Schauder Bases and Operator Theory II: (SI) Schauder Operators

TL;DR

This paper demonstrates that injective operators with dense range can be transformed into strongly irreducible operators via invertible operators, with applications to Schauder matrices, advancing the understanding of operator structure.

Contribution

It shows the existence of invertible operators transforming certain operators into strongly irreducible ones, including a specific construction involving unitary and compact operators.

Findings

Existence of invertible operators making T strongly irreducible

Strongly irreducible operators exist in the orbit of Schauder matrices

Construction of invertible operators as sum of unitary and compact operators

Abstract

In this paper, we will show that for an operator $T$ which is injective and has dense range, there exists an invertible operator $X$ (in fact we can find $U+K$, where $U$ is an unitary operator and $K$ is a compact operator with norm less than a given positive real number) such that $XT$ is strongly irreducible. As its application, strongly irreducible operators always exist in the orbit of Schauder matrices.