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On the Sum of Generalized Frames in Hilbert Spaces

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Let \({\mathcal {H}}\) be a separable Hilbert space. It is known that the finite sum of Bessel sequences in \({\mathcal {H}}\) is still a Bessel sequence. But the finite sum of generalized notions of frames does not necessarily remain stable in its initial form. In this paper, for a prescribed Bessel sequence \(F=\{f_n\}_{n=1}^\infty \), we introduce and study \({\mathcal {KF}}\), the set consisting of all operators \(K\in {\mathcal {B}}({\mathcal {H}})\), such that \(\{f_n\}_{n=1}^\infty \) is a K-frame. We show that \({\mathcal {KF}}\) is a right ideal of \({\mathcal {B}}({\mathcal {H}})\). We indicate by an example that \({\mathcal {KF}}\) is not necessarily a left ideal. Moreover, we provide some sufficient conditions for the finite sum of K-frames to be a K-frame. We also use some examples to compare our results with existing ones. These examples demonstrate that our achievements do not depend on the available results. Furthermore, we study the same subject for K-g-frames and controlled frames and get some similar significant results.

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The authors express their sincere gratitude to the reviewer for his/her very careful reading of the manuscript and constructive comments on the manuscript, which substantially helped us to improve the quality of the paper. His/her invaluable comments, especially simplifying the proof of the results, improved the presentation of the manuscript significantly.

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Correspondence to F. Abtahi.

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Abtahi, F., Kamali, Z. & Keyshams, Z. On the Sum of Generalized Frames in Hilbert Spaces. Mediterr. J. Math. 18, 178 (2021).

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