Sesquilinear forms associated to sequences on Hilbert spaces

Abstract

The possibility of defining sesquilinear forms starting from one or two sequences of elements of a Hilbert space is investigated. One can associate operators to these forms and in particular look for conditions to apply representation theorems of sesquilinear forms, such as Kato's theorems. The associated operators correspond to classical frame operators or weakly-defined multipliers in the bounded context. In general some properties of them, such as the invertibility and the resolvent set, are related to properties of the sesquilinear forms. As an upshot of this approach new features of sequences (or pairs of sequences) which are semi-frames (or reproducing pairs) are obtained.

25 pages

References in corpus (4)

• Biorthogonal vectors, sesquilinear forms and some physical operators
• Reproducing pairs and Gabor systems at critical density
• Representation Theorems for Solvable Sesquilinear Forms
• U-cross Gram matrices and their invertibility

Cited by in corpus (8)

• Generalized frame operator, lower semi-frames and sequences of translates
• Frames and weak frames for unbounded operators
• Some notes about distribution frame multipliers
• Localization of the spectra of dual frames multipliers
• Lower semi-frames and metric operators
• Some perturbation results for quasi-bases and other sequences of vectors
• On some dual frames multipliers with at most countable spectra
• Weak $A$-frames and weak $A$-semi-frames