By P.R. Halmos

From the Preface: "This booklet was once written for the lively reader. the 1st half contains difficulties, usually preceded by means of definitions and motivation, and occasionally by way of corollaries and historic remarks... the second one half, a really brief one, comprises hints... The 3rd half, the longest, involves recommendations: proofs, solutions, or contructions, reckoning on the character of the problem....

This isn't really an creation to Hilbert area thought. a few wisdom of that topic is a prerequisite: a minimum of, a research of the weather of Hilbert house thought may still continue at the same time with the interpreting of this book."

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**Extra resources for A Hilbert Space Problem Book**

**Example text**

All these algebraic statements are trivialities; the generalization of the principle of uniform boundedness from linear functionals to linear transformations is somewhat subtler. The generalization can be formulated almost exactly the same way as the special case: a pointwise bounded collection of bounded linear transformations is uniformly bounded. The assumption of pointwise boundedness can be formulated in a "weak" manner and a "strong" one. A set Q of linear transformations (from H into K) is weakly bounded if for each fin H and each g in K there exists a positive constant a (j,g) such that I(Af,g) I ~ a(j,g) for all A in Q.

What is the spectrum of the bilateral shift, and what are its parts (point spectrum, compression spectrum, and approximate point spectrum)? What are the answers to the same questions for the adjoint of the bilateral shift? 69. Spectrum of a functional multiplication. Every operator studied so far has been a multiplication, either in the legitimate sense (on an V) or in the extended sense (on a functional Hilbert space). The latter kind is usually harder to study; it does, however, have the advantage of having a satisfactory characterization in terms of its spectrum.

Is invariant under U, and from this, in turn, it follows that U is a diagonal operator. Since the diagonal entries of a unitary diagonal matrix have modulus 1, it follows that, for each n, the effect of A on en can differ from that of B by a factor of modulus 1 only. This settles the unitary equivalence theory for weighted shifts with non-zero weights; what about similarity? Problem 76. If A and B are unilateral weighted shifts, with nonzero weights {an I and {j3n I, then a necessary and sufficient condition that A and B be similar is that the sequence of quotients be bounded away from °and from 00.