Frames and bases of subspaces in Hilbert spaces

In this paper we develop the frame theory of subspaces for separable Hilbert spaces. We will show that for every Parseval frame of subspaces {Wi}_i∈I for a Hilbert space H, there exists a Hilbert space K⊇H and an orthonormal basis of subspaces {Ni}_i∈I for K such that Wi=P(Ni), where P is the orthogonal projection of K onto H. We introduce a new definition of atomic resolution of the identity in Hilbert spaces. In particular, we define an atomic resolution operator for an atomic resolution of the identity, which even yield a reconstruction formula.