An Example of J-unitary Operator. Solving a Problem Stated by M.G. Krein

الأصل · EN

Theorem 1. Given a number c >= 1, there exists a J-unitary operator V, such that: (a) r(V)= r(V⁻¹)= c; (b) S(c⁻¹V)=S(c⁻¹V⁻¹) =S(c⁻¹V*⁻¹) = S(c⁻¹V*)=0 (c) there exist maximal strictly positive and strictly negative V± ¹-invariant subspaces L₊, L₋, such that they are mutually J-orthogonal and L₊ + L₋ is dense in the space. (d₁) if L₁ is non-zero V-invariant subspace, then r(V|L₁)=r(V) (d₂) if L₂ is non-zero V⁻¹-invariant subspace, then r(V⁻¹|L₂)=r(V⁻¹).