2025/10/20 by Hans-Olav Tylli, Tylli, Hans-Olav, Henrik Wirzenius +1
Mathematics · #46B28 #46H10 #47L10 #Advanced Banach Space Theory #Advanced Operator Algebra Research #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA)
paper · pdf · doi:10.48550/arxiv.2510.17310
openalex publication_date 2025/10/20 · openalex created_date 2025/10/22 · openalex updated_date 2026/07/28
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 ≥ 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 Mn)n∈ \mathbb N of closed subalgebras, where for all n∈\mathbb N the subalgebra \mathcal Mn 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.