Frames and weak frames for unbounded operators

Abstract

In 2012 Găvruţa 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 $A$.

24 pages

References in corpus (6)

• Biorthogonal vectors, sesquilinear forms and some physical operators
• Non-self-adjoint hamiltonians defined by generalized Riesz bases
• Hamiltonians defined by biorthogonal sets
• Sesquilinear forms associated to sequences on Hilbert spaces
• Reproducing pairs and Gabor systems at critical density
• Generalized frame operator, lower semi-frames and sequences of translates

Cited by in corpus (2)

• Some notes about distribution frame multipliers
• Frame-related Sequences in Chains and Scales of Hilbert Spaces