An Application of the Gamma Operator Functions to Semigroup Theory

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

The Generalized Asymtotic C-Resolvent of a Closed Linear Operator on a Banach Space(I)

Tanaka and Okazawa 建立了算子A 的漸進C-豫族理論並且將之應用於局部C-半群的生成定理。在2000 年，Takenaka and Piskarev 修正上面的定義以用來研究局部C-正弦函數以及在X 上的n-次積分正弦函數理論。這是一個兩年研究計畫。在第一年裡，我們將再修正並放寬Tanaka and Okazawa的定義並且考慮下面的研究方向：(1) 建立一個不需要A 有稠密性的局部n-次積分C-半群的生成定理。(2) 應用(1) 的結果來解在X 上的一階抽像Cauchy 問題。在第二年裡，我們將考慮下面的研究方向：(3) 建立一個不需要A 有稠密性的局部n-次積分C-正弦函數的生成定理。(4) 應用(3) 的結果來解在X 上的二階抽像Cauchy 問題。在以上的這些研究基礎上，我們也將考慮應用到近似理論以及其他領域的可能性

Some Approximation Theorems of Almost Positive Operators

令ℓ1 表示所有有界複數值數列的åŠåˆä½µè³¦äºˆç¯„數sup-norm。我們說作用在ℓ1 的正線性泛函是一個Banach limit 如果ä»-滿足ϕ({an+k}) = Ï•({an}) for all k = 1, 2, 3, . . .且當limn!1an 存在時有ϕ({an}) = limn!1an。我們以πLet â„“1 denote the space of all bounded sequences in the comple

Study on Invariant Means on a Bounded and Vector-Valued Functions

令X 是一個布æ-¼è¤‡æ•¸é«”çš„å·´æ°ç©ºé-“。我們以XLet X be a Banach space over the comple