Complex Analysis Seminar

Thursday 4pm-5pm at UH 4010

To give a talk in this seminar please write to Sonmez Sahutoglu (sonmez.sahutoglu @ utoledo.edu) or
Zeljko Cuckovic (zcuckovi @ math.utoledo.edu)

Use of Firefox is recommended for this page.
SPRING 2010
April 22, 2010
John P. D'Angelo (University of Illinois- Urbana Champaign)
Complex variable analogues of Hilbert's 17th Problem

Abstract: Hilbert's 17th problem, solved affirmatively by Artin in 1925, asks whether a nonnegative polynomial in several real variables is necessarily a sum of squares of rational functions. In this talk I describe several analogues for Hermitian symmetric polynomials. Proof techniques use the Bergman projection and simple facts about compact operators. If time permits I will give some applications to CR Geometry.
April 15, 2010
Ozgur Martin (Bowling Green State University)
Disjoint hypercyclic and supercyclic composition operators

Abstract: In 2007, Bernal, Bonilla, and Calderon showed that for most sequences of composition operators induced by automorphic symbols of a simply connected domain the notions of hypercyclicity and supercyclicity coincide. In this talk, we will show that this result holds for all sequences and also can be generalized to the notion of disjointness in hypercyclicity introduced by Bernal (2007) and also independently by Bes and Peris (2007).
April 8, 2010

Abstract: One of the major questions in the theory of Toeplitz operators on the Bergman space over the unit disk D in the complex plane ℂ is a complete description of the commutant of a given Toeplitz operator, that is the set of all Toeplitz operators that commute with it. In a previous paper the first author (I. Louhichi, Powers and roots of Toeplitz operators, Proc. Amer. Math. Soc. 135, (2007), 1465-1475) obtained a complete description of the commutant of Toeplitz operator $T$ with any quasihomogeneous symbol $\phi(r)e^{ip\theta}, p > 0$ in case it has a Toeplitz p-th root $S$ with symbol $\psi(r)e^{i\theta},$ namely, commutant of $T$ is the closure of the linear space generated by powers $S^n$ which are Toeplitz. But the existence of p-th root was known until now only when $\phi(r) = r^m , m \geq 0.$ In this paper we will show the existence of p-th roots for a much larger class of symbols, for example, it includes such symbols for which $\phi(r)=\sum_{i=1}^k r^{a_i}(\ln r)^{b_i} , 0 \leq a_i , b_i$ for all $1\leq i\leq k.$
April 1, 2010
Mehrdad Simkani (University of Michigan--Flint)
Green's potential and the degree of rational approximation

Abstract: We will see a connection between the geometric rate of convergence of some sequence of rational functions and the minimum value of Green's potential of some measure, where the minimum is taken over the natural (meromorphic) boundary of the limiting function. The measure in question reflects the limiting behavior of the poles of the approximating rational functions. The approximation takes place over a closed subset of the extended complex plane. This talk relates rational approximation to the classical potential theory over the extended complex plane.
March 4, 2010

Abstract: We will discuss the interplay between Berezin's operator calculus and the higher order Schwarz-Pick lemma in the study of higher order Caratheodory-Reiffen metric. This is part of a joint work with Y. Pan.
February 25, 2010
Alexander Izzo (Bowling Green State University)
Function algebras invariant under group action

Abstract: We will answer a question raised by Ronald Douglas in connection with his work on a conjecture in operator theory due to William Arveson. Let $S$ denote the unit sphere in $ℂ^n.$ If $A$ is a function algebra on $S$ that contains the ball algebra $A(S)$ and whose maximal ideal space is $S$, and if $A$ is invariant under the action of the $n$-torus on $S$, does it follow that $A = C(S)$? When $n=1$, Wermer's maximality theorem gives immediately that the answer is yes. Surprisingly, in higher dimensions the answer depends on the dimension. The proof is related to a peak point theorem of John Anderson and the speaker and counterexamples to the peak point conjecture due to Richard Basener and the speaker.
We will also present a related result of a more general nature concerning function algebras that are invariant under a transitive group action.
February 18, 2010
Raul E. Curto (University of Iowa)
Spectral pictures of 2-variable weighted shifts

Abstract: In joint work with Jasang Yoon, we study the spectral pictures of (jointly) hyponormal 2-variable weighted shifts with commuting subnormal components. By contrast with all known results in the theory of (single and 2-variable) weighted shifts, we show that the Taylor spectrum can be disconnected. We do this by obtaining a simple sufficient condition that guarantees disconnectedness, based on the norms of the horizontal slices of the shift. We also show that for every $k\geq 1$ there exists a $k$-hyponormal 2-variable weighted shift whose horizontal and vertical slices have 1- or 2-atomic Berger measures, and whose Taylor spectrum is disconnected.
We use tools and techniques from multivariable operator theory, from our previous work on the Lifting Problem for Commuting Subnormals, and from the groupoid machinery developed by the author and P. Muhly to analyze the structure of $C^{\ast }$-algebras generated by multiplication operators on Reinhardt domains. As a by-product, we show that, for 2-variable weighted shifts, the Taylor essential spectrum is not necessarily the boundary of the Taylor spectrum.
February 4, 2010
Crystal Anne Zeager (University of Michigan)
Invariant metrics

