AN OPERATOR INFORMATION QUANTITY OF A SEMIGROUP AND ASSOCIATED DIFFERENTIAL EQUATIONS

In this paper, we present recent developments on the operator entropy. In particular, we introduce an operator information quantity of a semigroup by extending the operator entropy of a semigroup, and prove characterizations of the operator information quantity and entropy in terms of operator-valued differential equations. As an application, by further extending those to the white noise theory, we give a stochastic expression of the operator entropy. Moreover, by the Gauss-Poisson correspondence operator K introduced in [31] which transfers regular white noise functionals to functionals of exponential white noise, we give a relationship between the operator information quantity of an infinite dimensional Fourier-Mehler transform and the L´evy Laplacian through the operator K.