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.

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.

Reviewer: G.O.Okikiolu (London)

##### 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 |