〈2, 1〉-Compact operators

Abstract

We consider the class of the continuous L2,1 linear operators in L2 that are sums of the operators of multiplication by bounded measurable functions and the operators sending the unit ball of L2 into a compact subset of L1. We prove that a functional equation with an operator from L2,1 is equivalent to an integral equation with kernel satisfying the Carleman condition. We also prove that if T ∈ L2,1 and VTV−1 ∈ L2,1 for all unitary operators V in L2 then T = α1 + C, where α is a scalar, 1 is the identity operator in L2, and C is a compact operator in L2.