Stone's theorem on one-parameter unitary groups

Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis which establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space H and one-parameter families of unitary operators

which are strongly continuous, that is

and are homomorphisms:

Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups. The theorem is named after Marshall Stone who formulated and proved this theorem in 1932.

The formal statement is as follows:

Theorem. Let A be a self-adjoint operator on a Hilbert space H. Then

is a strongly continuous one-parameter family of unitary operators. The infinitesimal generator of {U_t}_t is the operator i A. This mapping is a bijective correspondence.

A will be a bounded operator iff the operator-valued function t → U_t is norm continuous.

Example. The family of translation operators

is a one-parameter unitary group of unitary operators; the infinitesimal generator of this family is an extension of the differential operator

defined on the space of complex-valued continuously differentiable functions of compact support on R. Thus

Stone's theorem has numerous applications in quantum mechanics. For instance, given an isolated quantum mechanical system, with Hilbert space of states H, time evolution is a strongly continuous one-parameter unitary group on H. The infinitesimal generator of this group is the system Hamiltonian.

The Hille-Yosida theorem is a generalization of Stone's theorem to strongly continuous one-parameter semigroups of contractions on a Banach spaces

References

• M. H. Stone, On one-parameter unitary groups in Hilbert Space, Annals of Mathematics 33, 643-648, (1932).

• K. Yosida, Functional Analysis, Springer-Verlag, (1968)