  This manuscript is aimed at addressing several long standing limitations of
dynamic mode decompositions in the application of Koopman analysis. Principle
among these limitations are the convergence of associated Dynamic Mode
Decomposition algorithms and the existence of Koopman modes. To address these
limitations, two major modifications are made, where Koopman operators are
removed from the analysis in light of Liouville operators (known as Koopman
generators in special cases), and these operators are shown to be compact for
certain pairs of Hilbert spaces selected separately as the domain and range of
the operator. While eigenfunctions are discarded in the general analysis, a
viable reconstruction algorithm is still demonstrated, and the sacrifice of
eigenfunctions realizes the theoretical goals of DMD analysis that have yet to
be achieved in other contexts. However, in the case where the domain is
embedded in the range, an eigenfunction approach is still achievable, where a
more typical DMD routine is established, but that leverages a finite rank
representation that converges in norm. The manuscript concludes with the
description of two Dynamic Mode Decomposition algorithms that converges when a
dense collection of occupation kernels, arising from the data, are leveraged in
the analysis.
