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JOURNALS // Vladikavkazskii Matematicheskii Zhurnal // Archive

Vladikavkaz. Mat. Zh., 2023 Volume 25, Number 1, Pages 48–63 (Mi vmj847)

On stability of retro Banach frame with respect to $b$-linear functional in $n$-Banach space

P. Ghosha, T. K. Samantab

a Department of Pure Mathematics, University of Calcutta, 35 Ballygunge Circular Road, Kolkata 700019, West Bengal, India
b Department of Mathematics, Uluberia College, Uluberia, Howrah 711315, West Bengal, India

Abstract: We introduce the notion of a retro Banach frame relative to a bounded $b$-linear functional in $n$-Banach space and see that the sum of two retro Banach frames in $n$-Banach space with different reconstructions operators is also a retro Banach frame in $n$-Banach space. Also, we define retro Banach Bessel sequence with respect to a bounded $b$-linear functional in $n$-Banach space. A necessary and sufficient condition for the stability of retro Banach frame with respect to bounded $b$-linear functional in $n$-Banach space is being obtained. Further, we prove that retro Banach frame with respect to bounded $b$-linear functional in $n$-Banach space is stable under perturbation of frame elements by positively confined sequence of scalars. In $n$-Banach space, some perturbation results of retro Banach frame with the help of bounded $b$-linear functional in $n$-Banach space have been studied. Finally, we give a sufficient condition for finite sum of retro Banach frames to be a retro Banach frame in $n$-Banach space. At the end, we discuss retro Banach frame with respect to a bounded $b$-linear functional in Cartesian product of two $n$-Banach spaces.

Key words: frame, Banach frame, retro Banach frame, stability, $n$-Banach space, $b$-linear functional.

UDC: 517.982.22

MSC: 42C15, 46C07, 46M05, 47A80

Received: 09.11.2021

Language: English

DOI: 10.46698/o3961-3328-9819-i



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