ABSTRACT
Differential operators have been introduced, the usual procedure is to specify an operator expression, i.e., a differential expression, and an appropriate set of differentiable functions as the domain for the operator expression to act on. Quite often the specification of an intuitively obvious domain would lead to a symmetric operator which is not selfadjoint. The adjoint operator would often have the same differential expression but acting on a bigger domain. In many cases it is possible to choose a suitable enlargement of the domain of a symmetric operator to obtain a new operator with a bigger domain which is selfadjoint operator. The resulting selfadjoint operator is called a selfadjoint extension of the symmetric operator. In many applications we would make use of such an essential selfadjoint restriction since the domain of an essentially selfadjoint operator is easier to specified.
