⊂ T∗∗ ⊂ T∗. If T is closed and symmetric then T = T∗∗ ⊂ T∗. If T is self-adjoint then T = T∗∗ = T∗. If T is essentially self-adjoint then T ⊂ T∗∗ = T∗. Importance of self-adjoint operators The class of self-adjoint operators is especially important in mathematical physics. Every self-adjoint operator is densely defined, closed and symmetric. The converse holds for bounded operators but fails in general. Self-adjointness is substantially more restricting than these three properties. The famous spectral theorem holds for self-adjoint operators. In combination with Stone's theorem on one-parameter unitary groups it shows that self-adjoint operators are precisely the infinitesimal generators of strongly continuous one-parameter unitary groups, see Self-adjoint operator#Self-adjoint extensions in quantum mechanics. Such unitary groups are especially important for describing time evolution in classical and quantum mechanics. See also Hilbert space#Unbounded operators Stone–von Neumann theorem Bounded operator Notes References (see Chapter 12 "General theory of unbounded operators in Hilbert spaces"). (see Chapter 5 "Unbounded operators"). (see Chapter 8 "Unbounded operators"). Linear operators Operator theory de:Linearer Operator#Unbeschränkte lineare Operatoren