2024 HCMC Focus Program on Function Theory, Operator Theory and Applications
KIAS 8101, 8309
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| Titles/Abstracts | Home > Titles/Abstracts |
â—¾Schedule:
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Friday March 08 [KIAS 8101; 3:00pm – 5:00pm] |
Short course on composition operators on RKHS Boo Rim Choe (Korea Univ) Hyungwoon Koo (Korea Univ) |
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Friday March 22 [KIAS 8101; 3:00pm – 5:00pm] |
Short course on operator algebras Hun Hee Lee (Seoul Nat’l Univ) Ja A Jeong (Seoul Nat’l Univ) |
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Friday April 05 [KIAS 8101; 3:00pm – 5:00pm] |
Short course on quantum information theory Kyung Hoon Han (Univ of Suwon) Seung-Hyeok Kye (Seouul Nat’l Univ) |
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Friday April 26 [KIAS 8101; 3:00pm – 5:00pm] |
Short course on vector-valued function theory In Sung Hwang (Sungkyunkwan Univ) Sumin Kim (Sungkyunkwan Univ) |
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Friday May 03 [KIAS 8101; 2:00pm – 5:00pm] |
Strategy Formulation Workshop on the Invariant Subspace Problem Il Bong Jung (Kyungpook Nat’l Univ) Jaeseong Heo (Hanyang Univ) Hyoung Joon Kim (Seoul Nat’l Univ) |
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Friday May 17 [KIAS 8309; 1:00pm – 3:00pm] |
Tutorial: Spectral theory In Hyoun Kim (Incheon Nat’l Univ) Dong-O Kang (Chungnam Nat’l Univ) |
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Friday May 17 [KIAS 8101; 3:00pm – 5:00pm] |
Short course on weighted composition operators Inyoung Park (Ehwa Womans Univ) Koeun Choi (Ehwa Womans Univ) |
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Friday May 31 [KIAS 8101; 1:00pm – 5:00pm] |
Hot topics workshop on RKHS and operators Ji Eun Lee (Sejong Univ) Yoenha Kim (Ajou Univ) Mee-Jung Lee (Kookmin Univ) Jongrak Lee (Sungkyunkwan Univ) |
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June 20-22 [KIAS 8101; 9:00am – 6:00pm] |
2024 HCMC International Workshop on Function Theory, Operator Theory and Applications |
â—¾Titles and Abstracts
Boo Rim Choe (Korea Univ): The Shapiro-Sundberg Problems
Abstract: The Shapiro-Sundberg Problems in the title refers to the two problems, concerning composition operators acting on the Hardy space, raised by Shapiro and Sundberg in 1990. One is the path component problem and the other is the compact difference problem. For the past three decades these problems have been studied by many experts. I, together with collaborators, have quite recently succeeded in obtaining two different solutions to the compact difference problem, but the path component problem is still open. In this talk I will present background, motivation, progress, our contributions and further problems related to the Shapiro-Sundberg problems.
Hyungwoon Koo (Korea Univ): Composition operators in several variables
Abstract: In this talk, we survey the progress in the research on composition operators in several variables and talk about unsettled problems. We consider Bergman, Hardy and holomorphic Sobolev spaces over several domains: the unit ball, the unit polydisk, strictly pseudo-convex domains and the finite type domains. We focus on the case when the symbol map is smooth up to the boundary and discuss several jump phenomena. We also give some recent results on the unit polydisk.
Hun Hee Lee (Seoul Nat’l Univ): Lie group representations and operators in abstract harmonic analysis
Abstract: Lie groups are central objects in many areas of pure mathematics. In this talk we focus on its connection to functional analysis through its representation theory. When the group is non-compact we often witness a rich structure of operators acting on infinite dimensional Hilbert spaces. We will examine various operators appearing in the study of weighted Fourier algebras on Lie groups including the "ax+b" group.
Ja A Jeong (Seoul Nat’l Univ): Z-stability of simple crossed products of commutative C*-algebras by Hilbert bimodules
Abstract: We show that the Cuntz--Pimsner algebras associated to minimal homeomorphisms twistd by line bundles, along with their orbit-breaking subalgebras, are Z-stable whenever the underlying dynamical system has mean dimension zero. This entails that this class is classified by the Elliott invariant. If time permits, we will also discuss a recursive homogeneous algebra structure of the orbit-breaking subalgebras.
