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**Quasi-uniformly positive operators in Krein space.**
*(English)*
Zbl 0883.47020

Gohberg, I. (ed.) et al., Operator theory and boundary eigenvalue problems. Proceedings of the international workshop held in Vienna, Austria, July 27-30, 1993. Basel: Birkhäuser. Oper. Theory, Adv. Appl. 80, 90-99 (1995).

Definitizable operators in Krein spaces have spectral properties similar to those of selfadjoint operators in Hilbert spaces. A sufficient condition for definitizability of a selfadjoint operator \(A\) with a nonempty resolvent set \(\rho(A)\) in a Krein space \(({\mathcal H},[\cdot|\cdot])\) is the finiteness of the number of negative squares of the form \([Ax|y]\).

In this note, we consider a more restrictive class of operators which we call quasi-uniformly positive. A closed symmetric form \(s\) is called quasi-uniformly positive if its isotropic part \({\mathcal N}_s\) is finite-dimensional and the space \(({\mathcal D}(s),s(\cdot,\cdot))\) is a direct sum of a Pontryagin space with a finite number \(\pi(s)\) of negative squares and \({\mathcal N}_s\). The number \(\kappa(s):= \dim{\mathcal N}_s+ \pi(s)\) is the number of nonpositive squares of \(s\); it is called the negativity index of \(s\). A selfadjoint operator \(A\) in a Krein space \(({\mathcal H},[\cdot|\cdot])\) is quasi-uniformly positive if the form \(a(x,y)= [Ax|y]\) defined on \({\mathcal D}(A)\) is closable and its closure \(\overline a\) is quasi-uniformly positive. The number \(\kappa(A):= \kappa(\overline a)\) is the negativity index of \(A\).

It turns out that this class of operators is stable under relatively compact perturbations, see Corollaries 1.2 and 2.3. The perturbations as well as the operators are usually defined as forms, so the above definition is natural.

Most of the results in this note are known. In particular, the perturbation results from Section 2 are consequences of the results of [P. Jonas, J. Oper. Theory 25, No. 1, 183-211 (1991; Zbl 0796.47024)]. We have found it useful to state the results in the framework of quadratic forms and quasi-uniformly positive operators since the proofs and the statements are simpler but still sufficiently general for several important applications.

For the entire collection see [Zbl 0826.00022].

In this note, we consider a more restrictive class of operators which we call quasi-uniformly positive. A closed symmetric form \(s\) is called quasi-uniformly positive if its isotropic part \({\mathcal N}_s\) is finite-dimensional and the space \(({\mathcal D}(s),s(\cdot,\cdot))\) is a direct sum of a Pontryagin space with a finite number \(\pi(s)\) of negative squares and \({\mathcal N}_s\). The number \(\kappa(s):= \dim{\mathcal N}_s+ \pi(s)\) is the number of nonpositive squares of \(s\); it is called the negativity index of \(s\). A selfadjoint operator \(A\) in a Krein space \(({\mathcal H},[\cdot|\cdot])\) is quasi-uniformly positive if the form \(a(x,y)= [Ax|y]\) defined on \({\mathcal D}(A)\) is closable and its closure \(\overline a\) is quasi-uniformly positive. The number \(\kappa(A):= \kappa(\overline a)\) is the negativity index of \(A\).

It turns out that this class of operators is stable under relatively compact perturbations, see Corollaries 1.2 and 2.3. The perturbations as well as the operators are usually defined as forms, so the above definition is natural.

Most of the results in this note are known. In particular, the perturbation results from Section 2 are consequences of the results of [P. Jonas, J. Oper. Theory 25, No. 1, 183-211 (1991; Zbl 0796.47024)]. We have found it useful to state the results in the framework of quadratic forms and quasi-uniformly positive operators since the proofs and the statements are simpler but still sufficiently general for several important applications.

For the entire collection see [Zbl 0826.00022].

### MSC:

47B50 | Linear operators on spaces with an indefinite metric |

46C20 | Spaces with indefinite inner product (Kreĭn spaces, Pontryagin spaces, etc.) |

47A55 | Perturbation theory of linear operators |