Operator and System Theory Seminar
SPEAKER: Paul Fuhrmann, BGU
TITLE:   On invariant subspaces and intertwining maps
DATE:    Monday, February 6, 2012 
PLACE:   Room 201(NOTICE THE UNUSUAL LOCATION!), Mathematics Building, BGU
TIME:	 14:10
The present paper can be considered as a follow up to Fuhrmann and Helmke [2012], filling in gaps left open in that paper. As in the above mentioned paper, the context in which we work is that of polynomial models, introduced in Fuhrmann [1976]. There are several advantages in taking a polynomial approach. First and foremost, it is an efficient one and allows us to pass easily from the level of arithmetic of polynomial matrices to the geometric level of invariant subspaces. The polynomial model theory not only provides a characterization of the commutant of a given transformation as well as all maps intertwining two given ones, but at the same characterizes, in terms of coprimeness of polynomial matrices, the invertibility properties of these maps. The main topic of the present paper is the representation of invariant subspaces of a linear transformation as kernels and images of maps commuting with it. This extends a result of Halmos [1971]. See also Domanov [2010], who presented a short proof based on elementary matrix calculations and a clever choice of coordinates, and the references therein. However, the present paper has a much broader scope, treating also the embeddability of quotient modules into the model, the relation between the invariant factors of a polynomial model and those of its submodules and quotient modules. We focus also on the study of how complementarity of invariant subspaces is related to the invertibility of linear maps. That such a connection exists is not surprising as both properties can be characterized in terms of coprimeness of polynomial matrices. This analysis connects to the concept of skew-primeness, introduced in Wolovich [1978] as well as to a theorem of Roth [1952]. For a geometric interpretation of skew-primeness, see Khargonekar, Georgiou and ĻOzgĻuler [1983]. Fuhrmann [1994] contains an infinite dimensional generalization of skew-primeness. This opens up the possibility of establishing the analog of Halmos's theorem in the context of backward shift invariant subspaces. A different approach, based on dimension arguments, to Roth's theorem is given in Flanders and Wimmer [1977].

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