Abstract
In order to make this monograph self-contained, we summarize in this chapter some basic definitions and results for unbounded linear operators in a Hilbert space. In Section 1.1, we recall the definitions of C*-algebras and von Neumann algebras. In Section 1.2, we define and investigate the notion of closedness, the closure and the adjoint of an unbounded linear operator in a Hilbert space. Section 1.3 is devoted to the Cayley transform approach to the self-adjointness of a symmetric operator. Section 1.4 deals with the self-adjoint extendability of a symmetric operator with help of the deficiency spaces. In Section 1.5, we extend to unbounded self-adjoint operators the spectral theorem and the functional calculus theorem for bounded self-adjoint operators. Section 1.6 is devoted to Stone’s theorem. In Section 1.7, the polar decomposition of bounded linear operators is extended to closed linear operators. Section 1.8 introduces Nelson’s analytic vector theorem for the selfadjointness of closed symmetric operators. Section 1.9, finally, deals with the form representation theorem and the Friedrichs self-adjoint extension theorem.
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Notes for Chapter 1
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© 2002 Springer Science+Business Media Dordrecht
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Antoine, JP., Inoue, A., Trapani, C. (2002). Unbounded Linear Operators in Hilbert Spaces. In: Partial *-Algebras and Their Operator Realizations. Mathematics and Its Applications, vol 553. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0065-8_1
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DOI: https://doi.org/10.1007/978-94-017-0065-8_1
Publisher Name: Springer, Dordrecht
Print ISBN: 978-90-481-6176-8
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