A survey on solvable sesquilinear forms

Abstract

The aim of this paper is to present a unified theory of many Kato type representation theorems in terms of solvable forms on Hilbert spaces. In particular, for some sesquilinear forms Ω on a dense domain D one looks for an expressionwhere T is a densely defined closed operator with domain D(T ) ⊆ D.There are two characteristic aspects of solvable forms. Namely, one is that the domain of the form can be turned into a reflexive Banach space need not be a Hilbert space. The second one is the existence of a perturbation with a bounded form which is not necessarily a multiple of the inner product.Mathematics Subject Classification (2010). Primary 47A07; Secondary 47A10, 47A12.Keywords. Kato's representation theorems, q-closed/solvable sesquilinear forms.Expression (1.1) holds for every bounded sesquilinear form Ω and for some bounded operator by Riesz's classical representation theorem. The situation in the unbounded case is more complicated. One of the earliest result of this topic is formulated by Kato in [8].Kato's first representation theorem. Let Ω be a densely defined closed sectorial form with domain D ⊆ H. Then there exists a unique m-sectorial operator T , with domain D(T ) ⊆ D, such that Ω(ξ, η) = T ξ, η , ∀ξ ∈ D(T ), η ∈ D.( 1.2)

An equivalent formulation of 0-closed sesquilinear forms

“…In the last decades, several theorems about the representation (1.2) have been given [1][2][3][4][5]9,11,[13][14][15][16][17]. The topic of the representation is connected to the Lebesgue decomposition (see [6][7][8]12,19]) as motivated in [8].…”