every satisfying the above property for is a restriction of
An adjoint does not need to exist in general.
An unbounded operator is symmetric if and for all (one also writes ).
The domain of is the set of all vectors such that the linear functional is bounded on .
The graph satisfies where denotes the orthogonal complement and denotes the transposition of the direct summands changing the sign of one of the factors, i.e. . An unbounded operator is closed if is closed subspace of . An operator is a closure of an operator if is a closure of operator . It is said that is an extension of and one writes if . The closure of an unbounded operator does not need to exist.
For any unbounded operator with a dense , if the adjoint operator exists, is closed, and if exists then it coincides with a closure of .