Fourier Integral Operators of Boutet de Monvel Type

Given two compact manifolds $X,Y,$ with boundary and a boundary preserving symplectomorphism $\chi :T^{\ast }Y\setminus 0\to T^{\ast }X\setminus 0$, which is one-homogeneous in the fibers and satisfies the transmission condition, we introduce Fourier integral operators of Boutet de Monvel type associated with $\chi$. We study their mapping properties between Sobolev spaces, develop a calculus and prove a Egorov type theorem. We also introduce a notion of ellipticity which implies the Fredholm property. Finally, we show how – in the spirit of a classical construction by A. Weinstein – a Fredholm operator of this type can be associated with $\chi$ and a section of the Maslov bundle. If $\dim Y>2$ or the Maslov bundle is trivial, the index is independent of the section and thus an invariant of the symplectomorphism.