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Stone's theorem on one-parameter unitary groups

In mathematics, 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

{U_t}_{t in mathbb{R}}

which are strongly continuous, that is

lim_{t rightarrow t_0} U_t xi = U_{t_0} xi quad forall t_0 in mathbb{R}, xi in H

and are homomorphisms:

U_{t+s} = U_t U_s. quad

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.

Formal statement

Let U be a strongly continuous 1-parameter unitary group, then there exists a unique self-adjoint operator A such that

U_t := e^{i t A} quad t in mathbb{R}.

Conversely, let A be a self-adjoint operator on a Hilbert space H. Then

U_t := e^{i t A} quad t in mathbb{R}

is a strongly continuous one-parameter family of unitary operators.

The infinitesimal generator of {Ut}t is the operator iA. This mapping is a bijective correspondence. A will be a bounded operator iff the operator-valued function t mapsto U_t is norm continuous.


The family of translation operators

[T_t psi](x) = psi(x + t) quad

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

frac{d}{dx} = i frac{1}{i} frac{d}{dx}

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

T_t = e^{t , {d}/{dx}}. quad

Applications and generalizations

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 generalizes Stone's theorem to strongly continuous one-parameter semigroups of contractions on Banach spaces.


  • 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)
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