A new approach to operator ideals on Hilbert space and their traces
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 ideals, especially for researchers who need to use the machinery of operator ideals in other areas of mathematics.
scientific article; zbMATH DE number 6832767
scientific article