The following results (and one or two others like them) are proved in [3]; the continuous-parameter analogues will appear in [4] and some applications will be discussed in [5]. The proofs are based on Sz.-Nagy's group of theorems [9] which show how a discreteor continuous-parameter semigroup of contraction operators on Hilbert space can be "dilated" to a similar semigroup (actually a group) of ...

Here we study the notions of lattice-valued finite state machine and lattice-valued fuzzy transformation semigroup when the lattice consists of all the closed subintervals contained in [0, 1]. So, we can apply techniques of interval-valued fuzzy sets to fuzzy automata. In particular, we prove that Atanassov’s Kα operators allow us to retrieve the information given by the transition functions of...

1. Basic Assumptions. Assume that (i) {Tt} is a strongly-continuous semigroup of bounded linear operators on the Banach space B = C0(I) = { f continuous on I = [0, 1] : f(0) = f(1) = 0 } (ii) If f(x) ≥ 0 and f ∈ B, then Ttf(x) ≥ 0 and Ttf(x) ≤ maxy∈I f(y). In general, the infinitesimal generator of a strongly-continuuous semigroup of linear operators Tt on any Banach space B is defined by Af = ...

A generalized Mehler semigroup (Ornstein-Uhlenbeck semigroup) associated with some strongly continuous semigroup of linear operators on a real separable Hilbert space may be defined by using a skew convolution semigroup. Under a mild moment assumption, it is proved that the characteristic functional of any centered skew convolution semigroup is absolutely continuous and characterizations are gi...