Duality is established for new spaces of entire functions in two infinite dimensional variables with certain growth rates determined by Young functions. These entire functions characterize the symbols of generalized Fock space operators. As an application, a proper space is found for a solution to a normal-ordered white noise differential equation having highly singular coefficients.Keywords: Fock space; operator symbol; infinite-dimensional holomorphy; normalordered white noise differential equation; quantum white noise.
Motivated by the notion of P-functional, we introduce a notion of α-completely positive map between ∗-algebras which is a Hermitian map satisfying a certain positivity condition, and then a α-completely positive map which is not completely positive is constructed. We establish the Kasparov-Stinespring-Gelfand-Naimark-Segal constructions of C∗-algebra and ∗-algebra on Krein C∗-modules with α-completely positive maps.
Communicated by O. SmolyanovWe introduce, for each a ∈ R + , the Brownian motion associated to the distribution derivative of order a of white noise. We prove that the generator of this Markov process is the exotic Laplacian of order 2a, given by the Cesàro mean of order 2a of the second derivatives along the elements of an orthonormal basis of a suitable Hilbert space (the Cesàro space of order 2a). In particular, for a = 1/2 one finds the usual Lévy Laplacian, but also in this case the connection with the 1/2-derivative of white noise is new. The main technical tool, used to achieve these goals, is a generalization of a result due to Accardi and Smolyanov 5 extending the well-known Cesàro theorem to higher order arithmetic means. These and other estimates allow to prove existence of the heat *
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