Complex Analysis and Operator Theory
April 26, 2018 - ROOM CHANGE to UH 4010
Positivity of Toeplitz Operators on Certain Hilbert Spaces of Holomorphic Functions
Krishna Subedi (University of Toledo)
Abstract. We study the relationship between the positivity of Toeplitz operators and Berezin transforms on certain Hilbert spaces like Bergman, Fock, and Model spaces. Also, we study the necessary condition on symbols when the self-commutator of Toeplitz operators is positive on the weighted Bergman space.
April 19, 2018
Compactness of Hankel operators on the Bergman spaces of some classes of bounded pseudoconvex domains in $\mathbb{C}^n$
Timothy G. Clos (University of Toledo)
Abstract. This talk will survey some previous results and the techniques we used to prove our main results. We partially generalized a compactness result on the Bergman spaces of bounded convex Reinhardt domains in $\mathbb{C}^2$ to the Bergman spaces of bounded convex domains in $\mathbb{C}^n$, $n\geq2$, bounded pseudoconvex complete Reinhardt domains in $\mathbb{C}^n$, $n\geq 2$, and also bounded pseudoconvex Lipschitz domains in $\mathbb{C}^2$. We consider symbols that are continuous up to the closure of the domain. We will also discuss obstructions to proving the converse of these theorems. This is a joint work with Sonmez Sahutoglu and Mehmet Celik.
March 29, 2018
Complex Symmetry of Weighted Composition Operators, Part II
Uthpala Nawalage (University of Toledo)
Abstract. We will introduce the idea of complex symmetry of bounded operators on a complex Hilbert space and discuss some results of complex symmetry of weighted composition operators on the Hardy and Bergman spaces.
March 22, 2018
Complex Symmetry of Weighted Composition Operators
Uthpala Nawalage (University of Toledo)
Abstract. We will introduce the idea of complex symmetry of bounded operators on a complex Hilbert space and discuss some results of complex symmetry of weighted composition operators on the Hardy and Bergman spaces.
March 1, 2018
Wolff's Proof of the Corona Theorem
Sonmez Sahutoglu (University of Toledo)
Abstract. I will present a proof of the Corona Theorem by Theodore W. Gamelin.
February 15, 2018
Carleson Measures and Douglas’ Question on the Bargmann-Fock Space, Part II
Anthony Vasaturo (University of Toledo)
Abstract. Motivated by Douglas' question about the invertibility of Toeplitz operators on the Hardy Space and by our previous work on the weighted Bergman space of the unit disk, we study a related question concerning the Berezin transform and averaging function of a Carleson measure for the Bargmann-Fock space. As a consequence, we obtain a necessary and sufficient condition for the invertibility of Toeplitz operators whose symbols are averaging functions of these Carleson measures.
February 8, 2018
Carleson Measures and Douglas’ Question on the Bargmann-Fock Space, Part I
Anthony Vasaturo (University of Toledo)
Abstract. Motivated by Douglas' question about the invertibility of Toeplitz operators on the Hardy Space and by our previous work on the weighted Bergman space of the unit disk, we study a related question concerning the Berezin transform and averaging function of a Carleson measure for the Bargmann-Fock space. As a consequence, we obtain a necessary and sufficient condition for the invertibility of Toeplitz operators whose symbols are averaging functions of these Carleson measures.
February 1, 2018
Compactness of Hankel Operators on the Bergman Spaces of Some Domains in $\mathbb{C}^n$, Part II
Tim Clos (University of Toledo)
Abstract. I will survey some compactness results for Hankel operators on the Bergman spaces of various bounded domains in higher dimensions. Specifically I will focus on domains with certain boundary geometry and discuss the interplay between the boundary geometry and compactness of Hankel operators with symbols belonging to different regularity classes.
January 25, 2018
Compactness of Hankel Operators on the Bergman Spaces of Some Domains in $\mathbb{C}^n$, Part I
Tim Clos (University of Toledo)
Abstract. I will survey some compactness results for Hankel operators on the Bergman spaces of various bounded domains in higher dimensions. Specifically I will focus on domains with certain boundary geometry and discuss the interplay between the boundary geometry and compactness of Hankel operators with symbols belonging to different regularity classes. This is the first of two talks, and will be accessible to graduate students.
December 7, 2017
On Carleson Embedding Theorem
Zeljko Cuckovic (University of Toledo)
Abstract. In the second talk we will derive the Bezout formulation of the Corona Problem. One of Carleson's powerful ideas in his proof of the Corona Problem is the notion of Carleson measure. We will discuss the Carleson Embedding Theorem which gives an integral characterization of Carleson measures.
November 30, 2017
A brief introduction to the Corona Problem
Zeljko Cuckovic (University of Toledo)
Abstract. I will develop a necessary background to introduce the Corona Problem. Its solution involved some of the greatest Analysts of the 20th century: Lennart Carleson, Lars Hormander and Tom Wolff. The talk is aimed at graduate students and non-experts who would like to learn about this classical problem that is still open in higher dimensions.
