X-valued Smirnov classes and t-basicity of Faber polynomials

Abstract

In this work we consider X-valued Lebesgue space (so called Bochner space)L-p(Gamma;X), 1 < p < + infinity,where Gamma isaclosed,simpleLya-punov or Radon curve on complex plane. We introduce the concept of at-basis forLp(Gamma;X), and investigate systems of generalized Faber polynomials corresponding to the domainsint Gamma andext Gamma. Some operatorsare defined that act from the Hardy-Bochner classes H-p(+/-)(X) to L-p(Gamma;X), where X is a UMD space. We prove the invertibility of these operatorsand, using the results oft-basicity of parts of the exponential systemfor H-p(+/-)(X) spaces. Additionally, we define the X-valued Smirnov classes E-p(+/-)(X) corresponding to the domainsint Gamma andext Gamma, respectively. It isproved that Faber polynomials form at-basis for Smirnov-Faber classes,and using this fact, the t-basisness of double the system of Faber polynomials for Bochner spaces is established. Some properties of Smirnov-Bochner classes are also studied.