• Speaker: Soumyashant Nayak, University of Pennsylvania
• Title: Analyticity in Operator Algebras
• Date: 08/20/2018
• Time: 11:00 AM - 12:00 PM
• Place: C517 Wells Hall

The title of this talk is borrowed from a seminal paper by Arveson discussing non-commutative analogues of the Hardy space H^∞(T) via the so-called subdiagonal algebras. Subdiagonal algebras are a family of non-self-adjoint operator algebras which give a common perspective to the study of some triangular operator algebras (for example, the algebra of block upper triangular matrices in M_n(C)), Dirichlet function algebras, etc. The first part of the talk will be about a non-commutative version of inner-outer factorization in finite maximal subdiagonal algebras. We will then discuss a proof of a version of Jensen's inequality in this setting which relates to some classical results by Szegö.

• Speaker: Richard Kollar, Comenius University, Bratislava, Slovakia
• Title: Krein signature - three unexpected lessons
• Date: 09/19/2018
• Time: 4:10 PM - 5:00 PM
• Place: C517 Wells Hall

Krein signature is an algebraic quantity characterizing purely imaginary eigenvalues of linearized Hamiltonian systems. Instabilities growing from a stable state in these systems are caused by Hamiltonian-Hopf bifurcations, i.e. events when two purely imaginary eigenvalues collide and split off the imaginary axis. The necessary condition for such an event is that the colliding eigenvalues must have mixed signature. In the talk we present three elegant results related to Krein signature - graphical Krein signature and its use to simplify proofs, a connection to stability in general extended systems, and ability to characterize the nature of the eigenvalue collisions directly from the reduced dispersion relation.

• Speaker: N. K. Nikolski, University of Bordeaux
• Title:  Dilated systems and multivariable analysis
• Date: 10/10/2018
• Time: 4:10 PM - 5:00 PM
• Place: C517 Wells Hall

Geometric L^2(0,1) properties of dilated function systems D(f)= {f(kx): k= 1,2,...} are discussed, as completeness, Riesz basis property, etc. (Completeness of D(f) for f(x)= 1/x-[1/x] is equivalent to the Riemann Hypothesis). The Bohr's lift techniques permit to explain (all) known results and show some new, as well as to discuss open problems.

• Speaker: A. N. Karapetyants, Southern Federal University, Rostov on Don, Russia
• Title: On Bergman type spaces of functions of nonstandard growth and some related questions.
• Date: 10/24/2018
• Time: 4:10 PM - 5:00 PM
• Place: C517 Wells Hall

We study various Banach spaces of holomorphic functions on the unit disc and half plane. As a main question we investigate the boundedness of the corresponding holomorphic projection. We exploit the idea of V.P.Zaharyuta, V.I.Yudovich (1962) where the boundedness of the Bergman projection in Lebesgue spaces was proved using Calderon-Zygmund operators. We treat the cases of variable exponent Lebesgue space, Orlicz space, Grand Lebesgue space and variable exponent generalized Morrey space. The major idea is to show that the approach can be applied to a wide range of function spaces. This opens a door in a sense for introducing and studying new function spaces of Bergman type in complex analysis. We also study the rate of growth of functions near the boundary in spaces under consideration and their approximation by mollifying dilations.