Factorization of Singular Integral Operators with a Carleman Backward Shift: The Vector Case

Abstract

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 T_A,B can be obtained in terms of that factorization, in particular the dimensions of the kernel and of the cokernel.