Heat Kernel Analysis and Cameron–Martin Subgroup for Infinite Dimensional Groups

Abstract

The heat kernel measure + t is constructed on GL(H), the group of invertible operators on a complex Hilbert space H. This measure is determined by an infinite dimensional Lie algebra g and a Hermitian inner product on it. The Cameron Martin subgroup G CM is defined and its properties are discussed. In particular, there is an isometry from the L 2 +t -closure of holomorphic polynomials into a space H t (G CM ) of functions holomorphic on G CM . This means that any element from this L 2 +t -closure of holomorphic polynomials has a version holomorphic on G CM . In addition, there is an isometry from H t (G CM ) into a Hilbert space associated with the tensor algebra over g. The latter isometry is an infinite dimensional analog of the Taylor expansion. As examples we discuss a complex orthogonal group and a complex symplectic group. Academic Press

Square integrable holomorphic functions on infinite-dimensional Heisenberg type groups

“…Gordina [10][11][12] considered the Taylor isomorphism in the context of Hilbert-Schmidt groups, while Cecil [2] considered the Taylor isomorphism for path groups over stratified Lie groups. The nilpotentcy of the Heisenberg like groups studied in this paper allow us to give a more complete description of the square integrable holomorphic function spaces than was possible in [10][11][12] for the Hilbert-Schmidt groups.…”

Holomorphic functions and subelliptic heat kernels over Lie groups

“…The Taylor map isomorphism has also been proven for some infinite-dimensional groups: in [11] and [10] M. Gordina found a precise analog of this unitary isomorphism for the infinite-dimensional complex Hilbert-Schmidt orthogonal group, and in [12] she proved the analog for the group of invertible operators in a factor of type I I 1 . Also M. Cecil, in [3], has shown that a unitary Taylor isomorphism holds for path groups over stratified nilpotent Lie groups.…”

TAYLOR MAP ON GROUPS ASSOCIATED WITH A II1-FACTOR

“…In this work we continue to study the type of heat kernel analysis on infinitedimensional groups developed in [4], [5]. These papers dealt with so called HilbertSchmidt (infinite-dimensional) complex groups of operators on a Hilbert space.…”

“…The main difference from the case of Hilbert-Schmidt groups studied in [4] and [5] is that the heat kernel measure lives in a space of unbounded operators, which makes the case of the II 1 -factor more complicated. Some of the new features are addressed in Section 6.…”

“…Then this operator is in general an unbounded linear operator although its mathematical expectation is a bounded operator. In [4], [5] we proved that the stochastic process analogous to X t actually lives in the group of invertible elements of B(H) for a Hilbert space H. In the case of II 1 -factor the situation is more complicated, since X t takes values in L 2 (C), not C itself. We will show that the elements X t (ω) have right inverses in L 2 (C) defined by another stochastic differential equation.…”