Berger Measure Research Articles

The p-th power problem for Berger measures

In this paper we discuss the p-th power problem for measures which is, for a given measure ν and p>0, to find, if it exists, a measure μ such that ∫tndμ(t)=(∫tndν(t))p. Bergman shifts have some results for this problem and we research the problem for generalizations of Bergman shifts. We investigate the Berger measures for the class of weighted shifts S(a,b,c,d) with a weight sequence {an+bcn+d}n=0∞ (a,b,c,d>0 and ad>bc). Using the square root measure, we discuss also the Berger measures of the Aluthge transform. This problem is related to subnormal weighted shifts and these results add to the extremely small library of subnormal weighted shifts for which the Berger measure is known concretely.

The matricial subnormal completion problem

Given a truncated sequence of positive numbers a=(aj)j=0m, the subnormal completion problem (originally posed and geometrically solved by J. Stampfli) asks whether or not there exists a subnormal weighted shift operator on ℓ2 whose initial weights are given by a. Subsequently a concrete solution based on a solution of the truncated Stieltjes moment problem was discovered by Curto and Fialkow. In this paper we will consider a matricial analogue of the subnormal completion problem, where the truncated sequence of positive numbers (aj)j=0m is replaced by a truncated sequence of positive definite matrices. We will provide concrete conditions for a solution based on the parity of m and make a connection with a matricial truncated Stieltjes moment problem. We will also put forward a certain canonical subnormal completion which is minimal in the sense of the norm of the corresponding weighted shift operator and describe the support of the corresponding matricial Berger measure in terms of the zeros of a matrix polynomial which describes the initial positive rank preserving extension.

Polynomial Embeddings of Unilateral Weighted shifts in 2–Variable Weighted Shifts

Given a bounded sequence $$\omega $$ of positive numbers, and its associated unilateral weighted shift $$W_{\omega }$$ acting on the Hilbert space $$\ell ^2(\mathbb {Z}_+)$$ , we consider natural representations of $$W_{\omega }$$ as a 2–variable weighted shift, acting on the Hilbert space $$\ell ^2(\mathbb {Z}_+^2)$$ . Alternatively, we seek to examine the various ways in which the sequence $$\omega $$ can give rise to a 2–variable weight diagram, corresponding to a 2–variable weighted shift. Our best (and more general) embedding arises from looking at two polynomials p and q nonnegative on a closed interval $$I \subseteq \mathbb {R}_+$$ and the double-indexed moment sequence $$\{\int p(r)^k q(r)^{\ell } d\sigma (r)\}_{k,\ell \in \mathbb {Z}_+}$$ , where $$W_{\omega }$$ is assumed to be subnormal with Berger measure $$\sigma $$ such that $${\text {supp}}\; \sigma \subseteq I$$ ; we call such an embedding a (p, q)–embedding of $$W_{\omega }$$ . We prove that every (p, q)–embedding of a subnormal weighted shift $$W_{\omega }$$ is (jointly) subnormal, and we explicitly compute its Berger measure. We apply this result to answer three outstanding questions: (i) Can the Bergman shift $$A_2$$ be embedded in a subnormal 2–variable spherically isometric weighted shift $$W_{(\alpha ,\beta )}$$ ? If so, what is the Berger measure of $$W_{(\alpha ,\beta )}$$ ? (ii) Can a contractive subnormal unilateral weighted shift be always embedded in a spherically isometric 2–variable weighted shift? (iii) Does there exist a (jointly) hyponormal 2–variable weighted shift $$\Theta (W_{\omega })$$ (where $$\Theta (W_{\omega })$$ denotes the classical embedding of a hyponormal unilateral weighted shift $$W_{\omega }$$ ) such that some integer power of $$\Theta (W_{\omega })$$ is not hyponormal? As another application, we find an alternative way to compute the Berger measure of the Agler j–th shift $$A_{j}$$ ( $$j\ge 2$$ ). Our research uses techniques from the theory of disintegration of measures, Riesz functionals, and the functional calculus for the columns of the moment matrix associated to a polynomial embedding.

Solution of the reconstruction-of-the-measure problem for canonical invariant subspaces

We study the Reconstruction-of-the-Measure Problem (ROMP) for commuting 2-variable weighted shifts \(W_{(\alpha ,\beta )}\), when the initial data are given as the Berger measure of the restriction of \(W_{(\alpha ,\beta )}\) to a canonical invariant subspace, together with the marginal measures for the 0–th row and 0–th column in the weight diagram for \(W_{(\alpha ,\beta )}\). We prove that the natural necessary conditions are indeed sufficient. When the initial data correspond to a soluble problem, we give a concrete formula for the Berger measure of \(W_{(\alpha ,\beta )}\). Our strategy is to build on previous results for back-step extensions and one-step extensions. A key new theorem allows us to solve ROMP for two-step extensions. This, in turn, leads to a solution of ROMP for arbitrary canonical invariant subspaces of \(\ell ^2({\mathbb {Z}}_+^2)\).

