Acta ApplicandaeMathematicae25: 99-103, 1991.
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Book Review J. B. Conway and B. B. Morrel (editors): Surveys of Some Recent Results in Operator Theory, Volume 1, Pitman Research Notes in Mathematics Series, No. 171, Longman Scientific & Technical, Harlow, Essex, copublished in the U.S. with John Wiley & Sons, New York, 1988, paperback, 259 pp., ISBN 0-582-00519-1 (0-47021030-3 in U.S.A.). This book contains survey papers based on series of lectures given at Indiana University as part of the Special Year of Operator Theory in the 1985-86 academic year. It is the first of two volumes and is concerned primarily with single linear operators, whereas the second volume will focus on operator algebras. To quote from the editors' Preface, As originallyenvisioned,each of the lecture series in the Special Year would developa topic of current interest in Operator Theory,starting from a point accessibleto advancedgraduate students in the area, and proceeding rapidly to current and future researchers. We feel that the results have far exceeded our expectations. The first paper is 'Bergman spaces and their operators' by S. Axler. While the theory of HP-spaces and of Toeplitz and Hankel operators on H 2 is now well understood and has become classical, its area-measure counterpart, i.e. Bergrnan spaces and Toeplitz or Hankel operators on them, is far less tractable, and the results are fewer though a lot of work has been done. These spaces and operators are encountered in quantum mechanics, where they emerge in connection with quantization procedures. (This point, however, is not touched upon in Axler's paper; an interested reader can consult, e.g., I-2] or [5] as a general reference on quantization and [3] for Hankel and Toeplitz operators.) The author quickly introduces the subject and then proceeds to the latest results on duality, on the Bloch and little Bloch spaces, which seem to play the role of BMO and VMO from the Hardy space theory, on the atomic decomposition in the spirit of Coifman-Rochberg-Weiss, on Toeplitz operators and algebras generated by them, and on Hankel operators. Some unsolved problems are brought to attention. The paper is quite engaging and excellently readable and might well serve as a textbook on Bergman spaces and their Toeplitz and Hankel operators (it would be the only one the reviewer would know of, by the way). The second article is J. A. Bali's 'Nevanlinna-Pick interpolation: Generalizations and applications'. Its central topic is the Nevanlinna-Pick interpolation problem, which, in its simplest form, may be formulated as follows: given points zl, z2,..., z n inside the unit disc D = {z~C: Izl < l} and complex numbers wl, w2,..., wn, does there exist a function f analytic in D, with modulus bounded by 1 and such that f(zi) = wi, i = 1, 2. . . . . n? Moreover, one would like to describe the set of all such functions. This problem can be generalized in many ways: the points zi may be
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allowed to be on the boundary of the unit disc, or values of derivatives of f at z i may be prescribed, or, finally, one can consider vector- or matrix-valued analytic functions. The problem, which was originally of purely mathematical interest, has recently found many applications in electrical engineering, systems theory, and even medicine (see [6] for more about the applications). In the present paper, after showing the classical approach of Nevanlinna, Pick, and Schur, the author expounds a powerful indefinite inner product space technique, which he and J. W. Helton have developed to tackle the problem. Later on, this technique is shown to also apply to a very general Axiomatic Nevanlinna-Pick Problem; several concrete examples are then presented, leading to interesting results (e.g. a connection with the Arveson distance formula, etc.). The whole machinery is shown to work in the case of boundary interpolation as well, and practical computational aspects are also discussed. The paper is well written and makes an excellent introduction both to the 'big' works [1] and to papers like [4] which are closer to the engineer's point of view. A small error has intruded into the references: No. 4 was not written by Abrahamse, but by the authors of No. 5 (Adamjan, Arov, and Krein). E. Berkson's paper 'A Fourier analysis theory of abstract spectral decompositions' presents the results of Berkson, Gillespie, and Muhly connected with some weakened forms of 'orthogonality' outside the Hilbert space setting. The starting point is a suitable generalization of the notion of spectral measure and, consequently (via the spectral theorem) of the selfadjoint operator. The resulting theory enhances many important operators of harmonic analysis (e.g. the translation operators