Then z can be only real, otherwise, the selfadjoint operator corresponding to (1.7), (1.8) will have nonreal eigenvalues, which is impossible.
For that case, we describe the spectrum of the selfadjoint operator S(U) in terms of structural properties of U. In the model, U will be realized as a unitary scaling operator of the formf(x)↦f(cx), c>1, and the spectrum of S Uc) is then computed in terms of the given number c.
Let be a selfadjoint operator on the Hilbert space and assume that for some scalars with If is a convex function on then (MP).
Let be a selfadjoint operator on the Hilbert space and assume that for some scalars If and are continuous on and and then (3.5).
Let H⩾1 be a selfadjoint operator in H, let J be a linear and bounded operator from (D H1/2),∥H1/2⋅∥) to Haux and for β>0 let HJβ be the nonnegative selfadjoint operator in H satisfying ∥(HβJ 1/2f∥2="∥H1/2f∥2+β∥Jf∥aux2,f∈D((HβJ 1/2)="D H1/2).Let DJ∞ be the limit of the operators DβJ= H-1-(HβJ)-1.
Let H be a nonnegative selfadjoint operator, E the closed quadratic form associated with H, and P a nonnegative quadratic form such that E+P is closed and D(P ⊃D(H).
Let be an interval and a convex and differentiable function on (the interior of whose derivative is continuous on If is a selfadjoint operator on the Hilbert space with then (2.1).
Let be an interval and a convex and differentiable function on (the interior of whose derivative is continuous on. If is a selfadjoint operator on the Hilbert space with then (3.2). for any with.
We prove in this paper resolvent estimates for the boundary values of resolvents of selfadjoint operators on a Krein space: if H is a selfadjoint operator on a Krein space H, equipped with the Krein scalar product ⟨⋅|⋅⟩, A is the generator of a C0-group on H and I⊂R is an interval such that:.
for each with (b) If is a selfadjoint operator on, then we have the inequality (2.11) .
