Gohberg-Krupnik Localisation for Discrete Wiener-Hopf Operators on Orlicz Sequence Spaces

Abstract

Let $\Phi$ be an $N$-function whose Matuszewska-Orlicz indices satisfy $1<\alpha_\Phi\le\beta_\Phi<\infty$. Using these indices, we introduce ``interpolation friendly" classes of Fourier multipliers $M_{[\Phi]}$ and $M_{\langle\Phi\rangle}$ such that $M_{[\Phi]}\subset M_{\langle\Phi\rangle}\subset M_\Phi$, where $M_\Phi$ is the Banach algebra of all Fourier multipliers on the reflexive Orlicz sequence space $\ell^\Phi(\mathbb{Z})$. Applying the Gohberg-Krupnik localisation in the corresponding Calkin algebra, the study of Fredholmness of the discrete Wiener-Hopf operator $T(a)$ with $a\in M_{\langle\Phi\rangle}$ is reduced to that of $T(a_\tau)$ for certain, potentially easier to study, local representatives $a_\tau\in M_{[\Phi]}$ of $a$ at all points $\tau\in[-\pi,\pi)$.