Search results for " 47A07"
showing 2 items of 2 documents
Sesquilinear forms associated to sequences on Hilbert spaces
2019
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.
Maximal Operators with Respect to the Numerical Range
2018
Let $\mathfrak{n}$ be a nonempty, proper, convex subset of $\mathbb{C}$. The $\mathfrak{n}$-maximal operators are defined as the operators having numerical ranges in $\mathfrak{n}$ and are maximal with this property. Typical examples of these are the maximal symmetric (or accretive or dissipative) operators, the associated to some sesquilinear forms (for instance, to closed sectorial forms), and the generators of some strongly continuous semi-groups of bounded operators. In this paper the $\mathfrak{n}$-maximal operators are studied and some characterizations of these in terms of the resolvent set are given.