November 16, 2017
The Leray operator on two dimensional model domains
Luke Edholm (University of Michigan, Ann Arbor)
Abstract. One major difference between complex analysis in one and several variables is the lack of a true analogue to the one-variable Cauchy transform, $\mathbf{C}$. However, by looking at domains satisfying a convexity condition, we are able to construct the Leray transform, $\mathbb{L}$, which shares many of $\mathbf{C}$'s familiar properties. A significant amount of recent work has been done to study the mapping properties of $\mathbb{L}$ in various settings. This talk will focus on a family of model domains in $\mathbb{C}^2$, and discuss new techniques used in the analysis of the Leray operator. These models can be used to locally approximate a very general class of domains, and it is expected that the theorems in the model case will carry over to the general case. I will also discuss what these results mean in terms of dual CR structures on hypersurfaces in projective space. This is joint work with David Barrett.
November 9, 2017
Schatten class Hankel and $\overline{\partial}$-Neumann operators, Part II
Sonmez Sahutoglu (University of Toledo)
Abstract. Let $\Omega$ be a $C^2$-smooth bounded pseudoconvex domain in $\mathbb{C}^n$ for $n\geq 2$ and let $\varphi$ be a holomorphic function on $\Omega$ that is $C^2$-smooth on the closure of $\Omega$. We prove that if $H_{\overline{\varphi}}$ is in Schatten $p$-class for $p\leq 2n$ then $\varphi$ is a constant function. As a corollary, we show that the $\overline{\partial}$-Neumann operator on $\Omega$ is not Hilbert-Schmidt. This is joint work with Nihat Gökhan Göğüş.
October 26, 2017
Schatten class Hankel and $\overline{\partial}$-Neumann operators, Part I
Sonmez Sahutoglu (University of Toledo)
Abstract. Let $\Omega$ be a $C^2$-smooth bounded pseudoconvex domain in $\mathbb{C}^n$ for $n\geq 2$ and let $\varphi$ be a holomorphic function on $\Omega$ that is $C^2$-smooth on the closure of $\Omega$. We prove that if $H_{\overline{\varphi}}$ is in Schatten $p$-class for $p\leq 2n$ then $\varphi$ is a constant function. As a corollary, we show that the $\overline{\partial}$-Neumann operator on $\Omega$ is not Hilbert-Schmidt. This is joint work with Nihat Gökhan Göğüş.
October 19, 2017
Approximation of smooth functions on $[1,\infty)$ and $\mathbb{R}\backslash (-1,1)$ by entire functions of exponential type
Alexander (Oleksandr) V. Tovstolis (University of Central Florida)
October 5, 2017
On finite rank perturbations of Toeplitz products, Part II
Damith Thilakarathna (University of Toledo)
Abstract. Let $f$ and $g$ be polynomials in $z$. We will prove necessary and sufficient conditions for the product $T_f T_{\bar{g}}$ to be a finite rank perturbation of a Toeplitz operator.
September 28, 2017
On finite rank perturbations of Toeplitz products
Damith Thilakarathna (University of Toledo)
Abstract. Let $f$ and $g$ be polynomials in $z$. We will prove necessary and sufficient conditions for the product $T_f T_{\bar{g}}$ to be a finite rank perturbation of a Toeplitz operator.
September 21, 2017
A sharp estimate for norms of Hankel operators
Trieu Le (University of Toledo)
Abstract. A few years ago, J-F. Olsen and M.C. Reguera obtained a curious sharp estimate for the norm of Hankel operators with conjugate holomorphic symbols on the Bergman space over the unit disk: $$\|H_{\bar{\psi}}\| \leq \dfrac{\|\psi'\|_{2}}{\sqrt{2}}.$$ Here, $\psi$ is a holomorphic function on the unit disk. The equality occurs when $\psi(z)=z$. Their approach is elementary but computational. I shall discuss their proof with the hope of generalizing it to more general settings.
September 13, 2017 (SPECIAL DAY)
Reproducing kernels and distinguished metrics
(Lecture 3 in the Shoemaker Lecture Series)
Miroslav Englis (Mathematics Institute, Czech Academy of Sciences -- Prague)
Abstract. Two classical distinguished Hermitian metrics on a complex domain are the Bergman metric, coming from the reproducing kernel of the space of square-integrable holomorphic functions, and the Poincare metric, i.e. a Kähler-Einstein metric with prescribed (natural) behaviour at the boundary. In the setting of compact Kähler manifolds rather than domains, the so-called balanced metrics were introduced some time ago by S. Donaldson, building on earlier works on S.T. Yau and G. Tian. The talk will discuss the questions of existence and uniqueness of balanced metrics on (noncompact) complex domains, where some answers are yet unknown nowadays even for the simplest case of the unit disc.
September 5, 2017, 10-11am (SPECIAL DAY and TIME)
Reducing subspaces and commutant algebra of some Toeplitz operators
Caixing Gu (California Polytechnic State University)
Abstract. In this talk, I will give a short survey of several recent results on reducing subspaces of Toeplitz operators on Hardy, Bergman and Dirichlet spaces in one and several variables. Then I will focus on the reducibility of the $N$-th power of a truncated Toeplitz operator (or so-called a model operator). We form a conjecture and try to prove the conjecture for small $N$.

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