Generalized atomic subspaces for operators in Hilbert spaces
aa r X i v : . [ m a t h . F A ] F e b Generalized atomic subspaces for operators in Hilbert spaces
Prasenjit Ghosh
Department of Pure Mathematics, University of Calcutta,35, Ballygunge Circular Road, Kolkata, 700019, West Bengal, Indiae-mail: [email protected]
T. K. Samanta
Department of Mathematics, Uluberia College,Uluberia, Howrah, 711315, West Bengal, Indiae-mail: mumpu − [email protected] Abstract
We introduce the notion of a g-atomic subspace for a boundedlinear operator and construct several useful resolutions of the identityoperator on a Hilbert space using the theory of g-fusion frames. Alsowe shall describe the concept of frame operator for a pair of g-fusionBessel sequences and some of their properties.
Keywords:
Frame, atomic subspace, g-fusion frame, K-g-fusion frame.
Primary 42C15; Secondary 46C07.
Frames for Hilbert spaces were first introduced by Duffin and Schaeffer [6] in1952 to study some fundamental problems in non-harmonic Fourier series. Later on,after some decades, frame theory was popularized by Daubechies, Grossman, Meyer[4]. At present, frame theory has been widely used in signal and image processing,filter bank theory, coding and communications, system modeling and so on. Severalgeneralizations of frames namely, K -frames, g -frames, fusion frames etc. have beenintroduced in recent times. K -frames were introduced by L. Gavruta [8] to study the atomic system withrespect to a bounded linear operator. Using frame theory techiques, the authoralso studied the atomic decompositions for operators on reporducing kernel Hilbertspaces [9]. Sun [15] introduced a g -frame and a g -Riesz basis in complex Hilbertspaces and discussed several properties of them. Huang [11] began to study K - g -frame by combining K -frame and g -frame. P. Casazza [2] was first to introduce thenotion of fusion frames or frames of subspaces and gave various ways to obtain aresolution of the identity operator from a fuison frame. The concept of an atomicsubspace with respect to a bounded linear operator were introduced by A. Bhandariand S. Mukherjee [1]. Construction of K - g -fusion frames and their dual were pre-sented by Sadri and Rahimi [14] to generalize the theory of K -frame, fusion frame Prasenjit Ghosh & T. K. Samanta and g -frame. P. Ghosh and T. K. Samanta [10] studied the stability of dual g -fusionframes in Hilbert spaces.In this paper, we present some useful results about resolution of the identityoperator on a Hilbert space using the theory of g -fusion frames. We give the notionof g -atomic subspace with respect to a bounded linear operator. The frame operatorfor a pair of g -fusion Bessel sequences are discussed and some properties are goingto be established.The paper is organized as follows; in Section 2, we briefly recall the basic def-initions and results. Various ways of obtaining resolution of the identity operatoron a Hilbert space in g -fusion frame are studied in Section 3. g -atomic subspacesare introduced and discussed in Section 4. In Section 5, frame operator for a pair of g -fusion Bessel sequences are given and establish various properties.Throughout this paper, H is considered to be a separable Hilbert space withassociated inner product h · , · i and { H j } j ∈ J are the collection of Hilbert spaces,where J is subset of integers Z . I H is the identity operator on H . B ( H , H )is a collection of all bounded linear operators from H to H . In particular B ( H )denote the space of all bounded linear operators on H . For T ∈ B ( H ), we denote N ( T ) and R ( T ) for null space and range of T , respectively. Also, P V ∈ B ( H )is the orthonormal projection onto a closed subspace V ⊂ H . Define the space l (cid:16) { H j } j ∈ J (cid:17) = { f j } j ∈ J : f j ∈ H j , X j ∈ J k f j k < ∞ with inner product is given by h { f j } j ∈ J , { g j } j ∈ J i = P j ∈ J h f j , g j i H j . Clearly l (cid:16) { H j } j ∈ J (cid:17) is a Hilbert space with the pointwise operations [14]. Theorem 2.1. ( Douglas’ factorization theorem ) [5] Let
U, V ∈ B ( H ) . Thenthe following conditions are equivalent:( 1 ) R ( U ) ⊆ R ( V ) .( 2 ) U U ∗ ≤ λ V V ∗ for some λ > .( 3 ) U = V W for some bounded linear operator W on H . Theorem 2.2. [12] The set S ( H ) of all self-adjoint operators on H is apartially ordered set with respect to the partial order ≤ which is defined as for T, S ∈ S ( H ) T ≤ S ⇔ h T f , f i ≤ h