Inverse problems with non-compact operators

In the recent studies of inverse problems with random noise, one of most common assumptions is that the linear operator A is compact. This hypothesis is natural and has fine statistical properties. However, dealing with compact operators is not necessary. By use of the Spectral Theorem we may extend results for compact operators to any linear continuous operator. Using the method of unbiased risk estimation, we prove some oracle inequality for this estimator. As examples of non-compact operators, the problem of convolution on R and the estimation of the derivative of some function are studied.