Conditioned invariance and detectability subspaces in the behavioral approach * *This work was supported in part by the Portuguese Foundation for Science and Technology (FCT-Fundação para a Ciencia e a Tecnologia), through CIDMA - Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2013 and also by Project POCI-01-0145-FEDER-006933 - SYSTEC - Research Center for Systems and Technologies - funded by FEDER funds through COMPETE2020 - Programa Operacional Competitividade e Internacionalizaçao (POCI) - and by national funds through FCT - Fundao para a Ciencia e a Tecnologia.