Resumen de Frames of subspaces and approximation of the inverse frame operator

Amir Khosravi, Mohammad Sadegh Asgari

• A frame of subspaces in a Hilbert space H allows that identity operator on H to be written as a sum of some bounded operators on H. This family of bounded operators on H is called an atomic resolution of the identity on H. We show the atomic resolution of the identity associated to a frame of subspaces have a certain minimum property relative to its associated norm. We further show that under extra condition every atomic resolution of the identity provides a frame of subspaces for H. We consider direct sum of frames of subspaces with respect to the same family of weights which is a frame of subspaces for their direct sum space. Frame theory of subspaces describes how one can choose the corresponding atomic resolution of the identity, which is interesting from mathematical point of view, but for applications it is a problem that requires to know the inverse frame operator S-1W,v on H. If the underlying Hilbert space is infinite dimensional it is hard to invert the frame operator SW,v. We show how the inverse of SW,v can be approximated by using the methods of linear algebra.