Book Review: Subnormal operators

Abstract

The theory of subnormal operators is the symbiotic interplay between operator theory and the theory of rational approximation. To see the connection, recall first some definitions. A (bounded linear) operator T on a (complex) Hilbert space % is called normal if T commutes with its adjoint, T*. The operator T is called subnormal if it is possible to imbed % in a larger Hilbert space % and to find a normal operator N on % leaving % invariant (i.e., Ni G %, £ G %) such that T is the restriction of N to %. If T is any operator, and if K is its spectrum ( a compact subset of the plane), then in a familiar way one may form r(T) where r is a rational function with poles located in the complement of K. One says that T is rationally cyclic if there is a vector £ 0 in the Hilbert space of T 7 , %, such that % is the closed linear span of the vectors r(T)£ 0 where r runs through the rational functions with poles off K. The starting point of the theory of subnormal operators, and its link with approximation theory, is the observation that if T is a rationally cyclic subnormal operator on %, then there is a finite positive measure ju on the spectrum, K, of T and there is a Hilbert space isomorphism V from % onto R 2 (K, /x), the closure in L 2 (K, /A) of the space of rational functions with poles off K, such that {VTV~xi){z) = z£(z) for all £ G R 2 (K, /A). This transform of T by V is usually denoted by M z . While a lot of attention has been paid to general subnormal operators, it is fair to say that most of the work in the subject to date, and the deepest, has been devoted to understanding M z on R 2 (K, JU).At first glance, it may appear that the representation of a rationally cyclic subnormal operator in this fashion makes the study of subnormal operators trivial. After all, how could anything so concrete be inscrutable? It turns out, however, that the representation is no panacea. Some of the most fundamental facts require deep analysis and many basic problems remain to be solved. For example, the problem of deciding if a subnormal operator has a proper invariant subspace was settled (in the affirmative) only in 1978 by Scott Brown [6]. His work, in turn, was based on Sarason's penetrating analysis [19] of P°°(ju), the weak-* closure of the space of polynomials in L°°(/A), where JU is a compactly supported measure in the plane. Except for (essentially only) one example, the invariant subspace structure of M z on R 2 (K, /i) is a complete mystery. The exception is when K is the closed unit disc and JU is arclength measure on the boundary. The description of the invariant subspaces of M z in this context is Beurling's famous theorem [5] which asserts that each invariant subspace is of the form 6R 2 (K 9 JU) where 6 is a so-called inner function. Beurling's result is definitive .because the structure of inner functions is so completely well understood. There is also an important generalization of