Isometric and contractive operators in Kre\v\i n spaces
Algebra i analiz, Tome 3 (1991) no. 3, pp. 110-126.

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Let $T$ be a continuous isometric linear operator on a Krein space $\mathcal K$. In general, $T$ is not isometric with respect to a norm on $\mathcal K$ whose metric topology is the Mackey topology on $\mathcal K$. In this note we give a sufficient condition that a norm exists which preserves an isomerty or contraction. We apply this result to prove that, under a certain assumption, the main transformation of a linear system is similar to a Hilbert space contraction. A slight modification of this result is used to give a new proof of a theorem of Davis and Foias. It says that an operator in a Hilbert space is similar to a contraction if a corresponding transfer function is bounded on the open unit disk. As another application it is used to generalize the Beurling-Lax theorem to Krein spaces which are contained continuously and contractively in a space of square summable power series with coefficients in a Krein space.
@article{AA_1991_3_3_a5,
     author = {Manfred M\"oller},
     title = {Isometric and contractive operators in {Kre\v\i} n spaces},
     journal = {Algebra i analiz},
     pages = {110--126},
     publisher = {mathdoc},
     volume = {3},
     number = {3},
     year = {1991},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/AA_1991_3_3_a5/}
}
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Manfred Möller. Isometric and contractive operators in Kre\v\i n spaces. Algebra i analiz, Tome 3 (1991) no. 3, pp. 110-126. http://geodesic.mathdoc.fr/item/AA_1991_3_3_a5/