On C-algebras of singular integral operators with PQC coefficients and shifts with fixed points

A Fredholm representation on a Hilbert space, whose kernel coincides with the ideal of compact operators, is constructed for the -algebra generated by all multiplication operators by piecewise quasicontinuous (PQC) functions, by the Cauchy singular integral operator and by the unitary weighted shift...

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Bibliographic Details
Published in:Complex variables and elliptic equations 2022-03, Vol.67 (3), p.581-614
Main Authors: Bastos, M. Amélia, Fernandes, Cláudio A., Karlovich, Yuri I.
Format: Article
Language:English
Subjects:
Algebra
Calculus
fixed points
Fixed points (mathematics)
Fredholmness
Hilbert space
Mathematical analysis
Multiplication
Operators (mathematics)
piecewise quasicontinuous function
representation of a
Singular integral operator with shifts
Topology
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Summary:A Fredholm representation on a Hilbert space, whose kernel coincides with the ideal of compact operators, is constructed for the -algebra generated by all multiplication operators by piecewise quasicontinuous (PQC) functions, by the Cauchy singular integral operator and by the unitary weighted shift operators , acting on the space over the unit circle . Here G denotes a discrete amenable group of orientation-preserving piecewise smooth homeomorphisms with finite sets of discontinuities for their derivatives , which acts topologically freely on , where is the interior of the nonempty closed set composed by all common fixed points for all shifts , with boundary of zero Lebesgue measure. A Fredholm symbol calculus for the -algebra is constructed and a Fredholm criterion for the operators is established.
ISSN:1747-6933
1747-6941
DOI:10.1080/17476933.2021.1944778