For a start, A has a domain D(A) (which is a dense subspace of H). It has an adjoint A* with domain D(A*), such that for all f in D(A) and g in D(A*). To say that A is symmetric means that and A*(f)=A(f) for all f in D(A).
The notation gra(A) means the graph of A, which is the set .
The spaces are defined by . The kernel of an adjoint operator is always the orthogonal complement of the range of the operator, so (because is the adjoint of ).
The spaces are by definition contained in the graph of A*. To see that they are orthogonal to the graph of A, let and . Then
(definition of inner product in direct sum)
(conjugate-linear property of inner product)
A similar argument shows that gra(A) and are orthogonal.
The fact that and are orthogonal is more or less immediate, because if and then