The Maslov index in symplectic Banach spaces. (English) Zbl 07000096
Memoirs of the American Mathematical Society 1201. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2800-6/print; 978-1-4704-4371-9/ebook). x, 122 p. (2018).
Summary: We consider a curve of Fredholm pairs of Lagrangian subspaces in a fixed Banach space with continuously varying weak symplectic structures. Assuming vanishing index, we obtain intrinsically a continuously varying splitting of the total Banach space into pairs of symplectic subspaces. Using such decompositions we define the Maslov index of the curve by symplectic reduction to the classical finite-dimensional case. We prove the transitivity of repeated symplectic reductions and obtain the invariance of the Maslov index under symplectic reduction, while recovering all the standard properties of the Maslov index.
As an application, we consider curves of elliptic operators which have varying principal symbol, varying maximal domain and are not necessarily of Dirac type. For this class of operator curves, we derive a desuspension spectral flow formula for varying well-posed boundary conditions on manifolds with boundary and obtain the splitting formula of the spectral flow on partitioned manifolds.
As an application, we consider curves of elliptic operators which have varying principal symbol, varying maximal domain and are not necessarily of Dirac type. For this class of operator curves, we derive a desuspension spectral flow formula for varying well-posed boundary conditions on manifolds with boundary and obtain the splitting formula of the spectral flow on partitioned manifolds.
Keywords:
Banach bundles; Calderón projection; Cauchy data spaces; elliptic operators; Fredholm pairs; desuspension spectral flow formula; Lagrangian subspaces; Maslov index; symplectic reduction; unique continuation property; variational properties; weak symplectic structure; well-posed boundary conditionsReferences:
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