Kyung Hoon Han (Univ of Suwon): Duality and geometry in quantum states and positive maps
Abstract: We will survey the basic notions in entanglement theory to include the following:
- separable states / entangled states
- Schmidt number
- block positive matrices
- Choi-matrices
- positive partial transpose
- k-positive maps / k-superpositive maps
We focus on mathematical aspects without physical meaning, in particular, their duality and geometric aspects.
Seung-Hyeok Kye (Seouul Nat’l Univ): Choi matrices; infinite dimensional cases and examples
Abstract: The Choi matrix of a linear map between matrix algebras gives rise to a block matrix, which is a bi-partite states with normalization if and only if the original map is completely positive. In this talk, we consider possible variants of Choi matrices, and infinite dimensional analogues as follows: For a given normal completely bounded map between von Neumann factor which acts on a Hilbert space H with a separating and cyclic vector, we define the linear map which plays ampliation. When this map extends to the whole B(H), we associate a bounded operator and a trace class operator, which play the role of Choi matrix. In the case of type I factor, we use this construction to characterize the infinite dimensional analogues of Schmidt numbers, entanglement breaking maps and superpositive maps. We also give concrete examples to distinguish k-positivity. This is infinite dimensional analogues of Tomiyama maps, which play important roles to analyze isotropic state and Werner states. This talk is based on the papers:
- Choi matrices revisited, J. Math. Phys. 63 (2022), 092202.
- Choi matrices revisited. II, J. Math. Phys. 64 (2023), 102202. (with K. H. Han)
- Infinite dimensional analogues of Choi matrices, arXiv 2311.18240. (with K. H. Han and E. Stormer)
- Positive maps on the infinite dimensional type I factors (in preparation)
In Sung Hwang (Sungkyunkwan Univ): Operator-valued Hardy spaces
Abstract: Classically, the problem asks whether for each function h on the unit disk, there exists a “boundary function” bh on the unit circle such that the mapping bh → h is an isometric isomorphism between Hardy spaces of the unit circle and the unit disk with values in some Banach space. For the case of bounded linear operator-valued functions, we construct a Hardy space of the unit circle such that its elements are SOT measurable, and their norms are integrable: indeed, this new space is isometrically isomorphic to the Hardy space of the unit disk via a “strong Poisson integral”
Sumin Kim (Sungkyunkwan Univ): An operator-valued version of Abrahamse's Theorem
Abstract: In this talk, we discuss vector-valued function theory on strong L^p-spaces, a meromorphic pseudo-continuations of bounded type, and operator poles. Beyond developing this theory, as an interplay between function theory and operator theory, we formulate and prove an operator-valued version of Abrahamse's Theorem. This is the most interesting affirmative answer to Halmos' Problem 5, which can be stated as: Is every subnormal Toeplitz operator either normal or analytic ?
Il Bong Jung (Kyungpook Nat’l Univ): Some aspects of the invariant subspace problem
Abstract: Let X be a complex Banach space and let B(X) be the algebra of all bounded operators on X. A subspace M of X is called an invariant subspace for T in B(X) if TM ⊆
M. Many problems in analysis are related to the classification of the invariant subspaces for bounded operators on X. In particular, there has been considerable interest in the invariant subspace problem: does there exist a nontrivial invariant subspace M for a given T in B(X)? In 1950, N. Aronszajn proved that every compact operator on a Hilbert space has a nontrivial invariant subspace, and he communicated his proof to J. von Neumann, who replied that he proved the same theorem in the early of 1930s (but had never published it). This was the beginning of invariant subspace problem. In 1987 Enflo obtained a Banach space X and an operator T in B(X) that does not have any nontrivial invariant subspace. For the case of Hilbert spaces, this problem still remains unsolved. In this lecture we will discuss the studies on the invariant subspace problem with several topics by era.
Jaeseong Heo (Hanyang Univ): Transitive algebra problem and von Neumann algebras
Abstract: In this talk, we consider Kadison's transitive algebra problem which is the infinite dimensional analogue of Burnside's theorem. We review some partial results on the transitive algebra problem and discuss the invariant subspace problem relative to von Neumann algebras. Finally, we introduce some trials for finding non-trivial transitive algebras.
