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**In Sung Hwang (Sungkyunkwan Univ): Vector-valued function theory 1**

**- The Beurling-Lax-Halmos theorem**

**Abstract:** Often, some argue that the purpose of extending the scalar-valued function theory into the cases of vector-valued functions is not clear in general. However, the vector-valued function theory is highly useful for understanding scalar-valued functions. This is just like how certain features are better visible in 2-dimensions when viewed from a 3-dimensional perspective. One prominent example is the Nagy-Foias model theory (1970). Indeed, the Beurling-Lax-Halmos theorem was soon revealed to be very important. That is just because without the Beurling-Lax-Halmos theorem, the Nagy-Foias model theorem could not have been effectively explained. Roughly speaking, the Nagy-Foias model theorem says that a general operator in a scalar-valued world is nothing but a shadow of an action of backward-shifting in the vector-valued world.

**In Sung Hwang (Sungkyunkwan Univ): Vector-valued function theory 2**

**- Circle companions of Hardy space of the unit disc.**

**Abstract:** A study on the boundary values of functions in Banach-space-valued Hardy spaces of the unit disk was initiated in 1976 by A.V. Bukhvalov. Since then, many researchers

have studied the spaces of boundary values of functions in Banach-space-valued Hardy spaces of the unit disk. In particular, in 1982 A.V. Bukhvalov and A.A. Danilevich showed that if a Banach space has the analytic Radon-Nikodym property, then the space of boundary values of functions in Banach-space-valued Hardy spaces of the unit disk is the Banach-space-valued Hardy spaces of the unit cicle.

**In Sung Hwang (Sungkyunkwan Univ): Vector-valued function theory 3**

**- Invariant subspaces of Toeplitz operators with inner symbols.**

**Abstract:** The Beurling's Theorem characterizes the invariant subspaces of the shift operator on the Hardy space. The shift operator is a special case of Toeplitz operators with inner symbol, that is, the symbols that are inner functions. However, the invariant subspaces of Toeplitz operators with inner symbols were not characterized until now. Thus the following problem seems to be interesting and challengeable: Describe all invariant subspaces of a Toeplitz operator with inner symbol .

**Jaeseong Heo (Hanyang Univ): Von Neumann algebras 1**

**- Von Neumann algebra and similarity problem 1**

**Abstract:** Motivated by quantum mechanics and group representation theory, John von Neumann introduced in the early 30's certain algebras of bounded operators on a Hilbert space, the so-called von Neumann algebras. Applications to the foundations of quantum mechanics were a motivation for von Neumann's interest. In his double commutant theorem he showed that these algebras can be characterized either in purely topological or in purely algebraic terms, a fact, that has numerous beautiful and deep consequences. Today, the subject is one of the most dynamic areas of research in modern mathematics. The theory of operator algebras has many fruitful interrelations and fruitful connections with other areas of mathematics and physics. In this lecture, I review the von Neumann algebra theory and some results on the similarity problem on C*-algebras.

**Jaeseong Heo (Hanyang Univ): Von Neumann algebras 2**

**- Von Neumann algebra and similarity problem 2**

**Abstract:** This lecture is devoted to the background necessary to understand the similarity problem, to the solutions that are known in some special cases and to numerous related concepts and results. One of approaches to the similarity problem is to use the key concept of "complete boundedness". In some sense, completely bounded maps can be viewed as spaces of "coefficients" of C*-algebraic representations, if we allow "B(H) valued coefficients", this is the content of the fundamental factorization property of these maps, which plays a central role in this lecture. I explain that the similarity problem can all be formulated as asking whether "boundedness" implies "complete boundedness" for linear maps satisfying certain additional conditions.

**Hyungwoon Koo (Korea Univ): Composition operators 1 **

**Abstract:** In this lecture, we provide the details of the well-known characterization of the compactness of a single composition operator or the difference of two composition operators on the Bergman space over the unit disc. We prove theorems which are used as basic tools for the proof of the compact difference: Littlewood’s Subordination Principle, Theorem of Julia-Caratheodory, Carleson measure for the Bergman spaces. Using these tools we provide the full details of the proof of the compact difference.

**Hyungwoon Koo (Korea Univ): Composition operators 2**

**Abstract:** : In this lecture, we discuss the compactness of a single composition operator or of the difference of two composition operators on Hardy space over the unit disc. We prove theorems which are used as basic tools to handle composition operators for the Hardy spaces: Carleson measure theorem for Hardy space, Frostan’s theorem, Nevanlinna counting function. We first prove the compactness of a single composition operator which was proved by Shapiro. We prove a measure theoretic characterization of the compact difference and give some open problems.