The Square Root Problem and Aluthge transforms of weighted shifts

AbstractIn this paper we consider the following question. When does there exist a square root of a probability measure supported on ? This question is naturally related to subnormality of weighted shifts. The main result of this paper is that if μ is a finitely atomic probability measure having at most 4 atoms, then μ has a square root, i.e., there exists a measure ν such that (* means the convolution) if and only if the Aluthge transform of a subnormal weighted shift with Berger measure μ is subnormal. As an application of them, we give non‐trivial, large classes of probability measures having a square root. We also prove that there are 6 and 7‐atomic probability measures which don't have any square root. Our results have a connection to the following long‐open problem in Operator Theory: characterize the subnormal operators having a square root.

SUBNORMALITY OF S2(a, b, c, d) AND ITS BERGER MEASURE

We introduce a 2-variable weighted shift, denoted by <TEX>$S_2$</TEX>(a, b, c, d), which arises naturally from analytic function space theory. We investigate when it is subnormal, and compute the Berger measure of it when it is subnormal. And we apply the results to investigate the relationship among 2-variable subnormal, hyponormal and 2-hyponormal weighted shifts.

Berger Measure for Some Transformations of Subnormal Weighted Shifts

A subnormal weighted shift may be transformed to another shift in various ways, such as taking the p-th power of each weight or forming the Aluthge transform. \ We determine in a number of cases whether the resulting shift is subnormal, and, if it is, find a concrete representation of the associated Berger measure, directly for finitely atomic measures, and using both Laplace transform and Fourier transform methods for more complicated measures. \ Alternatively, the problem may be viewed in purely measure-theoretic terms as the attempt to solve moment matching equations such as $(\int t^n \, d\mu(t))^2 = \int t^n \, d\nu(t)$ ($n=0, 1, \ldots$) for one measure given the other.

BACKWARD EXTENSION OF SUBNORMAL 2-VARIABLE WEIGHTED SHIFTS

Abstract. We study backward extensions of 2-variable weightedshifts with nite atomic Berger measure. We provide a necessaryand sucient condition for the subnormality of such extensions.As an application, we give a simple counterexample for the Curto-Muhly-Xia conjecture [10]. 1. IntroductionLet Hbe a complex Hilbert space and let B(H) denote the algebraof bounded linear operators on H. We say that T2B(H) is normal ifTT= TT, subnormal if T= Nj H , where Nis normal and N(H) H,and hyponormal if TTTT. For S;T2B(H), let [S;T] := ST TS.We say that an n-tuple T = (T 1 ; ;T n ) of operators on His (jointly)hyponormal if the operator matrix[T;T] :=0BBB@[T 1 ;T 1 ] [T 2 ;T 1 ] [T n ;T 1 ][T 1 ;T 2 ] [T 2 ;T] [T n ;T]......... ...[T 1 ;T n ] [T 2 ;T n ] [T n ;T n ]1CCCAis positive semide nite on the direct sum of ncopies of H(cf. [1], [2],[10], [14]). The n-tuple T is said to be normal if T is commuting andeach T i is normal, and T is subnormal if T is the restriction of a normaln-tuple to a common invariant subspace. Clearly, normal )subnormal)hyponormal.

On the Aluthge Transforms of Weighted Shifts with Moments of Fibonacci Type. Application to Subnormality

In this paper, we investigate the properties of the Aluthge transforms of weighted shifts whose moment sequence is a generalized Fibonacci sequence of order r (or of Fibonacci type). We provide sufficient conditions under which the Aluthge transform has also a moment sequence of Fibonacci type. Notably, we detail the case when the moment sequence is a generalized Fibonacci sequence of order r ≤ 4. This allows us to recover the main result given by S. H. Lee et al. concerning the subnormality of Aluthge transforms of weighted shifts with two atomic Berger measure. We end this paper by considering the extended case of the subnormal weighted shift of a finite atomic Berger measure, with a view to whether its Aluthge transforms is again subnormal. As a result, we give a negative answer to a recent conjecture posed by S. H. Lee et al. in its general setting. Even so, under some mild assumptions we give a positive answer to this conjecture.

$k$-hyponormality and $n$-contractivity for Agler-type shifts

s of KOTAC Volume 10(2008), 10–10 k-hyponormality and n-contractivity for Agler-type shifts George Exner Bucknell University, Lewisburg, PA 17837, USA exner@bucknell.edu The well-known Bran-Halmos condition for subnormality of Hilbert space operators gives rise to the classes of k-hyponormal operators, k = 1, 2, · · · . The Agler-Embry condition for subnormality of a contraction uses the n-contractive classes, n =, 1, 2, · · · . The comparative study of these classes has been fruitful: for example, if a contraction is khyponormal it is 2k-contractive. We consider some back-step extensions of Agler model weighted shifts for which an n-contractivity condition guarantees (in some cases, is equivalent to) a k-hyponormality one. Elements of the study include the Berger measure of a subnormal shift and orthogonal polynomials.

