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 $M_{⟨\Phi ⟩}$ such that $M_{[\Phi ]}\subset M_{⟨\Phi ⟩}\subset M_{\Phi }$, 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 $a\in M_{⟨\Phi ⟩}$ 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 $\tau \in [-\pi ,\pi )$.

28 pages