Download A Hilbert Space Problem Book by P.R. Halmos PDF

By P.R. Halmos

From the Preface: "This booklet used to be written for the lively reader. the 1st half contains difficulties, usually preceded by way of definitions and motivation, and infrequently via corollaries and old remarks... the second one half, a truly brief one, contains hints... The 3rd half, the longest, involves suggestions: proofs, solutions, or contructions, counting on the character of the problem....

This isn't really an advent to Hilbert house conception. a few wisdom of that topic is a prerequisite: no less than, a learn of the weather of Hilbert house idea should still continue simultaneously with the interpreting of this book."

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An assertion such as that if A is an operator and p is a polynomial, then A(P(A)) = P(A(A)) (see Halmos [1951, p. 53J) is called a spectral mapping theorem; other instances of it have to do with functions other than polynomials, such as inversion, conjugation, and wide classes of analytic functions (Dunford-Schwartz [1958, p. 569J). Problem 59. Is the spectral mapping theorem for polynomials true with ITo, or IT, or r in place of A? What about the spectral mapping theorem jor inversion (P(s) = liz wf-um z ~ 0), applied to invertible operators, ulith ITo, or ll, or r?

An early and still useful reference for functional Hilbert spaces is Aronszajn [1950]. 30. Kernel functions. If H is a functional Hilbert space, over X say, then the linear functionalf ~ fey) on H is bounded for each y in X, and, consequently, there exists, for each y in X, an element KII of H such thatf(y) = (j,KII ) for all]. The function K on X X X, defined by K (x,y) = KlI (x), is called the kernel function or the reproducing kernel of H. Problem 30. If {ejl is an orthonormal basis for a functional Hilbert space H, then the kernel function K of H is given by K(x,y) = 2::ei(x)ej(Y)*.

It should not be too surprising now that the theory of such functions enters the study of operators in every case (whether the dimension is finite or infinite). Problem 73. Every operator has a non-empty spectrum. 74. Spectral radius. r ( A), is defined by The spectral radius of an operator A, in symbols rCA) = sup{iAI:AEACA)}. Clearly 0 ~ r (A) ~ II A II; the spectral mapping theorem implies also that r(An) = (r(A»n for every positive integer n. It frequently turns out that the spectral radius of an operator is easy to compute even when it is hard to find the spectrum; the tool that makes it easy is the following assertion.

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