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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References:

 [1] DOI: 10.1007/978-3-0348-7881-4_14 [2] DOI: 10.1090/S0002-9947-06-03930-4 · Zbl 1115.47014 [3] Kato, Perturbation Theory for Linear Operators (1966) · Zbl 0148.12601 [4] DOI: 10.1007/BF02054965 · Zbl 0043.32603 [5] DOI: 10.1002/mana.19981900106 · Zbl 0898.34018 [6] DOI: 10.1007/s00020-009-1678-x · Zbl 1200.47003 [7] Fleige, Spectral Theory of Indefinite Krein–Feller Differential Operators (1996) · Zbl 0856.34085 [8] DOI: 10.1017/S0308210509000286 · Zbl 1187.47006 [9] Esteban, Mathematical Results in Quantum Mechanics pp 41– (2008) [10] DOI: 10.1007/3-7643-7516-7_6 [11] Edmunds, Spectral Theory and Differential Operators (1989) · Zbl 0664.47014 [12] Fleige, Act. Sci. Math. (Szeged) 66 pp 633– (2000) [13] DOI: 10.1007/BF01204699 · Zbl 0572.47023 [14] DOI: 10.1007/BF01203080 · Zbl 0922.47014 [15] DOI: 10.2140/pjm.1983.104.241 · Zbl 0457.47002 [16] DOI: 10.1007/978-3-642-65567-8 [17] Birman, Spectral Theory of Selfadjoint Operators in Hilbert Space (1987) [18] Azizov, Linear Operators in Spaces with an Indefinite Metric (1989) · Zbl 0714.47028 [19] DOI: 10.1017/S0308210500023283 · Zbl 0865.34017 [20] DOI: 10.7153/oam-02-19 · Zbl 1198.47022 [21] DOI: 10.1016/S0024-3795(99)00157-3 · Zbl 0982.15022 [22] DOI: 10.1007/BF01617872 · Zbl 0349.47014 [23] DOI: 10.1090/S0002-9904-1970-12526-5 · Zbl 0198.16403 [24] McIntosh, J. Math. Mech. 19 pp 1027– (1970) [25] Langer, Functional Analysis pp 1– (1982) [26] Krein, Interpolation of Linear Operators (1982) [27] DOI: 10.1007/BF01270924 · Zbl 1056.47021
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