On restricted diagonalization

CHIUMIENTO, EDUARDO; MASSEY, PEDRO

JOURNAL OF FUNCTIONAL ANALYSIS

ACADEMIC PRESS INC ELSEVIER SCIENCE

Lugar: Amsterdam; Año: 2022 vol. 282

0022-1236

Let H be a separable infinite-dimensional complex Hilbertspace, B(H) the algebra of bounded linear operators actingon H and J a proper two-sided ideal of B(H). Denote byUJ (H) the group of all unitary operators of the form I + J .Recall that an operator A ∈ B(H) is diagonalizable if thereexists a unitary operator U such that UAU∗ is diagonal withrespect to some orthonormal basis. A more restrictive notionof diagonalization can be formulated with respect to a fixedorthonormal basis e = {en}n≥1 and a proper operator idealJ as follows: A ∈ B(H) is called restrictedly diagonalizableif there exists U ∈ UJ (H) such that UAU∗ is diagonal withrespect to e. In this work we give a sufficient condition fora diagonalizable operator to be restrictedly diagonalizable.This condition becomes a characterization when the idealis arithmetic mean closed. Then we obtain results on thestructure of the set of all restrictedly diagonalizable operators.In this way we answer several open problems recently raisedby Beltiţă, Patnaik and Weiss