Complex Analysis and Operator Theory
Thursday 4pm-5pm (Place: UH 4170)

Where to find us? The University of Toledo's address is at 2801 W. Bancroft St, Toledo, OH 43606. Please check Google Map and Campus Map. The Department of Mathematics and Statistics is located in University Hall (UH).

Parking for speakers: after your car has been registered in the parking system, you may park anywhere where an "A" permit is allowed. Suggested parking lots for UH can be found in this parking map.
April 18, 2019
Lattices of invariant subspaces of the shift plus Volterra operator, Part II
Zeljko Cuckovic (University of Toledo)
Abstract. The invariant subspace problem for separable Hilbert spaces is a long standing open problem in operator theory. However for certain classes of operators, there is a characterization of all their invariant subspaces. We will review the classical results of Beurling, Sarason, Rudin, Korenblum and others and then focus on the lattice of invariant subspaces of the shift plus Volterra operator on the Hardy space of the unit disk. This is a joint work with Bhupendra Paudyal.
March 28, 2019
Lattices of invariant subspaces of the shift plus Volterra operator, Part I
Zeljko Cuckovic (University of Toledo)
Abstract. The invariant subspace problem for separable Hilbert spaces is a long standing open problem in operator theory. However for certain classes of operators, there is a characterization of all their invariant subspaces. We will review the classical results of Beurling, Sarason, Rudin, Korenblum and others and then focus on the lattice of invariant subspaces of the shift plus Volterra operator on the Hardy space of the unit disk. This is a joint work with Bhupendra Paudyal.
March 21, 2019
Weighted $L^p$ norm estimate for the Bergman projection on the Hartogs triangle, II
Zhenghui Huo (University of Toledo)
Abstract. The boundedness of the Bergman projection on the weighted $L^p$ space of the unit ball was first studied by Bekoll\'e and Bonami in the 80s. Using modern dyadic harmonic analysis techniques, sharp weighted $L^p$ estimates were obtained for the Bergman projection on the upper half plane by Pott and Reguera in 2012, and on the unit ball in $\mathbb C^n$ by Rahm, Tchoundja, and Wick in 2016. In this talk, I will introduce the dyadic operator technique used for these results and give a sharp weighted $L^p$ norm estimate for the Bergman projection on the Hartogs triangle. This work is joint with Brett Wick.
March 14, 2019
Weighted $L^p$ norm estimate for the Bergman projection on the Hartogs triangle
Zhenghui Huo (University of Toledo)
Abstract. The boundedness of the Bergman projection on the weighted $L^p$ space of the unit ball was first studied by Bekoll\'e and Bonami in the 80s. Using modern dyadic harmonic analysis techniques, sharp weighted $L^p$ estimates were obtained for the Bergman projection on the upper half plane by Pott and Reguera in 2012, and on the unit ball in $\mathbb C^n$ by Rahm, Tchoundja, and Wick in 2016. In this talk, I will introduce the dyadic operator technique used for these results and give a sharp weighted $L^p$ norm estimate for the Bergman projection on the Hartogs triangle. This work is joint with Brett Wick.
February 21, 2019
The local Hausdorff dimension and local walk dimension on compact metric spaces and fractals
John Dever (Bowling Green State University)
Abstract. We discuss two scaling exponents that may be defined on any compact metric space: the local Hausdorff dimension and the local walk dimension. Intuitively, the local Hausdorff dimension determines how the volume of metric balls scales with the radius of the ball, and the local walk dimension determines how fast a diffusion process leaves a ball on average relative to its radius. Spaces that have walk dimension not equal to 2 exhibit what is called anomalous diffusion. We give examples of fractals that have continuously variable Hausdorff dimension and continuously variable walk dimension. Moreover, we explain how these dimensions may be useful in approximating diffusion on a compact metric space.
February 14, 2019
Hankel operators on the Bergman spaces of Reinhardt domains and foliations of analytic disks, Part II
Timothy G. Clos (Bowling Green State University)
Abstract. Let $\Omega\subset \mathbb{C}^2$ be a bounded pseudoconvex complete Reinhardt domain with a smooth boundary. We study the behavior of analytic structure in the boundary of $\Omega$ and obtain a compactness result for Hankel operators on the Bergman space of $\Omega$. This talk is graduate student friendly.
February 7, 2019
Hankel operators on the Bergman spaces of Reinhardt domains and foliations of analytic disks, Part I
Timothy G. Clos (Bowling Green State University)
