  For an arbitrary self-adjoint operator $B$ in a Hilbert space $H$, we present
direct and inverse theorems establishing the relationship between the degree of
smoothness of a vector $x \in H$ with respect to the operator $B$, the rate of
convergence to zero of its best approximation by exponential-type entire
vectors of the operator $B$, and the $k$-modulus of continuity of the vector
$x$ with respect to the operator $B$. The results are used for finding a priori
estimates for the Ritz approximate solutions of operator equations in a Hilbert
space.