Abstract: Biholomorphically invariant metrics can be used as a tool to study holomorphic functions. For example, one proof that the ball and polydisk are not biholomorphic uses the boundary behavior of the Bergman metric on those domains. Also, Fefferman's results on extending holomorphic functions to the boundary rely on understanding geodesics of the Bergman metric near the boundary. I will discuss how del-bar theory can be used to study invariant metrics, and also some joint work with Lina Lee and Hyunsuk Kang.
January 28, 2010

Abstract: Given a complex polynomial in one variable, I will discuss the centralizer of its Toeplitz operator in the algebra generated by Toeplitz operators of bounded functions on the Bergman space of the unit disc.
January 21, 2010
N. V. Rao (University of Toledo)
Range of Berezin transform

Abstract: Let $dA = \frac{dxdy}{\pi}$ denote the normalized Lebesgue area measure on the unit disk $D$ and $u$, a summable function on $D.$ The Berezin transform of $u$ is defined as $ B(u)(z) = \int_{D} u(\xi)\frac{(1-|z|^2)^2}{|1-\xi \overline z|^4} dA(\xi)$. Ahern described all the possible functions of the form $B(u)$ for which $B(u)(z) = f(z) \overline{g(z)}$ where both $f, g$ are holomorphic in $D.$ The natural next question was to describe all functions in the range of Berezin Transform which are of the form $\sum_{j=1}^Nf_j \overline{g_j}$ where $f_j , g_j$ are all holomorphic in $D.$ We shall describe all $u$ for which $B(u) = \sum_{j=1}^Nf_j \overline{g_j}$ where $f_j , g_j$ are all holomorphic in $D.$ Further we give very simple proof of the result of Ahern.
FALL 2009
December 10, 2009

Abstract: We establish some basic properties of BMO${}^p$ for $p\geq 1$ and complete the characterization of bounded and compact Toeplitz operators with BMO${}^1$ symbols on the Segal-Bargmann space of Gaussian square-integrable entire functions on $ℂ^n$. This is a joint work with L. A. Coburn and J. Isralowitz.
December 3, 2009
Yun-Su Kim (University of Toledo)
Algebraic elements and invariant subspaces

Abstract: If T is a contraction, then
(Case 1) T is a completely non-unitary contraction with a non-trivial algebraic element, or
(Case 2) T is a completely non-unitary contraction without a non-trivial algebraic element, that is, every non-zero element in H is transcendental with respect to T, or
(Case 3) T is not completely non-unitary.
We discuss the invariant subspace problem for operators of (Case 1) or (Case 3).
November 19, 2009

Abstract: I will present part of the paper by Arazy and Englis with the same title, published in the Ann. Inst. Fourier (Grenoble), 2001.
November 12, 2009

Abstract: T. Carleman (1927) showed that for any continuous function $f$ on the real line and any strictly positive and continuous error function $e$, there exists an entire function h such that $| h(x)-f(x) | \leq e(x)$ for all real $x$. We show how this result can be extended to approximation on totally real sets in Stein manifolds.
November 5, 2009
Sonmez Sahutoglu (University of Toledo)
On regularity of Bergman projections

Abstract: The following question will be discussed: Is the Bergman projection ``regular" on smooth bounded pseudoconvex domains in $ℂ^n$? This question has been answered by several authors in $ℂ^2.$ A partial answer will be given on domains in $ℂ^n$ for $n\geq 3.$
October 29, 2009

Abstract: In this talk, an affirmative answer is given to one of the questions, posed by P. R. Halmos.
October 22, 2009

Abstract: In this talk, an affirmative answer is given to one of the questions, posed by P. R. Halmos.
October 15, 2009

Abstract: Let $f$ be a function that is continuous on the closed polydisc $\overline{D}^n$. I will discuss the compactness of the Toeplitz operator $T_f$ and of the Hankel operator $H_f$ on the Bergman space $A^2(D^n)$.
October 8, 2009
Yunus Zeytuncu (Ohio State University)
$L^p$ mapping properties of weighted Bergman projections

Abstract: We give examples of weights, on the unit disc, for which the weighted Bergman projection is only bounded on $L^p$ for a finite range values of $p$. This range is more than just $p=2$ but smaller than the interval $(1,\infty)$.
October 2, 2009 (SPECIAL TIME & PLACE: Friday at 2pm, UH3008)

Abstract: I will introduce few types of extension problems for holomorphic functions on complex manifolds and explain relationships between them.
September 24, 2009

Abstract: In this talk plurisubharmonic hull, existence of analytic discs, and the ∂̅-problem will be defined. Furthermore, some relations between them will be mentioned.
September 17, 2009
Malgorzata Marciniak (University of Toledo)
Hartogs type extension phenomena

Abstract: I will introduce few types of extension problems for holomorphic functions on complex manifolds and explain relationships between them.
September 10, 2009
ORGANIZATIONAL MEETING
Powered by MathJax