Invariant Subspaces for Commuting Operators on a Real Banach Space

详细

It is proved that the commutative algebra A of operators on a reflexive real Banach space has an invariant subspace if each operator T ∈ A satisfies the condition

\({\left\| {1 - \varepsilon {T^2}} \right\|_e} \leqslant 1 + o\left( \varepsilon \right)as\varepsilon \searrow 0,\)

where ║ · ║_e denotes the essential norm. This implies the existence of an invariant subspace for any commutative family of essentially self-adjoint operators on a real Hilbert space.