Quotient algebra of compact-by-approximable operators on Banach spaces failing the approximation property

We initiate a study of structural properties of the quotient algebra $\mathcal K(X)/\mathcal A(X)$ of the compact-by-approximable operators on Banach spaces $X$ failing the approximation property. Our main results and examples include the following: (i) there is a linear isomorphic embedding from $c_0$ into $\mathcal K(Z)/\mathcal A(Z)$, where $Z$ belongs to the class of Banach spaces constructed by Willis that have the metric compact approximation property but fail the approximation property, (ii) there is a linear isomorphic embedding from a non-separable space $c_0(\Gamma)$ into $\mathcal K(Z_{FJ})/\mathcal A(Z_{FJ})$, where $Z_{FJ}$ is a universal compact factorisation space arising from the work of Johnson and Figiel.Comment: 21 page

On universal operators and universal pairs

We study some basic properties of the class of universal operators on Hilbert spaces, and provide new examples of universal operators and universal pairs.Peer reviewe

Exotic closed subideals of algebras of bounded operators

We exhibit a Banach space $Z$ failing the approximation property, for which there is an uncountable family $\mathcal F$ of closed subideals contained in the Banach algebra $\mathcal K(Z)$ of the compact operators on $Z$, such that the subideals in $\mathcal F$ are mutually isomorphic as Banach algebras. This contrasts with the behaviour of closed ideals of the algebras $\mathcal L(X)$ of bounded operators on $X$, where closed ideals $\mathcal I \neq \mathcal J$ are never isomorphic as Banach algebras. We also construct families of non-trivial closed subideals in the strictly singular operators $\mathcal S(X)$ for the classical spaces $X = L^p$ with $p \neq 2$, where the subideals are not pairwise isomorphic.Comment: 14 page

Closed ideals in the algebra of compact-by-approximable operators

We construct various examples of non-trivial closed ideals of the compact-by-approximable algebra U-X := K(X)/A(X) on Banach spaces X failing the approximation property. The examples include the following: (i) if X has cotype 2, Yhas type 2, U-X not equal {0} and U-Y not equal {0}, then U-X circle plus Y has at least 2 closed ideals, (ii) there are closed subspaces X subset of l(p) for 4 Y, where X subset of l(p) and Y subset of l(q) are closed subspaces for p not equal q. (c) 2021 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).Peer reviewe

The quotient algebra of compact-by-approximable operators on Banach spaces failing the approximation property

We initiate a study of structural properties of the quotient algebra K (X)/A(X) of the compact-by-approximable operators on Banach spaces X failing the approximation property. Our main results and examples include the following: (i) there is a linear isomorphic embedding from c(0) into K (Z)/A(Z), where Z belongs to the class of Banach spaces constructed by Willis that have the metric compact approximation property but fail the approximation property, (ii) there is a linear isomorphic embedding from a nonseparable space c(0)(Gamma) into K (Z(FJ))/A(Z(FJ)), where Z(FJ) is a universal compact factorisation space arising from the work of Johnson and Figiel.Peer reviewe

Optimal growth of harmonic functions frequently hypercyclic for the partial differentiation operator

available as electronic pre-print arXiv:1708.08764Peer reviewe

Hypercyclicity Properties of Commutator Maps

We investigate the hypercyclic properties of commutator operators acting on separable Banach ideals of operators. As the main result we prove the commutator map induced by scalar multiples of the backward shift operator fails to be hypercyclic on the space of compact operators on . We also establish several necessary conditions which identify large classes of operators that do not induce hypercyclic commutator maps.Peer reviewe

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

Peer reviewe

Rigidity of composition operators on the Hardy space H-P

Let phi be an analytic map taking the unit disk ID into itself. We establish that the class of composition operators f bar right arrow C-phi(f) = f o phi exhibits a rather strong rigidity of non-compact behaviour on the Hardy space H-P, for 1 H-P, (ii) C-phi fixes a (linearly isomorphic) copy of l(P) in H-P, but C-phi does not fix any copies of l(2) in H-P, (iii) C-phi fixes a copy of l(2) in H-P. Moreover, in case (iii) the operator C-phi actually fixes a copy of L-P(0, 1) in H-P provided p > 1. We reinterpret these results in terms of norm-closed ideals of the bounded linear operators on H-P, which contain the compact operators k(H-P). In particular, the class of composition operators on H-P does not reflect the quite complicated lattice structure of such ideals. (C) 2017 Elsevier Inc. All rights reserved.Peer reviewe