Convegno
“ADVANCED COURSES IN OPERATOR THEORY AND COMPLEX ANALYSIS-ACOTCA2026”

Lecture 1. l^2 and the shift in l^2. The Hardy space H^2. The shift in H^2. Inner and outer functions; Beurling’s decomposition. Beurling’s theorem. Abstract interpretation of Beurling’s theorem. Shifts in Hilbert spaces. Index of a shift. Equivalence of shifts. Lecture 2. Rota’s universality theorem. Relation with the invariant subspace problem: maximal shift-invariant subspaces. Shift in l^2(H^ 2). The Beurling-Lax theorem. The Beurling-Lax matrix of a shift-invariant subspace of l^2(H^2). Determinantal operators and determinantal subspaces. Lecture 3. Shift-invariant subspaces in l^2(H^2) of finite rank are infinite intersection of determinantal subspaces. Limit of shift-invariant subspaces. Shift-invariant subspaces in l^2(H^2) are limit of infinite intersection of determinantal subspaces. Maximal shift-invariant subspaces in finite direct sum of H^2 and in l^2(H^2).

Lecture I: The invariant form of the Schwarz lemma can be interpreted to say that every holomorphic function from the disk D to itself is distance reducing in the pseudohyperbolic metric, an extremely useful property. Caratheodory (1927) showed how to port the pseudohyperbolic metric to any domain U, in one or several variables, by considering all holomorphic maps from U to D. Kobayashi later (1967) dualized the construction, by considering maps from D to U. Every holomorphic map from U_1 to U_2 is distance reducing with respect to both the the Caratheodory distance and the Kobayashi distance. In 1981, Laszlo Lempert proved the wonderful theorem that on convex domains, both these distances are equal to each other. Lecture II: We will discuss Agler's 1990 operator theory proof of Lempert's theorem, which involves a very careful analysis of two point interpolation problems, and the fact that certain two dimensional representations are contractive if and only if they are completely contractive. Lecture III: If V is a lower dimensional subset of a domain U, it has both its intrinsic Caratheodory and Kobayashi distances, and the ones it inherits from U.When are these the same? This is a complex analogue to asking when a submanifold, or subvariety, of a larger manifold is totally geodesic. The question lies somewhere on the nexus of Pick interpolation problems, von Neumann inequalities, contractive distances, and complex geometry. We will discuss what is known, including some recent results with L. Kosinski in the case that U is the polydisk.

We focus on providing detailed proofs of the quantitative characterizations for a broad class of finite products (words) of these paraproducts act- ing on standard weighted Bergman spaces. These specific cases serve to illustrate the fundamental tools and techniques required to obtain a boundedness characterization for arbitrary words.

TBA

I will discuss how to use the classical Schur's algorithm for analytic functions as a tool for solving inverse spectral problems. The connection between Schur's algorithm and Spectral theory was found very recently. Among other things, it gives sharp spectral stability results for Dirac operators with potentials of class L^2. I will discuss these results and formulate some open questions.

A classical result of Hardy and Littlewood states that the uncentred Hardy–Littlewood maximal operator (UHLMO) on balls is bounded on L p (R n ) for all p > 1 and it is of weak type (1, 1). It is well known that, replacing R n with either homogenous trees of degree at least three or the hyperbolic plane, the UHLMO on balls is bounded on L p if and only if p > 2, and it is of restricted weak type (2, 2). Loosely speaking, the different behaviour of the UHLMO on R n and on the hyperbolic plane depends on the fact that (maximal) fam- ilies of balls with large radii have “worse” overlapping properties in the latter case than in the former. In this talk we consider the UHLMO with respect to half balls, and prove that, perhaps surprisingly, this operator is bounded on L p for all p > 1 and it is not of weak type (1, 1) both on homogeneous trees of degree at least three and on the hyperbolic plane. We shall also discuss generalisations of this result to Damek–Ricci spaces. This is joint work with Nikos Chalmoukis (Milano–Bicocca), Effie Papageorgiou (Paderborn) and Federico Santagati (Politecnico di To- rino).

Motivated by applications in the control theory of infinite-dimensional systems, several authors have investigated the boundedness of the Laplace transform $\mathcal{L}: L^p(0, \infty) \rightarrow L^q( \mathbb{C}_+, \mu)$ in terms of properties of the measure $\mu$, with the case $q=p=2$ being the classical Carleson Embedding Theorem. We will cover the case $p >2$, thus completing the picture to all $p,q$ with $1/p + 1/q \le 1$. In case $p >2$, the Laplace-Carleson Embedding Theorem can also be seen as an appropriate replacement of the Hausdorff-Young inequality. This is joint work with Eskil Rydhe (Lund).

We obtain a characterization of complete interpolating sequences in a class of Fock-type spaces with radial weights for which such sequences exist. Our criterion is formulated in terms of logarithmic separation and controlled perturbations of a reference sequence satisfying an Avdonin-type condition. This provides a geometric description of complete interpolating sequences and extends previous results of Borichev–Lyubarskii and Baranov–Belov–Borichev on Riesz bases of reproducing kernels in Fock-type spaces. It also yields explicit density criteria for sampling and interpolating sequences. This is joint work with Y. Omari

Classical Clark measures are singular measures on the unit circle defined via inner functions that are closely tied to important topics in operator theory and complex analysis (for example, model spaces, compressed shifts, and composition operators). In this talk, we’ll consider an analogous definition for Clark measures associated with two-variable inner functions. For certain classes of such functions, we’ll give exact formulas for these Clark measures, characterize when associated Clark embeddings are unitary, and obtain nice unitary perturbations of pairs of compressed shift operators. This is joint work with John Anderson, Palak Arora, Linus Bergqvist, Joseph Cima, Conni Liaw, and Alan Sola.