mix-class unilateral weighted shifts on in terms of their weight sequences. In contrast to the previously mentioned result we show that a bilateral weighted shift on cannot be a J-class operator. During this research the second author was fully supported by SFB 701 “Spektrale Strukturen und Topologische Methoden in der Mathematik" at the University of Bielefeld, Germany. He would also like to express his gratitude to Professor H. Abels for his support.     

Complete hyperexpansivity, subnormality and inverted boundedness conditions

Athavale introduced in [3] the notion of a completely hyperexpansive operator. In this paper some results concerning powers of completely (alternatingly) hyperexpansive operators (not necessarily bounded) are extended tok-hyperexpansive ones. A semispectral measure is associated with a subnormal contraction as well as with a completely hyperexpansive operator, and an operator version of the Levy-Khinchin representation is obtained. Passing to the Naimark dilation of the semispectral measure, such an operator is related to a positive contraction in a natural way. New characterizations of a completely hyperexpansive operator and a subnormal contraction are given. The power bounded completely hyperexpansive operators are characterized. All these are illustrated using weighted shifts.     

Positive Operator Majorization and <Emphasis Type="Italic">p</Emphasis>-hyponormality

In this note we examine the relationships between p-hyponormal operators and the operator inequality . This leads to a method for generating examples of p-hyponormal operators which are not q-hyponormal for any . Our methods are also shown to have implications for the class of Furuta type inequalities.     

On Aluthge Transforms of p-hyponormal Operators

In this note we give an example of an ∞-hyponormal operator T whose Aluthge transform is not (1+ɛ)-hyponormal for any ɛ > 0 and show that the sequence of interated Aluthge transforms of T need not converge in the weak operator topology, which solve two problems in [6].     

Common Extensions for Locally Compact Markov Shifts

 We extend a result of W. Parry by showing that two locally compact transitive Markov shifts have the same Gurevic entropy iff they have a common entropy preserving extension by a locally compact transitive Markov shift. Additionally, the factor maps can be made countable-to-1 biclosing. (Received 12 September 2000; in revised form 16 January 2001)     

Hyperexpansive completion problem via alternating sequences; an application to subnormality

Truncations of completely alternating sequences are entirely characterized. The completely hyperexpansive completion problem is solved for finite sequences of (positive) numbers in terms of positivity of attached matrices. Solutions to the problem are written explicitly for sequences of two, three, four, five and six numbers. As an application, an explicit solution of the subnormal completion problem for five numbers is given.     

Weakly compact operators and interpolation

The class of weakly compact operators is, as well as the class of compact operators, a fundamental operator ideal. They were investigated strongly in the last twenty years. In this survey, we have collected and ordered some of this (partly very new) knowledge. We have also included some comments, remarks and examples.     

Quadratically hyponormal weighted shifts with two equal weights

We present an extensive analysis ofpositively quadratically hyponormal weighted shiftsW with ₀ = ₁ = 1. Our main result states that such weighted shifts abound! Specifically, by focusing on recursively generated weighted shifts of the form, we establish that the planar set is positively quadratically hyponormal} is a closed convex, set with nonempty interior. In addition, we are able to describe in detail the boundary of Research partially supported by NSF grants DMS-9401455 and DMS-9800931Research partially supported by KOSEF grant 971-0102-006-2 and by TGRC-KOSEF     

A construction of unitary linear systems

A construction is made of a unitary linear system whose transfer function is a given power seriesB(z) with operator coefficients such that multiplication byB(z) is an everywhere defined transformation in the space of square summable power series with vector coefficients. A condition is also given for the existence of an observable linear system with such a transfer function. For both constructions properties of the spaces are given which imply essential uniqueness of linear systems with given transfer functions. A canonical conjugate-isometric linear system is uniquely determined by its transfer function whenever the state space is a Pontryagin space.