Banach algebras of quasi-triangular operators are spectrally regular

Let f be an analytic function on some bounded Cauchy domain Δ with values in some Banach algebra of quasi-triangular operators and suppose that the contour integral of the logarithmic derivative f′(λ)f-1(λ) along the positively oriented boundary ∂Δ vanishes. We prove that then f takes invertible values on all of Δ. This means that Banach algebras of quasi-triangular operators are spectrally regular.