Todd Kemp

Ph.D. (2005) Cornell University

First Position

Escobar Assistant Professor, Cornell University

Dissertation

Hypercontractivity in Non-commutative Holomorphic Spaces

Advisor:

Leonard Gross

Research Area:

functional analysis, quantum probability theory

Abstract: In this dissertation, non-commutative holomorphic spacesH_q (for –1 ≤ q ≤ 1) are introduced. These spaces naturally generalize the algebra of entire functions (identified with H₁) in the context of the q-Gaussian von Neumann algebras Gamma_q of Bozejko and Speicher. A non-commutative Segal-Bargmann transformS_q — an isometric isomorphism L²(Gamma_q)\to L²(H_q) which commutes with the number operator N_q, and which canonically generalizes the classical Segal-Bargmann transform — is constructed. The following theorem, which is the main result, is proved: for even integers r, the semigroup generated by N_q is a contraction L²(H_q)\to L^r(H_q) precisely when e^–2t≤ 2/r. This strong hypercontractivity theorem generalizes (a special case of) the complex hypercontractivity result of Janson to the algebras H_q.