On restricted diagonalization

Abstract

Let H be a separable infinite-dimensional complex Hilbert space, B(H) the algebra of bounded linear operators acting on H and J a proper two-sided ideal of B(H). Denote by UJ(H) the group of all unitary operators of the form I+J. Recall that an operator A∈B(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 e={en}n≥1 and a proper operator ideal J as follows: A∈B(H) is called restrictedly diagonalizable if there exists U∈UJ(H) such that UAU⁎ is diagonal with respect to e. In this work we give a sufficient condition for a diagonalizable operator to be restrictedly diagonalizable. This condition becomes a characterization when the ideal is arithmetic mean closed. Then we obtain results on the structure of the set of all restrictedly diagonalizable operators. In this way we answer several open problems recently raised by Beltiţă, Patnaik and Weiss.