In this work, we show that a strongly continuous semigroup generated by a normal operator N is conjugate to the semigroup generated by the real part of N , provided zero is not an eigenvalue of the real part of N . We also show that in case N satisfies a certain sectorial property, the homeomorphism establishing the conjugacy, as well as its inverse, is locally Hölder continuous. Moreover, in c...

This paper addresses extensions of the complex Ornstein-Uhlenbeck semigroup to operator algebras in free probability theory. If a1, . . . , ak are ∗-free R-diagonal operators in a II1 factor, then Dt(ai1 · · · ain) = e −ntai1 · · · ain defines a dilation semigroup on the non-self-adjoint operator algebra generated by a1, . . . , ak. We show that Dt extends (in two different ways) to a semigroup...

In this paper. (1) We determine the complex-valued solutions of the following variant of Van Vleck's functional equation $$int_{S}f(sigma(y)xt)dmu(t)-int_{S}f(xyt)dmu(t) = 2f(x)f(y), ;x,yin S,$$ where $S$ is a semigroup, $sigma$ is an involutive morphism of $S$, and $mu$ is a complex measure that is linear combinations of Dirac measures $(delta_{z_{i}})_{iin I}$, such that for all $iin I$, $z_{...

We begin with a definition Definition 1. A one-parameter family T (t) for 0 ≤ t < ∞ of bounded linear operators on a Banach space X is a C 0 (or strongly continuous) Semigroup on X if 1. T (0) = I (the identity on X). 2. T (t + s) = T (t)T (s) (semigroup property) 3. lim t↓0 T (t)v = v for all v ∈ X (This the Strong Continuity at t = 0, i.e., continuous at t = 0 in the strong operator topology)...

In this paper, we introduce the notion of fuzzy ideal,fuzzy weak ideal,fuzzy weakly prime left ideal and fuzzy prime left ideal system of a near-subtraction semigroup. We characterize fuzzy weak ideal and fuzzy ideal of a near-subtraction semigroup X through weak ideal and ideal of X respectively. We have shown that a fuzzy left ideal μ of a nearsubtraction semigroup X is a fuzzy weakly prime l...

Let ū(t) be a control that satisfies the infinite-dimensional version of Pontryagin’s maximum principle for a linear control system, and let z(t) be the costate associated with ū(t). It is known that integrability of ‖z(t)‖ in the control interval [0, T ] guarantees that ū(t) is time and norm optimal. However, there are examples where optimality holds (or does not hold) when ‖z(t)‖ is not integ...

We consider the non-linear impulsive evolution equation u′(t) = Au(t) + f(t, u(t), Tu(t), Su(t)), 0 < t < T0, t 6= ti, u(0) = u0, ∆u(ti) = Ii(u(ti)), i = 1, 2, 3, . . . , p. in a Banach space X, where A is the infinitesimal generator of a C0 semigroup. We study the existence and uniqueness of the mild solutions of the evolution equation by using semigroup theory and then show that the mild solu...

Let {r(Z):Z>0} be a C0-semigroup on a Banach space X with generator A , and let x € X. If a (A) n ;'R is empty and t »-> T(t)x is uniformly continuous, then ||7"(Z)jc|| —> 0 as t —» oo . If the semigroup is sun-reflexive, o(A)CiiR is countable, Pa(A)DiS. is empty, and 1 >-> T(t)x is uniformly weakly continuous, then T(t)x —► 0 weakly as t —» oo . Questions of almost periodicity and of stabiliza...

We construct the freest idempotent-pure expansion of an inverse semigroup, generalizing an expansion of Margolis and Meakin for the group case. We also generalize the Birget-Rhodes prefix expansion to inverse semigroups with an application to partial actions of inverse semigroups. In the process of generalizing the latter expansion, we are led to a new class of idempotent-pure homomorphisms whi...