Wiener-Hopf Factorisation and Fredholmness of Toeplitz Operators on Abstract Hardy Spaces

Let X be a Banach function space over the unit circle T T , let $$X'$$ X ′ be its associate space, and let H [ X ] and $$H[X']$$ H [ X ′ ] be abstract Hardy spaces built upon X and $$X'$$ X ′ , respectively. Suppose the Riesz projection P is bounded from X onto H [ X ]. We say that a function a invertible in L^∞ L ∞ admits a Wiener-Hopf factorisation in X if it can be written as a=a₋e_κ a₊ a = a - e κ a + , where the middle factor is e_κ (t)=t^κ e κ ( t ) = t κ with t∈ T t ∈ T and κ ∈ Z κ ∈ Z , the outmost factors satisfy a₋̄∈ H[X], a₋⁻¹∈ H[X'], a₊∈ H[X'], a₊⁻¹∈ H[X], a - ¯ ∈ H [ X ] , a - - 1 ¯ ∈ H [ X ′ ] , a + ∈ H [ X <mml:m