Fredholm index and spectral flow in non-self-adjoint case
Spectral flow is a topological invariant introduced by Atiyah, Patodi and Singer for a family of self-adjoint elliptic operators \(\{A(t)\}_{t\in\mathbb{R}}\) of positive order on a closed odd dimensional manifold \(X\). Assuming the invertibility of the limiting operators \(A_{\pm}=\lim_{t\to\pm\infty}A(t)\), the spectral flow \(\mathrm{Sf}(A(t))\) is given by the net number of eigenvalues of \(A(t)\) that cross the origin as \(t\) runs from \(-\infty\) to \(\infty\). Demonstrated by \textit{M. F. Atiyah, V. K. Patodi} and \textit{I. M. Singer} [Math. Proc. Camb. Philos. Soc. 77, 43--69 (1975; Zbl 0297.58008)], the elliptic operator \(D_A=\frac{d}{dt}-A(t)\) on the product manifold \(X\times\mathbb{R}\) is Fredholm and the spectral flow \(\mathrm{Sf}(A(t))\) is equal to the Fredholm index of \(D_A\). Statement of this type is referred to as ``Fredholm index=spectral flow'' theorem and is very useful in some aspect of Floer homology. The main result of the paper is a ``Fredholm index=spectral flow'' theorem when the assumption of \(A(t)\) being self-adjoint is dropped. Here, the author employs the definition of the spectral flow extended to non-self-adjoint paths of elliptic operators [\textit{C. Zhu}, Papers of a workshop in honor of Krzysztof P. Wojciechowski on his 50th birthday, Roskilde, Denmark, May 20--22, 2005. Hackensack, NJ: World Scientific. 493--540 (2006; Zbl 1121.58013)]. Given a family of first order elliptic differential operators (non-self-adjoint) \(\{A(t)\}_{t\in\mathbb{R}}\) on \(X\), the theorem states that \(D_A=\frac{d}{dt}-A(t)\) is Fredholm and its index agrees with the spectral flow of \(\{A(t)\}_{t\in\mathbb{R}}\) provided the following conditions are satisfied: (1) The coefficients of \(A(t)\) depend smoothly on \(t\) and their derivatives are bounded on \(X\times \mathbb{R}\); (2) \(D_A\) is elliptic on \(X\times\mathbb{R}\); (3) The limits \(\lim_{t\to\pm\infty}A(t)\) exist and are hyperbolic, i.e., The spectrum of \(A\) and that of its principal symbol over each co-vector (excluding the zero sections of \(T^*X\)) do not intersect the imaginary axis. The theorem is proved using functional analysis. The author firstly shows that these conditions are sufficient to ensure the Fredholmness of \(D_A\) as well as well-definedness of the spectral flow in the sense of Zhu. Then, as the spectral flow and the Fredholm index are homotopy invariant, the author perturbs \(\{A(t)\}_{t\in\mathbb{R}}\) by a homotopy and obtains a path of hyperbolic operators, where the ``Fredholm index=spectral flow'' theorem is easier to show.
scientific article; zbMATH DE number 6162879