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.


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.)
Full Text: DOI arXiv


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