일정:
12월 14일 오후 1시 - 오후 4시 Preliminaries/Discussion (기초강의 및 토론, 이훈희)
12월 15일 오전 10시 - 12시 (이종락 1강)
12월 15일 오후 2시 - 4시 (이종락 2강)
12월 16일 오전 10시 - 12시 (지운식 1강)
12월 16일 오후 2시 - 4시 (이종락 3강)
12월 17일 오전 10시 - 12시 (지운식 2강)
12월 17일 오후 1시 - 3시 (지운식 3강)
12월 17일 오후 강의 종료후 해산
참가에 관심있는 분들은 이훈희(E-mail: hunheelee_at_snu_dot_ac_kr)에게 메일로 문의바랍니다.
강연안내
초록:
1. Hyponormal Toeplitz operators on the weighted Hardy spaces.
- In this talk, we remark the hyponormal Toeplitz operators on the Hardy spaces. Furthermore, we study necessary and sufficient conditions for the hyponormality of Toeplitz operators on the weighted Hardy spaces. Next, we consider hyponormality of Toeplitz operators with non-harmonic symbols.
2. Hyponormality of Toeplitz operators on the several Hilbert spaces.
- In this talk, we study the hyponormality of Toeplitz operators on the several Hilbert spaces. Especially, we consider the basic properties of Toeplitz operators on the weighted Bergman spaces, Dirichlet spaces, Fock spaces and Newton spaces. Next, we stduy the normality and hyponormality of Toeplitz operators on the spaces above.
3. Complex symmetric Toeplitz operators.
- In this talk, we introduce some conjugations and complex symmetric Toeplitz operators on the weigthed Hardy spaces. Next, we give basic properties of complex symmetric Toeplitz operators. Finally, we investigate a complex symmetric Toeplitz operators with respect to the conjugations.
초록:
Since the quantum stochastic calculus initiated by Hudson and Parthasarathy
(so called the Hudson-Parthasarathy (HP) quantum stochastic calculus)
as a quantum counterpart of (classical) Ito calculus,
then it has been developed extensively with wide applications to mathematical physics, quantum optics and so on.
In the HP quantum stochastic calculus, the annihilation, creation and conservation processes play important roles
as fundamental quantum stochastic (noise) processes,
and then the unique solution of a certain quantum stochastic differential equation
provides a stochastic dilation, called the Hudson-Parthasarathy dilation,
of a uniformly continuous completely positive semigroup.
This intensive seminar on quantum stochastic calculus, mainly to understand the HP quantum stochastic calculus,
is organized into three parts:
I. (Quantum Stochastic Processes) As the most basic notions of the HP quantum stochastic calculus,
we introduce the annihilation, creation and conservation operators in the Boson Fock space over a Hilbert space.
We discuss the Weyl representation and commutation relations (intertwining properties)
of the annihilation, creation and conservation operators.
We also discuss the quantum stochastic processes as Fock space operator-valued processes
and their adaptedness, and then we study the annihilation,
creation and conservation processes as the fundamental quantum (noise) processes.
II. (Quantum Stochastic Integrals and Quantum Ito Formula) Based on the Boson canonical commutation relations,
we discuss the quantum stochastic integrals against the fundamental quantum stochastic (noise) processes,
annihilation, creation and conservation processes.
Then we study the quantum Ito (product) formula for the quantum stochastic integrals
as the quantum extension of the classical It\^o formula.
The quantum Ito formula is the most basic tool in the quantum stochastic calculus.
III. (Quantum Evolutions and Stochastic Dilations of CP Semigroups) The quantum stochastic evolutions are described by quantum stochastic differential equations (QSDEs)
and then we discuss the unique existence of solution to a QSDE.
Then we discuss the unitarity conditions for the solution of QSDE
from which we establish the (stochastic) unitary dilation of uniformly continuous completely positive semigroup.
If time permits, then we discuss on the recent development of quantum stochastic calculus for quadratic quantum white noises.