Invariant subspaces for operators whose polynomials satisfy the (G1)-condition

In this note, we show that if T∈B(H) is such that p(T) satisfies the (G1)-condition for every polynomial p and if the outer boundary of the spectrum of T is a finite union of Jordan curves, then T has a nontrivial invariant subspace. In addition, we give a connection between operator-valued Poisson kernels and invariant subspaces.