Schedule for: 21w5124 - Multivariable Operator Theory and Function Spaces in several Variables (Online)

I will present Davidson and Kennedy’s theory of noncommutative convexity and noncommutative Choquet theory, which appeared in a preprint two years ago. I will compare to older notions of convexity, such as matrix convexity, and illustrate what it can do for us.

We will discuss the paper of Helton, Klep, and Volčič with this title (and present some relevant background). It concerns the zero locus of a noncommutative polynomial. If p is a noncommutative polynomial in d variables, its zero locus is defined to be the set of d-tuples of square matrices X, of all sizes, for which det(p(X))=0. It is proved (among other things) that p is irreducible if and only if the zero locus (at size n) is an irreducible variety for sufficiently large n. A key step in the proof is an irreducibility result for linear pencils.

I will try to discuss some of the basic properties of the Drury-Arveson space from a different point of view: that is, to view the Drury Arveson space as an analytic function space that is $L^2$ integrable with a distribution. This is based on several papers by Shalit, Arcozzi, Rochberg, Sawyer, etc

A classical result of Herglotz and F. Riesz says that any bounded holomorphic function on the unit disk admits a factorization into a product of an inner Blaschke product, an inner singular function and an outer function. We will discuss an extension of this result, due to Jury, Martin and Shamovich, to free analytic functions on the non-commutative unit ball. Time permitting, we will showcase some examples coming from nc rational functions.

The class of Hardy Sobolev spaces in the unit ball of C^n is a family of spaces including the Hardy, Drury Arveson, Bergman and Dirichlet space. In this talk we will focus on questions such as characterization of multipliers, interpolating sequences and exceptional sets, mostly presenting earlier work of Ahern, Work, Verbitsky and others. In particular we find that a common factor of all these problems is an abstract potential theory due to Adams and Hedberg adapted to the setting of Hardy Sobolev spaces. We shall make an effort to highlight the limitations of the techniques that have been used so far and present some open problems that might be of interest.

A crucial assumption of the classical monodromy theorem states that the underlying domain must be simply connected. Recent work by J.E. Pascoe has established the surprising fact that, in the non-commutative free setting, “simply connected” can be replaced with merely “connected.” This talk is based on Pascoe’s associated paper “Non-commutative Free Universal Monodromy, Pluriharmonic Conjugates, and Plurisubharmonicity” and will provide both the geometric intuition behind his monodromy theorem as well as a number of interesting applications.

In this talk, I will present the works of Clouatre and Dor-On and Clouatre and Ramsey. These works define and study residual finite-dimensionality for non-self-adjoint operator algebras. In particular, we will explore the residual finite-dimensionality of the maximal C^*-cover of an RFD operator algebra. I will connect these notions to noncommutative function theory. Time permitting, I will discuss the notion of coactions of semigroups on operator algebras, and in particular, RFD coactions.

The main goal of the talk is to give an overview of some known arguments that relates interpolating sequences in a multi-variable setting to Riesz system type conditions on reproducing kernel Hilbert spaces. The first part of the talk reviews Agler’s and McCarthy’s characterization of interpolating sequences in the bidisc, and it highlights how some of those techniques apply also to a generalized interpolating problem, in which the nodes are d-tuples of commuting square matrices (of any dimension). The second part of the talk deals with the case of sequences of eventually non commuting matrices. We review the robust theory of noncommutative function theory on the noncommutative unit ball, and we see how a noncommutative version of the Pick property enjoyed by the NC Drury-Arveson space gives a characterization of interpolating sequences in this non commutative setting.

We study the boundary behavior of rational inner functions (RIFs) in dimensions three and higher from both analytic and geometric viewpoints. On the analytic side, we use the critical integrability of the derivative of a rational inner function of several variables to quantify the behavior of a RIF near its singularities, and on the geometric side we show that the unimodular level sets of a RIF convey information about its set of singularities. We then specialize to three-variable degree (m,n,1) RIFs and conduct a detailed study of their derivative integrability, zero set and unimodular level set behavior, and non-tangential boundary values. Our results, coupled with constructions of non-trivial RIF examples, demonstrate that much of the nice behavior seen in the two-variable case is lost in higher dimensions.

We shall discuss some aspects of the theory of invariant functions and their applications to Nevanlinna Pick interpolation and extension problems

Real algebraic geometry as a discipline was born out of Hilbert's 17th problem, presented at the 1900 ICM. In it, the primary goal is to succinctly describe the set of polynomials which are non-negative on a semi-algebraic set (that is, one described by a finite set of polynomial inequalities). Until the 1980s, the field was predominantly studied via logic and algebra. Konrad Schmüdgen then discovered a deep connection to analysis. More recently, analysts have focused on (freely) non-commutative versions of the area's now classical problems. We emphasize the latter, especially a few key papers of Helton and McCullough, along with some of the work following on.

A theorem of Crouzeix implies that, given a Hilbert space operator $T$, its numerical range $W(T)$ is necessarily a spectral set. In other words, upon endowing the space of polynomials with the supremum norm over $W(T)$, the functional calculus $$p\mapsto p(T), \quad p\in \mathbb{C}[z]$$ is a bounded homomorphism. What is the norm of this homomorphism? To this day, the precise answer is yet unknown, although in 2007 Crouzeix conjectured it to be at most $2$. In this talk, I will describe a recent approach to the conjecture, proposed by Ostermann and Ransford. Surprisingly, this approach is very general and almost purely algebraic: it is concerned with the interaction between finite-dimensional representations and certain conjugate-linear self-maps of a uniform algebra.