By P.R. Halmos
From the Preface: "This publication used to be written for the energetic reader. the 1st half includes difficulties, often preceded by way of definitions and motivation, and occasionally via corollaries and historic remarks... the second one half, a truly brief one, involves hints... The 3rd half, the longest, involves ideas: proofs, solutions, or contructions, reckoning 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: at the least, a examine of the weather of Hilbert house concept may still continue at the same time with the interpreting of this book."
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Extra info for A Hilbert Space Problem Book
The exact determination of the spectrum and its parts for arbitrary weighted shifts is a non-trivial problem. Here is a useful fragment. Problem 78. Find all the eigenvalues of all unilateral weighted shifts (with non-zero weights) and of their adjoints. The possible presence of 0 among the weights is not a genuine difficulty but a nuisance. A unilateral weighted shift, one of whose weights vanishes, becomes thereby the direct sum of a finite-dimensional operator and another weighted shift. The presence of an infinite number of zero weights can cause some interesting trouble (d.
The theory of operator matrices does not become trivial even if the number of direct summands is small (say two) and even if all the direct summands are identical. The following situation is the one that occurs most frequently: a Hilbert space H is given, the role of what was H in the preceding paragraph is played now by the direct sum H $ H, and operators on that direct sum are expressed as two-by-two matrices whose entries are operators on H. Problem 55. If A, B, C, and D are pairwise commutative operators on a Hilbert space, then a necessary and sufficient condition that the operator matrix be invertible is that the formal determinant AD - BC be invertible.
Norms are not only kept within reasonable bounds, but they are preserved exactly; the unilateral shift is an isometry. The range of U is not l2 but a proper subspace of l2, the subspace of vectors with vanishing first coordinate. The existence of an isometry whose range is not the whole space is characteristic of infinite-dimensional spaces. If en is the vector (~o, ~l, ~2, ••• ) for which ~n = 1 and ~i = whenever i ¢ n (n = 0, 1, 2, ... ), then the en's form an orthonormal basis for 12. The effect of U on this basis is described by ° (n = 0, 1, 2, ...
A Hilbert Space Problem Book by P.R. Halmos