Injo Hur (Chonnam Nat’l Univ): 1D Schrodinger operators and de Branges spaces

Abstract: In this talk, we apply de Branges theory on (the spectral theory of) Schrodinger operators. More precisely, we first compare (old) de Branges spaces and Sub-Hardy spaces and then see the connection between them. After this, for 1D Schrodinger operators on $L^2(0,∞)$ defined by $H =−frac{d2}{dx2} +V(x)$, we show the one-to-one correspondence between (real-valued) potentials V and inner products on one specific de Branges space which does not depend on V as a set (and so which can be naturally expected)

Jaeseong Heo (Hanyang Univ): Free entropy and its application 

Abstract: We briefly review free probability theory and introduce the definition of free entropy. We examine applications of free entropy for some problems in von Neumann algebras. We discuss methods for estimating free entropy by calculating the volume of convex sets in Euclidean space.

Seonguk Yoo (Gyeongsang Nat’l Univ): Truncated Moment Problems with an Extension of Multiplication Operators

Abstract:  The moment problem is a fundamental mathematical problem with significant implications in probability theory, statistics, physics, and engineering. The application of this problem is vast, ranging from characterizing the distribution of a random variable to analyzing system response characteristics, image reconstruction, and solving problems in operator theory and optimization. The moment problem mathematically focuses on reconstructing the original probability distribution or measure from a given sequence of moments. To address the truncated moment problem, various techniques are employed, such as verifying the existence of a flat extension of the moment matrix, analyzing the properties of orthogonal polynomials corresponding to the moment sequence, and checking the commutativity of multiplication operators in the Hilbert space generated by the moment sequence. In this talk, we will explore an algorithm for verifying the commutativity of multiplication operators in a Hilbert space associated with a bivariate truncated moment sequence. We will also discuss the dimension stability of the Hilbert space.

Seung-Hyeok Kye (Seoul Nat’l Univ): Detecting Schmidt numbers of bi-partite quantum states

Abstract: Schmidt numbers of bi-partite states measure the degree of entanglement; the higher Schmidt number of a state is, the more entangled it is. It is very hard to determine the Schmidt numbers of bi-partite states. In fact, it is known to be NP hard, and we need Schmidt number witnesses in order to determine Schmidt numbers of bi-partite states. Such witnesses are Choi matrices of $k$-positive maps between matrix algebras and they detect Schmidt numbers through bilinear pairing between mapping spaces and tensor products. In this contexts, we begin the talk to introduce various kinds of positive maps between matrix algebras, and use Choi matrices and bi-linear pairing to explain what are Schmidt numbers and how witnesses detect Schmidt numbers. We explore recent criteria to determine if a given Hermitian matrix is Schmidt number $k$ witnesses or not, and use supporting hyperplanes for witnesses to determine Schmidt numbers. The main parts of this talk will be based on two papers with Kyung Hoon Han:

-- Global locations of Schmidt number witnesses, Phys. Rev. A 112 (2025), 032426. arXiv 2505.10288

-- Supporting hyperplanes for Schmidt numbers and Schmidt number witnesses, Open Syst. Inf. Dyn. 32 (2025), 2550008. arXiv 2506.03733

Young Joo Lee (Chonnam Nat’l Univ): Zero sums of dual Toeplitz products

Abstract:  In this talk, we consider dual Toeplitz operators acting on the orthogonal complements of the Bergman, Fock and Dirichlet spaces, and discuss recent progress on the problem of when a sum of several dual Toeplitz products equals zero.

Yun-Ho Kim (Sangmyung Univ): Elliptic problems with unbalanced growth and Hardy potential

Abstract: This talk is devoted to demonstrating existence, a-priori bounds and uniqueness of nontrivial solutions to nonlinear elliptic problems with unbalanced growth and Hardy potential. From a mathematical point of view, such elliptic problems with singular nonlinearities have some technical difficulties that arise in the absence of the compactness condition of Palais-Smale and the weak lower semicontinuity of the energy functional. To provide the existence result, we first overcome the lack of compactness of the Euler-Lagrange functional by applying the cut-off function method. Next, we present the $L^{infty}$-bound for any possible weak solution by exploiting the De Giorgi iteration method and a truncated energy technique. Finally,  we utilize the Lions type Concentration-Compactness principle to get the weak lower semicontinuity of the energy functional. By using this semicontinuity property and considering eigenvalue problems, we obtain the existence of the positive solution. Moreover, we get the uniqueness result by applying several versions of the discrete Picone's inequality as the main tool. 
 

