Fredholm discrete Wiener–Hopf operators on reflexive rearrangement-invariant Banach sequence spaces are one-sided invertible

Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>X</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ℤ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {X(Z)} be a reflexive rearrangement-invariant Banach sequence space and let a be a periodic distribution generating the bounded Laurent operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>L</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>a</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {L(a)} on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>X</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ℤ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {X(Z)} . We prove that if the discrete Wiener–Hopf operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>T</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>a</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {T(a)} is Fredholm on the subspace <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>X</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>ℤ</m:mi> <m:mo>+</m:mo> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mi>f</m:mi> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:msub> <m:mi>f</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo stretchy="false">}</m:mo> </m:mrow> <m:mrow> <m:mi>k</m:mi> <m:mo>∈</m:mo> <m:mi>ℤ</m:mi> </m:mrow> </m:msub> <m:mo>∈</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>ℤ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>:</m:mo> <m:mrow> <m:msub> <m:mi>f</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>⁢</m:mo> <m:mtext> for </m:mtext> <m:mo>⁢</m:mo> <m:mi>k</m:mi> </m:mrow> <m:mo><</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:mrow> </m:math> {X(Z₊)=\{f=\{fₖ\}ₖ∈ X(Z):fₖ=0% for k<0\}} and has index κ, then <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>T</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>a</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {T(a)} is left-invertible if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>κ</m:mi> <m:mo>≤</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {κ≤ 0} , right-invertible if <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>κ</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math>