Hyoung Joon Kim (Seoul Nat’l Univ): Hyperinvariant subspaces for operators having a part
Abstract: Let T be a nonscalar operator of the upper triangular form. It is well known that if both the diagonal entries are normal operators, then T has a nontrivial hyperinvariant subspace. In this talk, we will consider the hyperinvariant subspace problem of the upper triangular operator when the upper left corner is a nonscalar normal operator.
In Hyoun Kim (Incheon Nat’l Univ): Isometric and symmetric commuting d-tuples of Banach space operators
Abstract: Generalising the definition to commuting -tuples of operators, a number of authors have considered structural properties of -isometric, -symmetric and -isosymmetric commuting -tuples in the recent past. In this talk, we attempt to take the mystique out of this extension and show how a large number of these properties follow from the more familiar arguments used to prove the single operator version of these properties.
Dong-O Kang (Chungnam Nat’l Univ): More consideration on kernels of block Hankel operators on Hardy spaces
Abstract: We first take a look back on some known facts about kernels of block Hankel operators on the vector-valued Hardy spaces and then introduce some problems that are being considered currently, or, can be considered later.
Inyoung Park (Ehwa Womans Univ): Linear connection between composition operators on the Hardy space
Abstract: We consider the space of all composition operators, acting on the Hardy space over the unit disk, in the uniform operator topology. We obtain a characterization for linear connection between composition operators. As applications, we obtain the operator norm of the difference of composition operators and prove that the set of all compact composition operators is a polygonally connected component. This result has made some progress in the known fact that all compact composition operators are properly contained in a path connected component.
Koeun Choi (Ehwa Womans Univ): Difference of weighted composition operators on the weighted Bergman spaces
Abstract: In this talk, we introduce basic concepts about weighted composition operators. We also survey the progress in the research on difference of weighted composition operators in one or several variables. Finally, we consider the differences of weighted composition operators with (un)bounded and (non)holomorphic weights acting on weighted Bergman spaces. Thus, we obtain complete characterizations in terms of Carleson measures and reproducing kernel thesis is added to the characterizations.
Ji Eun Lee (Sejong Univ): On Toeplitz operators on the Newton space
Abstract: In this talk, we study the Newton space N^2(P) which has the Newton polynomials as an orthonormal basis. We first explore some relations between the orthonormal basis {z^n} of the Hardy space H^2(D) and the orthonormal basis {N_n} of the Newton space N^2(P). Also, we investigate the truncated Toeplitz operator on N^2(P) and we examine the product of N_m and N_n and the orthogonal projection P of overline{N_n}N_m. Finally, we find the matrix representation of Toeplitz operators with respect to such an orthonormal basis on the Newton space N^2(P). Furthermore, we investigate properties of expansive and contractive Toeplitz operators with analytic and co-analytic symbols on a Newton space. (This is jointed work Eungil Ko and Jongrak Lee.)
Yoenha Kim (Ajou Univ): On square roots of self-adjoint weighted composition operators on the Hardy space
Abstract: In this talk, we study square roots of self-adjoint weighted composition operators on the Hardy space. We investigate several properties of such square roots. We also provide that square roots of self-adjoint weighted composition operators may be other, nonself-adjoint weighted composition operators and have nontrivial invariant subspaces. This is a joint work with Sungeun Jung and Eungil Ko.
Mee-Jung Lee (Kookmin Univ): On properties of C-normal operators
Abstract: The class of C-normal operator includes C-symmetric operators and C-skew-symmetric operators. In this talk, We study various properties of C-normal operators and consider the local spectral properties of C-normal operators. Some more associated for this class of operators are obtained. (This is jointed work Eungil Ko and Ji Eun Lee.)
Jongrak Lee (Sungkyunkwan Univ): Hyponormality of Toeplitz operators over the bidisk
Abstract: In this talk, we consider the hyponormality of Toeplitz operators over the bidisk. First, we study hyponormal Toeplitz operators on the Drury-Arveson spaces. In particular, we will discuss whether the sum of two symbols that are hyponormal in Hardy space is hyponormal. Next, we consider the hyponormality of Toeplitz operators on the weighted Bergman spaces over the bidisk.