By Ruben A. Martinez-Avendano, Peter Rosenthal

The topic of this ebook is operator idea at the Hardy house H^{2}, also known as the Hardy-Hilbert house. this can be a renowned quarter, partly as the Hardy-Hilbert area is the main traditional surroundings for operator thought. A reader who masters the cloth coated during this e-book may have bought a company beginning for the examine of all areas of analytic services and of operators on them. The objective is to supply an straight forward and interesting advent to this topic that might be readable by way of every body who has understood introductory classes in advanced research and in sensible research. The exposition, mixing strategies from "soft" and "hard" research, is meant to be as transparent and instructive as attainable. the various proofs are very dependent.

This ebook advanced from a graduate direction that was once taught on the collage of Toronto. it may end up compatible as a textbook for starting graduate scholars, or maybe for well-prepared complicated undergraduates, in addition to for self sustaining examine. there are lots of routines on the finish of every bankruptcy, besides a short consultant for extra learn along with references to purposes to issues in engineering.

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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.

Let χn be the characteristic function of En (which clearly is in L2 ). Then Aχn 2 = φχn 2 = 1 2π |φ(eiθ )|2 dθ ≥ n2 m(En ). En Also, χn 2 = 1 2π dθ = m(En ). En Thus Aχn 2 ≥ n2 χn 2 . Therefore if n > A , then χn = 0, so m(En ) = 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 .

4. Let A be a bounded linear operator. (i) If 1 − A < 1, then A is invertible. (ii) The spectrum of A is a nonempty compact subset of C. 22 1 Introduction (iii) If A is an invertible operator, then 1 : λ ∈ σ(A) . λ σ(A−1 ) = (iv) If A∗ denotes the Hilbert space adjoint of A, then σ(A∗ ) = λ : λ ∈ σ(A) . (v) The spectral radius formula holds: r(A) = lim n→∞ An 1/n . In particular, r(A) ≤ A . (vi) If A is an operator on a ﬁnite-dimensional space, then σ(A) = Π0 (A) (for operators on inﬁnite-dimensional spaces, Π0 (A) may be the empty set).