摘要
In this paper the author proves a new fundamental lemma of Hardy-Lebesgue class H<sup>2</sup>（σ） and by this lemma obtains some fundamental results of exponential stability of C<sub>0</sub>-semigroup of bounded linear operators in Banach spaces. Specially, if ω<sub>s</sub>=sup{Reλ; λ∈σ（A）}【0 and sup{‖（λ-A）<sup>-1</sup>‖; Reλ≥σ}【∞, where σ∈（ω<sub><</sub>sup>,</sup> 0） and A is the infinitesimal generator of a G<sub>0</sub>-semigroup e<sup>tA</sup> in a Banach space X, then （a）integral from n=0 to ∞(e<sup>-σt</sup>|f(e<sup>tA</sup>x)|f(e<sup>tA</sup>x)|dt【∞, f∈X<sup>*</sup> and x∈X; （b） there exists M】0 such that ‖e<sup>tA</sup>x‖≤Ne<sup>σt</sup>‖Ax‖, x∈D（A）; （c） there exists a Banach space X such that ‖e<sup>tA</sup>x‖≤e<sup>σt</sup>‖x‖, x∈X.
In this paper the author proves a new fundamental lemma of Hardy-Lebesgue class H^2(σ) and by this lemma obtains some fundamental results of exponential stability of C_0-semigroup of bounded linear operators in Banach spaces. Specially, if ω_s=sup{Reλ; λ∈σ(A)}<0 and sup{‖(λ-A)^(-1)‖; Reλ≥σ}<∞, where σ∈(ω_~, 0) and A is the infinitesimal generator of a G_0-semigroup e^(tA) in a Banach space X, then (a)integral from n=0 to ∞(e^(-σt)|f(e^(tA)x)|f(e^(tA)x)|dt<∞, f∈X~* and x∈X; (b) there exists M>0 such that ‖e^(tA)x‖≤Ne^(σt)‖Ax‖, x∈D(A); (c) there exists a Banach space X such that ‖e^(tA)x‖≤e^(σt)‖x‖, x∈X.
