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Adjoint of sums and products of operators in Hilbert spaces. (English) Zbl 1374.47003
The authors develop a two-by-two matrix technique on the range of an (unbounded) operator matrix of type $M_{S,T}=\left[\begin{matrix} I & -T \\ S & I \end{matrix}\right]$ to gain new characterizations of closed, selfadjoint and essentially selfadjoint operators. More precisely, they provide some sufficient and necessary conditions guaranteeing $$(A+B)^*=A^*+B^*$$ and $$(AB)^*=B^*A^*$$ for densely defined unbounded operators $$A, B$$ between Hilbert spaces; for some sufficient conditions, see M. S. Birman and M. Z. Solomyak [Spectral theory of self-adjoint operators in Hilbert space (1987); translation from Leningrad: Leningrad. Univ., 264 p. (loose errata) (1980; Zbl 0744.47017)]. Moreover, they improve the perturbation theory of some operator classes due to E. Nelson [Ann. Math. (2) 70, 572–615 (1959; Zbl 0091.10704)], T. Kato [Perturbation theory for linear operators. 2nd ed. Berlin-Heidelberg-New York: Springer-Verlag (1976; Zbl 0342.47009)] and F. Rellich [Math. Ann. 116, 555–570 (1939; JFM 65.0510.01)].

##### MSC:
 47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47A55 Perturbation theory of linear operators 47B25 Linear symmetric and selfadjoint operators (unbounded)
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