## Representation theorems for indefinite quadratic forms revisited.(English)Zbl 1272.47004

Let $$X$$ be a complex Hilbert space with inner product $$\langle\cdot,\cdot\rangle$$ and consider the class of forms given by $b[x,y] = \langle A^{1/2}x,HA^{1/2}y \rangle \;\;\text{ for } x,y \in \text{Dom}[b] = \text{Dom}(A^{1/2}),$ where $$A$$ is a uniformly positive self-adjoint operator in the Hilbert space $$X$$ and $$H$$ is a bounded, not necessarily positive, self-adjoint operator in $$X$$. The following representation theorems are natural generalizations to indefinite forms of the first and second representation theorems for semi-bounded sesquilinear forms.
(i) If $$H$$ has a bounded inverse, then there is a unique self-adjoint boundedly invertible operator $$B$$ with $$\text{Dom}(B) \subset \text{Dom}[b]$$ associated with the form $$b$$, in the sense that $b[x,y] = \langle x,By \rangle \;\;\text{ for all } x \in \text{Dom}[b] \text{ and } y \in \text{Dom}(B).$ (ii) If, in addition, the domains of $$|B|^{1/2}$$ and $$A^{1/2}$$ agree, then the form $$b$$ is represented by $$B$$, in the sense that $b[x,y] = \langle |B|^{1/2}x,\text{sign}(B)|B|^{1/2}y \rangle \;\;\text{ for all } x,y \in \text{Dom}[b].$
In the paper under review, the authors provide new and straightforward proofs of the representation theorems (i) and (ii) based on functional-analytic ideas. As a consequence of (i), they prove the first representation theorem for block operator matrices defined as quadratic forms, provided that certain conditions are satisfied. They also obtain a number of necessary and sufficient conditions for the second representation theorem to hold and present a new simple and explicit example of a self-adjoint operator for which the second representation theorem fails to hold.

### MSC:

 47A07 Forms (bilinear, sesquilinear, multilinear) 47A55 Perturbation theory of linear operators 15A63 Quadratic and bilinear forms, inner products 46C20 Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.)