A Kato's second type representation theorem for solvable sesquilinear forms

TL;DR

This paper extends Kato's second representation theorem to a broader class of solvable sesquilinear forms, including non-symmetric and indefinite cases, establishing a new correspondence with associated operators.

Contribution

It introduces the concept of hyper-solvable forms and proves a one-to-one correspondence with operators, generalizing existing theorems to more general forms.

Findings

Established a correspondence between hyper-solvable forms and operators.

Generalized Kato's second theorem to non-symmetric, indefinite forms.

Applicable to closed semibounded forms and beyond.

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.