New results are added to the paper (Di Bella and Trapani in J Math Anal Appl 451:64-83, 2017) about q-closed and solvable sesquilinear forms. The structure of the Banach space defined on the domain of a q-closed sesquilinear form is unique up to isomorphism, and the adjoint of a sesquilinear form has the same property of q-closure or of solvability. The operator associated to a solvable sesquilinear form is the greatest which represents the form and it is self-adjoint if, and only if, the form is symmetric. We give more criteria of solvability for q-closed sesquilinear forms. Some of these criteria are related to the numerical range, and we analyse in particular the forms which are solvable with respect to inner products. The theory of solvable sesquilinear forms generalises those of many known sesquilinear forms in literature

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 weaklydefined 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.

Few years ago Gȃvruţa gave the notions of K-frame and atomic system for a linear bounded operator K in a Hilbert space H in order to decompose R(K), the range of K, with a frame-like expansion. These notions are here generalized to the case of a densely defined and possibly unbounded operator on a Hilbert space A in a continuous setting, thus extending what have been done in a previous paper in a discrete framework.

Abstract. Kato's second representation theorem is generalized to solvable sesquilinear forms. These forms need not be non-negative nor symmetric. The representation considered holds for a subclass of solvable forms (called hyper-solvable), precisely for those whose domain is exactly the domain of the square root of the modulus of the associated operator. This condition always holds for closed semibounded forms, and it is also considered by several authors for symmetric sign-indefinite forms. As a consequence, a one-to-one correspondence between hyper-solvable forms and operators, which generalizes those already known, is established.

Given a sequence of elements ξ = {ξ n } n∈N of a Hilbert space, an operator T ξ is defined as the operator associated to a sesquilinear form determined by ξ. This operator is in general different to the classical frame operator but possesses some remarkable properties. For instance, T ξ is self-adjoint (in an specific space), unconditionally defined and, when ξ is a lower semi-frame, T ξ gives a simple expression of a dual of ξ. The operator T ξ and lower semi-frames are studied in the context of sequences of integer translates.