Invariant subspaces for operators whose polynomials satisfy the -condition

In this note, we show that if T ∈ B ( H ) is such that p ( T ) satisfies the ( G 1 ) -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.