Abstract
We prove that if the Quojection-Fréchet space $X$ is a Grothendieck space with the Dunford-Pettis property, then every $C_ {0}$-cosine family is necessarily uniformly continuous and therefore its infinitesimal generator is a continuous linear operator.
Highlights
Let X be a sequentially complete locally convex Hausdorff space and ΓX a system of continuous seminorms determining the topology of X
We say that {C(t)}t≥0 is exponentially equicontinuous of order ω if the set {e−ωtC(t)}t≥0 is equicontinuous, i.e. ∀p ∈ ΓX, ∃q ∈ ΓX, ∃M ≥ 0 such that: p(C(t)x) ≤ M eωtq(x), for all x ∈ X, and t ≥ 0
It is known that every C0-cosine family in a Banach space is necessary exponentially equicontinuous [8,12]
Summary
Let X be a sequentially complete locally convex Hausdorff space and ΓX a system of continuous seminorms determining the topology of X. The strong topology τ s in the space L(X), of all continuous linear operators from X into itself, is determined by the family of seminorms: qx(S) = q(Sx), S ∈ L(X), for each x ∈ X and q ∈ ΓX , in this case we write Ls(X). Let {C(t)}t≥0 be a family of continuous linear operators in X. It is known that every C0-cosine family in a Banach space is necessary exponentially equicontinuous [8,12]. The following proposition generalizes this result in locally convex spaces. Let {C(t)}t≥0 be a locally equicontinuous C0-cosine family in locally convex space X. i.e. The previous proposition show that if the C0-cosine family {C(t)}t≥0 is locally equicontinuous, the following conditions are equivalent: i) C(t) −→ I in Ls(X) as t −→ 0+. Every locally equicontinuous C0-cosine family {C(t)}t≥0 commute. Let {C(t)}t≥0 be a locally equicontinuous C0-cosine family in X and A its infinitesimal generator.
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