Abstracts, Operator and Spectral Theory

12:30-12:50 K.-M. Perfekt: “Poincaré fundamental functions
Abstract: I will discuss the spectral decomposition of the double layer potential for the Laplacian, operating within the framework of symmetrisation. I will also indicate a recent result on the spectrum of this operator for domains in 3D with conical points. Based on joint work with Johan Helsing.

13:00-13:50 A. Pushnitski, “Schmidt subspaces of Hankel operators
Abstract: A Schmidt subspace of a bounded (not necessarily self-adjoint) operator A in a Hilbert space is an eigenspace of |A|. In this talk, I will describe the structure of Schmidt spaces of Hankel operators. This is recent joint work with Patrick Gerard (Orsay).

14:00-14:50 L. Scardia, “Equilibrium measures for nonlocal energies: The effect of anisotropy
Abstract: Nonlocal energies are continuum models for large systems of particles with long-range interactions. Under the assumption that the interaction potential is radially symmetric, several authors have investigated qualitative properties of energy minimisers​. But what can be said in the case of anisotropic kernels? Motivated by the example of dislocation interactions in materials science, we pushed the methods developed for nonlocal energies beyond the case of radially symmetric potentials, and discovered surprising connections with random matrices and fluid dynamics.​

15:30-16:20 J.R. Partington, “Properties of restricted shift operators”
Abstract: We discuss numerical ranges of restricted shift operators and their unitary dilations, and we link these with the theory of truncated Toeplitz operators (TTO) and Hankel operators. Further results on the norm of a TTO are derived, and a conjecture on the existence of continuous symbols for compact TTO is resolved. This is joint work with Pamela Gorkin (Bucknell) and others.

16:30-17:20 O.F. Brevig, “Hilbert–type inequalities and operator theory in the Hardy space of Dirichlet series
Abstract: We present two instances where inequalities related to the classical Hilbert inequality appear in the operator theory of the Hardy space of (ordinary) Dirichlet series with square summable coefficients. After some preliminaries, we will discuss a canonical Hankel form which is a multiplicative analogue of the Hilbert matrix. We will then explain how a family of Hilbert–type inequalities are related to sharp norm estimates for composition operators. Some related open problems will also be discussed. Some of the material presented is based on joint work with F. Bayart and joint work with K.-M. Perfekt, K. Seip, A. G. Siskakis and D. Vukotić.

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