Bernstein functions, complete hyperexpansivity and subnormality. I.
type: Item
instance of: scholarly article
title: Bernstein functions, complete hyperexpansivity and subnormality. I.
zbMATH Open document ID: 1052.47014
DOI: 10.1007/BF01255562
author: Ameer Athavale
author: Abhijit Ranjekar
published in: Integral Equations and Operator Theory
publication date: 2002-01-01T00:00:00Z
review text: The notion of subnormal operator was introduced in [Summa Brasil. Math. 2, 125--134 (1950; Zbl 0041.23201)] by \textit{P. R. Halmos}, while the notion of a completely hyperexpansive operator was introduced in [Proc. Am. Math. Soc. 124, 3745--3752 (1996; Zbl 0863.47017)] by \textit{A. Athavale}. Let us recall that by definition, a bounded operator \(T\in B(H)\) is completely hyperexpansive if \( \sum_{0\leq p\leq n}(-1)^p \binom np T^{*p}T^p\leq 0\) for all \(n\geq 1\). It is well-known that subnormal operators are closely related to the theory of positive definite functions on the abelian semigroup \(({\mathbb{N}},+,n^*=n)\), while completely hyperexpansive operators correspond to negative definite functions on \(({\mathbb{N}},+,n^*=n)\). In the paper under review, the authors characterize minimal Levy sequences and prove that the composition of a completely alternating function with a Bernstein function is a completely alternating function. Using this, the authors show that the weight sequence of any completely hyperexpansive weighted shift is a Hausdorff moment sequence. They also observe that the weighted sequence of a completely hyperexpansive weighted shift with the weight sequence \(\{\alpha_n\}_{n=0}^\infty\) gives rise to a subnormal weighted shift with weight sequence \(\{\alpha_{n+1}/\alpha_n\}_{n=0}^\infty\).
reviewed by: Zenon Jan Jabłoński
Mathematics Subject Classification ID: 47B20
Mathematics Subject Classification ID: 47B37
Mathematics Subject Classification ID: 47B39
Mathematics Subject Classification ID: 43A35
zbMATH DE Number: 1802505
zbMATH Keywords: Bernstein function
zbMATH Keywords: Stieltjes function
zbMATH Keywords: completely hyperexpansive operator
zbMATH Keywords: subnormal operator
zbMATH Keywords: Hausdorff sequence
zbMATH Keywords: minimal Levy sequences
MaRDI profile type: Publication
cites work: On completely hyperexpansive operators
cites work: Alternatingly hyperexpansive operator tuples
cites work: Completely hyperexpansive operator tuples
cites work: Q3229718
cites work: Q4000087
cites work: Q3671491
cites work: Intégrales de résolvantes et calcul symbolique. (Integrals of resolvents and symbolic calculus.)
cites work: Q4529385
cites work: What is the Laplace Transform?
full work available at URL: https://doi.org/10.1007/bf01255562
OpenAlex ID: W4230732611
description: scientific article; zbMATH DE number 1802505