| Abstract: It is shown that lattice of invariant subspace of the operator of multiplication by a cyclic element of a Banach algebra consists of closed ideals of this algebra. As a application, lattice of invariant subspaces of composition operator acting on Hardy space, whose symbol is parabolic non-automorpshism, is found. In particular each invariant subspace always consists of closed span of a set of eigenfunctions. |
April 18, 2013
| Abstract: We will show that every separable infinite-dimensional Frechet space supports an arbitrarily large finite disjoint mixing collection of operators. We will also show the existence of d-universal functions of exponential type zero for arbitrary finite tuples of pairwise distinct translation operators. |
April 11, 2013
| Abstract: Invariant subspaces are a natural topic in linear algebra and operator theory. In some rare cases, the restrictions of operators to different invariant subspaces are unitarily equivalent, such as certain restrictions of the unilateral shift on the Hardy space of the disk. A composition operator with symbol fixing 0 has a nested sequence of invariant subspaces, and if the symbol is linear fractional and extremally noncompact, the restrictions to these subspaces all have the same norm and spectrum. Despite this evidence, we will use semigroup techniques to show many cases where the restrictions are still not unitarily equivalent. |
April 4, 2013
Thomas Garrity
(Williams College)
Equivalences of higher codimensional CR structures: a merging of moving frames and some algebraic geometry
Equivalences of higher codimensional CR structures: a merging of moving frames and some algebraic geometry
| Abstract: The best examples of CR structures are real submanifolds of a complex space. The natural notion of equivalence captures the idea of when one of these real submanifolds can be mapped to another under a biholomorphic change of coordinates. CR structures are of interest in several complex variables, partial differential equations and differential geometry. The behavior of CR structures depends heavily on their codimension (which for real submanifolds of a complex space just means the real codimension of the manifold with respect to the ambient complex space). We will see that the equivalence problem for CR structures of codimension three or greater has a particularly rich structure. In particular, to solve the equivalence problem for CR structures of codimension three or greater, we will need to use two big machines: geometric invariant theory of algebraic geometry and Cartan's method of moving frames. This talk will provide an introduction to CR geometry, with an emphasis on higher codimensional CR equivalence problems. We will close with a number of open questions. |
March 28, 2013
Amila Appuhamy
(University of Toledo)
Compactness of Hankel Product $H^*_{\psi} H_{\varphi}$ on the weighted Bergman space on the unit bidisk (Part III)
Compactness of Hankel Product $H^*_{\psi} H_{\varphi}$ on the weighted Bergman space on the unit bidisk (Part III)
| Abstract: This is a continuation of the previous two talks. |
March 21, 2013
| Abstract: The proof for the existence of a holomorphic, conformal, bijective map from any open, simply connected, proper subset of the complex plane will be discussed briefly. Then, I will discuss two results: Let the mapping $T$ be defined by $T(z)=\frac{R(z-z_0)}{R^2-\overline{z_0}z}$. Then for $|z_0|<{R}$, $T$ maps the open disk $D(z_0,R) $ bijectively to the open unit disk centered at the origin such that $T(z_0)=0$. Also, I will show that given $\{f_n\}$, a sequence of uniformly bounded holomorphic functions, with the domain being the open unit disk centered at the origin, then a subsequence of $\{f_n\}$ converges uniformly on compact subsets to an analytic function on the open disk $D(0,1)$. |
February 28, 2013
| Abstract: The WAT conjecture is the statement that every composition operator on the Hardy space $H^2$, is weakly asymptotically Toeplitz, or WAT. It can be shown that the conjecture is true if a certain sequence of Fourier coefficients associated to the symbol of the operator converges to zero. In this talk, I give a short survey of the known results, along with a new contribution involving improvement, sharpening, and extension of those results. |
February 21, 2013
Amila Appuhamy
(University of Toledo)
Compactness of Hankel Product $H^*_{\psi} H_{\varphi}$ on the weighted Bergman space on the unit bidisk (Part II)
Compactness of Hankel Product $H^*_{\psi} H_{\varphi}$ on the weighted Bergman space on the unit bidisk (Part II)
| Abstract: We are interested in the Hankel product $H^*_{\psi} H_{\varphi}$ on the weighted Bergman space $A^2(\mathbb{D}^2, dV_{\alpha, \beta})$, where \[dV_{\alpha, \beta}=(1+\alpha)(1+\beta)(1-|z|^2)^\alpha(1-|w|^2)^\beta dA(z)dA(w)\] and $\alpha, \beta\in \mathbb{N}$. For symbols $\varphi, \psi\in C(\overline{\mathbb{D}^2})$, I will show a necessary condition for this operator to be compact. |
February 21, 2013
Amila Appuhamy
(University of Toledo)
Compactness of Hankel Product $H^*_{\psi} H_{\varphi}$ on the weighted Bergman space on the unit bidisk (Part II)
Compactness of Hankel Product $H^*_{\psi} H_{\varphi}$ on the weighted Bergman space on the unit bidisk (Part II)
