On restricted diagonalization

TL;DR

This paper characterizes when a diagonalizable operator on an infinite-dimensional Hilbert space can be made diagonal using unitary operators close to the identity within a specific operator ideal, addressing open problems in the field.

Contribution

It provides necessary and sufficient conditions for restricted diagonalizability of operators, especially when the ideal is arithmetic mean closed, and explores the structure of such operators.

Findings

Characterization of restricted diagonalizable operators

Conditions for operators to be restricted diagonalizable when ideal is arithmetic mean closed

Resolution of open problems related to restricted diagonalization

Abstract

Let $\mathcal{H}$ be a separable infinite-dimensional complex Hilbert space, $\mathcal{B}(\mathcal{H})$ the algebra of bounded linear operators acting on $\mathcal{H}$ and $\mathcal{J}$ a proper two-sided ideal of $\mathcal{B}(\mathcal{H})$. Denote by $\mathcal{U}_\mathcal{J}(\mathcal{H})$ the group of all unitary operators of the form $I+\mathcal{J}$. Recall that an operator $A \in \mathcal{B}(\mathcal{H})$ is diagonalizable if there exists a unitary operator $U$ such that $UAU^*$ is diagonal with respect to some orthonormal basis. A more restrictive notion of diagonalization can be formulated with respect to a fixed orthonormal basis $\mathrm{e}=\{ e_n\}_{n\geq 1}$ and a proper operator ideal $\mathcal{J}$ as follows: $A \in \mathcal{B}(\mathcal{H})$ is called restricted diagonalizable if there exists $U\in \mathcal{U}_\mathcal{J}(\mathcal{H})$ such that $UAU^*$ is diagonal with respect to $\mathrm{e}$. In this work we give necessary and sufficient conditions for a diagonalizable operator to be restricted diagonalizable. Our conditions become a characterization of those diagonalizable operators which are restricted diagonalizable when the ideal is arithmetic mean closed. Then we obtain results on the structure of the set of all restricted diagonalizable operators. In this way we answer several open problems recently raised by Belti\c{t}$\breve{\text{a}}$, Patnaik and Weiss.