  The paper focuses on numerical solution of parametrized diffusion equations
with scalar parameter-dependent coefficient function by the stochastic
(spectral) Galerkin method. We study preconditioning of the related discretized
problems using preconditioners obtained by modifying the stochastic part of the
partial differential equation. We present a simple but general approach for
obtaining two-sided bounds to the spectrum of the resulting matrices, based on
a particular splitting of the discretized operator. Using this tool and
considering the stochastic approximation space formed by classical orthogonal
polynomials, we obtain new spectral bounds depending solely on the properties
of the coefficient function and the type of the approximation polynomials for
several classes of block-diagonal preconditioners. These bounds are guaranteed
and applicable to various distributions of parameters. Moreover, the conditions
on the parameter-dependent coefficient function are only local, and therefore
less restrictive than those usually assumed in the literature.
