Aktualne ogłoszenie
Current announcement

26 marca 2024 / March 26, 2024
The seminar takes the hybrid form, you may attend in person or join via Zoom (for the Zoom link consult DC).
André Ran (VU Amsterdam and North West University, South Africa)
A class of unbounded Toeplitz operators
Abstract. In this talk a class of unbounded Toeplitz operators is discussed, namely Toeplitz operators with a symbol that is rational with one or more poles on the unit circle. A classical Toeplitz operator with a symbol $\phi(z)$ which is continuous on the unit circle is defined by $T_\phi f=\mathbb{P}(\phi f)$ for $f\in H^2$, the Hardy space of analytic functions on the disc with $L^2$ boundary values. A similar definition for the case where $\phi$ is not continuous, but rational with a pole on the unit circle requires a lot more care with the domain. The talk will start with an overview of the classical theory for continuous symbols. After that, the talk will present (parts of) the current state of the art regarding our knowledge of such unbounded Toeplitz operators.
This is joint work with Gilbert Groenewald (NWU), Sanne ter Horst (NWU) and Jacob Jaftha (UCT).
The work is supported by the National Research Foundation, South Africa.

Patryk Pagacz (Jagiellonian University, Kraków)
On four (quasi-)singular classes of linear pencils.
Abstract: During my talk I will discuss the relation between the following four kinds of linear pencils:
(i) singular pencils, i.e. linear pencils with the spectrum equal to the whole complex plane,
(ii) linear pencils with the numerical range equal to the whole complex plane,
(iii) linear pencils such that (0,0) belongs to the Taylor spectrum of their coefficients,
(iv) linear pencils such that (0,0) belongs to the joint numerical range of their coefficients.
The direct motivation for our research was two questions: "Is it true that the class (i) is equal to (iii)?" arise in [1] and "Is it true that the class (ii) is equal to (iv)?" arise in [2]. The answers of these questions differ depending on the specific assumptions. For example there are no equivalences relations between (i)-(iv) for linear pencils of operators. We will also show that linear pencils from class (ii) do not have to belong to (iv), even if matrix coefficients commute. However, for matrix linear pencils with coefficients that have positive semidefinite Hermitian parts conditions (ii) and (iv) means the same. Moreover, we will show that the first question has an affirmative answer also for matrix polynomials. The talk is based on the joint works with Vadym Koval.

[1] S. V. Djordjevic, J. Kim and J. Yoon, Spectra of the spherical Aluthge transform, the linear pencil, and a commuting pair of operators, Linear and Multilinear Algebra, 70:2533-2550, 2022.
[2] C. Mehl, V. Mehrmann and M. Wojtylak, Matrix pencils with coefficients that have positive semidefinite Hermitian part, SIAM J. Matrix Anal. Appl., 43(3):1186-1212, 2022.

Przyszłe spotkania
Incoming meetings

9 kwietnia / April 9: TBA