By Joseph A. Cima

ISBN-10: 0821820834

ISBN-13: 9780821820834

Shift operators on Hilbert areas of analytic services play an immense function within the examine of bounded linear operators on Hilbert areas on the grounds that they generally function versions for numerous periods of linear operators. for instance, ``parts'' of direct sums of the backward shift operator at the classical Hardy area $H^2$ version particular types of contraction operators and in all likelihood have connections to figuring out the invariant subspaces of a common linear operator. This publication is an intensive therapy of the characterization of the backward shift invariant subspaces of the well known Hardy areas $H^{p}$. The characterization of the backward shift invariant subspaces of $H^{p}$ for $1 < p < \infty$ was once performed in a 1970 paper of R. Douglas, H. S. Shapiro, and A. Shields, and the case $0 < p \le 1$ was once performed in a 1979 paper of A. B. Aleksandrov which isn't popular within the West. This fabric is pulled jointly during this unmarried quantity and contains the entire worthy history fabric had to comprehend (especially for the $0 < p < 1$ case) the proofs of those effects. numerous proofs of the Douglas-Shapiro-Shields outcome are supplied so readers can get familiar with varied operator idea and thought strategies: purposes of those proofs also are supplied for figuring out the backward shift operator on a number of different areas of analytic features. the implications are completely tested. different beneficial properties of the amount comprise an outline of functions to the spectral houses of the backward shift operator and a therapy of a few normal real-variable strategies that aren't taught in ordinary graduate seminars. The ebook comprises references to works via Duren, Garnett, and Stein for proofs and a bibliography for extra exploration within the parts of operator conception and practical research.

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**Example text**

Can be writte n using th e truncation 1. For each g G BMO, / -> lim / fhp(g) the linear functional dm is continuous on 3W71. 2. For g G BMO A and f G H1 lim / fr~gdm— r-+l~ J lim / fhp(g) P^ooJ dm. P R O O F . Since hp(g) is a bounded B M O function, th e linear functional f - • / fhp{g) dm is continuous on JRiJ 1 . 20) lirn^ f fhp(g) C Lp dm = A / £ dm. To show t h a t th e following limit lim / fhp(g) dm exists for every / G 3W/ 1 , we let pn —> 00 and £ > 0 be given. 18) and t h e fact t h a t fs —• / in i^ 1 we can fix 5 G (0,1) so t h a t {sup|Ms)IL}ll/a-/l|i

If f e N+, then p(frJ) -> 0 as r -> 1~. P R O O F . Since fr —> f almost everywhere, we can apply Egorov's theorem to say that for a given e > 0, there is a closed set E C T such that |JE7| > 1 — e {dm is normalized) and fr—>f uniformly on E. Write p ( / r , / ) as P(fr, /) = f l0g(l + \fr ~ f\)dm = [ + [ JT JE JT\E and notice that the first integral goes to zero as r —•> 1~. For the second integral, / < / log 2 dm + / < £log2+ / < £log2 + c£ , log+|/r-/| log+|/r-/|dm where c£ —> 0 as e —> 0. 8).

H1 and bounded mean oscillation. 5), (iJ 1 )* is isometrically isomorphic to L°° /(H1)1- = L°° /HQ°. Moreover, by our previous analysis with the Hahn-Banach extension theorem, every £ G (if 1 )* is of the form t(f) = Jfgdm feH1 for some g G L°°. However, we are unable use the Riesz projection as before to assume that g G H°° (since PL00 (jL H°°). Instead, with a slightly different linear functional, g can be taken to belong to BMO A, the analytic functions of bounded mean oscillation. A good general reference for BMO and VMO are the notes of Sarason [81].

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