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.     

Recursively generated weighted shifts and the subnormal completion problem,II

Both authors were partially supported by grants from NSF Raúl Curto was partially supported by a University of Iowa faculty scholar award     

Recursively generated weighted shifts and the subnormal completion problem

Both authors were partially supported by grants from NSF Raúl Curto was partially supported by a University of Iowa faculty scholar award     

k-Hyponormality of multivariable weighted shifts

We characterize joint k-hyponormality for 2-variable weighted shifts. Using this characterization we construct a family of examples which establishes and illustrates the gap between k-hyponormality and (k+1)-hyponormality for each k?1. As a consequence, we obtain an abstract solution to the Lifting Problem for Commuting Subnormals.     

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.     

On unbounded hyponormal operators II

The paper deals with unbounded hyponormal operators. Among others it is proved that any closed hyponormal operator with spectrum contained in a parabola generates a cosine function.     

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.     

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.      

Essential norm of differences of weighted composition operators between weighted-type spaces on the unit ball

We find an asymptotically equivalent expression to the essential norm of differences of weighted composition operators between weighted-type spaces of holomorphic functions on the unit ball in C^N. As a consequence we characterize the compactness of these operators. The boundedness of these operators is also characterized.     

Dilation to the unilateral shifts

The classical result of Foias says that an operator power dilates to a unilateral shift if and only if it is aC ₀ contraction. In this paper, we consider the corresponding question of dilating to a unilateral shift. We show tht for contractions with at least one defect index finite, dilation and power dilation to some unilateral shift amount to the same thing. The only difference is on the minimum multiplicity of the unilateral shift to which the contraction can be (power) dilated. We also obtain a characterization of contractions which are finite-rank perturbations of a unilateral shift, generalizing the rank-one perturbation result of Nakamura.     

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].     

On hyponormal toeplitz operators with polynomial and symmetric-type symbols

This paper treats the hyponormality of Toeplitz operators that have polynomial symbols with symmetric-type sets of coefficients.     

On hyponormal Toeplitz operators with polynomial and circulant-type symbols

This paper characterises those hyponormal Toeplitz operators on the Hardy space of the unit circle among all Toeplitz operators that have polynomial symbols with circulant-type sets of coefficients.Supported in part by The Natural Sciences and Engineering Research Council of CanadaSupported in part by BSRI-96-1420 and KOSEF 94-0701-02-01-3     

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.     

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.