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 by way of definitions and motivation, and infrequently via corollaries and old remarks... the second one half, a truly brief one, comprises hints... The 3rd half, the longest, comprises strategies: proofs, solutions, or contructions, counting on the character of the problem....

This isn't really an creation to Hilbert house concept. a few wisdom of that topic is a prerequisite: no less than, a research of the weather of Hilbert area conception should still continue at the same time with the analyzing of this book."

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The result about the unitary equivalence of weighted shifts has two useful consequences. veighted shift with weights I O'n \. Since unitarily equivalent operators are "abstractly identical", there is never any loss of generality in restricting attention to weighted shifts whose weights are non-negative; this is what really justifies the use of the word "weight". Second, if A is a weighted shift and if 0' is a complex number of modulus 1, then, since O'A is a weighted shift, whose weights have the same moduli as the corresponding weights of A, it follows that A and O'A are unitarily equivalent.

A necessary and sufficient condition that an operator A on a Hilbert space H be representable as a multiplication on a functional Hilbert space is that the eigenvectors of A * span H. Caution: as the facts for multiplications on V spaces show (d. Solution 66) this characterization is applicable to functional Hilbert spaces only. The result seems to be due to P. R. Halmos and A. L. Shields. 43 PROBLEMS 71 70. Relative spectrum of shift. An operator A is relatively invertible if there exists an operator B such that ABA = A.

The correspondence preserves not only the familiar algebraic operations, but also the conjugation; that is, if rp - A, then rp* - A *. If the measure is u-finite, the correspondence preserves the norm also (see Solution 49). The role played by the range of a sequence is played, in the general case, by the essential range of a function rp; by definition, that is the set of all complex numbers " such that for each neighborhood N of " the set rp-l(N) has positive measure. Problem 52. The multiplication operator on V (jor au-finite measure) induced by rp is an invertible operator if and only if rp is an invertible function.

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