Completion, extension, factorization, and lifting of operators

Abstract

The famous results of M.G. Kreȋn concerning the description of selfadjoint contractive extensions of a Hermitian contraction T 1 and the characterization of all nonnegative selfadjoint extensions A of a nonnegative operator A via the inequalities A K ≤ A ≤ A F , where A K and A F are the Kreȋn-von Neumann extension and the Friedrichs extension of A, are generalized to the situation, where A is allowed to have a fixed number of negative eigenvalues. These generalizations are shown to be possible under a certain minimality condition on the negative index of the operators I − T * 1 T 1 and A, respectively; these conditions are automatically satisfied if T 1 is contractive or A is nonnegative, respectively.The approach developed in this paper starts by establishing first a generalization of an old result due to Yu.L. Shmul'yan on completions of 2 × 2 nonnegative block operators. The extension of this fundamental result allows us to prove analogs of the above mentioned results of M.G. Kreȋn and, in addition, to solve some related lifting problems for J-contractive operators in Hilbert, Pontryagin and Kreȋn spaces in a simple manner. Also some new factorization results are derived, for instance, a generalization of the well-known Douglas factorization of Hilbert space operators. In the final steps of the treatment some very recent results concerning inequalities between semibounded selfadjoint relations and their inverses turn out to be central in order to treat the ordering of non-contractive selfadjoint operators under Cayley transforms properly.

Completion and extension of operators in Kreĭn spaces

“…Namely, we consider classes of "quasicontractive" symmetric operators T 1 in a Kreȋn space with ν − (I − T * 1 T 1 ) < ∞ and we describe all possible selfadjoint (in the Kreȋn space sense) extensions T of T 1 which preserve the given negative index ν − (I −T * T ) = ν − (I −T * 1 T 1 ). This problem is close to the completion problem studied in [5] and has a similar description for its solutions. For further history behind this problem see also [1,2,3,7,8,9,10,11,12,14,15,16,20].…”

“…Moreover, there is an equality κ = κ − + κ + (see [5,Lemma 5.1]). We recall the results for the operator T 11 from the paper [5] and after that reformulate them for the operator T 11 . We recall completion problem and its solutions that was investigated in a Hilbert space setting in [5].…”

“…In paper [5] we studied classes of "quasi-contractive" symmetric operators T 1 allowing a finite number of negative eigenvalues for the associated defect operator I − T * 1 T 1 , i.e., ν − (I − T * 1 T 1 ) < ∞ as well as "quasi-nonnegative" operators A with ν − (A) < ∞ and the existence and description of all possible selfadjoint extensions T and A of them which preserve the given negative indices ν − (I − T 2 ) = ν − (I − T * 1 T 1 ) and ν − ( A) = ν − (A), and proved precise analogs of the above mentioned results of M.G. Kreȋn under a minimality condition on the negative indices ν − (I − T * 1 T 1 ) and ν − (A), respectively.…”

Operators on Anti-dual pairs: Self-adjoint Extensions and the Strong Parrott Theorem

“…The question whether a self-adjoint extension exists arises naturally in various situations when a partially defined (bounded or unbounded) symmetric operator is given. For classical results we refer the reader to [2,6,8,16] and the references therein, for more recent results see for example [3,11]. In our previous paper [19], we have developed a Krein-von Neumann type extension theory for positive operators acting on anti-dual pairs.…”

Schur complements of selfadjoint Krein space operators

“…In Section 4 we present our main results concerning the Schur complement of a Krein space operator. This section includes three subsections: the first deals with the notion of weak complementability on Krein spaces; the second presents an application inspired on some completion problems previously considered in Hilbert and Krein spaces by Baidiuk and Hassi in [6] and [7]; in the last subsection our notion of Schur complement in the Krein space setting is compared to those in [18] and [19].…”