Continuous frames for unbounded operators

TL;DR

This paper extends the concept of frames to unbounded operators in Hilbert spaces, enabling continuous decompositions of their ranges, thus generalizing previous discrete and bounded operator frameworks.

Contribution

It introduces continuous frames for densely defined unbounded operators, broadening the scope of frame theory in Hilbert spaces.

Findings

Generalization of $K$-frames to unbounded operators

Development of continuous frame theory for unbounded operators

Extension of discrete frame concepts to continuous settings

Abstract

Few years ago G\u{a}vru\c{t}a gave the notions of $K$-frame and atomic system for a linear bounded operator $K$ in a Hilbert space $\mathcal{H}$ in order to decompose $\mathcal{R}(K)$, the range of $K$, with a frame-like expansion. These notions are here generalized to the case of a densely defined and possibly unbounded operator on a Hilbert space $A$ in a continuous setting, thus extending what have been done in a previous paper in a discrete framework.