Frame-normalizable Sequences

Abstract

Let $H$ be a separable Hilbert space and let $\{x_n\}$ be a sequence in $H$ that does not contain any zero elements. We say that $\{x_n\}$ is a \emph{Bessel-normalizable} or \emph{frame-normalizable} sequence if the normalized sequence $\{\frac{x_n}{\|x_n\|}\}$ is a Bessel sequence or a frame for $H$, respectively. In this paper, several necessary and sufficient conditions for sequences to be frame-normalizable and not frame-normalizable are proved. Perturbation theorems for frame-normalizable sequences are also proved. As applications, we show that the Balazs-Stoeva conjecture %\cite{BS11} holds for Bessel-normalizable sequences. Finally, we apply our results to partially answer the open question raised by Aldroubi et al.\ %\cite{ACMCP16} as to whether the iterative system $\{\frac{A^n x}{\|A^nx\|}\}_{n\geq 0,\, x\in S}$ associated with a normal operator $A\colon H\rightarrow H$ and a countable subset $S$ of $H$, is a frame for $H$. In particular, if $S$ is finite, then we are able to show that $\{\frac{A^n x}{\|A^nx\|}\}_{n\geq 0,\, x\in S}$ is not a frame for $H$ whenever $\{A^nx\}_{n\geq 0,\,x\in S}$ is a frame for $H$.