2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/102019Given a row contraction of operators on Hilbert space and a family of projections on the space which stabilize the operators, we show there is a unique minimal joint dilation to a row contraction of partial isometries which satisfy natural relations. For a fixed row contraction the set of all dilations forms a partially ordered set with a largest and smallest element. A key technical device in our analysis is a connection with directed graphs. We use a Wold Decomposition for partial isometries to describe the models for these dilations, and discuss how the basic properties of a dilation depend on the row contraction.12 pages, preprintFunctional AnalysisOperator Algebras47A20, 47A45text