By Ruben A. Martinez-Avendano, Peter Rosenthal

The topic of this publication is operator concept at the Hardy house H^{2}, also known as the Hardy-Hilbert area. it is a well known sector, in part as the Hardy-Hilbert house is the main normal atmosphere for operator conception. A reader who masters the fabric coated during this booklet may have got a company beginning for the learn of all areas of analytic features and of operators on them. The aim is to supply an uncomplicated and fascinating advent to this topic that may be readable through each person who has understood introductory classes in complicated research and in useful research. The exposition, mixing concepts from "soft"and "hard" research, is meant to be as transparent and instructive as attainable. the various proofs are very dependent.

This publication advanced from a graduate path that was once taught on the collage of Toronto. it may turn out compatible as a textbook for starting graduate scholars, or perhaps for well-prepared complex undergraduates, in addition to for autonomous research. there are various workouts on the finish of every bankruptcy, besides a short consultant for extra research consisting of references to purposes to issues in engineering.

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A0 , 0, 0, 0, . . ) ∈ M, 44 2 The Unilateral Shift and Factorization of Functions since M is a subspace. Dividing by a0 , we see that (1, 0, 0, 0, . . ) is in M; that is, e0 ∈ M. Since en = U n e0 for every n, it follows that M contains every basis vector en , so M = 2 . The bilateral shift, on the other hand, has many reducing subspaces. 25) to begin by determining the operators that commute with the bilateral shift. 2. The commutant of a bounded linear operator A is the set of all bounded linear operators that commute with A.

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 ).

1. On 2 , we deﬁne the unilateral shift operator U by U (a0 , a1 , a2 , a3 , . . ) = (0, a0 , a1 , a2 , a3 , . . ) for (a0 , a1 , a2 , a3 , . ) ∈ 2 . 2. , U f for all f ∈ 2 ). = f (ii) The adjoint, U ∗ , of the unilateral shift has the following form: U ∗ (a0 , a1 , a2 , a3 , . . ) = (a1 , a2 , a3 , . . ) for (a0 , a1 , a2 , a3 , . . ) 2 . (The operator U ∗ is the backward unilateral Proof. To prove (i), we must show that (a0 , a1 , a2 , . . ) = (0, a0 , a1 , a2 , . . ) . ∞ ∞ But this is trivial since k=0 |ak |2 = |0|2 + k=1 |ak−1 |2 .