A Hilbert Space Problem Book by P.R. Halmos

By P.R. Halmos

From the Preface: "This booklet used to be written for the energetic reader. the 1st half contains difficulties, usually preceded through definitions and motivation, and occasionally through corollaries and old remarks... the second one half, a really brief one, includes hints... The 3rd half, the longest, involves strategies: proofs, solutions, or contructions, counting on the character of the problem....

This isn't really an advent to Hilbert area conception. a few wisdom of that topic is a prerequisite: at least, a examine of the weather of Hilbert house concept should still continue simultaneously with the analyzing of this book."

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If A, B, and Dare operators on a Hilbert space H, then the operator matrix M = (~ ~) induces (is) an operator on H ED H, and (d. Problem 56) if both A and D are invertible, then M is invertible. The converse (if M is invertible, then A and D are) is not true (see Problem 56 again). Operator matrices define operators on direct sums of Hilbert spaces whether the direct summands are identical or not. In at least one special case of interest the converse that was false in the preceding paragraph becomes true.

The definition of the resolvent is very explicit; this makes it seem plausible that the resolvent is a well-behaved function. To formulate this precisely, consider, quite generally, functions cp whose domains are open sets in the complex plane and whose values are operators on H. Such a function cp will be called analytic if, for each j and g in H, the numerical function A -- (cp(A)j,g) (with the same domain as cp) is analytic in the usual sense. ) In case the function if; defined by if;(A) = cp(l/A) can be assigned a value at the origin so that it becomes analytic there, then (just as for numerical functions) cp will be called analytic at 00, and cp is assigned at 00 the value of if; at O.

As for symbols: the spectrum is often IT (or ~) instead of A. The best way to master these concepts is, of course, through illuminating special examples, but a few general facts should come first; they help in the study of the examples. The most useful things to know are the relations of spectra to the algebra and topology of the complex plane. Perhaps the easiest algebraic questions concern conjugation. Problem 58. What happens to the point spectrum, the compression spectrum, and the approximate point spectrum when an operator is replaced by its adjoint?

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