TY - JOUR
ID - 164487
TI - A study of Bessel sequences and frames via perturbations of constant multiples of the identity
JO - Khayyam Journal of Mathematics
JA - KJM
LA - en
SN -
AU - Movahed, Sima
AU - Hosseini Giv, Hossein
AU - Ahmadi Ledari, Alireza
AD - Department of Mathematics, Faculty of
Mathematics, Statistics and Computer Science, University of Sistan and Baluchestan, P. O. Box
98135-674, Zahedan, Iran
Y1 - 2023
PY - 2023
VL - 9
IS - 1
SP - 102
EP - 115
KW - frame
KW - Bessel sequence
KW - Compact-tight frame
KW - Finite-rank-tight frame
KW - Reverse Schwarz inequality
DO - 10.22034/kjm.2022.355505.2629
N2 - 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.
UR - https://www.kjm-math.org/article_164487.html
L1 - https://www.kjm-math.org/article_164487_14305fcfe5e3748748467f9717be45eb.pdf
ER -