On multiplier analogues of the algebra C+H∞ on weighted rearrangement-invariant sequence spaces

Let X(Z) be a reflexive rearrangement-invariant Banach sequence space with nontrivial Boyd indices αX,βX and let w be a symmetric weight in the intersection of the Muckenhoupt classes A1/αX(Z) and A1/βX(Z). Let MX(Z,w) denote the collection of all periodic distributions a generating bounded Laurent operators L(a) on the space X(Z,w)={φ:Z→ℂ:φw∈X(Z)}. We show that MX(Z,w) is a Banach algebra. Further, we consider the closure of trigonometric polynomials in MX(Z,w) denoted by CX(Z,w) and HX(Z,w)∞,±={a∈MX(Z,w):â(±n)=0 for n<0}. We prove that CX(Z,w)+HX(Z,w)∞,± are closed subalgebras of MX(Z,w).