On weighted slant Hankel operators

Abstract

Inthispaper, weintroduceandstudythenotionofweightedslantHankeloperatorK β ϕ , ϕ ∈ L ∞ (β) on the space L 2 (β), β = {βn}n∈Z being a sequence of positive numbers with β0 = 1. In addition to some algebraic properties, the commutant and the compactness of these operators are discussed. 1. Preliminaries and Introduction Laurent operators (8) or multiplication operators Mϕ(f 7→ ϕf) on L 2 (T) induced by ϕ ∈ L ∞ (T), T being the unit circle, play a vital role in the theory of operators with their tendency of inducing various classes of operators. In the year 1911, O. Toeplitz (15) introduced the Toeplitz operators given as Tϕ = PMϕ, where P is an orthogonal projection of L 2 (T) onto H 2 (T) and later in 1964, Brown and Halmos (4) studied algebraic properties of these operators. We refer (1, 2, 4, 7, 9 and 12) for the applications and extensions of study to Hankel operators, slant Toeplitz operators, slant Hankel operators and k th -order slant Hankel operators. In the mean time, the notions of weighted sequence spaces H 2 (β) and L 2 (β) also gained momentum. Shield (14) made a systematic study of the Laurent operators on L 2 (β). We prefer to call the Laurent operator on L 2 (β) as weighted Laurent operator. Weighted Toeplitz operators, Slant weighted Toeplitz operators and weighted Hankel operaots on L 2 (β) are discussed in (10), (3) and (5, 6) respectively. In this paper, we extend the study to a new class of operators namely, weighted slant Hankel operators and describe its algebraic properties. We now begin with the notations and preliminaries that are needed in the paper. We consider the space L 2 (β) of all formal Laurent series f(z) = ∑ n∈Z a nz n , an ∈ C, (whether or not the series