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Hans-Olav Tylli

Publications and source records attributed to Hans-Olav Tylli.

12 recordsLinked to original sources
arXiv2025

Structure of closed subideals of $\mathcal L(X)$

The closed subalgebra $\mathcal J$ of the Banach algebra $\mathcal L(X)$ of bounded linear operators on the Banach space $X$ is a non-trivial closed $\mathcal I$-subideal of $\mathcal L(X)$ if $\mathcal I$ is a closed ideal of $\mathcal L(X)$ and $\mathcal J$ is an ideal of $\mathcal I$, but $\mathcal J$ is not an ideal of $\mathcal L(X)$. We obtain a variety of examples of non-trivial closed subideals of $\mathcal L(X)$ for different spaces $X$, which highlight further significant differences compared to the class of closed ideals. We study the concept of a closed $n$-subideal of $\mathcal L(X)$ for $n \ge 3$, which is a natural generalization of that of a closed subideal. In particular, we find explicit spaces $X$ for which $\mathcal L(X)$ contains a decreasing sequence $(\mathcal M_n)_{n\in \mathbb N}$ of closed subalgebras, where for all $n\in\mathbb N$ the subalgebra $\mathcal M_n$ is an $(n+1)$-subideal of $\mathcal L(X)$ but not an $n$-subideal. Moreover, we construct closed $n$-subideals contained in the compact operators $\mathcal K(X)$ for certain Banach spaces $X$ which fail the approximation property.

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arXiv2023

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 $\mathscr F$ of closed subideals contained in the Banach algebra $\mathcal K(Z)$ of the compact operators on $Z$, such that the subideals in $\mathscr F$ 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 contained in the strictly singular operators $\mathcal S(X)$ for classical spaces such as $X = L^p$ with $p \neq 2$, where pairwise isomorphic as well as pairwise non-isomorphic subideals occur.

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arXiv2021

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 $\mathfrak{A}_X =:\mathcal K(X)/\mathcal A(X)$ on Banach spaces $X$ failing the approximation property. The examples include the following: (i) if $X$ has cotype $2$, $Y$ has type $2$, $\mathfrak{A}_X \neq \{0\}$ and $\mathfrak{A}_Y \neq \{0\}$, then $\mathfrak{A}_{X \oplus Y}$ has at least $2$ closed ideals, (ii) there are closed subspaces $X \subset \ell^p$ for $4 < p < \infty$ and $X \subset c_0$ such that $\mathfrak{A}_X$ contains a non-trivial closed ideal, (iii) there is a Banach space $Z$ such that $\mathfrak{A}_Z$ contains an uncountable lattice of closed ideal having the reverse order structure of the power set of the natural numbers. Some of our examples involve non-classical approximation properties associated to various Banach operator ideals. We also discuss the existence of compact non-approximable operators $X \to Y$, where $X \subset \ell^p$ and $Y \subset \ell^q$ are closed subspaces for $p \neq q$.

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arXiv2018

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 $c_0(Γ)$ 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.

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arXiv2017

Structural rigidity of generalised Volterra operators on $H^p$

We show that the non-compact generalised analytic Volterra operators $T_g$, where $g \in \mathit{BMOA}$, have the following structural rigidity property on the Hardy spaces $H^p$ for $1 \le p < \infty$ and $p \neq 2$: if $T_g$ is bounded below on an infinite-dimensional subspace $M \subset H^p$, then $M$ contains a subspace linearly isomorphic to $\ell^p$. This implies in particular that any Volterra operator $T_g\colon H^p \to H^p$ is $\ell^2$-singular for $p \neq 2$.

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arXiv2016

Hypercyclicity Properties of Commutator Maps

We investigate the hypercyclic properties of commutator maps acting on separable 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 $\ell^2$. We also establish some necessary conditions which identify large classes of operators that do not induce hypercyclic commutator maps.

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arXiv2016

Rigidity of composition operators on the Hardy space $H^p$

Let $ϕ$ be an analytic map taking the unit disk $\mathbb{D}$ into itself. We establish that the class of composition operators $f \mapsto C_ϕ(f) = f \circ ϕ$ exhibits a rather strong rigidity of non-compact behaviour on the Hardy space $H^p$, for $1\le p < \infty$ and $p \neq 2$. Our main result is the following trichotomy, which states that exactly one of the following alternatives holds: (i) $C_ϕ$ is a compact operator $H^p \to H^p$, (ii) $C_ϕ$ fixes a (linearly isomorphic) copy of $\ell^p$ in $H^p$, but $C_ϕ$ does not fix any copies of $\ell^2$ in $H^p$, (iii) $C_ϕ$ fixes a copy of $\ell^2$ in $H^p$. Moreover, in case (iii) the operator $C_ϕ$ 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 $\mathcal K(H^p)$. In particular, the class of composition operators on $H^p$ does not reflect the quite complicated lattice structure of such ideals.

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arXiv2015

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

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 infinite-dimensional 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 BMOA 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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arXiv2010

Compact and weakly compact composition operators on BMOA

We show that a composition operator induced by an analytic self-map of the unit disc in the complex plane is weakly compact on the space BMOA precisely when the operator is compact on BMOA. As a crucial step we simplify the compactness criterion due to Smith for composition operators on BMOA and show that his condition on the Nevanlinna counting function alone characterizes compactness. In addition, other equivalent compactness criteria are established for composition operators on both BMOA and its subspace VMOA.

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arXiv2003

Weakly compact approximation in Banach spaces

The Banach space $E$ has the weakly compact approximation property (W.A.P. for short) if there is a constant $C < \infty$ so that for any weakly compact set $D \subset E$ and $ε> 0$ there is a weakly compact operator $V: E \to E$ satisfying $\sup_{x\in D} || x - Vx || < ε$ and $|| V|| \leq C$. We give several examples of Banach spaces both with and without this approximation property. Our main results demonstrate that the James-type spaces from a general class of quasi-reflexive spaces (which contains the classical James' space $J$) have the W.A.P, but that James' tree space $JT$ fails to have the W.A.P. It is also shown that the dual $J^*$ has the W.A.P. It follows that the Banach algebras $W(J)$ and $W(J^*)$, consisting of the weakly compact operators, have bounded left approximate identities. Among the other results we obtain a concrete Banach space $Y$ so that $Y$ fails to have the W.A.P., but $Y$ has this approximation property without the uniform bound $C$.

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