By Ruben A. Martinez-Avendano, Peter Rosenthal

ISBN-10: 0387354182

ISBN-13: 9780387354187

ISBN-10: 0387485783

ISBN-13: 9780387485782

The topic of this e-book is operator idea at the Hardy house H^{2}, often known as the Hardy-Hilbert house. this can be a well known quarter, partly as the Hardy-Hilbert house is the main ordinary atmosphere for operator thought. A reader who masters the cloth lined during this ebook could have got an organization beginning for the research of all areas of analytic features and of operators on them. The aim is to supply an trouble-free and interesting advent to this topic that would be readable by way of every body who has understood introductory classes in advanced research and in practical research. The exposition, mixing concepts from "soft" and "hard" research, is meant to be as transparent and instructive as attainable. a few of the proofs are very based.

This ebook advanced from a graduate direction that used to be taught on the college of Toronto. it may turn out appropriate as a textbook for starting graduate scholars, or maybe 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 additional research together with references to purposes to issues in engineering.

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

**Example text**

9. e. is an inner function. 10. If φ is a nonconstant inner function, then |φ(z)| < 1 for all z ∈ D. Proof. 17. The deﬁnition of inner functions requires that the functions be in H ∞ . e. 11. Let φ ∈ H 2 . , then φ is an inner function. Proof. 24. 12 (Beurling’s Theorem). Every invariant subspace of the unilateral shift other than {0} has the form φH 2 , where φ is an inner function. Proof. The unilateral shift is the restriction of multiplication by eiθ to H 2 , so if M is an invariant subspace of the unilateral shift, it is an invariant subspace of the bilateral shift contained in H 2 .

9. The operator Mz on H 2 is unitarily equivalent to the unilateral shift. Proof. If V is the unitary operator mapping 2 onto H 2 given by ∞ an z n , V (a0 , a1 , a2 , . . ) = n=0 it is trivial to verify that V U = Mz V . Thus Mz is a representation of the unilateral shift as an operator on H 2 ; we often refer to Mz as U when no confusion is possible. Notice that Mz ek = ek+1 for k = 0, 1, 2, . . , where ek (z) = z k . The bilateral shift has an analogous representation on L2 . 10. The operators Meiθ and Me−iθ are deﬁned on L2 by (Meiθ f )(eiθ ) = eiθ f (eiθ ) and (Me−iθ f )(eiθ ) = e−iθ f (eiθ ).

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. The next corollary provides further justiﬁcation for this identiﬁcation. 28. If f ∈ H 2 , then lim f (reiθ ) = f (eiθ ) for almost all θ. r→1− Proof. 21). e. 29. If f ∈ H ∞ , then f ∈ L∞ . Proof. 28) that the essential supremum of f is at most f ∞ .

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

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