Some questions on the space with an indefinite metric. (English) Zbl 0631.47025
Let \(\Pi\) be a Krein space and let U be a unitary operator in \(\Pi\). First the author considers the decomposition of \(\Pi\) into spectral subspaces of U corresponding to a decomposition of the spectrum of U into spectral sets [cf. e.g. H. Langer, Lect. Notes Math. 948, 1-46, § I.3 (1982; Zbl 0511.47023)]. Then some examples related to the spectrum of operators in Krein spaces are given: an invertible, bounded, quasinilpotent selfadjoint operator, a selfadjoint operator with empty spectrum, a selfadjoint operator A with non-densely defined \(A^ 2\) and others. Furthermore, the maximal nonnegative subspaces of (H,[A\(\cdot,\cdot])\), where (H,[\(\cdot,\cdot])\) is a Hilbert space and A is an invertible, bounded selfadjoint operator in (H,[\(\cdot,\cdot])\), are characterized in terms of their angular operators [cf. e.g. J. Bognár, Indefinite inner product spaces (1974; Zbl 0286.46028), Theorem II.11.7].
Reviewer: P.Jonas
MSC:
47B50 | Linear operators on spaces with an indefinite metric |
46C20 | Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.) |
47A10 | Spectrum, resolvent |
Keywords:
unitary operator; decomposition; spectral subspaces; spectrum of operators in Krein spaces; quasinilpotent selfadjoint operator; maximal nonnegative subspaces; angular operatorsReferences:
[1] | Yan Shaozong, Unitary Operators on SpaceII (III),Acta Math. Sinica,25 (1982), 610–616. · Zbl 0522.47032 |
[2] | Yan Shaozong, The Structure ofII Spaces (I),Ann. of Math. China, Ser. B,5 (1984), 91–100. · Zbl 0548.46021 |
[3] | Yan Shaozong, On Operators in Indefinite Metric SpaceII k ,Scientia Sinica,24 (1981), 1615–1625. · Zbl 0489.47021 |
[4] | Yan Shaozong, Tong Yusun, Unbounded Self-Adjoint Operator inII k Space,Ann. of Math. China,2 (1981), 157–180. · Zbl 0476.47027 |
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