on spaces L p, p # 2); however, it rather differs from its Hilbert space counterpart. In the first sections, well-bounded and trigonometrically well-bounded operators, which are the above-mentioned Banach space generalizations of selfadjoint and unitary operators, respectively, are introduced and investigated. Then the exposition proceeds to such topics as power-bounded well-bounded operators, the General Transference Principle for convolution operators, etc. Various applications of these results are also given, e.g. to the theory of U M D (unconditionality property for martingale differences) Banach spaces, or to Helson's theory of invariant subspaces and generalized analycity. Though the paper is amply interspersed with 'philosophical' comments and observations, it is, nevertheless, a little harder to read than the first two. The next survey is 'Some remarks on principal currents and index theory for single and several commuting operators' by R. W. Carey. Unlike previous papers, this one does not 'start from the beginning', but rather assumes some familiarity with such things as homology theory, geometric measure theory, complex varieties (the notion of Lelong number is needed in the last section), etc. In addition to a survey of results, both recent and those of longer standing, of Carey, Pincus, Helton, Howe and others, it also poses many problems that might stimulate further research in this area. The most impressive result is perhaps the following: although any Lebesgue integrable function is the principal function of some bounded linear operator, the principal function of a subnormal operator must assume integral values almost everywhere. This paper is also a little harder to read, and minor annoyances are encountered
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occasionally - e.g. in Section 7, the reader is more or less left to guess what is exactly 'the space Ae' of operator ranges; in Proposition 5.7, it is not clear what is 'a bounded evaluation for [an operator] T' (perhaps a bounded evaluation for R2(X, v)), or the measure with respect to which the integral on the next line is taken. The article 'Hyponormal and subnormal Toeplitz operators', by C. C. Cowen, is devoted to 'past. present and future of Problem 5 of Halmos's 1970 lectures: Is every subnormal Toeplitz operator either normal or analytic?' Besides proving the answer (which is negative), the reader is shown particular cases when the above assertion holds, his attention is drawn to many related problems, and, on the whole, he becomes more familiar with the subject than he would perhaps expect from the shortness of the paper. Next comes L. A. Fialkow's extensive paper on 'Closures of similarity orbits of Hilbert space operators'. Through a brief motivation (the Weyl-von Neumann-Berg theorem, some Brown-Douglas-Fillmore theory) and discussion of four types of 'unitary equivalence', it quickly proceeds to the latest results of Voiculescu, Herrero, Apostol, Foias, Fialkow, and others. To mention just a few, it is the determination of the closure of nilpotent operators, of algebraic operators, the characterization of operators with closed similarity orbit, of universal quasinilpotent operators (i.e. those whose closed similarity orbit consists of all quasinilpotents). Proofs are frequently given, even the more difficult ones. The paper is rather long and a little technical at some places, but well worth reading. The paper of V. I. Paulsen, 'Toward a Theory of K-spectral Sets', treats the problem of spectral sets and related questions. The motivation for these questions stems from an old theorem of von Neumann: a Hilbert space operator T is a contraction if and only if, for every polynomial p, [Ip(T)II ~< IIPlID. If we allow p to be, more generally, a rational function with poles off D, the latter condition becomes the definition of a spectral set for the operator T. The most transparent proof of von Neumann's theorem uses the dilation theory of Sz.-Nagy. The paper discusses some questions which now naturally arise: the (im)possibility of extending Sz.-Nagy's theory to other domains than the unit disc D; to extend von Neumann's theorem to the case of several commuting operators (which, surprisingly, comes through for two operators, but not for three or more); finally, some generalizations and variants of the definition of a spectral set are also considered. A small mistake seems to be present in the text of Theorem 3.6 on page 228; the first sentence should read "Let T1, T2 be commuting contractions on H". Ignoring this, the article is neatly written, requiring few prerequisites, and also reveals interesting connections with uniform algebras, completely positive mappings (also termed mappings of positive type by some authors), power bounded operators, etc. Many interesting unsolved problems are brought to the reader's attention. The last paper, 'Invariant Subspaces for Subnormal Operators' by J. E. Thomson, is concerned with one of the celebrated problems of operator theory - namely, the invariant subspace problem. While in the full generality, the problem remains still unanswered, it was shown by S. Brown that every subnormal operator possesses a
