Deformation of Bargmann-Fock representation in non-commutative probability theory

2020 Fiscal Year Final Research Report

• Project/Area Number: 16K05175
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Aichi University of Education
• Principal Investigator: Asai Nobuhiro 愛知教育大学, 教育学部, 教授 (60399029)
• Project Period (FY): 2016-04-01 – 2021-03-31
• Keywords: フォック空間 / 非可換確率論 / 直交多項式論 / 確率分布 / B型ガウス・ポアソン作用素

Outline of Final Research Achievements

We considered deformations of the Bargmann-Fock space, Gaussian/Poisson operators, and their probability distributions in terms of non-commutative probabilistic approach. (1) The radial Bargmann density function associated with the Fock space of type B is explicitly obtained. (2) Gaussian-Poisson type operators on the type B space are introduced. As a result, we showed that a q2-Meixner class of probability measures can be treated within the framework of type B. (3) We constructed a weighted q-Fock space and presented unknown and non-trivial examples in previous works.

Free Research Field

非可換確率論，直交多項式論，

Academic Significance and Societal Importance of the Research Achievements

本研究において，従来のフォック空間ならびにガウス・ポアソン型作用素の変形・補間理論の拡張により，確率論や非可換確率論では馴染みの薄い非自明な確率分布とより強い非可換性との関係解明への足掛かりになっていることが意義深い．