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