Quasi-uniformly Positive Operators in Krein Space

BRANKO CURGUS and BRANKO NAJMAN Deenitizable operators in Krein spaces have spectral properties similar to those of selfadjoint operators in Hilbert spaces. A suucient condition for deenitizability of a selfadjoint operator A with a nonempty resolvent set (A) in a Krein space (H; j ]) is the niteness of the number of negative squares of the form Axjy] (see 10, p. 11]). 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 N s is nite dimensional and the space (D(s); s(;)) is a direct sum of a Pontryagin space with a nite number (s) of negative squares and N s. The number (s) := dimN s + (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 (H; j ]) is quasi-uniformly positive if the form a(x; y) = Axjy] deened on D(A) is closable and its closure a is quasi-uniformly positive. The number (A) := (a) is the negativity index of A: Such operators often appear in applications, see 3, 4, 5] and Section 3 of this note. 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 deened as forms, so the above deenition 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 7]. 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 suuciently general for several important applications. As an illustration of these results we consider the operator associated with the Klein-Gordon equation " @ @t ieq 2 X j @ @x j ieA j 2 + m 2 # u = 0: Setting u 1 = u; u 2 = i @ @t eq u we get a system of equations for (u 1 ; u 2): The associated operator is quasi-uniformly positive in a Krein space suggested by the physical interpretation of the equation. The obtained