We propose to run a one-week workshop on the topic “Analytic Function Spaces and Operators on Them” in the fall of 2015. This is a research area that has been very active in the past few decades. Major research themes in the area include:

(a)  Theory of Bergman spaces: the function theory of Hardy spaces and operators on them had been well understood for a long time. But progress on the close relatives, the Bergman spaces, was lacking for a long time, until the 1980s when several breakthroughs took place. This included Seip’s characterization of interpolating and sampling sequences, Hedenmalm’s introduction and subsequent studies on contractive zero divisors, the formulation and proof of a Beurling type theorem by Aleman, Richter, and Sundberg, and the systematic study of Hankel and Toeplitz operators by Zhu and his collaborators. Despite substantial progress in this area of analysis, many outstanding problems remain, including the structure of the invariant subspace lattice of the Bergman shift, the description of cyclic vectors for the Bergman space, and the characterization of zero sets for the Bergman space.

(b)  The theory of composition operators: the study of composition operators was motivated by Littlewood’s subordination principle in geometric function theory. There is now a complete theory for composition operators on the Hardy and Bergman spaces of the unit disk, with major contributions by Cowen, Luecking, and Shapiro. However, the theory is only rudimentary in higher dimensions (even in the simple cases of the unit ball and polydisk) and many fundamental problems remain open.

(c)   Nevanlinna-Pick interpolation and reproducing kernel Hilbert spaces: this is an extremely interesting area of research that exhibits beautiful connections between function theory, operator theory, and functional analysis. Much of the recent progress has been summarized in Agler and McCarthy’s momograph “Pick Interpolation and Hilbert Function Spaces”. But again, many fundamental problems remain open and the interest in the area is expected to continue for years to come.

(d)  The Corona problem and related operator theory: one of the most famous problems in function-theoretic operator theory is the structure of the invariant subspace lattice of the Bergman shift, which is known to be connected to so-called “invariant subspace problem” (whether or not every bounded linear operator on a Hilbert space has an invariant subspace). Another equally important problem is the Corona Problem in higher dimensions. The problem remains open for the unit ball in dimension 2 or higher, and many approaches have been devised to attack the problem, including hard PDE estimates and abstract operator-theoretic techniques. To show the continued interest in this area, we mention a 2013 workshop at the Fields Institute that was exclusively devoted to this topic and a 2014 workshop at Oberwolfach that was closely related to this.

(e)  Fock spaces and applications: this is an area of current research that is intimately related to many physical applications. Zhu’s 2012 monograph “Analysis on Fock Spaces” summarizes most of the mathematical results obtained in the past few decades. But the potential and far-reaching applications of this field are still being overlooked by mathematicians. The Fock space is of course one of the significant mathematical tools in quantum physics, but even in pure mathematics, it has shown up in numerous problems. For example, via the so-called Bargmann transform, Gabor frames are equivalent to atomic decomposition or sampling in the Fock space, the Fourier transform is equivalent to the extremely simple operator f(z)àf(iz) on the Fock space, pseudo-differential operators are unitarily equivalent to Toeplitz operators on the Fock space, and the Hilbert transform is a special case of a family of very intriguing integral operators on the Fock space. Also, the reproducing kernel of the Fock space is closely related to the heat kernel, and the Berezin transform on the Fock space provides a fundamental solution to the heat equation. Many problems remain open in the area as well.

(f)    Other function spaces and operators: several other holomorphic function spaces have attracted much attention in recent years, including Qp and Qk spaces (with Wulan playing a significant role in its development), spaces defined by Dirichlet series (which has a very natural connection to the Riemann Hypothesis), Morrey spaces, and the Drury-Arveson space (which is one of the model spaces in multivariable operator theory). In addition to the much studied Hankel operators, Toeplitz operators, and composition operators, there are also several other natural classes of operators that are being studied on holomorphic function spaces, including Cesaro type operators, Cowen-Douglass operators, and core operators defined on invariant subspaces of the Hardy and Bergman spaces.

We think there is sufficient interest and significance to run a workshop in this active area of analysis. Most of the principal contributors work in the US and Europe (most notably France, Norway, Finland, Sweden, and Spain). A workshop like this, where we intend to invite a significant number of young researchers from China, will undoubtedly benefit the research and training of analysts in China. It will also serve to reflect on progress on major open problems in the area, raise the visibility of Chinese mathematicians on the world stage, and increase the level of collaboration between Chinese and western mathematicians.