A new approach to operator ideals on Hilbert space and their traces
type: Item
instance of: scholarly article
title: A new approach to operator ideals on Hilbert space and their traces
zbMATH Open document ID: 1464.47016
author: Albrecht Pietsch
published in: Integral Equations and Operator Theory
publication date: 2018-01-31T00:00:00Z
review text: The paper under review constitutes an elegant presentation of the theory of operator ideals (on an infinite-dimensional separable Hilbert space \(H\)) from the point of the so-called shift invariant ideals of \(\ell_\infty(\mathbb N_0)\). It builds upon a series of papers published not long ago by the author. Let \(\mathcal{A}\) be an ideal of \(\mathcal{L}(H)\), the \(C^*\)-algebra of all (bounded, linear) operators on \(H\). As proved by \textit{J. W. Calkin} [Ann. Math. (2) 42, 839--873 (1941; Zbl 0063.00692)], only \(\{0\}\), the ideal of compact operators \(\mathcal{K}(H)\), and \(\mathcal{L}(H)\) itself are ideals of \(\mathcal{L}(H)\) that are closed in the operator norm. The theory of operator ideals on \(\mathcal{L}(H)\) is therefore concerned with sub-ideals of \(\mathcal{K}(H)\), often endowed with semi-norms that are stronger than the operator norm. A semi-norm \(\|\cdot \|_{\mathcal{A}}\) is called stable whenever \[ \|BTA\|_{\mathcal{A}} \leqslant \|B\|_{\text{op}}\cdot \|T\|_{\mathcal{A}}\cdot \|A\|_{\text{op}} \qquad (A,B\in \mathcal{L}(H),\, T\in \mathcal{A}). \] The Frobenius and Hilbert-Schmidt norms are paradigmatic examples of such (semi-)norms. The leitmotif of this expository paper is the equivalence between ideals endowed with a stable semi-norm and ideals \(\mathfrak{s}\) of \(\ell_\infty(\mathbb N_0)\) with the property that, if a sequence \((a_n)_{n=1}^\infty\) from \(\ell_\infty(\mathbb N_0)\) is dominated by a sequence \((b_n)_{n=1}^\infty\) from \(\mathfrak{s}\) in the following sense: \[ \sup_{h\geqslant k} |a_h| \leqslant \sup_{h\geqslant k} |b_h| \quad (k \in \mathbb N), \] then it necessarily belongs to \(\mathfrak{s}\). This approach is particularly useful as it offers a handy reduction to diagonal operators belonging to the given operator ideal and a possibility of traversing between commutative and non-commutative objects. The paper is very well written and it could serve as a mild introduction to the vast theory of operator idea…
reviewed by: Tomasz Kania
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 47B37
Mathematics Subject Classification ID: 46B45
Mathematics Subject Classification ID: 46A45
Mathematics Subject Classification ID: 47-02
zbMATH DE Number: 6832767
zbMATH Keywords: operator ideal
zbMATH Keywords: sequence ideal
zbMATH Keywords: quasi-norm
zbMATH Keywords: shift
zbMATH Keywords: trace
MaRDI profile type: Publication
full work available at URL: s00020-017-2410-x
OpenAlex ID: W2769811972
cites work: Completeness of quasi-normed operator ideals generated by \(s\)-numbers
cites work: Singular traces. Theory and applications
cites work: Traces and Shift Invariant Functionals
cites work: Dixmier traces of operators on Banach and Hilbert spaces
cites work: Traces on operator ideals and related linear forms on sequence ideals. I
cites work: Traces on operator ideals and related linear forms on sequence ideals. II
cites work: Traces on operator ideals and related linear forms on sequence ideals. III
DOI: 10.1007/S00020-017-2410-X
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description: scientific article; zbMATH DE number 6832767