A Riemann Type Theorem for Unconditional Convergence of Operators

Abstract

We prove that if a series of bounded linear operators is compactly conditionally convergent in the strong operator topology, that is, each of its partial sums converge, in the strong operator topology to a compact operator, then the series converges in the uniform (operator norm) topology; although not necessarily absolutely. In case the operators are all mutually diagonalizable, then under the same hypothesis, the series converges absolutely uniformly.