Abstract. Let $\Omega\subset \mathbb{C}^2$ be a bounded pseudoconvex complete Reinhardt domain with a smooth boundary. We study the behavior of analytic structure in the boundary of $\Omega$ and obtain a compactness result for Hankel operators on the Bergman space of $\Omega$. This talk is graduate student friendly.
November 15, 2018
Boundedness and compactness of operators on the Bergman space of the Thullen domain
Zhenghui Huo (University of Toledo)
Abstract. In a variety of classical function spaces, some properties of a given operator can be analyzed by examining only its behavior on the normalized reproducing kernel. In this talk, we consider conditions for the boundedness and compactness of operators on the Bergman space of the Thullen domain. We give a sufficient condition for the boundedness. Under this sufficient condition, we show that the behavior of operator on the normalized Bergman kernels determines the compactness of the operator. We will discuss several results about Toeplitz operators and Hankel operators along this line. This work is joint with Brett Wick.
November 1, 2018
Some classes of invariant subspaces in the polydisc
Beyaz Basak Koca(Istanbul University and Michigan State University)
Abstract. An important open problem in multivariable operator theory and function theory of several complex variables is the problem of classification or an explicit description (in some sense) of all invariant subspaces of the Hardy space $H^2(D^n)$ on the polydisc $D^n$ given by W. Rudin [1, p.78]. This problem has been extensively studied by various authors in different context, but it is still open. Hence we need good examples to help us understand the structure of invariant subspaces of $H^2(D^n)$. In this talk, we completely classify the singly-generated invariant subspace and define two types of invariant subspaces of $H^2(D^n)$. Then, we give a characterization of these types invariant subspaces in view of the Beurling-Lax-Halmos Theorem.
References
[1] W. Rudin, Function Theory in Polydisks, W.A.Benjamin, Inc., New York-Amsterdam, 1969.
October 25, 2018
Restriction operator II
Sonmez Sahutoglu (University of Toledo)
Abstract. This is an introductory talk about the restriction operator on Bergman spaces and its relation to Hankel operators. This is joint work with Debraj Chakrabarti.
October 18, 2018
Laurent series
Debraj Chakrabarti (Central Michigan University)
Abstract. We consider Laurent series of holomorphic functions on Reinhardt domains and obtain a condition for the density of Laurent polynomials in Banach spaces of holomorphic functions on Reinhardt domains. Using the method of proof for this condition, we obtain a generalization of a result of Sibony on the behavior of holomorphic functions smooth up to the boundary on the Hartogs triangle.
October 4, 2018
Restriction operator
Sonmez Sahutoglu (University of Toledo)
Abstract. This is an introductory talk about the restriction operator on Bergman spaces and its relation to Hankel operators. This is joint work with Debraj Chakrabarti.
September 27, 2018
Berezin symbols and Borel summability
Ulaş Yamanci (Suleyman Demirel University (Turkey), currently visiting University of Toledo)
Abstract. We will prove in terms of so-called Berezin symbols some theorems for Borel summability method for sequences and series of complex numbers, and give a new Tauberian type theorem for Borel summability.
September 20, 2018
Complex symmetry of composition/weighted composition operators on the unit ball II
Uthpala Nawalage (University of Toledo)
Abstract. We will define complex symmetry of bounded operators on a complex Hilbert space and discuss conditions for a composition/weighted composition operator to be complex symmetric with respect to a specific conjugation on Hardy and Bergman spaces.
September 13, 2018
Complex symmetry of composition/weighted composition operators on the unit ball I
Uthpala Nawalage (University of Toledo)
Abstract. We will define complex symmetry of bounded operators on a complex Hilbert space and discuss conditions for a composition/weighted composition operator to be complex symmetric with respect to a specific conjugation on Hardy and Bergman spaces.
September 6, 2018
A construction of inner functions on weighted Hardy spaces
Trieu Le (University of Toledo)
Abstract. An inner function is a bounded holomorphic function on the unit disk whose values have modulus one almost everywhere on the unit circle. Inner functions play an important role in function theory and operator theory. Recent work of Beneteau et al. studies inner functions in the more general setting of weighted Hardy spaces. Motivated by their results, we shall discuss a construction of such inner functions.
August 30, 2018
ORGANIZATIONAL MEETING

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