# Frames and weak frames for unbounded operators

## TL;DR

This paper extends the concepts of frames and weak frames to unbounded operators in Hilbert spaces, providing new approaches for decomposing the range of such operators using different types of sequences.

## Contribution

It introduces two new frameworks for frames related to unbounded operators, one with non-Bessel sequences and another with Bessel sequences, considering different continuity conditions.

## Key findings

- Extended frame concepts to unbounded operators
- Developed two approaches with different sequence types
- Provided conditions for continuous coefficient sequences

## Abstract

In 2012 G\u{a}vru\c{t}a introduced the notions of $K$-frame and of atomic system for a linear bounded operator $K$ in a Hilbert space $\mathcal{H}$, in order to decompose its range $\mathcal{R}(K)$ with a frame-like expansion. In this article we revisit these concepts for an unbounded and densely defined operator $A:\mathcal{D}(A)\to\mathcal{H}$ in two different ways. In one case we consider a non-Bessel sequence where the coefficient sequence depends continuously on $f\in\mathcal{D}(A)$ with respect to the norm of $\mathcal{H}$. In the other case we consider a Bessel sequence and the coefficient sequence depends continuously on $f\in\mathcal{D}(A)$ with respect to the graph norm of $A$.