= Logarithm from a spectral slit in a Banach algebra
{title2=$x=\exp(\Theta_x(\log))$}
If a <compact> <spectrum of an element> avoids zero and zero belongs to its unbounded complementary component, choose a simple arc from zero to infinity avoiding the <Banach algebra spectrum>. Its complement is a <simply connected> neighborhood of the <Banach algebra spectrum>, admitting a <holomorphic logarithm>. Apply <holomorphic functional calculus> to that logarithm and the <composition rule for holomorphic functional calculus> to the exponential to obtain the displayed identity. An invertible <matrix> over the <complex numbers> always satisfies the hypothesis because its finite <Banach algebra spectrum> cannot separate zero from infinity.
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