  We study regularization of ill-posed equations involving multiplication
operators when the multiplier function is positive almost everywhere and zero
is an accumulation point of the range of this function. Such equations
naturally arise from equations based on non-compact self-adjoint operators in
Hilbert space, after applying unitary transformations arising out of the
spectral theorem. For classical regularization theory, when noisy observations
are given and the noise is deterministic and bounded, then non-compactness of
the ill-posed equations is a minor issue. However, for statistical ill-posed
equations with non-compact operators less is known if the data are blurred by
white noise. We develop a regularization theory with emphasis on this case. In
this context, we highlight several aspects, in particular we discuss the
intrinsic degree of ill-posedness in terms of rearrangements of the multiplier
function. Moreover, we address the required modifications of classical
regularization schemes in order to be used for non-compact statistical
problems, and we also introduce the concept of the effective ill-posedness of
the operator equation under white noise. This study is concluded with
prototypical examples for such equations, as these are deconvolution equations
and certain final value problems in evolution equations.
