Abstract
Commutative Toeplitz-composition subalgebras of the Calkin algebra on the classical Hardy space are analyzed with piecewise quasicontinuous symbols and a linear fractional non-automorphism fixing a boundary point. For the parabolic case the fiber structure of the maximal ideal space of the C*-subalgebra is explicitly described, resulting in essential spectrum and essential norm formulas. For the non-parabolic case, the Shilov boundary of a non-self-adjoint subalgebra is identified and essential spectra obtained. The subalgebra, although not maximal commutative, preserves spectra in the Calkin algebra. Fredholm index formulas are obtained for both cases.
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