An Invertibility Criterion in a C*-Algebra Acting on the Hardy Space with Applications to Composition Operators
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In this paper, we prove an invertibility criterion for certain operators which is given as a linear algebraic combination of Toeplitz operators and Fourier multipliers acting on the Hardy space of the unit disc. Very similar to the case of Toeplitz operators, we prove that such operators are invertible if and only if they are Fredholm and their Fredholm index is zero. As an application, we prove that for “quasi-parabolic” composition operators the spectra and the essential spectra are equal.
KeywordsComposition operators Hardy spaces essential spectra
Mathematics Subject Classification47B33
The authors would like show their sincere gratitude to Prof. Aydın Aytuna for pointing out a serious flaw in the proof of main theorem which had been overcome in the second version of this manuscript. We also would like to thank the referee for a very detailed report on the paper which pointed out an ambiguity in the proof of the main theorem that has been overcome in this final version of the paper.
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