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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.)
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