if , then will no longer be closed, but it will be closable, with the closure being its extension defined on . Symmetric operators and self-adjoint operators An operator T on a Hilbert space is symmetric if and only if for each x and y in the domain of we have . A densely defined operator is symmetric if and only if it agrees with its adjoint T∗ restricted to the domain of T, in other words when T∗ is an extension of . In general, if T is densely defined and symmetric, the domain of the adjoint T∗ need not equal the domain of T. If T is symmetric and the domain of T and the domain of the adjoint coincide, then we say that T is self-adjoint. Note that, when T is self-adjoint, the existence of the adjoint implies that T is densely defined and since T∗ is necessarily closed, T is closed. A densely defined operator T is symmetric, if the subspace (defined in a previous section) is orthogonal to its image under J (where J(x,y):=(y,-x)). Equivalently, an operator T is self-adjoint if it is densely defined, closed, symmetric, and satisfies the fourth condition: both operators , are surjective, that is, map the domain of T onto the whole space H. In other words: for every x in H there exist y and z in the domain of T such that and . An operator T is self-adjoint, if the two subspaces , are orthogonal and their sum is the whole space This approach does not cover non-densely defined closed operators. Non-densely defined symmetric operators can be defined directly or via graphs, but not via adjoint operators. A symmetric operator is often studied via its Cayley transform. An operator T on a complex Hilbert space is symmetric if and only if its quadratic form is real, that is, the number is real for all x in the domain of T. A densely defined closed symmetric operator T is self-adjoint if and only if T∗ is symmetric. It may happen that it is not. A densely defined operator T is called positive (or nonnegative) if its quadratic form is nonnegative, that is, for all x in the domain of T. Such operator is necessarily symmetric. The operator T∗T is self-adjoint and positive for every densely defined, closed T. The spectral theorem applies to self-adjoint operators and moreover, to