Extension Theorems for Contraction Operators on Kreĭn Spaces

Decompositions of a Krein space in regular subspaces invariant under a uniformly bounded C0-semigroup of bi-contractions

“…The statement about the bi-contractive property follows from the fact that a contraction on a Krein space is a bi-contraction if and only if it maps a maximal uniformly negative subspace onto a maximal uniformly negative subspace (see, for example, [DR,Theorem 1.3.6]). For a proof of the statement concerning power boundedness we refer to the proof of [ABDJ,Theorem 2.5].…”

“…We assume that the reader is familiar with the geometry of indefinite inner product spaces and the corresponding operator theory; see the books [AI,Bo,IKL] and also [A2,DR]. We briefly recall some of the notions, also in order to make clear the notations used in this paper.…”

An Extension Theorem for a Sequence of Krein Space Contractions

“…It is shown in [2] (see also [3]) that if is a contraction, then there exist contractive row and column extensions of where the extension space is a Hilbert space. Contractive 2 × 2 matrix extensions of a contractive operator : H → K where the extending space E is a Hilbert space are thoroughly discussed in [2] where Lemma 1, Theorem 2, and Lemma 3 can be found. Results more general than those provided by Lemma 1 and Theorem 2 can be found in [1] while Lemma 3 can also be found in [4].…”

“…In particular, the commutant lifting theorem in the Hilbert space case which was obtained by Sz.-Nagy and Foias has been used to solve extension problems like the ones of Nevanlinna-Pick, Nudelman, Nehari, and many others. Extensions of this theorem to an indefinite setting are given in [1][2][3][4][5]. In [6] (see also [7,8]), a time-variant version of the commutant lifting theorem is developed.…”

Abstract Hankel Operators in Kreĭn Spaces

“…We refer to [1], [4], [6], [9] and [13] as authoritative sources of information about indefinite inner product spaces and operators on them, and to [8] for the treatment of the Kreȋn space extensions of the Hilbert space notions of defect and Julia operators, minimal isometric dilations and minimal unitary dilations.…”