Fredholm criteria for the algebra generated by Wiener–Hopf operators with continuous symbols

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:msub> <m:mi>ℝ</m:mi> <m:mo>+</m:mo> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {X(R₊)} be a Lorentz space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>ℝ</m:mi> <m:mo>+</m:mo> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {Lp,q(R₊)} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo><</m:mo> <m:mi>p</m:mi> </m:mrow> <m:mo>,</m:mo> <m:mrow> <m:mi>q</m:mi> <m:mo><</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:mrow> </m:math> {1<p,q<∞} or a reflexive Orlicz space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mi mathvariant="normal">Φ</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msub> <m:mi>ℝ</m:mi> <m:mo>+</m:mo> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {LΦ(R₊)} . We establish a Fredholm criterion for operators from the Banach algebra generated by all Wiener–Hopf operators with continuous symbols on the space <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:msub> <m:mi>ℝ</m:mi> <m:mo>+</m:mo> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {X(R₊)} .