Properties of restrictions of the operator of multiplication by a continuous function

Abstract

For the operatorA of multiplication by a continuous functiona (t) in the Hilbert spaceL ₂[0, b]=H, we give a description of two sets of infinite-dimensional subspaces with infinite codimensions:I(A)={N⊂H:A/N is an isomorphism},K(A)={M⊂H: A/M is a compact mapping}. As an application, we consider the problem of determining whether the sequence {a(t)e_n(t)}, where {e_n(t)} is an orthonormal basis in L₂[0,b], is an unconditional basis.