You are here

The Theory of H(b) Spaces, Volume 1

Emmanuel Fricain and Javad Mashreghi
Cambridge University Press
Publication Date: 
Number of Pages: 
New Mathematical Monographs 20
[Reviewed by
Steven Deckelman
, on

An \(\mathcal{H}(b)\) space is a sub-Hardy Hilbert space; that is, a Hilbert subspace of \(H^2\) in the unit disc. These spaces lie at the confluence of function theory and operator theory and have been a rich source of fruitful interaction since at least the 1960s. This two volume set is an encyclopedic monograph on the theory of \(\mathcal{H}(b)\) spaces. These spaces were introduced by L. de Branges and J. Rovnyak in Square Summable Power Series.

Let \(\mathcal{H}_1\) and \(\mathcal{H}_2\) be Hilbert spaces and \(A:\mathcal{H}_1\to \mathcal{H}_2\) a bounded linear operator. It’s not assumed that the Hilbert space structures are the same, even in the case where \(\mathcal{H}_1\subset\mathcal{H}_2\). \(A\) induces a Hilbert space structure on its range, \(\mathcal{R}(A)\subset \mathcal{H}_2\) by means of the first homomorphism theorem: \[ \langle Ax,Ay \rangle := \langle x+ ker (A), y+ ker (A) \rangle. \] The Hilbert space defined in this way is denoted by \(\mathcal{M}(A)\). In the case where \(A\) is a contraction, the complementary space of \(\mathcal{M}(A)\), denoted by \(\mathcal{H}(A)\), is defined as \(\mathcal{M}((I-AA^*)^\frac12)\). \(\mathcal{H}(A)\) may be thought of as a generalization of the orthogonal complement of \(\mathcal{M}(A)\). \(\mathcal{M}(A)\cap \mathcal{H}(A)\) can be nontrivial and is called the overlapping space. Indeed it turns out to be \(\mathcal{H}(A^*)\) where \(A^*\) is the adjoint.

Let \(u\in L^\infty(\partial U)\) where \(\partial U\) is the unit circle. Now consider the Toeplitz operators with symbols \(u\) and \(\bar{u}\), denoted by \(T_u\) and \(T_{\bar{u}}\) respectively. \(\mathcal{M}(T_u) \) and \(\mathcal{M}(T_{\bar{u}})\) are then subspaces of \(H^2\) and for simplicity denoted by \(\mathcal{A}(u)\) and \(\mathcal{A}(\bar{u})\). When \(u\) is a nonconstant analytic function in the unit ball of \(H^\infty\) (traditionally denoted by \(b\)), the corresponding complementary space is denoted by \(\mathcal{H}(b)\). To wit,

  1. \(b\in H^\infty\),
  2. \(b\) is nonconstant,
  3. \(||b||_{\infty} \leq 1\).

As an example, when \(b\) is an inner function, \(\mathcal{H}(b)\) gives the invariant subspaces of the unilateral shift operator on \(H^2\).

This two volume monograph is a compendium of the \(\mathcal{H}(b)\) spaces that will be of interest to both graduate students and practicing mathematicians interested in function-theoretic operator theory. There are 31 chapters between the two volumes and a detailed bibliography consisting of 766 entries. The first volume is devoted to general function-theoretic operator theory (and indeed is a useful reference in its own right) while the second volume is more specialized and contains an in-depth survey of \(\mathcal{H}(b)\) theory and related ideas.

Steven Deckelman is a professor of mathematics at the University of Wisconsin-Stout, where he has been since 1997. He received his Ph.D from the University of Wisconsin-Madison in 1994 for a thesis in several complex variables written under Patrick Ahern. Some of his interests include complex analysis, mathematical biology and the history of mathematics.

List of figures
List of symbols
Important conventions
1. *Normed linear spaces and their operators
2. Some families of operators
3. Harmonic functions on the open unit disc
4. Analytic functions on the open unit disc
5. The corona problem
6. Extreme and exposed points
7. More advanced results in operator theory
8. The shift operator
9. Analytic reproducing kernel Hilbert spaces
10. Bases in Banach spaces
11. Hankel operators
12. Toeplitz operators
13. Cauchy transform and Clark measures
14. Model subspaces KΘ
15. Bases of reproducing kernels and interpolation