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 that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space and one-parameter families

of unitary operators that are strongly continuous, i.e.,

and are homomorphisms, i.e.,

Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups.

The theorem was proved by Marshall Stone (1930, 1932), and Neumann (1932) showed that the requirement that be strongly continuous can be relaxed to say that it is merely weakly measurable, at least when the Hilbert space is separable.

This is an impressive result, as it allows to define the derivative of the mapping which is only supposed to be continuous. It is also related to the theory of Lie groups and Lie algebras.

Formal statementEdit

The statement of the theorem is as follows.[1]

Theorem. Let   be a strongly continuous one-parameter unitary group. Then there exists a unique (possibly unbounded) operator  , that is self-adjoint on   and such that
The domain of   is defined by
Conversely, let   be a (possibly unbounded) self-adjoint operator on   Then the one-parameter family   of unitary operators defined by
is a strongly continuous one-parameter group.

In both parts of the theorem, the expression   is defined by means of the spectral theorem for unbounded self-adjoint operators.

The operator   is called the infinitesimal generator of   Furthermore,   will be a bounded operator if and only if the operator-valued mapping   is norm-continuous.

The infinitesimal generator   of a strongly continuous unitary group   may be computed as


with the domain of   consisting of those vectors   for which the limit exists in the norm topology. That is to say,   is equal to   times the derivative of   with respect to   at  . Part of the statement of the theorem is that this derivative exists—i.e., that   is a densely defined self-adjoint operator. The result is not obvious even in the finite-dimensional case, since   is only assumed (ahead of time) to be continuous, and not differentiable.


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 continuously differentiable complex-valued functions with compact support on   Thus


In other words, motion on the line is generated by the momentum operator.


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  . The infinitesimal generator of this group is the system Hamiltonian.

Using Fourier transformEdit

Stone's Theorem can be recast using the language of the Fourier transform. The real line   is a locally compact abelian group. Non-degenerate *-representations of the group C*-algebra   are in one-to-one correspondence with strongly continuous unitary representations of   i.e., strongly continuous one-parameter unitary groups. On the other hand, the Fourier transform is a *-isomorphism from   to   the  -algebra of continuous complex-valued functions on the real line that vanish at infinity. Hence, there is a one-to-one correspondence between strongly continuous one-parameter unitary groups and *-representations of   As every *-representation of   corresponds uniquely to a self-adjoint operator, Stone's Theorem holds.

Therefore, the procedure for obtaining the infinitesimal generator of a strongly continuous one-parameter unitary group is as follows:

  • Let   be a strongly continuous unitary representation of   on a Hilbert space  .
  • Integrate this unitary representation to yield a non-degenerate *-representation   of   on   by first defining
and then extending   to all of   by continuity.
  • Use the Fourier transform to obtain a non-degenerate *-representation   of   on  .
  • Then   is the infinitesimal generator of  

The precise definition of   is as follows. Consider the *-algebra   the continuous complex-valued functions on   with compact support, where the multiplication is given by convolution. The completion of this *-algebra with respect to the  -norm is a Banach *-algebra, denoted by   Then   is defined to be the enveloping  -algebra of  , i.e., its completion with respect to the largest possible  -norm. It is a non-trivial fact that, via the Fourier transform,   is isomorphic to   A result in this direction is the Riemann-Lebesgue Lemma, which says that the Fourier transform maps   to  


The Stone–von Neumann theorem generalizes Stone's theorem to a pair of self-adjoint operators,  , satisfying the canonical commutation relation, and shows that these are all unitarily equivalent to the position operator and momentum operator on  

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


  1. ^ Hall 2013 Theorem 10.15


  • Hall, B.C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, 267, Springer, ISBN 978-1461471158
  • Neumann, J. von (1932), "Über einen Satz von Herrn M. H. Stone", Annals of Mathematics, Second Series (in German), Annals of Mathematics, 33 (3): 567–573, doi:10.2307/1968535, ISSN 0003-486X, JSTOR 1968535
  • Stone, M. H. (1930), "Linear Transformations in Hilbert Space. III. Operational Methods and Group Theory", Proceedings of the National Academy of Sciences of the United States of America, National Academy of Sciences, 16 (2): 172–175, doi:10.1073/pnas.16.2.172, ISSN 0027-8424, JSTOR 85485, PMC 1075964, PMID 16587545
  • Stone, M. H. (1932), "On one-parameter unitary groups in Hilbert Space", Annals of Mathematics, 33 (3): 643–648, doi:10.2307/1968538, JSTOR 1968538
  • K. Yosida, Functional Analysis, Springer-Verlag, (1968)