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1). The idea that sum rules occur as Taylor coefficients of suitable analytic functions recurs throughout this book. In the infinite-dimensional case, there are convergence and other issues. Let X be a Banach space. A bounded linear map A : X → X is called finite rank if Ran(A) is finite-dimensional. 4) j =1 N ∗ For some { j }N j =1 ⊂ X and {xj }j =1 ⊂ X. 4) (essentially by the invariance of trace in the finite-dimensional case). 6) A 1 = inf j X∗ xj X A = j (·)xj ⎩ ⎭ j =1 The nuclear operators, N(X), are the completion of the finite-rank operators in · 1 .

N 2 )1/n then converges to lim n 2 . 4. 13) Remarks and Historical Notes. Szeg˝o’s great 1920–1921 paper [430] was the first systematic exploration of OPUC, although he had earlier discussed OPs on curves [429]. 8 VERBLUNSKY’S FORM OF SZEGO’S In this section, we give the final reformulation of Szeg˝o’s theorem as a sum rule and see that it implies a gem of spectral theory. The first element we need is the recursion relation obeyed by the monic OPUC, n (z), that will give us the parameters of the direct problem.

The basic strategy here was invented to prove the Killip–Simon theorem by them [225] and honed by Simon–Zlatoš [410] and Simon [396]. Parts of it appear applied to the Szeg˝o theorem in Chapter 2 of [399]. There is some overlap with ideas in Verblunsky’s proof [453]. 34) n=0 where |·| is Lebesgue measure. 35) holds for all p > 2. 11]. e. c. spectrum. If 2 > 1, µ is pure point, and if 2 ≤ 1, the spectrum is purely singular continuous of Hausdorff dimension 1 − 2 . While the OPUC case is from [400], it is motivated by an OPRL paper of Kiselev, Last, and Simon [227]; see [400] for earlier papers on OPRL with decaying random potentials.

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A Fundamental Theorem on One-Parameter Continuous Groups of Projective Functional Transformations by Kennison L. S.


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