By P.R. Halmos
From the Preface: "This ebook was once written for the energetic reader. the 1st half comprises difficulties, usually preceded through definitions and motivation, and occasionally by means of corollaries and historic remarks... the second one half, a truly brief one, includes hints... The 3rd half, the longest, includes options: proofs, solutions, or contructions, counting on the character of the problem....
This isn't an creation to Hilbert area idea. a few wisdom of that topic is a prerequisite: at least, a learn of the weather of Hilbert house concept should still continue simultaneously with the analyzing of this book."
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Extra resources for A Hilbert Space Problem Book
If neither A nor B is invertible, then, in the infinite-dimensional case, the two products need not have the same spectrum (many examples occur below), but their spectra cannot differ by much. Here is the precise assertion. Problem 61. The non-zero elements of A(AB) and A(BA) are the same. 62. Closure of approximate point spectrum. Problem 62. Is the approximate point spectrum always closed? 63. Boundary of spectrum. Problem 63. The boundary oj the spectrum of an operator is included in the approximate point spectrum.
54. Multipliers of functional Hilbert spaces. Suppose that H is a functional Hilbert space over a set X. A function
Spectral parts of a diagonal operator. The spectrum of a diagonal operator was determined (Problem 48) as the closure of its diagonal; the determination of the fine structure of the spectrum requires another look. Problem 65. For each diagonal operator, find its point spectrum, compression spectrum, and approximate point spectrum. 66. Spectral parts of a multiplication. Problem 66. For each multiplication, find its point spectrum, compression spectrum, and approximate point spectrum. 67. Unilateral shift.
A Hilbert Space Problem Book by P.R. Halmos