One-parameter Groups Of Operators Research Articles

A SIMILARITY BETWEEN THE GROSS LAPLACIAN AND THE LÉVY LAPLACIAN

In this paper we present a construction of an infinite dimensional separable Hilbert space associated with a norm induced from the Lévy trace. The space is slightly different from the Cesàro Hilbert space introduced in Ref. 1. The Lévy Laplacian is discussed with a suitable domain which is constructed by a rigging of Fock spaces based on a rigging of Hilbert spaces with the Lévy trace. Then the Lévy Laplacian can be considered as the Gross Laplacian acting on a certain countable Hilbert space. By constructing one-parameter group of operators of which the infinitesimal generator is the Lévy Laplacian, we study the existence and uniqueness of solution of heat equation associated with the Lévy Laplacian. Moreover we give an infinite dimensional stochastic process generated by the Lévy Laplacian.

One-Parameter Groups Arising from Real Subspaces of Self-Dual HilbertW*-Moduli

The purpose of this paper is to generalize the results of M. A. Rieffel and A. van Daele [8, §§ 1, 2, 3] for Hilbert W*-moduli over commutative W*-Algebras. Some special real subspaces of such Hilbert W*-moduli and the related operators are investigated. Particularly, the relation is established between *weakly continuous unitary one-parameter groups of operators arising from them and the generalized K.M.S. condition. All key definitions are formulated without any commutativity supposition for the underlying W*-algebra. The interpretation of these results is given for sets of continuous sections of “self-dual” locally trivial Hilbert bundles over hyperstonian compact spaces. At the end of this paper some aspects of the general noncommutative case are discussed.

A remark on the space of testing random variables in the white noise calculus

The first author and S. Takenaka introduced the structure of a Gel’fand triplet ℋ ⊂ (L2) ⊂ ℋ* into Hida’s calculus on generalized Brownian functionals [4-7]. They showed that the space ℋ of testing random variables has nice properties. For example, ℋ is closed under multiplication of two elements in ℋ, each element of ℋ is a continuous functional on the basic space ℰ*, in addition it can be considered as an analytic functional, and moreover exp [tΔv] (Δv is Volterra’s Laplacian) is real analytic in t ∊ R as a one-parameter group of operators on ℋ, etc.

Spectral and ergodic properties of the analytic generators

In this paper similarity situations between one-parameter groups of operators are characterized in terms of analytic generators and spectral subspaces, and ergodic properties of one-parameter groups are studied. These topics permit us to give “noncommutative” extensions of classical results from the theory of Hardy's Hp spaces. As major applications we mention the possibility of use of the analyticity notion from Arveson (Amer. J. Math.89 (1967), 578–642) for one-parameter groups of *-automorphisms of operator algebras with “good” ergodic properties (see Section 3), and a procedure to “improve” implementing groups of unitaries for one-parameter groups of *-automorphisms of operator algebras (see Section 5).

On cosine operator functions and one-parameter groups of operators

On cosine operator functions and one-parameter groups of operators

A Remark on Integration of Almost-Periodic Functions

A result proved by Favard for scalar-valued almost-periodic functions has an immediate extension to Banach space valued functions (see [2] and [3] for explicit details).The result says that integrals of almost-periodic functions whose 'spectrum' is at positive distance from 0 are again almost-periodic. Our aim here is to indicate a more general formulation of this result using strongly almost-periodic oneparameter groups of operators in Banach spaces.

On one-parameter groups of operators

On one-parameter groups of operators