Algebras of Convolution Type Operators with Continuous Data do Not Always Contain All Rank One Operators

摘要

Let X(R) be a separable Banach function space such that the Hardy-Littlewood maximal operator is bounded on X(R) and on its associate space X'(R). The algebra C-X((R) over dot) of continuous Fourier multipliers on X(R) is defined as the closure of the set of continuous functions of bounded variation on (R) over dot = R boolean OR {infinity} with respect to the multiplier norm. It was proved by C. Fernandes, Yu. Karlovich and the first author [11] that if the space X(R) is reflexive, then the ideal of compact operators is contained in the Banach algebra A(X(R)) generated by all multiplication operators aI by continuous functions a is an element of C((R) over dot) and by all Fourier convolution operators W-0(b) with symbols b is an element of C-X((R) over dot)). We show that there are separable and non-reflexive Banach function spaces X(R) such that the algebra A(X(R)) does not contain all rank one operators. In particular, this happens in the case of the Lorentz spaces L-p,(1) (R) with 1 < p < infinity.