Abstract
We consider plus-operators in Krein spaces and generated operator linear fractional relations of the following form: . We study some special type of factorization for plus-operators T, among them the following one: T = BU, where B is a lower triangular plus-operator, U is a J-unitary operator. We apply the above factorization to the study of basical properties of relations (1), in particular, convexity and compactness of their images with respect to the weak operator topology. Obtained results we apply to the known Koenigs embedding problem, the Krein-Phillips problem of existing of invariant semidefinite subspaces for some families of plus-operators and to some other fields.
Highlights
During World War II, one of the burning problems was the firing accuracy of the new rocket weapon called “Katyusha”.In 1943, S
We study some special type of factorization for plus-operators T, among them the following one: T = BU, where B is a lower triangular plus-operator, U is a J-unitary operator
Where B is an upper triangular operator, C is a lower block triangular operator, and U and U are J-unitary operators. It follows from the results presented below that each strict plus-operator admits factorization (1)
Summary
During World War II, one of the burning problems was the firing accuracy of the new rocket weapon called “Katyusha”. Where B is an upper triangular operator, C is a lower block triangular operator, and U and U are J-unitary operators It follows from the results presented below that each strict plus-operator (the definition is given below) admits factorization (1). We show that this is not the case for factorization (2). Factorizations (1) and (2) are rather useful tools for studying both the operators in spaces with indefinite metric and the so-called linear-fractional relations of operator balls. There does not exist any direct generalization of the mapping factorization theory to the case of linear-fractional relations. A significant generalization of the factorization theory (to arbitrary linear operators) opens new ways for studying the relations.
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