@article {
author = {Movahed, Sima and Hosseini Giv, Hossein and Ahmadi Ledari, Alireza},
title = {A study of Bessel sequences and frames via perturbations of constant multiples of the identity},
journal = {Khayyam Journal of Mathematics},
volume = {9},
number = {1},
pages = {102-115},
year = {2023},
publisher = {Department of Pure Mathematics, Ferdowsi University of Mashhad (in cooperation with the Center of Excellence in Analysis on Algebraic Structures)},
issn = {2423-4788},
eissn = {2423-4788},
doi = {10.22034/kjm.2022.355505.2629},
abstract = {We study those Bessel sequences $\{f_k\}_{k=1}^{\infty}$ in an infinite-dimensional, separable Hilbert space $H$ for which the operator $S$ defined by $Sf:=\sum_{k=1}^{\infty} \langle f,f_k\rangle f_k$ is of the form $cI+T$, for some real number $c$ and a bounded linear operator $T$, where $I$ denotes the identity operator. We use a reverse Schwarz inequality to provide conditions on $T$ and $c$ that allow $\{f_k\}_{k=1}^{\infty}$ to be a frame. Moreover, we introduce and study frames whose frame operators are compact (respectively, finite-rank) perturbations of constant multiples of the identity, frames to which we refer as compact-tight (respectively, finite-rank-tight) frames. As our final result, we prove a theorem on the weaving of certain compact-tight frames.},
keywords = {frame,Bessel sequence,Compact-tight frame,Finite-rank-tight frame,Reverse Schwarz inequality},
url = {https://www.kjm-math.org/article_164487.html},
eprint = {https://www.kjm-math.org/article_164487_14305fcfe5e3748748467f9717be45eb.pdf}
}