Lower semi-frames and metric operators

TL;DR

This paper explores how metric operators can transform weakly measurable functions in Hilbert spaces into continuous frames, with a focus on lower semi-frames and their relation to generalized frame operators.

Contribution

It introduces conditions under which weakly measurable functions can be converted into continuous frames using metric operators, especially for lower semi-frames.

Findings

A dense domain of the analysis operator is necessary for transformation.

Generalized frame operators have better properties than usual frame operators.

Lower semi-frames can be converted into Parseval frames with a suitable metric operator.

Abstract

This paper deals with the possibility of transforming a weakly measurable function in a Hilbert space into a continuous frame by a metric operator, i.e., a strictly positive self-adjoint operator. A necessary condition is that the domain of the analysis operator associated to the function be dense. The study is done also with the help of the generalized frame operator associated to a weakly measurable function, which has better properties than the usual frame operator. A special attention is given to lower semi-frames: indeed if the domain of the analysis operator is dense, then a lower semi-frame can be transformed into a Parseval frame with a (special) metric operator.