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

By P.R. Halmos

ISBN-10: 1461599768

ISBN-13: 9781461599760

ISBN-10: 1461599784

ISBN-13: 9781461599784

From the Preface: "This ebook was once written for the energetic reader. the 1st half comprises difficulties, often preceded by means of definitions and motivation, and infrequently via corollaries and historic remarks... the second one half, a really brief one, contains hints... The 3rd half, the longest, includes strategies: proofs, solutions, or contructions, looking on the character of the problem....

This isn't an advent to Hilbert area thought. a few wisdom of that topic is a prerequisite: no less than, a learn of the weather of Hilbert area conception may still continue simultaneously with the studying of this book."

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Example text

A sufficient condition of the same kind is that the family of all entries be square-summable; if, that is, LiL; I (Xi; 12 < 00, then there exists an operator A such that ai; = (Aej,ei). ) This condition is not necessary. ) There are no elegant and usable necessary and sufficient conditions. It is perfectly possible, of course, to write down in matricial terms the condition that a linear transformation is everywhere defined and bounded, but the result is neither elegant nor usable. This is the first significant way in which infinite matrix theory differs from the finite version: every operator corresponds to a matrix, but not every matrix corresponds to an operator, and it is hard to say which ones do.

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. Consequence: the spectrum of a multiplication is the essential range of the multiplier.

Halmos [1951, p. 110]. Let A be an operator on a Hilbert space H; the construction itself takes place in the direct sum H $ H. , the set of all vectors of the form (f,Af). It is trivial to verify that both M and N are subspaces of H $ H. When does (f,g) belong to M + N? The answer is if and only if it has the form (u,O) + (v,Av) = (u + v,Av); since u and v are arbitrary, a vector in H $ H has that form if and only if its second coordinate belongs to the range R of the operator A. (In other words, M + N = 27 PROBLEMS 43 H EB R) Is M + N closed?

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A Hilbert Space Problem Book by P.R. Halmos


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