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Erschienen in:
2018 | OriginalPaper | Buchkapitel
Factorization of Singular Integral Operators with a Carleman Backward Shift: The Vector Case
verfasst von : Amarino B. Lebre, Juan S. Rodríguez
Erschienen in: Operator Theory, Operator Algebras, and Matrix Theory
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On the vector Lebesgue space on the unit circle $${L}^{n}_{p} ({p} \in (1, \infty), {n} \in \mathbb{N}$$ we consider singular integral operators with a Carleman backward shift of linear fractional type, of the form $${T}_{A,B} = {AP}_{+} + {BP}_{-}$$ with A = aI + bU, B = cI + dU, where $${a, b, c, d} \in {L}^{n \times n}, {P}_{\pm} = \frac{1}{2}({I} {\pm} {S})$$ are the Cauchy projectors in $${L}^{n}_{p}$$ defined componentwise, and U is an involutory shift operator associated with the given Carleman backward shift also defined componentwise. By generalization to the vector case (n > 1) of the previously obtained results for the scalar case (n = 1), it is shown that whenever a certain 2n × 2n matrix function, associated with the original singular integral operator, admits a bounded factorization in $${L}^{2n}_{p}$$ the Fredholm characteristics of the paired operator TA,B can be obtained in terms of that factorization, in particular the dimensions of the kernel and of the cokernel.