论文标题
弗雷德·弗雷姆（Fredholm
The Index Bundle for Selfadjoint Fredholm Operators and Multiparameter Bifurcation for Hamiltonian Systems
论文作者
论文摘要
众所周知的事实是，自助式操作员的索引是零，即自助接合操作员的内核垂直于其范围。六十年代，阿蒂亚（Atiyah）和耶尼奇（Jänich）在六十年代被弗雷德霍尔姆（Fredholm）指数推广到家庭，很容易看出，在复杂的希尔伯特（Hilbert）空间上，这种所谓的索引束消失了，因为在单个操作员的情况下，弗雷德·弗雷姆（Fredholm）运营商的家庭消失了。本说明的第一个目的是指出，对于每个真正的希尔伯特空间和每个紧凑的拓扑空间$ x $，都有一个自助式弗雷姆操作员，由$ x \ times s^1 $参数，其中具有非平凡的索引束。此外，我们使用这种观察结果和PEJSACHOWICZ的家庭指数定理研究哈密顿系统的同型溶液的多参数分叉，在此我们概括了先前已知的示例类。
The index of a selfadjoint Fredholm operator is zero by the well-known fact that the kernel of a selfadjoint operator is perpendicular to its range. The Fredholm index was generalised to families by Atiyah and Jänich in the sixties, and it is readily seen that on complex Hilbert spaces this so called index bundle vanishes for families of selfadjoint Fredholm operators as in the case of a single operator. The first aim of this note is to point out that for every real Hilbert space and every compact topological space $X$ there is a family of selfadjoint Fredholm operators parametrised by $X\times S^1$ which has a non-trivial index bundle. Further, we use this observation and a family index theorem of Pejsachowicz to study multiparameter bifurcation of homoclinic solutions of Hamiltonian systems, where we generalise a previously known class of examples.