| Abstract: We are interested in the Hankel product $H^*_{\psi} H_{\varphi}$ on the weighted Bergman space $A^2(\mathbb{D}^2, dV_{\alpha, \beta})$, where \[dV_{\alpha, \beta}=(1+\alpha)(1+\beta)(1-|z|^2)^\alpha(1-|w|^2)^\beta dA(z)dA(w)\] and $\alpha, \beta\in \mathbb{N}$. For symbols $\varphi, \psi\in C(\overline{\mathbb{D}^2})$, I will show a necessary condition for this operator to be compact. |
February 14, 2013
| Abstract: The investigation of a question whether norm-attaining operators on a Banach space are dense has been parallel to the study of the denseness of numerical-radius attaining operators. We show how to obtain the Bishop-Phelps-Bollobas theorem for $\ell_1(\mathbb{C})$, a quantitative strengthening of the Bishop-Phelps theorem. Then we apply these constructions to show that $\ell_1(\mathbb{C})$ is one of the examples of spaces with the Bishop-Phelps-Bollobas property for numerical radius. (This is a joint work with Antonio J. Guirao) |
February 7, 2013
Debraj Chakrabarti
(Tata Institute of Fundamental Research)
Condition R and Proper Holomorphic mapping of piecewise smooth domains
Condition R and Proper Holomorphic mapping of piecewise smooth domains
| Abstract: We study the boundary regularity of proper holomorphic maps between certain classes of piecewise smooth domains including product domains. This leads to results on the biholomorphic inequivalence of such domains. This is joint work with Kaushal Verma. |
January 31, 2013
Amila Appuhamy
(University of Toledo)
Compactness of Hankel product $H^*_{\psi} H_{\varphi}$ on the weighted Bergman space on the unit bidisk
Compactness of Hankel product $H^*_{\psi} H_{\varphi}$ on the weighted Bergman space on the unit bidisk
| Abstract: We are interested in the Hankel product $H^*_{\psi} H_{\varphi}$ on the weighted Bergman space $A^2(\mathbb{D}^2, dV_{\alpha, \beta})$, where $\alpha, \beta\in \mathbb{N}$ and \[dV_{\alpha, \beta}=(1+\alpha)(1+\beta)(1-|z|^2)^\alpha(1-|w|^2)^{\beta} dA(z)dA(w).\] For some special symbols $\varphi, \psi\in L^\infty(\mathbb{D}^2)$, I will show a necessary condition for this operator to be compact. |
January 24, 2013
John Clifford
(University of Michigan --Dearborn)
Invertible Composition Operators: The product of a composition operator with the adjoint of a composition operator
Invertible Composition Operators: The product of a composition operator with the adjoint of a composition operator
| Abstract: I will introduce the two operators formed by multiplying a composition operator $C_{\varphi}$ with the adjoint of a composition operator $C^{*}_{\psi}$ on the Hardy space $H^2(\mathbb{D})$. I will characterize when the operator $C_{\varphi}C^{*}_{\psi}$ is invertible, isometric, and unitary, respectively. If one of the inducing maps $\varphi$ or $\psi$ is univalent, I will completely characterize when $C^{*}_{\psi}C_{\varphi}$ is invertible or unitary. |
January 17, 2013
Friedrich Haslinger
(University of Vienna)
Spectral properties of the $\overline{\partial}$-Neumann operator
Spectral properties of the $\overline{\partial}$-Neumann operator
| Abstract: We discuss spectral properties of the $\overline{\partial}$-Neumann operator. The spectrum of the $\overline{\partial}$-Neumann Laplacian on the Fock space is explicitly computed. It turns out that it consists of positive integer eigenvalues each of which is of infinite multiplicity. Spectral analysis of the $\overline{\partial}$-Neumann Laplacian on the Fock space is closely related to Schrodinger operators with magnetic field and to the complex Witten-Laplacian. |
FALL 2012
December 6, 2012
| Abstract: For various $L^p$-spaces ($1\leq p<\infty$) we investigate the minimum number of complex-valued functions needed to generate an algebra dense in the space. The results depend crucially on the regularity imposed on the generators. For $\mu$ a positive regular Borel measure on a compact Hausdorff space there always exists a single bounded measurable function that generates an algebra dense in $L^p(\mu)$. However, the situation is very different when the generators are required to be continuous or smooth. The most interesting case turns out to be that of continuous generators. This is joint work with Bo Li. |
November 29, 2012
| Abstract: The Segal-Bargmann space $H^2(\mathbb{C}^n)$ consists of entire functions on $\mathbb{C}^n$ that are square integrable with respect to the Gaussian measure. I will discuss recent results on the boundedness and spectral properties of composition operators between two such spaces. |
November 15, 2011 Joint with Shoemaker Lecture Series Special Place: ST 131
| Abstract: Monge-Ampere equations are second order partial differential equations whose leading term is the determinant of the Hessian of the unknown function. They are arguably the most natural fully non-linear equations, and occur frequently in geometry and applied science. They can be real or complex, depending on whether the variable is real or complex, in which case the unknown is required respectively to be convex or plurisubharmonic. The theory of Monge-Ampere equations has a long and distinguished history, with contributions from many of the leading geometers and analysts of our time. This lecture series is devoted to a survey of some aspects of this theory. In particular, we shall discuss examples of such equations and their applications to geometry; variational principles; notions of generalized solutions for both the real and the complex case; and a priori estimates. We shall also take the opportunity to discuss some major problems in Kahler geometry and partial differential equations which are currently attracting a lot of interest. |
