令X 為一佈於複數的巴氏空間且令B(X) 所有定義於X 上的有界線性算子所成的巴氏代數。我們稱在B(X) 裡的一有界算子族{T(t) | t ≥ 0}為一個(C0)-semigroupon X 如果它滿足下面條件：(C1) T(0) = I, 單位算子(the identity operator);(C2) T(s + t) = T(t)T(s), t, s ≥ 0;(C3) 對所有x ∈ X, T(·)x 在[0,∞) 為強連續。T(·) 的無窮生成元(infinitesimal generator) A 定義為：D(A) :={x ∈ X; limt→0T(t)x−xt存在}Ax := limt→0+T(t)x−xt for x ∈ D(A).關於(C0)-半群的無窮生成元的fractional powers 有很多表示公式。譬如： 如果A 是一個均勻有界(C0)-群或者是的生成元或者是指數有界的解析半群的生成元則A的fractional powers Aα 可以定義為(1) Aαx =sin παπ∫ ∞0tα−1(tI + A)−1Axdt for x ∈ D(A) and 0 0.其中T(·) 是指數有界的解析半群且Γ(α) 是伽瑪函數(gamma function)。我們引進一個Gamma 算子函數(Gamma operator function) 的新觀念，定義如下：一算子函數G : (0,∞) → B(X) 稱為一個Gamma operator function 如果它滿足下面條件：(G1) 對任意x ∈ X，G(·)x 在(0,∞) 是強連續；(G2) G(t)G(s) = B(t, s)G(t + s) for all t, s > 0， 其中B(·, ·) 是the Beta function;1(G3) 對所有x ∈ X， limt→0+tG(t)x = x.在本計畫中，我們的工作主要將考慮下面的問題：(i) Gamma 算子函數與(C0)-半群的關係。(ii) 將Gamma 算子應用到(C0)-半群的generation theorem。Let X be a complex Banach space. A family {T(t); t ≥ 0} of bounded linearoperators on X is said to be a (C0)-semigroup on X (cf.[12, pp.14]) if it satisfies(C1) T(0) = I, I is the identity operator on X;(C2) T(t + s) = T(t)T(s) for every t, s ≥ 0 (the semigroup property);(C3) T(·)x is strongly continuous on t ≥ 0 for every x ∈ X.We define S(t)x :=∫ t0 T(s)xds for all t ≥ 0 and x ∈ X. The infinitesimal generatorof T(·) is the linear operator A defined byD(A) :={x ∈ X; limt→0T(t)x−xt exists}Ax := limt→0+T(t)x−xt for x ∈ D(A).Fractional powers of the generator of a (C0)-semigroup have various representationformulae. For example, if −A is the generator of a uniformly bounded (C0)-group (cf.[4,p.62]) or an analytic (C0)-semigroup with||T(t)|| ≤ Me−δt for t ≥ 0and for some constant M, δ > 0(cf. [13, p.69]), then the fractional power Aα of A canbe defined as(1) Aαx =sin παπ∫ ∞0tα−1(tI + A)−1Axdt for x ∈ D(A) and 0 0.whenever T(·) is analytic with ||T(t)|| ≤ Me−δt for t ≥ 0 and Γ(α) is the gamma function[14].We introduce the new concept of Gamma operator-valued function on a Banachspace as following:1A family {G(t); t > 0} of bounded linear operators on a Banach space X is said tobe a Gamma operator function on X if it satisfies the following conditions.(G1) For any x ∈ X G(·)x is strongly continuous on (0,∞);(G2) G(t)G(s) = B(t, s)G(t + s) for all t, s > 0, where B(·, ·) is the Beta function;(G3) limt→0+tG(t)x = x for all x ∈ X.In this project, our work will study the following problems:(i) The relation between the Gamma operator functions and (C0)-semigroups.(ii) Apply the Gamma operator function to the generation theorem of (C0)-semigroups
