By Ruben A. Martinez-Avendano, Peter Rosenthal

The topic of this publication is operator conception at the Hardy house H^{2}, often known as the Hardy-Hilbert house. it is a renowned quarter, in part as the Hardy-Hilbert house is the main usual atmosphere for operator thought. A reader who masters the fabric lined during this e-book can have obtained a company beginning for the examine of all areas of analytic features and of operators on them. The target is to supply an trouble-free and interesting creation to this topic that may be readable by way of every person 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 dependent.

This ebook advanced from a graduate direction that was once taught on the collage of Toronto. it may turn out appropriate as a textbook for starting graduate scholars, or perhaps for well-prepared complicated undergraduates, in addition to for self reliant examine. there are many routines on the finish of every bankruptcy, in addition to a quick consultant for additional learn together with references to purposes to themes in engineering.

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Given a vector f and a bounded linear operator A, the invariant subspace generated by f is the subspace ∞ {An f }. n=0 We say that an invariant subspace M of A is cyclic if there is a vector g such ∞ that M = n=0 {An g}. If ∞ {An g} = H, n=0 we say that g is a cyclic vector for A. Clearly, the invariant subspace problem can be rephrased: does every bounded linear operator on Hilbert space have a noncyclic vector other than zero? It turns out that the collection of subspaces invariant under an operator (or any family of operators) is a lattice.

19. Prove that { An C } converges to 0 whenever C is a compact operator and {An } is a sequence of bounded operators such that {An f } converges to 0 for all vectors f . 20. , the measure of D is 1). The inner product of the functions f and g is deﬁned by (f, g) = 1 π f (x + iy)g(x + iy) dx dy. , prove the completeness of the inner product space deﬁned above). √ (ii) Show that the collection of functions { n + 1 z n }∞ n=0 forms an orthonormal basis for the Bergman space. (iii) Let Mz be the operator deﬁned by (Mz f )(z) = zf (z).

We claim that there exists a constant k > 0 and a positive integer M such that (A − λn )f ≥ k f for all f ∈ H whenever n ≥ M. If this were false, then for every ε > 0 and every natural M there would exist an n ≥ M and an fn of norm 1 such that (A − λn )fn < ε . 2 Given any ε > 0, choose a natural number M such that |λn − λ| < n ≥ M . With this ε and M , choose n and fn as above. Then ε 2 for all (A − λ)fn ≤ (A − λn )fn + (λn − λ)fn < ε. But this would imply that λ ∈ Π(A), which would be a contradiction, so the claim is proved.