Weighted shifts and composition operators on L2

1992 ◽  
pp. 10-17
Author(s):  
Alan Lambert
Keyword(s):  
Composition Operators ◽  
Weighted Shifts

scholarly journals The Wold-Type Decomposition for m-Isometries

2021 ◽  
Author(s):  
Jakub Kośmider
Keyword(s):  
Composition Operators ◽  
Directed Graphs ◽  
Equivalent Condition ◽  
Weighted Shifts ◽  
Type Decomposition

AbstractThe aim of this paper is to study the Wold-type decomposition in the class of m-isometries. One of our main results establishes an equivalent condition for an analytic m-isometry to admit the Wold-type decomposition for $$m\ge 2$$ m ≥ 2 . In particular, we introduce the k-kernel condition which we use to characterize analytic m-isometric operators which are unitarily equivalent to unilateral operator valued weighted shifts for $$m\ge 2$$ m ≥ 2 . As a result, we also show that m-isometric composition operators on directed graphs with one circuit containing only one element are not unitarily equivalent to unilateral weighted shifts. We also provide a characterization of m-isometric unilateral operator valued weighted shifts with positive and commuting weights.

scholarly journals $$\varvec{m}$$–Isometric Composition Operators on Directed Graphs with One Circuit

2021 ◽  
Vol 93 (5) ◽  
Author(s):  
Zenon Jan Jabłoński ◽  
Jakub Kośmider
Keyword(s):  
Composition Operators ◽  
Directed Graphs ◽  
Weighted Shifts ◽  
Completion Problem ◽  
Affirmative Solution

AbstractThe aim of this paper is to investigate m–isometric composition operators on directed graphs with one circuit. We establish a characterization of m–isometries and prove that complete hyperexpansivity coincides with 2–isometricity within this class. We discuss the m–isometric completion problem for unilateral weighted shifts and for composition operators on directed graphs with one circuit. The paper is concluded with an affirmative solution of the Cauchy dual subnormality problem in the subclass with circuit containing one element.

scholarly journals Chaos and frequent hypercyclicity for composition operators

2021 ◽  
pp. 1-19
Author(s):  
Udayan B. Darji ◽  
Benito Pires
Keyword(s):  
Recent Result ◽  
Large Class ◽  
Composition Operators ◽  
Intimate Relationship ◽  
Invertible Operator ◽  
Weighted Shift ◽  
Weighted Shifts ◽  
Linear Dynamics ◽  
Chaotic Operator

Abstract The notions of chaos and frequent hypercyclicity enjoy an intimate relationship in linear dynamics. Indeed, after a series of partial results, it was shown by Bayart and Ruzsa in 2015 that for backward weighted shifts on $\ell _p(\mathbb {Z})$ , the notions of chaos and frequent hypercyclicity coincide. It is with some effort that one shows that these two notions are distinct. Bayart and Grivaux in 2007 constructed a non-chaotic frequently hypercyclic weighted shift on $c_0$ . It was only in 2017 that Menet settled negatively whether every chaotic operator is frequently hypercylic. In this article, we show that for a large class of composition operators on $L^{p}$ -spaces, the notions of chaos and frequent hypercyclicity coincide. Moreover, in this particular class, an invertible operator is frequently hypercyclic if and only if its inverse is frequently hypercyclic. This is in contrast to a very recent result of Menet where an invertible operator frequently hypercyclic on $\ell _1$ whose inverse is not frequently hypercyclic is constructed.

scholarly journals Subnormal Weighted Shifts on Directed Trees and Composition Operators inL2-Spaces with Nondensely Defined Powers

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Piotr Budzyński ◽  
Piotr Dymek ◽  
Zenon Jan Jabłoński ◽  
Jan Stochel
Keyword(s):  
Positive Integer ◽  
Composition Operator ◽  
Measure Space ◽  
Composition Operators ◽  
Finite Measure ◽  
Weighted Shift ◽  
Weighted Shifts ◽  
Directed Tree ◽  
Finite Measure Space ◽  
Densely Defined

It is shown that for every positive integernthere exists a subnormal weighted shift on a directed tree (with or without root) whosenth power is densely defined while its (n+1)th power is not. As a consequence, for every positive integernthere exists a nonsymmetric subnormal composition operatorCin anL2-space over aσ-finite measure space such thatCnis densely defined andCn+1is not.

Essential norms of weighted differentiation composition operators between Zygmund type spaces and Bloch type spaces

Filomat ◽  
2017 ◽  
Vol 31 (9) ◽  
pp. 2877-2889 ◽  
Author(s):  
Amir Sanatpour ◽  
Mostafa Hassanlou
Keyword(s):  
Composition Operators ◽  
Sufficient Conditions ◽  
Essential Norm ◽  
Necessary And Sufficient Conditions ◽  
Norm Estimates ◽  
Type Spaces ◽  
Necessary And Sufficient

We study boundedness of weighted differentiation composition operators Dk?,u between Zygmund type spaces Z? and Bloch type spaces ?. We also give essential norm estimates of such operators in different cases of k ? N and 0 < ?,? < ?. Applying our essential norm estimates, we get necessary and sufficient conditions for the compactness of these operators.

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