November 8, 2012
| Abstract: I will give a brief introduction to complex manifold, Kahler manifold and Monge-Ampere equations. This is a preparatory talk for Duong H. Phong's Shoemaker's lecture next week. |
November 1, 2012
Zeljko Cuckovic
(University of Toledo)
Mapping properties of Toeplitz operators on strongly pseudoconvex domains 2
Mapping properties of Toeplitz operators on strongly pseudoconvex domains 2
| Abstract: Several years ago, McNeal and I proposed to study a special class of Toeplitz operators on strongly pseudoconvex domains that improve integrability. Very recently Abate, Raissy and Saracco generalized our result to Toeplitz operators whose symbols are Carleson measures. I plan to give an overview of both results. This is the second part of my previous talk. |
October 25, 2012
Sonmez Sahutoglu
(University of Toledo)
A short introduction to the $\overline{\partial}$-Neumann Problem 2
A short introduction to the $\overline{\partial}$-Neumann Problem 2
| Abstract: This is the second part of my previous talk. We will continue to discuss the basic set-up for the $\overline{\partial}$-Neumann problem. |
October 18, 2012
Mehmet Celik
(University of North Texas at Dallas)
Compactness of the $\overline{\partial}$-Neumann operator and of commutators of the Bergman projection with continuous functions
Compactness of the $\overline{\partial}$-Neumann operator and of commutators of the Bergman projection with continuous functions
| Abstract: We show that on a bounded pseudoconvex domain compactness of the $\overline{\partial}$-Neumann operator on square integrable forms, compactness of commutator operators (of the Bergman projection with functions continuous on $\overline{\Omega}$) on square integrable $\overline{\partial}$-closed forms, and compactness of the canonical solution operator of the non-homogeneous $\overline{\partial}$-equation on square integrable $\overline{\partial}$-closed forms are equivalent. (This is a joint work with Sonmez Sahutoglu.) |
October 11, 2012
Sonmez Sahutoglu
(University of Toledo)
A short introduction to the $\overline{\partial}$-Neumann Problem 1
A short introduction to the $\overline{\partial}$-Neumann Problem 1
| Abstract: The basic set-up for the $\overline{\partial}$-Neumann problem will discussed. This is a preparatory talk for Mehmet Celik's talk next week. |
October 4, 2012
Malgorzata (Margaret) Stawiska
(Mathematical Reviews)
A characterization of polynomials in holomorphic dynamics in one complex variable using potential theory
A characterization of polynomials in holomorphic dynamics in one complex variable using potential theory
| Abstract: In 1960's Hans Brolin initiated systematic application of potential-theoretic methods in the dynamics of holomorphic polynomials in one variable. Among other things, he proved the now-famous equidistribution theorem: for a polynomial $f$ of degree greater than 1 the preimages, under successive iterates of $f$, of a Dirac measure at an arbitrary point of the complex plane (except at most two so-called exceptional points) converge weakly to the equilibrium measure of the Julia set for $f$. In 1980's a similar result (about convergence of preimages of quite general probabilistic measures) was proved for a rational map $f$ of degree greater than 1. The limit measure obtained in this case (called the balanced measure) is also supported on the Julia set for $f$, but does not have to be its equilibrium measure. In fact, A.O. Lopes proved (using dynamical properties of Julia sets) that equality of these two measures (under suitable assumptions on $f$, also making precise the notion of the equilibrium measure for the Julia set) implies that $f$ is a polynomial. In this talk I present a proof of Lopes's theorem (under slightly weaker assumptions) using classical and weighted potential theory. It is joint work with Yusuke Okuyama from Kyoto Institute of Technology (Okuyama, Yusuke; Stawiska, Malgorzata: Potential theory and a characterization of polynomials in complex dynamics. Conform. Geom. Dyn. 15 (2011), 152-159). |
September 27, 2012
| Abstract: We will discuss some analogues of the Hilbert Nullstellensatz for Bergman spaces of smooth pseudoconvex domains and how to apply them to some questions involving Toeplitz operators. |
September 20, 2012
| Abstract: We will discuss some analogues of the Hilbert Nullstellensatz for Bergman spaces of smooth pseudoconvex domains and how to apply them to some questions involving Toeplitz operators. |
September 13, 2012
Zeljko Cuckovic
(University of Toledo)
Mapping properties of Toeplitz operators on strongly pseudoconvex domains 1
Mapping properties of Toeplitz operators on strongly pseudoconvex domains 1
| Abstract: Several years ago, McNeal and I proposed to study a special class of Toeplitz operators on strongly pseudoconvex domains that improve integrability. Very recently Abate, Raissy and Saracco generalized our result to Toeplitz operators whose symbols are Carleson measures. I plan to give an overview of both results. |
September 6, 2012
ORGANIZATIONAL MEETING
August 24, 2012 (Friday) in UH 4010
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Abstract:
This talk presents two independent parts of my Ph.D. thesis: |
