This paper contains the author's abstract of his Doctoral Dissertation. The dissertation is mainly concerned with best approximation in a certain class of elements of an in general unbounded operator by bounded linear operators. This problem is linked, among other extremal problems, with the problem of best restoration of operator values on elements of a class which are specified with a known error. The author obtains new relations and improves familiar relations between the modulus of continuity of an unbounded linear operator in the class of elements of a Banach space, the best approximation of the operator by bounded linear operators and ill- posed problems of restoration of operator values on inaccurately known elements of the class, and also between the first two problems and problems of approximation theory concerning the approximation of one class of elements by another. Most attention is paid to best approximation of a shift-invariant operator and, in particular, to the model problem of best approximation of the kth-order differentiation operator in the class of n-times differentiable functions. The results are used to solve some concrete problems.