The Invertibility of U -Fusion Cross Gram Matrices of Operators
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The Invertibility of U -Fusion Cross Gram Matrices of Operators Mitra Shamsabadi, Ali Akbar Arefijamaal
and Peter Balazs
Abstract. Finding matrix representations is an important part of operator theory. Calculating such a discretization scheme is equally important for the numerical solution of operator equations. Traditionally in both fields, this was done using bases. Recently, frames have been used here. In this paper, we apply fusion frames for this task, a generalization motivated by a block representation, respectively, a domain decomposition. We interpret the operator representation using fusion frames as a generalization of fusion Gram matrices. We present the basic definition of U -fusion cross Gram matrices of operators for a bounded operator U . We give necessary and sufficient conditions for their (pseudo-)invertibility and present explicit formulas for the (pseudo-)inverse. More precisely, our attention is on how to represent the inverse and pseudo-inverse of such matrices as U -fusion cross Gram matrices. In particular, we characterize fusion Riesz bases and fusion orthonormal bases by such matrices. Finally, we look at which perturbations of fusion Bessel sequences preserve the invertibility of the fusion Gram matrix of operators. Mathematics Subject Classification. Primary 42C15; Secondary 47A05. Keywords. Fusion frames, Gram matrices, representation of operators, pseudo-inverses.
1. Introduction and Motivation For the representation (and modification) of functions a standard approach is using orthonormal bases (ONBs). It can be hard to find a ’good’ orthonormal basis, in the sense that it sometimes cannot fulfill given properties, as formally expressed, e.g. in the Balian–Low theorem [35]. For solving this problem, frames were introduced by Duffin and Schaeffer [28] and widely developed by many authors [17,22,26,30]. In recent years, frames have been the focus of active research, both in theory [2,18,31] and applications [11,19,24,29]. Also, several generalizations have been investigated, e.g. [1,3,4,46,49], among them fusion frames [20,21,28,33], which are the topic of this paper. 0123456789().: V,-vol
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Recently, both in applications, e.g. numerical simulations in acoustics, and also in the mathematical theory it has been observed that frames are not only useful for analyzing signals or functions, but also for the matrix representation of operators. On an abstract level, it is well known that for orthonormal bases, operators can be uniquely described by a matrix representation [34]. We have shown that an analogous result holds for frames and their duals [8,10]. linear operator O and frames Ψ, Φ, the For a bounded, infinite matrix MΦ,Ψ (O) k,l = Oψl , φk acts as a bounded operator on 2 since MΦ,Ψ (O) = CΦ ODΨ . One question that remains is can we find a stable way to find a block representation of the operator? For a numerical treatment of operator equations, used, for example, for solving integral equations in acoustics [40] and stochastic [37] , the in
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