Introduction to Spectral Theory in Hilbert Space, by G. M. Helmberg. iii + 346 pages. Series in Applied Mathematics and Mechanics, North-Holland, Amsterdam, 6, 1969. U.S. $19.50.
J. C. Alexander · Canadian Mathematical Bulletin · 1972
The author states in his preface that "the aim of this book is to make the reader familiar with everything needed in order to understand, believe, and apply the spectral theorem for selfadjoint operators (not necessarily bounded) in Hilbert space".The reviewer feels that he achieves this handsomely.No previous algebraic knowledge is assumed, but a familiarity with Weierstrass' approximation theorem and Riemann-Stieltjes integration is required.Since the excellent collection of worked examples is largely concerned with the spaces L 2 (a, /?) where -ooxf(x) in L 2 (-oo, oo).Compact operators are discussed in Ch.V, the spectral decomposition of a compact selfadjoint operator is obtained, and there is a section on Fredholm integral equations.Ch. VI contains the representation of a bounded self-adjoint operator by the uniformly convergent Riemann-Stieltjes integral Jf?^ X dP(X) for a suitable spectral family of projections P(A).In a similar way the representationfor a unitary operator is obtained.Finally, the representation of a bounded normal