Schur product techniques for the subnormality of commuting 2-variable weighted shifts

In this paper, we study the subnormality of 2-variable weighted shifts using the Schur product techniques in matrices and the convolution of two continuous functions. As a consequence, we find the Berger measure of the subnormal weighted shift obtained from the Schur product of two subnormal weighted shifts. As applications, we first give non-trivial, large classes satisfying the Curto–Muhly–Xia conjecture (see the conjecture given below) for 2-variable weighted shifts. We next show when the 2-hyponormality of 2-variable weighted shifts becomes subnormality.

Berger measure for [formula omitted

In this note, we get the Berger measure of the weighted shift S(a,b,c,d) with weights αn:=an+bcn+d (a,b,c,d>0 and n⩾0) as well as its p-subshift. Then we will give examples from analytic function theory to illustrate that the necessary condition given by Curto and Yoon (2006) [12] for subnormality of 2-variable weighted shift is not sufficient. Furthermore, we will do some calculations about subnormal backward extension of S(a,b,c,d).

Subnormality of Aluthge Transforms of Weighted Shifts

In this note we study the k-hyponormality and the subnormality of Aluthge transforms of weighted shifts. It is shown that Aluthge transforms of weighted shifts need not preserve the k-hyponormality. Moreover, we show that if Wα is a subnormal weighted shift with 2-atomic Berger measure then its Aluthge transform \({\widetilde{W}_\alpha}\) is subnormal if and only if at least one of two atoms is zero.

Conglomerate Discounts and Their Biases: An International Survey

The Berger and Ofek (1995) excess value measure, comparing a conglomerate's actual market value to an imputed value based on standalones, has become the standard method to determine value effects of diversification. In this paper, we address a significant bias in this procedure stemming from the difference in cash holdings between diversified and standalone firms. Excess values are based on firm values, including corporate cash positions. As standalones hold significantly more cash, the imputed cash value is higher than the conglomerate's actual cash value, resulting in a downward biased excess value. We thus propose to calculate excess values based on enterprise values, replacing total debt by net debt. Based on an extensive U.S. sample, we show that there is significantly less evidence of a diversification discount when adjusting for the cash bias. In terms of average dollar losses, the firm value-based models overestimate the conglomerate discount by at least 25%. Apart from removing the cash bias, we propose a second modification to the excess value measure, arguing that standalone industry multipliers should be calculated using geometric mean aggregation instead of median aggregation.

A new approach to the 2-variable Subnormal Completion Problem

We study the Subnormal Completion Problem (SCP) for 2-variable weighted shifts. We use tools and techniques from the theory of truncated moment problems to give a general strategy to solve SCP. We then show that when all quadratic moments are known (equivalently, when the initial segment of weights consists of five independent data points), the natural necessary conditions for the existence of a subnormal completion are also sufficient. To calculate explicitly the associated Berger measure, we compute the algebraic variety of the associated truncated moment problem; it turns out that this algebraic variety is precisely the support of the Berger measure of the subnormal completion.

Disintegration‐of‐measure techniques for commuting multivariable weighted shifts

We employ techniques from the theory of disintegration of measures to study the Lifting Problem for commuting n‐tuples of subnormal weighted shifts. We obtain a new necessary condition for the existence of a lifting, and generate new pathology associated with bringing together the Berger measures associated to each individual weighted shift. For subnormal 2‐variable weighted shifts, we then find the precise relation between the Berger measure of the pair and the Berger measures of the shifts associated to horizontal rows and vertical columns of weights. 2000 Mathematics Subject Classification 47B20, 47B37, 47A13, 28A50 (primary), 44A60, 47‐04, 47A20 (secondary).

Effects of feedback on self-reinforcing behavior in relation to self-acceptance.

96 high and low self-accepting Ss were divided into 8 treatment groups: (a) a low group who received facilitating feedback for self-reinforcing responses; (b) a group comparable to the previous one but who also were given positive feedback on social comparison; (c) a low group who received inhibiting feedback for negative self-reinforcement responses; and (d) a group of low-control Ss. The remaining four groups were Ss high in self-acceptance; they were treated in a comparable fashion to the first four groups but instead given feedback designed to decrease their self-reinforcement rate. Ss were given the California Personality Inventory and the Berger measure of self-acceptance as well as self-ratings of creativity and intelligence before and after treatment. There were few significant changes for the post-measures following the conditioning treatment. The major outcome is questioning some hypotheses from past research concerning the relationship between self-reinforcement and self-acceptance.