Cosine算子值函数在t＞0上一致算子拓扑连续性的特征刻划

Charactcfization of Uniform Operator Topology Continuity of Cosine Operator Function for t>0

设Ｅ是一个Ｂａｎａｃｈ空间，考虑Ｅ上的强连续Ｃｏｓｈｅ算子值函数我们研究｛Ｃ（ｔ）｝－ｘ＜ｔ＜ｘ在ｔ＞０上一致算子拓扑划问题，本文证明了：｛Ｃ（ｔ）｝－ｘ＜ｔ＜ｘ在ｔ＞０上按一致算子拓扑连续的充分必要条件是其无穷小生成元为有界线性算子。进而得知：适定的Ｅ中的非完全二阶微分方程的Ｃａｕｃｈｙ问题（＊）．（Ａ为Ｅ上的一个闭稠定的线性算子）的解算子在ｔ＞０上按一致算子拓扑连续当且仅当Ａ有界，这种特征与Ｃ０类算子半群即适定的一阶抽象线性微分方程Ｃａｕｃｈｙ问题的解算子及适定的完全二阶及阶数ｎ≥３的高阶抽象线性微分方程Ｃａｕｃｈｙ问题解算于的性质迥然不同。

Let E be a Banach space Of concern is the strongly continuouus Cosine operator fonction{C(t)}-∞<t<∞on E。 We study the uniform operator topology continuity of {C(t)}-∞<t<∞for t>0.It isshown that {C(t)}-∞<t<∞is continuous in the uniform operator topology for t >0 iffits infinitesimal generator is bounded Accordingly,we know that the propagators of a wellpopsed Cauchy problem for theequation( * ):u”(t)=Au(t),t>0;u(0)=u0,u'(0)=u1,(where A is a densely defined closed linear op-erator on E) are continuous in the uniform operator topology for t > 0 iff A is bounded. Thischaracterization is rather different from that of C0 semigroup and the propagators of a well posed com-plete second order or higher order(n≥3) abstract Cauchy problem。