A study of Bessel sequences and frames via perturbations of constant multiples of the identity

Document Type : Original Article


Department of Mathematics, Faculty of Mathematics, Statistics and Computer Science, University of Sistan and Baluchestan, P. O. Box 98135-674, Zahedan, Iran



‎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.