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 the present paper for a large family of topological semigroups, namely foundation semigroups, for which topological groups and discrete semigroups are elementary examples, it is shown that ?(s) is the dual of a function algebra.

In this paper we discuss the characteristic property of the left invertible semigroups on general Banach spaces and admissibility of the observation operators for such semigroups. We obtain a sufficient and necessary condition about their generators. Further, for the left invertible and exponentially stable semigroup in Hilbert space we show that there is an equivalent norm under which it is co...

Denote byB(X) the set of all bounded linear operators acting on a Banach spaceX . For simplicity we assume that all Banach spaces are complex unless stated explicitly otherwise. However, the notions make sense also in real Banach spaces and most of the results remain true (with slight modifications) in the real case. Let X be a Banach space and let T be a bounded linear operator on X . By an or...

Abstract: Let X be a Banach space. We define the concept of a bi-parameter semigroup on X and its first and second generators. We also study bi-parameter semigroups on Banach algebras. A relation between uniformly continuous bi-parameter semigroups and σ-derivations is also established. It is proved that if {αt,s}t,s 0 is a uniformly continuous bi-parameter semigroup on a Banach algebra X , who...

Abstract. We consider the approximation of semigroups e and of the functions φj(τA) that appear in exponential integrators by resolvent series. The interesting fact is that the resolvent series expresses the operator functions e and φj(τA), respectively, in efficiently computable terms. This is important for semigroups, where the new approximation is different from well-known approximations by ...

In this note we relate two notions of compactness for strongly continuous semigroups of linear operators and cosine functions of linear operators. Specifically, if T denotes a strongly continuous semigroup of linear operators defined on a Banach space X, we will show that T is compact if and only if the set {(T(-)x : x £ X, \\x\\ < 1} is relatively compact in any space LP([Q, a]) ; X) for 1 < p...

The discrete case of the growth-fragmentation equation is not as popular in the literature as the continuous form of it, hence the need to explore the discrete version of this model for some interesting analysis. In this paper, We examine the discrete growth-fragmentation equations. The problem is treated as an infinite-dimensional differential equation, posed in a suitable Banach space. Pertur...

Convoluted C-cosine functions and semigroups in a Banach space setting extending the classes of fractionally integrated C-cosine functions and semigroups are systematically analyzed. Structural properties of such operator families are obtained. Relations between convoluted C-cosine functions and analytic convoluted C-semigroups, introduced and investigated in this paper are given through the co...