A note on the modulus of continuity for ill-posed problems in Hilbert space

The authors study linear ill-posed operator equations in Hilbert space. Such equations become conditionally well-posed by imposing certain smoothness assumptions, often given relative to the operator which governs the equation. Usually this is done in terms of general source conditions. Recently smoothness of an element was given in terms of properties of the distribution function of this element with respect to the self-adjoint associate of the underlying operator. In all cases the original ill-posed problem becomes well-posed, and properties of the corresponding modulus of continuity are of interest, specifically whether this is a concave function. The authors extend previous concavity results of a function related to the modulus of continuity, and obtained for compact operators in B. Hofmann, P. Mathé, and M. Schieck, Modulus of continuity for conditionally stable ill-posed problems in Hilbert space, J. Inverse Ill-Posed Probl. 16 (2008), no. 6, 567–585, to the general case of bounded operators in Hilbert space, and for recently introduced smoothness classes.