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Frames and weak frames for unbounded operators

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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.

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The authors warmly thank Prof. C. Trapani and the referees for their fruitful comments and remarks.


This work has been in part financially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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Correspondence to Giorgia Bellomonte.

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Communicated by: Tomas Sauer

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Bellomonte, G., Corso, R. Frames and weak frames for unbounded operators. Adv Comput Math 46, 38 (2020).

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Mathematics Subject Classification (2010)