  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.
