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Mathematics > Functional Analysis

arXiv:1804.00203 (math)
[Submitted on 31 Mar 2018 (v1), last revised 20 Mar 2019 (this version, v2)]

Title:U-cross Gram matrices and their invertibility

Authors:Peter Balazs, Mitra Shamsabadi, Ali Akbar Arefijamaal, Asghar Rahimi
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Abstract:The Gram matrix is defined for Bessel sequences by combining synthesis with subsequent analysis operators. If different sequences are used and an operator U is inserted we reach so called U-cross Gram matrices. This can be seen as reinterpretation of the matrix representation of operators using frames. In this paper we investigate some necessary or sufficient conditions for Schatten p-class properties and the invertibility of U-cross Gram matrices. In particular, we show that under mild conditions the pseudo-inverse of a U-cross Gram matrix can always be represented as a U-cross Gram matrix with dual frames of the given ones. We link some properties of U-cross Gram matrices to approximate duals. Finally, we state several stability results. More precisely, it is shown that the invertibility of U-cross Gram matrices is preserved under small perturbations.
Comments: 29 pages
Subjects: Functional Analysis (math.FA)
MSC classes: 41A58 (Primary), 43A35 (Secondary)
Cite as: arXiv:1804.00203 [math.FA]
  (or arXiv:1804.00203v2 [math.FA] for this version)
  https://doi.org/10.48550/arXiv.1804.00203
Journal reference: Journal of Mathematical Analysis and Applications, Vol. 476 (2), pp. 367-390 (2019)
Related DOI: https://doi.org/10.1016/j.jmaa.2019.03.045

Submission history

From: Ali Akbar Arefijamaal [view email]
[v1] Sat, 31 Mar 2018 18:42:05 UTC (30 KB)
[v2] Wed, 20 Mar 2019 20:50:29 UTC (37 KB)
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