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Hilbert space: Compact operators and the trace theorem. (English) Zbl 0783.47031
London Mathematical Society Student Texts. 27. Cambridge: Cambridge University Press,. xii, 131 p. (1993).
J. R. Retherfold’s book covers the theory of compact operators in Hilbert spaces culminating with Lidskij’s trace theorem. It is intended for graduate students at all levels, includes numerous exercises, and is written nicely and vivaciously. The first eight chapters cover classical material; general Hilbert space theory, spectral theory, selfadjoint operators, the Schur representation and the polar decomposition. Chapter 9 covers the Schatten \(p\)-classes \(S_ p\) and the weak Weyl-inequality for operators in \(S_ p\) as well as the Weyl-inequality. Chapter 10 is concerned with Hilbert-Schmidt and trace class operators. Using all those prerequisites, the main result – Lidskij’s trace theorem – is proved in chapter 11 in a way accessible to most graduate students.
Reviewer: H.König (Kiel)

47B07 Linear operators defined by compactness properties
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
47B50 Linear operators on spaces with an indefinite metric