Some results on U -cross Gram matrices by using K -frames
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Some results on U-cross Gram matrices by using K -frames Mitra Shamsabadi1 · Ali Akbar Arefijamaal1 Received: 22 November 2018 / Accepted: 12 May 2020 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2020
Abstract U -cross Gram matrices are produced by frames and Riesz bases. In this paper, we represent bounded operators as matrix operators using K -frames. We study the invertibility matrices respect to K -frames. Moreover, we apply the concept of K -Riesz bases in Hilbert space H to the concept of matrix induced by U with respect to K -Riesz bases. Keywords U -cross Gram matrix · Cross Gram matrix · K -frame · K -Riesz basis Mathematics Subject Classification Primary 41A58 · Secondary 43A35
1 Introduction, notation and motivation A unitary system is a set of unitary operators U acting on a Hilbert space H which contains the identity operator I of B(H). A Bessel generator for U is a vector x ∈ H with the property that U x := {U x : U ∈ U } is Bessel sequence for H. Many useful frames, which play an essential role in both theory and applications, can been considered as unitary systems, group-like unitary systems and atomic systems [16,18]. K -frames were recently introduced by Gavruta to study atomic systems with respect to a bounded operator K ∈ B(H). It is a generalization of frame theory such that the lower bound is only satisfied for the elements in the range of K [17]. It is shown that an atomic system for K is a K -frame and vice versa. For this reason, K -frames are a useful mathematical tool to study the structure of unitary systems. Another purpose of this paper is to study Gram Matrices. The operator equation U f = v where U ∈ B(H) does not have a smooth solution (i.e. have all derivatives continuous) in general. It can be rewritten of the form Ax = b
(1.1)
where Ai, j = U ei , e j and {ei }i∈I is an orthonormal basis of H . To solve linear systems (1.1) variational method can be applied for example [25]. Recently, frames, Riesz bases and
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Mitra Shamsabadi [email protected] Ali Akbar Arefijamaal [email protected]; [email protected]
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Department of Mathematics and Computer Sciences, Hakim Sabzevari University, Sabzevar, Iran
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M. Shamsabadi, A. Arefijamaal
g-frames are applied to obtain (1.1) [3,4,12]. In this paper, we apply K -frames to get (1.1) as atomic decompositions of elements in the range of K which may not be closed. Let H be a separable Hilbert space and K an operator from H to H. A sequence F := { f i }i∈I ⊆ H is called a K -frame for H, if there exist constants A, B > 0 such that AK ∗ f 2 ≤ | f , f i |2 ≤ B f 2 , ( f ∈ H). (1.2) i∈I
Clearly if K = IH , then F is an ordinary frame. The constants A and B in (1.2) are called lower and upper bounds of F, respectively. We call F a A-tight K -frame if AK ∗ f 2 = 2 i∈I | f , f i | and a 1-tight K -frame as Parseval K -frame. A K -frame is called an exact K -frame, if by removing any element, the reminder sequence is not a K -frame. Obviously, every K -frame i
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