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,

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

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.