An analytic Toeplitz operator that commutes with a compact operator and a related class of Toeplitz operators

Abstract

For a bounded analytic function, ƒ, on the unit disk, D, let T ƒ and M ƒ denote the operators of multiplication by ƒ on H 2( ∂D) and L 2( ∂D), respectively. In their 1973 paper, Deddens and Wong asked whether there is an analytic Toeplitz operator T ƒ that commutes with a nonzero compact operator, and whether every operator that commutes with an analytic Toeplitz operator has an extension that commutes with the corresponding multiplication operator on L 2. In the first part of this paper, we give an explicit example of an analytic Toeplitz operator T φ that settles both of these questions. This operator commutes with a nonzero compact operator (a composition operator followed by an analytic Toeplitz operator). The only operators in the commutant of T φ that extend to commute with M φ are analytic Toeplitz operators. Although the commutant of T φ contains more than just analytic Toeplitz operators, T φ is irreducible. The remainder of the paper seeks to explain more fully the phenomena incorporated in this example by introducing a class of analytic functions, including the function φ, and giving additional conditions on functions g in the class to determine whether T g commutes with nonzero compact operators, whether T g is irreducible, and which operators in the commutant of T g extend to the commutant of M g . In particular, we find representations for operators in the commutant and second commutant of T g .