A study of Bessel sequences and frames via perturbations of constant multiples of the identity
[摘要] 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.
[发布日期] [发布机构]
[效力级别] [学科分类] 公共、环境与职业健康
[关键词] frame;Bessel sequence;Compact-tight frame;Finite-rank-tight frame;Reverse Schwarz inequality [时效性]
