# Multivariate Operator Theory (15w5020)

Arriving in Banff, Alberta Sunday, April 5 and departing Friday April 10, 2015

## Organizers

(University of Waterloo)

(Texas A & M University)

Joerg Eschmeier (Universitat des Saarlandes)

J. William Helton (University of California at San Diego)

(University of California at Santa Barbara)

## Objectives

A workshop under the same name (10w5081) was held at BIRS in 2010. The extraordinary success of the meeting is partially revealed by the more than 250 research papers authored by the participants in the last few years; a good many of these works were prompted by discussions originating at the Banff workshop. The proposed meeting will not attempt to replicate the previous one but rather would build on it by emphasizing several new exciting and promising direction with many new people. Continuity would be maintained by previous participants, both young and experienced. Some new participants would be from areas somewhat remote from operator theory. Opportunities for such collaborations spring from the deep and mature techniques and results now being obtained in MVOT. The purpose is to exploit their knowledge and incite their curiosity for the many challenging problems now circulating under the general theme of MVOT. Finally, the previous workshop attracted several younger mathematicians, including a few women, to the subject. We plan to build on this feature of the previous workshop in 2015. MVOT (that is the study of more than one operator at a time) is, compared to the study of a single linear operator, a much younger subject. However, we are witnessing today the rapid maturing of some of its branches, and the planned workshop will explore each of them, separately and all together, in an organic symbiosis. Some studies of MVOT emphasize the analogy with analytic functions, with either commuting or noncommuting variables. A rather surprising recent development is the realization that the noncommutative case often parallels more closely the single operator case than does the commutative multivariable case and, moreover, often techniques and results from the non commutative case shed light on the more "intricate" commutative case. Another kind of noncommuativity involves something completely different, the study of "free variables." The development of this idea had to wait, for the most part, until the last couple of decades for relevant techniques in operator theory and operator algebra had been developed. Now it is a clear direction, pursued with vigor by several schools of mathematics, as the free analysis counterpart to classical function theory of several commuting complex variables. However, in many applications, the operators can't be assumed to commute. In free analysis, one studies the qualitative and quantitative properties of functions (that is, convergent power series) depending on elements of a free-* algebra. Motivations and surprising results in this directions abound; first is the free probability theory advocated by Voiculescu, Bercovici and their many followers. Then Gelu Popescu's study of free multivariate operator theory on the Fock space stands out for originality, depth and beauty. Bill Helton's program of linking intractable (due to large size) control theory problems to positivity questions in a free-* algebra has inspired several groups of researchers; notable are the efforts of Ball and Vinnikov in the direction of building on a systematic foundation the theory of functions of free variables (originating in the work of J. L. Taylor). Then we should mention the recent discoveries of Muhly and Solel on free function theory, complementary to those other studies. In short, the emerging field of free function theory will be well represented in the 2015 workshop and a priority will be inviting younger participants to gain a broader understanding of the field and its interconnections. Moreover, experts in classical (commutative, complex variables) function theory will be there, representing by now well established lines of investigation such as: bounded analytic interpolation, division and extension with bounds, approximation or singular integrals. These questions arise in the context of analytic Hilbert modules, a second principal topic of the workshop. A Hilbert module over an algebra of holomorphic functions such as the polynomials or entire functions depending on several (commuting) complex variables yields a variety of invariants, either with respect to topological isomorphism or the more rigid unitary equivalence. First, such a module is supported, in the algebraic sense of localization, by a compact subset of hermitian space, known as the Taylor joint spectrum. The Hochschild-type topological-homological theory of analytic Hilbert modules encodes the refined structure of the joint spectrum with local invariants such as the Fredholm index, the local analytical K-theoretic index, a Hilbert--Samuel polynomial, or, on restricted subsets of the joint spectrum, a Hermitian holomorphic vector bundle with canonical connection and related curvature. Applications range from a novel proof of the Atiyah--Singer index theorem, of Grauert's finiteness theorem in complex analytic geometry, Riemann-Roch theorem on spaces with singularities and to a classification of homogeneous Hermitian holomorphic vector bundles on classical domains of several complex variables. Further, many reproducing kernel Hilbert spaces of holomorphic functions actually define analytic Hilbert modules which is often the bridge between multivariate operator theory and the problems in complex function theory and geometry. Recent approaches to the Corona Theorem clarify this connection and indicate ways to bring ideas and techniques from harmonic analysis into the subject. Among the achievements in analytic Hilbert module theory of the years since the previous BIRS meeting on the subject we mention: - Douglas/Wang verified the Arveson conjecture for the closure of principal polynomial ideals in the Bergman space for the unit ball in $C^n$ using covering arguments from harmonic analysis - Kennedy/Shalit introduced new ideas involving subspace sums to the Arveson conjecture - Davidson/Ramsey/Shalit showed that the quotient modules in the setting of the Arveson conjecture provide universal noncommutative models for algebras on algebraic varieties in several complex variables - Fang discovered the correct additive formula for the Samuel multiplicity in the category of analytic Hilbert modules, - Biswas/Misra/Putinar have generalized Cowen-Douglas curvature invariant to Hilbert modules of finite, but non-locally constant, rank, - Costea/Sawyer/Wick found sharp estimates of Corona type problem in Besov type norms in several complex variables, - Rochberg and collaborators revealed new analytic extension theorems (from subvarieties) with precise Besov type norm bounds, - Agler/McCarthy/Knese/Young extended to the bidisk classical theorems of Caratheodory and Loewner concerning bounded analytic interpolation - Upmeier/Koranyi/Misra/Englis/Arazy vigorously advanced the study of Bergman-type spaces and homogeneous operators supported by classical symmetric domains. - Davidson/Ramsey/Shalit significantly developed the classification of the universal operator algebra associated to a variety in the $n$-ball. - Mittal/Paulsen developed a theory for an abstract algebra of multipliers for an arbitrary domain - Didas/Eschmeier/Everard studied abstract Toeplitz operators and applied their results to determine the essential commutant of the analytic Toeplitz operators on Hardy spaces over strictly pseudoconvex domains. - Andersson/Berndtsson extended residue theory with connections to singular spaces - Seip/Liubarskii applied interpolation theory on Fock spaces to signal processing theory - Sundberg/Richter studied Agler-convexity theory for several commuting operators. In addition to the specific problems and advances described above, there are many basic questions in the field whose answers would have significant application to the general field as well as related areas. As was mentioned earlier, much of the progress which has been made has depended on the development of techniques which often arose in connection with similar basic questions. The rigidity theory developed to classify submodules defined by ideals and the Hilbert--Samuel polynomial used to classify certain Hilbert modules defined by isometries are two examples as is the functional calculus developed by Taylor and other researchers to define the joint spectrum. Among the basic questions for which there has been significant progress but for which many questions remain are interpolation and division. In particular, the understanding of many problems rests on developing techniques in complex geometry involving uniform bounds. At present, all successful approaches to such questions rest on techniques from harmonic analysis but most questions are open. We plan to start the workshop with a full day of expository lectures, touching the main topics in a non-technical, introductory manner. Then we will have half-days of thematic sessions, complemented by open seminars aimed at raising questions of general interest and discussing new ideas. The last day will be entirely devoted to a seminar type forum leading to precise formulations of open problems and the identification of the main lines of future research.