Gram matrix associated to controlled frames

TL;DR

This paper introduces the controlled-Gram matrix for sequences in Hilbert spaces, providing diagnostic tools and structural insights into controlled frames, Riesz bases, and associated operators, enhancing analysis and computational efficiency.

Contribution

It defines the controlled-Gram matrix as a practical diagnostic tool and characterizes the structure and properties of controlled Riesz bases and related operators.

Findings

Controlled-Gram matrix can diagnose controlled Bessel, frame, or Riesz basis.

Explicit structure for controlled Riesz bases is provided.

Conditions for boundedness, invertibility, and operator class membership are discussed.

Abstract

Controlled frames have been recently introduced in Hilbert spaces to improve the numerical efficiency of interactive algorithms for inverting the frame operator. In this paper, unlike the cross-Gram matrix of two different sequences which is not always a diagnostic tool, we define the controlled-Gram matrix of a sequence as a practical implement to diagnose that a given sequence is a controlled Bessel, frame or Riesz basis. Also, we discuss the cases that the operator associated to controlled Gram matrix will be bounded, invertible, Hilbert-Schmidt or a trace-class operator. Similar to standard frames, we present an explicit structure for controlled Riesz bases and show that every $(U, C)$-controlled Riesz basis $\{f_{k}\}_{k=1}^{\infty}$ is in the form $\{U^{-1}CMe_{k}\}_{k=1}^{\infty}$, where $M$ is a bijective operator on $H$. Furthermore, we propose an equivalent accessible condition to the sequence $\{f_{k}\}_{k=1}^{\infty}$ being a $(U, C)$-controlled Riesz basis.