Necessary conditions for a given convergence rate of iterative methods for solution of linear ill-posed operator equations in a Banach space

We study the rate of convergence of iterative methods for solution of linear ill-posed equations with sectorial operators in the Banach space. It is stated that the power sourcewise representation condition on the initial discrepancy with an arbitrary positive exponent which is sufficient for the power estimate of the convergence rate with the same exponent is actually close to be necessary and thus cannot be essentially relaxed.