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Chapter 7 is devoted to a number of important technical tools and special topics for the study of closed operators and self-adjoint operators. We begin with the polar decomposition of a densely defined closed operator. Then the polar decomposition is applied to the study of the operator relation A ∗ A=AA ∗+I. Next, the bounded transform and its basic properties are developed. The usefulness of this transform has been already seen in the proofs of various versions of the spectral theorem in Chap. 5. Special classes of vectors (analytic vectors, quasi-analytic vectors, Stieltjes vectors) are studied in detail. They are used to derive criteria for the self-adjointness of symmetric operators (Nelson and Nussbaum theorems) and for the strong commutativity of self-adjoint operators. The last section of this chapter treats the tensor product of unbounded operators on Hilbert spaces.
KeywordsQuasi-analytic Vectors Strong Substitutability Polar Decomposition Important Technical Tool Symmetric Operator
- [Hl]Hille, E.: Functional Analysis and Semigroups. Am. Math. Soc. Coll. Publ., vol. 38. Am. Math. Soc., New York (1948) Google Scholar