# Sesquilinear forms associated to sequences on Hilbert spaces

## TL;DR

This paper explores how to define and analyze sesquilinear forms associated with sequences in Hilbert spaces, linking them to operators like frame operators and multipliers, and deriving new properties of semi-frames and reproducing pairs.

## Contribution

It introduces a framework for associating sesquilinear forms with sequences in Hilbert spaces and investigates conditions for applying representation theorems, revealing new features of semi-frames and reproducing pairs.

## Key findings

- Operators related to sesquilinear forms include classical frame operators.
- Invertibility and resolvent properties relate to sesquilinear form properties.
- New features of semi-frames and reproducing pairs are identified.

## 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.