In Sung Hwang (Sungkyunkwan Univ): Vector-valued measurable functions

Abstract: A function $f$ with values in a Banach space $X$ is said to be measurable if the pre-image $f^{-1}(B)$ of every Borel set $B$ in $X$ is measurable. As it turns out, in many respects this natural notion is not as useful as one might think, the reason being that the Borel $sigma$-algebra $B(X)$ is in general ‘too large’. In fact, the $sigma$-algebra generated by all bounded linear functionals on $X$ may be strictly smaller than $B(X)$. This presents an obstruction to applying the standard tools of functional analysis such as the Hahn-Banach theorem. If $X$ is a separable Banach space, this problem does not occur.

Jaeseong Heo (Hanyang Univ): Free probability and free entropy

Abstract: I introduce basic notions and theorems for free probability theory, for example noncommutative probability space, free independence, free central limit theorem, free convolutions. I will survey some of the basic ideas and results of free probability theory and introduce free entropy theory developed by Voiculescu, which is the free analogue of Shannon's entropy. I also discuss its application to large size random matrices and results of free probability theory.

Hyungwoon Koo (Korea Univ): Logarithmic order in analytic function spaces

Abstract: The weighted Bergman space over the unit disc $A^p_alpha(mathbb D)$ is the set of all analytic functions $f$ with $int_{mathbb D} |f(z)|^p (1-|z|^2)^alpha dA(z) -1$. It is well-known that the weight $alpha$ and the (fractional) derivative is exchangeable in the following sense; $$fin A^p_alpha(mathbb D) iff mathcal D^sf in A^p_{alpha +sp}(mathbb D)$$ where $mathcal D^sf$ is the fractional derivative of order $s$. In this talk we discuss various extensions of this relation which involves the logarithmic orders in the function and also in the weight.

Ji Eun Lee (Sejong Univ): Complex symmetric Toeplitz operators on the function space

Abstract: In this talk, we examine conjugations and complex symmetric Toeplitz operators on the function spaces. We begin by introducing a novel conjugation $C_{xi}$ on the weighted Hardy space $H_{rho}(Bbb D)$, defined as $C_{xi}f(z)=-k_{xi}(z)overline{f(psi_{xi}(overline{z}))}$ for some $fin H_{rho}(Bbb D)$, where $k_{xi}(z)$ is the normalized reproducing kernel and $psi_{xi}(z)=frac{xi-z}{1-overline{xi}z}$. In particular, we establish that $C_{xi}$ is unitarily equivalent to $C_{mu,lambda}$, previously  studied  in: E. Ko and J. E. Lee, On complex symmetric Toeplitz operators, J. Math. Anal. Appl. 434(2016), 20--34. Using this result, we investigate the complex symmetric Toeplitz operator $T_{varphi}$ with respect to the conjugation $C_{xi}$ on the weighted Hardy space $H_{rho}(Bbb D)$. Additionally, we explore the $C_{mu,lambda}$-invariance of the Berezin transform. Furthermore, we extend our analysis to block conjugations on the weighted Hardy space $H_{rho}({mathbb C}^n)$. Utilizing these conjugations, we examine the complex symmetry of block Toeplitz operators on the weighted Hardy space $H_{rho}({mathbb C}^n)$.

Hun Hee Lee (Seoul Nat’l Univ): Twisting in abstract harmonic analysis

Abstract: In this talk we will examine the concept of twisting of groups and its recent appearance in abstract harmonic analysis. More specifically, we will examine (1) the case of twisted Fourier-Stieltjes space and amenability; (2) the case of gaussian states in the general quantum kinematical systems.

Ja A Jeong (Seoul Nat’l Univ): C*-algebras of étale groupoids

Abstract: The C*-algebras associated with topological groupoids not only encompass classical C*-algebras such as group C*-algebras and C*-crossed products of commutative C*-algebras, but also play an important role in the recent study of operator algebras. In this talk, we review the construction of C*-algebras of étale groupoids and discuss their properties along with several examples. 

Jongrak Lee (Sungkyunkwan Univ): Hyponormality of Toeplitz operators on the Drury-Arveson spaces

Abstract: In this talk, we will consider the hyponormality of Toeplitz operators on the Drury–Arveson space. First, we present properties of hyponormal Toeplitz operators with two-variable symbols. Then, we extend the discussion to hyponormal Toeplitz operators with n-variable symbols.

Hong-Rae Cho (Pusan Nat’l Univ): Fock-Sobolev spaces and their applications

Abstract: Along with full derivatives, fractional derivatives of holomorphic functions play a crucial role in the function theory of holomorphic Sobolev spaces. We introduce the fractional derivative and the derivative of logarithmic order for an entire function and define the associated function spaces, the fractional Fock-Sobolev space and the logarithmic Fock-Sobolev space in $mathbb C^n$, respectively. We show the fractional Fock-Sobolev space coincides with the polynomial weighted Fock space by proving that the fractional order of derivative on an entire function can be exchanged with the fractional order on the polynomial weight, with their norms being equivalent. We further show that the logarithmic Fock-Sobolev space coincides with the logarithmic weighted Fock space by proving that the logarithmic order of derivative on a function can be exchanged with the logarithmic order on the weight, with their norms being equivalent. As applications, we can get Fock-Sobolev estimates for the solution of the cubic lowest Landau level equation. The lowest Landau level refers to the lowest energy state of a charged particle in a magnetic field in quantum mechanics. The cubic lowest Landau level equation refers to a nonlinear Schr"odinger equation constrained to the lowest Landau level subspace, where the nonlinearity is cubic.

Jasang Yoon (Univ of Texas - RGV): Invariant subspaces among the linear pencil, a pair of operators, and their operator transforms

Abstract: In this talk, we find and study explicit connections of invariant (hyperinvariant) subspaces among the linear pencil, a pair of operators, and their operator transforms. The linear pencil of a pair of operators is a kind of natural generalization of the multiparameter pencils of a pair of n-by-n complex matrices. It is known that multiparameter pencils find applications in a variety of fields, including primarily engineering, applied mathematics, and physics. We show that for two given operators, the linear combination has an invariant subspace if and only if the pair of the two operators has a joint invariant subspace. Next, we observe that if the two operators are commuting, then the results of the invariant subspaces can be extended to hyperinvariant subspaces. We also remove the commuting condition of the pair of operators in previous results in this topic, as the commuting condition is not required to define a joint invariant subspace of a pair of operators. To accomplish this, we employ a new technique adopting a horizontal asymptote of the linear combination of two operators. This talk is based on the joint work with Jaewoog Kim.

Inyoung Park (Korea Univ): Convergence of the powers of composition operators on weighted spaces of holomorphic functions

Abstract: We study the asymptotic behavior of the iterates of composition operators in various weighted spaces of holomorphic functions on the unit disc D. This approach allows us to extend all previous results in this area from classical function spaces - such as Bergman spaces, Dirichlet spaces, standard weighted Banach spaces with sup-norm, and Bloch spaces - to their corresponding weighted spaces induced by a doubling weight or a fast weight