Book Review: $H^\infty$ Operator theory and arithmetic in
Pei Yuan Wu · Bulletin of the American Mathematical Society · 1989
Linear algebra is supposed to be in every mathematician's toolbox.Somewhere in our undergraduate instruction, we learned that every finite matrix is similar to a uniquely determined Jordan form (resp. rational form), which reflects all its similarity-invariant properties.For operator theorists whose working environment is bounded linear operators on complex Hubert spaces, a suitable generalization to this context is very much desired.While the problem of developing a canonical model for general operators may be beyond the reach of present day operator theory, practitioners in this field try another approach by isolating classes of operators for which such models can be obtained.One such class is that of algebraic operators, that is, operators T for which p(T) = 0 for some nonzero polynomial p.That every finite matrix is algebraic is a consequence of the Cayley-Hamilton theorem.The Jordan canonical form for finite matrices can be generalized to this class: every algebraic operator is quasisimilar to a unique "Jordan operator" [7].Here the weaker notion of quasisimilarity replaces that of similarity in finite dimensions.Recall that two operators T and S are quasisimilar if there are operators X and Y which are infective and have dense range such that TX = XS and S Y = YT.To a large degree, these models reflect faithfully properties of the original operators.Recently, this was extended by K. R. Davidson and D. A. Herrero [2] to the class of bitriangular operators; these are operators T for which both T and T* have upper triangular matrices with respect to some (possibly different) orthonormal bases of the underlying space.Moreover, they showed that such operators form the largest class of operators for which a "Jordan model" can be constructed.The monograph under review is concerned with the structure and particularly the Jordan canonical form of another class of operators-that of Co contractions.A contraction T (\\T\\ (T) = 0 for some nonzero in the Hardy algebra H°° of bounded analytic functions on the unit disc.(Here the completely nonunitary assumption is required to guarantee that the Sz.-Nagy-Foia § functional calculus <p(T) is meaningful.)This class of operators was discovered by B. Sz.-Nagy and C. Foia § in 1964 [4] in their work on the functional model for contractions on Hubert space.In the more than two decades since then, a whole theory of these operators has been developed.Besides its two founders, the present author is a major contributor.Note that the theory of Q contractions contains properly that of algebraic operators since if T is algebraic then aT is of class Co for 0 < a < l/\\T\\.Their structure resembles closely that of operators on finite-dimensional spaces.The whole theory is rich in interplay between operator theoretic