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The backward shift on the Hardy space. (English) Zbl 0952.47029
Mathematical Surveys and Monographs. 79. Providence, RI: American Mathematical Society (AMS). xi, 199 p. (2000).
If $${\mathcal D}$$ denotes the open unit disk $$\{z:|z|<1\}$$, of the complex plane, let $${\mathfrak T}$$ denote the boundary (unit circle) $$\partial{\mathcal D}$$ of $${\mathcal D}$$, and if $$f$$ is an analytic function on $$D$$, and $$0< p<\infty$$, let $$\|f\|_p$$ be defined by $\|f\|_p= \sup_{0< r< 1} \Biggl\{{1\over 2\pi} \int^{2\pi}_0 |f(re^{i\theta})|^p d\theta\Biggr\}^{1/p}= \sup_{0< r< 1} \Biggl\{\int_{\mathfrak T}|f(r\phi)|^p dm(\phi)\Biggr\}^{1/p}.$ Then the Hardy space $$H^p= H^p({\mathcal D})$$ is the set of analytic functions on $${\mathcal D}$$ of the form $$f$$ for which the norm or metric expression $$\|f\|_p$$ is finite. If $$f$$ is expressed as a power series by $$f(z)= \sum^\infty_{j= 0} a_jz^j$$, then the backward shift transform $${\mathcal B}(f)$$ of $$f$$ is defined by ${\mathcal B}(f)(z)= z^{-1}(f(z)- f(0))= \sum^\infty_{j= 1}a_j z^{j- 1},$ and the closed linear subspace $${\mathfrak M}$$ of $$H^p$$ is said to be an invariant subspace if $${\mathcal B}{\mathfrak M}\subseteq{\mathfrak M}$$. If $$J$$ is an inner function on $${\mathcal D}$$, so that $$J$$ is bounded and satisfies $$|J(\phi)|= 1$$, $$\phi\in T$$, then the main results of this book as stated in the overview of the contents relate to the characterizations of the invariant subspace $$M$$ in forms which feature the expression $$H^p\cap \overline{JH^p_0}$$, where $$H^p_0= \{f\in H^p: f(0)= 0\}$$. In particular, the earlier sections of the book include the result of R. G. Douglas, H. S. Shapiro and A. L. Shields [Ann. Inst. Fourier 20, No. 1, 37-76 (1970; Zbl 0186.45302)] identifying $$M$$ with $$H^p\cap \overline{JH^p_0}$$, if $$1\leq p<\infty$$, and the result of A. B. Aleksandrov [J. Math. Sci., New York 92, No. 1, 3550-3559 (1998; Zbl 0907.30038)] identifying an inclusion operator $$i$$ which maps $$H^p\cap \overline{JH^p_0}$$ into the dual space $$H^{p'}\cap \overline{JH^{p'}_0}$$, if $$1<p'< p<\infty$$, where $$(1/p)+ (1/p')= 1$$. In the case $$0< p< 1$$, statements in the book include results of A. B. Aleksandrov [Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst. Steklova 113, 7-20 (1981; Zbl 0473.47014)], indicating that $$M$$ is of the form $${\mathcal E}^p(J,F,k)$$ consisting of functions with poles at sets of isolated points of a subset $$F$$ of $$T$$.
Consideration is given in Chapter 4 to Hardy space $$H^p({\mathfrak R})$$ in the upper half plane. Results stated in the chapter include theorems of C. Fefferman and E. M. Stein [Acta Math. 129, 137-193 (1972; Zbl 0257.46078)], indicating equivalent conditions that a tempered distribution $$L$$ is in $$H^p({\mathfrak R})$$, and theorems of R. R. Coifman [Stud. Math., 51, 269-274 (1974; Zbl 0289.46037)], which include statements that a distribution $$L$$ in $$H^p({\mathfrak R})$$, where $$0< p< 1$$, has the form $$L= \sum^\infty_{k=1}\lambda_k b_k$$, where $$\{b_k\}$$ are atoms.
In Chapter 5 there are applications of the results involving $$B$$-invariant spaces of $$H^p$$, where $$1\leq p<\infty$$, to considerations involving Bergmann spaces, spectral properties of the backward shift $$B$$, and the compactness of the inclusion operator mapping $$H^p\cap \overline{JH^p}$$ into $$H^{p'}\cap \overline{JH^{p'}}$$. The statements of Chapter 6 relate especially to ‘rational approximations’, and include results which indicate that spaces of the form $${\mathfrak H}^p_F\cap{\mathfrak H}^p({\mathfrak C}\setminus{\mathfrak R})$$, where $$0< p< 1$$, are contained in closures of the sets of rational functions.

MSC:
 47B38 Linear operators on function spaces (general) 47-02 Research exposition (monographs, survey articles) pertaining to operator theory 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 46E10 Topological linear spaces of continuous, differentiable or analytic functions 46E15 Banach spaces of continuous, differentiable or analytic functions 46-02 Research exposition (monographs, survey articles) pertaining to functional analysis