How many operators do there exist on a Banach space?
Th. Schlumprecht
TL;DR
This paper investigates whether every infinite-dimensional Banach space contains a subspace supporting a non-compact perturbation of scalar multiplication operators, providing partial results towards this open question.
Contribution
It offers partial results and insights into the existence of specific operators on subspaces of infinite-dimensional Banach spaces.
Findings
Partial results indicating conditions for the existence of such operators.
Identification of classes of Banach spaces where the question holds.
Open problems and directions for future research.
Abstract
We present partial results to the following question: Does every infinite dimensional Banach space have an infinite dimensional subspace on which one can define an operator which is not a compact perturbation of a scalar multiplication?
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Taxonomy
TopicsAdvanced Banach Space Theory · Holomorphic and Operator Theory · Advanced Topics in Algebra