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nontrivial invariant subspace. The author first presents a simple proof of Scott Brown's theorem (Theorem 1.6) and then exhibits an improvement of the original technique of S. Brown which leads to new results. En route, the reader becomes acquainted with many interesting topics, e.g. with D. Sarason's work on the w*closure of the polynomials in the space L~°(p), etc. A minor overlook that seems to be present is the absence of the definition of an analytic invariant subspace - the reader is left to guess what it can be, though the definitions of some more familiar notions (e.g. of a subnormal operator or a bounded point evaluation) are supplied. Ignoring this, the paper is neatly written, interesting, and very readable. Each of the surveys bring ample references to the literature suggested for further reading and information on the subject. What is to be regretted a little is that the book has neither name, nor subject index, nor a list of notation, but these may, perhaps, appear in the second volume. Another unpleasant thing is the period it took the book to appear - the results that were new in 1985-86 can hardly be new in 1988 when the first volume was published; but this is perhaps a thing that cannot be helped. As for the scope of each article, that is probably a matter of author's taste, and so should not be disputed. The reviewer feels that perhaps Carey's article would deserve to be longer (at least twice as long) or that the first paper could very briefly mention the connections with quantization and related topics, as indicated above (the works of Upmeier, Berezin, et al.). But, again, some of the latter will perhaps be remedied in Volume II, and so this rebuke may turn out to be unjust. The book does not contain more than the usual amount of misprints. Some of them betray the way in which the manuscripts were prepared, since they are typical TEX errors - such as S instead of §, which is written \S in TEX (e.g. on pages 102, 112, 116, 126, 137 and many others), or a theorem, the text of which is in italics at the beginning, but in Roman at the end (e.g. Proposition 2.2 on page 109). Others are just common oversights - e.g. 'Block space' instead of'Bloch' on page 12; Adamjan from pages 67 or 89 is spelled Adamian on pages 76 and 77; the orthogonal complement with respect to the J-inner product is denoted sometimes J, sometimes ', and someUmes Y (in the paper of J. Ball). Finally, there are some 'mathematical' misprints. In addition to those in the formulations of Theorem 5.7 on' page 140 (in Carey's paper) and of Theorem 3.6 on page 228 (in Paulsen's paper), which have already been pointed out above, it is, e.g., the missing 'dist' on page 186, line 4, or the expression Izl ~< 2R in the domain of integration of the double integral on page 242, line 3, which ought to read 121 ~< 2R. There are altogether about 110 misprints I have noticed; a list of them is available. On the whole, the reviewer agrees with the editors' sentence from the Preface that "the results have far exceeded our expectations". The surveys make a quick introduction to the topic, stress motivation, and, besides mere quotations of results, give proofs to an extent which is adequate for a proper insight into the topic, and draw the reader's attention to stimulating unsolved problems. Only a few places may be considered tiresome - mostly they are written in a neat, informal way and make excellent reading. I think the book is a very nice one, and certainly worth having. •
Prague, Czechoslovakia
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References 1. Ball, J. A. and Helton, J. W.: Beurling-Lax representations using classical Lie groups with many applications. II: Inte#ral Equations Operator Theory 7 (1984), 291-309; III: Amer. J. Math. 108 (1986), 95-174; IV: J. Funct. Anal. 69 (1986), 178-206. 2. Berezin, F. A.: Quantization in complex symmetric spaces, Math. USSR-Izv. 9 (1975), 341-379. 3. Berger, C. A. and Coburn, L. A.: Toeplitz operators on the Segal-Bargmann space, Trans. Amer. Math. Soc. 301 (1987), 813-829. 4. Francis, B. A., Helton, J. W., and Zames, G.: H~-optimal feedback controllers for linear multivariable systems. IEEE Trans. Automat. Control AC-29 (1984), 888-900. 5. Upmeier, H.: Weyl quantization of complex domains, preprint, 1989. 6. Young, N. J.: The Nevanlinna-Pick problem for matrix-valued functions, J. Operator Theory 15 (1980, 239-265.
