On Singular Critical Points of Positive Operators in Krein Spaces

We give an example of a positive operator B in a Krein space with the following properties: the nonzero spectrum of B consists of isolated simple eigenvalues, the norms of the orthogonal spectral projections in the Krein space onto the eigenspaces of B are uniformly bounded and the point ∞ is a singular critical point of B. An operator A in the Krein space (K, [ · , · ]) is said to be positive if [Ax, x] > 0 for all nonzero x in the domain of A. A bounded positive operator A in the Krein space (K, [ · , · ]) has a projection valued spectral function E with 0 being its only possible critical point (see [1, Theorem IV.1.5] or [5, Section II.3.]). Recall that, by [5, Proposition 5.6], the condition ‖E((−∞, α])‖ ≤ C− <∞ for all α < 0 (1) is equivalent to the existence of the limit limα↑0E((−∞, α]) in the strong operator topology. Similarly, ‖E([β,∞))‖ ≤ C+ <∞ for all β > 0 (2) is equivalent to the existence of the limit limβ↓0E([β,+∞)) in the strong operator topology. Since 0 is not an eigenvalue of a positive operator A, [5, Proposition 3.2] implies that (1) and (2) are equivalent. Also, if 0 is a critical point, it is said to be regular if one of the conditions (1) or (2) is fulfilled. If the critical point 0 is not regular, it is called singular. In the sequel the operator A considered will have a discrete spectrum outside 0. Examples of bounded positive operators inK having 0 as a singular critical point can be constructed as follows (see also the examples in [2, Section 1], [3], [4]). Consider a sequence of two-dimensional Krein spaces Kn = C with fundamental symmetry Jn = ( 1 0 0 −1 ) and positive operators An in Kn; denote by λn (λn , respectively) its positive (negative, respectively) eigenvalues and by P n (P − n , respectively) the orthogonal (in Kn) projection onto the corresponding eigenspace. Received by the editors October 15, 1998. 2000 Mathematics Subject Classification. Primary 47B50, 46C50.