By P.R. Halmos

From the Preface: "This e-book was once written for the lively reader. the 1st half involves difficulties, often preceded by way of definitions and motivation, and infrequently by way of corollaries and historic remarks... the second one half, a truly brief one, contains hints... The 3rd half, the longest, includes recommendations: proofs, solutions, or contructions, looking on the character of the problem....

This isn't really an advent to Hilbert house thought. a few wisdom of that topic is a prerequisite: a minimum of, a examine of the weather of Hilbert area idea should still continue at the same time with the examining of this book."

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**Example text**

The product of two functions in H2 is in HI. A kind of converse of this statement is true: it says that every function in HI is the product of two functions in H2. (See Hoffman [1962, p. ) The direct statement is more useful in Hilbert space theory than the converse, and the techniques used in the proof of the direct statement are nearer to the ones appropriate to this book. 28. Analytic characterization of H2. Iff E H2, with Fourier expansion f = L:~=o ane n , then L:~=o / an /2 < 00, and therefore the radius of convergence of the power series L:~=o an2zn is greater than or equal to 1.

Linearity is obvious, and boundedness is true with room to spare; the bilateral shift, like the unilateral one, is an isometry. Since the range of the bilateral shift is the entire space H, it is even unitary. If en is the vector ( ... , ~-l, (~o), 6, ... ) for which ~n = 1 and ~i = 0 whenever i 7fI= n (n = 0, ±1, ±2, ... ), then the en's form an orthonormal basis for H. The effect of W on this basis is described by (n = 0, ±1, ±2, ... ). Problem 68. What is the spectrum of the bilateral shift, and what are its parts (point spectrum, compression spectrum, and approximate point spectrum)?

42. Preservation of dimension. An important question about operators is what do they do to the geometry of the underlying space. It is familiar from the study of finite-dimensional vector spaces that a linear transformation can lower dimension: the transformation 0, for an extreme example, collapses every space to a O-dimensional one. , its kernel is to}), then it cannot lower dimension; since the same can be said about the inverse transformation (from the range back to the domain), it follows that dimension is preserved.