  This paper introduces a new theoretical and computational framework for a
data driven Koopman mode analysis of nonlinear dynamics. To alleviate the
potential problem of ill-conditioned eigenvectors in the existing
implementations of the Dynamic Mode Decomposition (DMD) and the Extended
Dynamic Mode Decomposition (EDMD), the new method introduces a Koopman-Schur
decomposition that is entirely based on unitary transformations. The analysis
in terms of the eigenvectors as modes of a Koopman operator compression is
replaced with a modal decomposition in terms of a flag of invariant subspaces
that correspond to selected eigenvalues. The main computational tool from the
numerical linear algebra is the partial ordered Schur decomposition that
provides convenient orthonormal bases for these subspaces. In the case of real
data, a real Schur form is used and the computation is based on real orthogonal
transformations. The new computational scheme is presented in the framework of
the Extended DMD and the kernel trick is used.
