In this paper we develop the frame theory of subspaces for separable Hilbert spaces. We will show that for every Parseval frame of subspaces {W i } i∈I for a Hilbert space H , there exists a Hilbert space K ⊇ H and an orthonormal basis of subspaces {N i } i∈I for K such that W i = P (N i ), 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.

The notion of frame has some generalizations such as frames of subspaces, fusion frames and g-frames. In this paper, we introduce fusion frames and g-frames in Hilbert C*-modules and we show that they share many useful properties with their corresponding notions in Hilbert space. We also generalize a perturbation result in frame theory to g-frames in Hilbert spaces. We also show that tensor product of fusion frames (g-frames) is a fusion frame (g-frame) and tensor product of resolution of identity is a resolution of identity.

Let G be a finite group. The main result of this paper is as follows: If G is a finite group, such that Γ(G) = Γ( 2 G 2 (q)), where q = 3 2n+1 for some n ≥ 1, then G has a (unique) nonabelian composition factor isomorphic to 2 G 2 (q). We infer that if G is a finite group satisfying |G| = | 2 G 2 (q)| and Γ(G) = Γ( 2 G 2 (q)) then G ∼ = 2 G 2 (q). This enables us to give new proofs for some theorems; e.g., a conjecture of W. Shi and J. Bi. Some applications of this result are also considered to the problem of recognition by element orders of finite groups.

Fusion frames and g-frames were considered recently as generalizations of frames in Hilbert spaces, in this paper we generalize some of the known results in frame theory to fusion frames and g-frames. We obtain new g-frames by considering g-frames for its components. We also obtain some results about alternate dual g-frames, excess of g-frames and stability of g-frames under small perturbations.

Abstract.In this article, we study tensor product of Hilbert C * -modules and Hilbert spaces. We show that if E is a Hilbert A-module and F is a Hilbert B-module, then tensor product of frames (orthonormal bases) for E and F produce frames (orthonormal bases) for Hilbert A ⊗ B-module E ⊗ F , and we get more results.For Hilbert spaces H and K, we study tensor product of frames of subspaces for H and K, tensor product of resolutions of the identities of H and K, and tensor product of frame representations for H and K.