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
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 was proved by Marshall Stone (1930, 1932). Von Neumann (1932) showed that the condition that Ut is strongly continuous can be relaxed to say that it is weakly measurable, at least when the Hilbert space is separable.
Read more about Stone's Theorem On One-parameter Unitary Groups: Formal Statement, Example, Applications, Generalizations
Famous quotes containing the words stone, theorem and/or groups:
“The subject of the novel is reality liberated from soul. The reader in complete independence presented with a structured process: let him evaluate it, not the author. The façade of the novel cannot be other than stone or steel, flashing electrically or dark, but silent.”
—Alfred Döblin (1878–1957)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)
“Instead of seeing society as a collection of clearly defined “interest groups,” society must be reconceptualized as a complex network of groups of interacting individuals whose membership and communication patterns are seldom confined to one such group alone.”
—Diana Crane (b. 1933)