Multiplications and elementary operators in the Banach space setting

Saksman, Eero; Tylli, Hans-Olav

doi:10.1017/cbo9780511721366.014

Cited by 15 publications

(13 citation statements)

References 66 publications

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“…The operator Φ A,B belongs to the larger class of so-called elementary operators the structural properties of which have been extensively studied in the literature (see the survey [26] and references therein).…”

Section: Elementary Operatorsmentioning

confidence: 99%

Solution formulas for the two-dimensional Toda lattice and particle-like solutions with unexpected asymptotic behaviour

Nilson

Schiebold

2019

JNMP

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The first main aim of this article is to derive an explicit solution formula for the scalar two-dimensional Toda lattice depending on three independent operator parameters, ameliorating work in [31]. This is achieved by studying a noncommutative version of the 2d-Toda lattice, generalizing its soliton solution to the noncommutative setting. The purpose of the applications part is to show that the family of solutions obtained from matrix data exhibits a rich variety of asymptotic behaviour. The first indicator is that web structures, studied extensively in the literature, see [4] and references therein, are a subfamily. Then three further classes of solutions (with increasingly unusual behaviour) are constructed, and their asymptotics are derived.

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Section: Elementary Operatorsmentioning

confidence: 99%

Solution formulas for the two-dimensional Toda lattice and particle-like solutions with unexpected asymptotic behaviour

Nilson

Schiebold

2019

JNMP

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“…It is known, see e.g. [44,Example 2.6], that for such operator compositions this yields the compactness of (C ϕ ) * | V on V . Hence C ϕ is compact H 2 → H 2 , so that (1) holds.…”

Section: Proposition 15mentioning

confidence: 99%

“…For instance, there is a well-developed theory of the composition operators S → A • S • B, where A and B are fixed bounded operators, on spaces of linear operators, see e.g. the survey [44]. Properties of such composition operators will actually be required in section 5 below.…”

Section: Introductionmentioning

confidence: 99%

Composition operators on vector-valued analytic function spaces: a survey

Laitila

Tylli

2014

ACUTM

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Abstract. We survey recent results about composition operators induced by analytic self-maps of the unit disk in the complex plane on various Banach spaces of analytic functions taking values in infinitedimensional Banach spaces. We mostly concentrate on the research line into qualitative properties such as weak compactness, initiated by Liu, Saksman and Tylli (1998), and continued in several other papers. We discuss composition operators on strong, respectively weak, spaces of vector-valued analytic functions, as well as between weak and strong spaces. As concrete examples, we review more carefully and present some new observations in the cases of vector-valued Hardy and BM OA spaces, though the study of composition operators has been extended to a wide range of spaces of vector-valued analytic functions, including spaces defined on other domains. Several open problems are stated.

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“…where σ J (τ A,B ) denotes the spectrum of τ A,B : J → J. Further details can be found, for instance, in the survey [39,Theorem 3.12].…”

Section: Classes Of Non-hypercyclic Derivationsmentioning

confidence: 99%

Dynamics of Generalised Derivations and Elementary Operators

Gilmore

2018

Complex Anal. Oper. Theory

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We identify concrete examples of hypercyclic generalised derivations acting on separable ideals of operators and establish some necessary conditions for their hypercyclicity. We also consider the dynamics of elementary operators acting on particular Banach algebras, which reveals surprising hypercyclic behaviour on the space of bounded linear operators on the Banach space constructed by Argyros and Haydon.

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Multiplications and elementary operators in the Banach space setting

Cited by 15 publications

References 66 publications

Solution formulas for the two-dimensional Toda lattice and particle-like solutions with unexpected asymptotic behaviour

Solution formulas for the two-dimensional Toda lattice and particle-like solutions with unexpected asymptotic behaviour

Composition operators on vector-valued analytic function spaces: a survey

Dynamics of Generalised Derivations and Elementary Operators

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