Download An Introduction to Operators on the Hardy-Hilbert Space by Ruben A. Martinez-Avendano, Peter Rosenthal PDF

By Ruben A. Martinez-Avendano, Peter Rosenthal

ISBN-10: 0387354182

ISBN-13: 9780387354187

The topic of this booklet is operator conception at the Hardy area H2, also referred to as the Hardy-Hilbert area. this can be a well known zone, partly as the Hardy-Hilbert house is the main usual environment for operator concept. A reader who masters the cloth coated during this publication could have got an organization origin for the research of all areas of analytic capabilities and of operators on them. The target is to supply an effortless and interesting creation to this topic that would be readable by way of everybody who has understood introductory classes in advanced research and in practical research. The exposition, mixing suggestions from "soft"and "hard" research, is meant to be as transparent and instructive as attainable. some of the proofs are very based.

This publication developed from a graduate path that was once taught on the college of Toronto. it's going to turn out compatible as a textbook for starting graduate scholars, or perhaps for well-prepared complicated undergraduates, in addition to for self reliant learn. there are lots of workouts on the finish of every bankruptcy, in addition to a short consultant for extra examine inclusive of references to purposes to subject matters in engineering.

Show description

Read or Download An Introduction to Operators on the Hardy-Hilbert Space PDF

Similar functional analysis books

Norm estimations for operator-valued functions and applications

Offering worthwhile new instruments for experts in sensible research and balance concept, this state of the art reference offers a scientific exposition of estimations for norms of operator-valued capabilities and applies the estimates to spectrum perturbations of linear operators and balance concept.

Almost Periodic Functions (Ams Chelsea Publishing)

Influenced via questions on which features will be represented by way of Dirichlet sequence, Harald Bohr based the speculation of just about periodic services within the Twenties. this pretty exposition starts with a dialogue of periodic services earlier than addressing the just about periodic case. An appendix discusses nearly periodic services of a fancy variable.

Hardy Spaces on Ahlfors-Regular Quasi Metric Spaces: A Sharp Theory

Systematically developing an optimum concept, this monograph develops and explores a number of ways to Hardy areas within the atmosphere of Alhlfors-regular quasi-metric areas. The textual content is split into major elements, with the 1st half delivering atomic, molecular, and grand maximal functionality characterizations of Hardy areas and formulates sharp types of easy analytical instruments for quasi-metric areas, equivalent to a Lebesgue differentiation theorem with minimum calls for at the underlying degree, a maximally tender approximation to the identification and a Calderon-Zygmund decomposition for distributions.

Extra info for An Introduction to Operators on the Hardy-Hilbert Space

Sample text

20 1 Introduction Therefore, if r ∈ [s, 1), we have |u(reit0 ) − L| < ε. We state the following special case of Fatou’s theorem for future reference. 27. Let φ be a function in L1 (S 1 , dθ). Define u by u(reit ) = 1 2π 2π Pr (θ − t)φ(eiθ ) dθ. e. Proof. Define α by θ φ(eix ) dx. e. 26) gives the result. The following corollary is an important application of Fatou’s theorem. It is often convenient to identify H 2 with H 2 ; in some contexts, we will refer to f and its boundary function f interchangeably.

Show that a function f analytic on D is in H 2 if there is a harmonic function u on D such that |f (z)|2 ≤ u(z) for all z ∈ D. 12. 7. Define H 1 to be the set of all functions in L1 (S 1 ) whose Fourier coefficients corresponding to negative indices are zero. Prove that the product of two functions in H 2 is in H 1 . 8. Let u be a real-valued function in L2 . Show that there exists a real-valued function v in L2 such that u + iv is in H 2 . 9. Let f be an even function of a real variable defined in a neighborhood of 0.

That is, φ ∈ L∞ . 46 2 The Unilateral Shift and Factorization of Functions We can now explicitly describe the reducing subspaces of the bilateral shift. 6. e. on E} for measurable subsets E ⊂ S 1 . Proof. e. on E}. If f (eiθ0 ) = 0, then eiθ0 f (eiθ0 ) = 0, so ME is invariant under W . Similarly, if f (eiθ0 ) = 0, then e−iθ0 f (eiθ0 ) = 0, so ME is invariant under W ∗ . 25, ME is reducing. 25. By the previous theorem, P = Mφ for some φ ∈ L∞ . Since P is a projection, P 2 = P and thus Mφ2 = Mφ .

Download PDF sample

Rated 4.06 of 5 – based on 39 votes