On invertibility of some classes of operators in weighted Hilbert spaces

This work reports a study of the invertibility of operators acting in a Hilbert couple. We consider the general conditions for operators in a Hilbert couple that use only the regularity of the couple as well as representing the Hilbert couple as a direct sum of quadratically summed vector-valued sequences. By considering the representation of the algebra of operators acting in a Hilbert couple in interpolation spaces, it becomes possible to study the invertibility of operators in these spaces. For a couple with sparse weights, it is shown that any operator whose matrix representation contains an “invertible” main diagonal will be invertible in some (central) interpolation space. Along the way, it was proved that any such operator belongs to the Neumann-Schatten class with an arbitrary positive index. By the representation of the operator in tandem with sparse weights, a criterion for the invertibility of the operator in a general Hilbert couple is obtained. This criterion is based on the invertibility of the main diagonal of the matrix corresponding to the operator as well as on the weight conditions which are superimposed on diagonals parallel to the main one.