Mingu Jung (Hanyang Univ): Group actions and the Radon-Nikodým property

Abstract: There exists a growing body of research in this direction, forming part of the emerging field of equivariant Banach space theory. From a categorical point of view, this area can be seen as an extension of the classical category of Banach spaces to an equivariant category, whose objects are Banach spaces endowed with continuous group actions and whose morphisms are equivariant linear operators. Equivariant Banach space theory draws upon the representation theory of group actions on Banach spaces—which is particularly well developed for compact groups—as well as on analytic group theory, which investigates the rigidity of such actions. Moreover, category-theoretic and homological methods have recently played an important role in the development of Banach space theory. In this talk, we will briefly discuss the group-equivariant analogue of the classical Radon–Nikodým property and several related properties.

◾Mini-Workshop on Hilbert Space Operators

Il Bong Jung (Kyungpook Nat’l Univ): Brownian-type operators: convergence of powers and stability

Abstract: In this talk, we introduce operators that are represented by upper triangular 2x2 block matrices having the (2,2)-entry X as a place holder for operators and satisfy some algebraic constraints. We call such operators "Brownian-type operators". These operators emerged from the study of Brownian isometries performed by Agler and Stankus via detailed analysis of the time shift operator of the modified Brownian motion process. If T is unitary (resp., isometric, normal, quasinormal, subnormal, hyponormal, etc.), then T is called B-unitary (resp., B-isometric, B-normal, B-quasinormal, B-subnormal, B-hyponormal, etc.). Our main goal of this study is to give criteria for the convergence of powers of a B-operator relative to three topologies, namely the weak operator topology, the strong operator topology and the operator norm topology. As a consequence, we obtain the complete characterizations of the stability of B-operators in terms of their entries relative to any of these topologies. It turns out that there are no strongly stable B-operators. In particular, we pay special attention to the case of B-subnormal operators.

Koeun Choi (Korea Univ): Schatten class difference of composition operators

Abstract: In this talk, we investigate the Schatten class membership of the difference of composition operators on the standard weighted Hilbert-Bergman space over the unit disk. Our main result is to characterize Schatten p-class differences of composition operators for the range p>2 and to provide a sufficient condition for the range p<2. Our characterization for p> 2 extends the known characterization for Hilbert-Schmidt differences. Our approach employs a discretization technique involving lattices and localized averaging over pseudohyperbolic disks. However, such an approach do not seem to work well for the range p<2 and whether the sufficient condition for p<2 is also necessary remains open.

Sumin Kim (Changwon Nat’l Univ): Composition operators on vector-valued Hardy spaces

Abstract: In this talk, we study the Poisson integral of vector-valued functions on the unit circle and composition operators on vector-valued function spaces. We establish norm estimates for composition operators on vector-valued Hardy spaces and describe several operator-theoretic properties of these operators on Hardy spaces on the unit circle and the open unit disk.

Yoenha Kim (Ajou Univ): On the products of self-adjoint weighted composition operators on the Hardy space

Abstract: In this talk, we study the products of two self-adjoint weighted composition operators on the Hardy space. In particular, we provide necessary and sufficient conditions for such products to be self-adjoint, unitary, or normal. This is a joint work with Eungil Ko.

◾Tutorials on the invariant subspaces of the model operator

Lecturer: In Sung Hwang (Sungkyunkwan Univ): 

Abstracts: In this tutorial series we explore the invariant subspaces of the model operators by using the invariant subspaces of the shift operator. We try to decompose the model operator into a sum of simpler arts, connecting such a decomposition with a refinement of the spectrum of the model operator. When a further refinement of the spectrum is not any longer possible, it will be necessary to single out a chain of invariant subspaces, which leads to integral representations for the operator which are analogous to the triangular form for matrices in linear algebra. It turns however out to be most difficult to reconstruct the properties of its restrictions to the terms of the decomposition. Several of the lectures will be devoted to this reconstruction.