Hille-yosida Theorem

A special case of the Hille-Yosida theorem is Stone's Theorem characterizing strongly continuous one-parameter groups of operators on a Hilbert Space .

FORMAL DEFINITIONS

If ''E'' is a Banach space, a one-parameter semigroup of operators on ''E'' is a family of operators indexed on the non-negative real numbers
{''T''_''t''}_{''t'' ∈ [0,∞)} such that

$T_0= I \quad$

$T_{s+t}= T_s \circ T_t, \quad orall t,s \geq 0$

strongly continuous

₀

t \mapsto T_t arphi

is continuous for all φ ∈ ''E'', where [0, ∞) has the usual topology and ''E'' has the norm topology.

Example. Consider the Banach space '''BUC'''[0, ∞) of bounded uniformly continuous complex-valued functions of the interval [0, ∞). Let
:

(x) = arphi(x+t), \quad x,t \in [0, \infty).

Then {''T''_''t''} is a strongly continuous one parameter semigroup. In this case the operators ''T''_''t'' have norm at most 1. The strong continuity property follows from the fact that the functions in the space BUC[0, ∞) are uniformly continuous. In fact, the family of translation operators defined by (1) on the larger space '''BC'''[0, ∞) of bounded continuous complex-valued functions on [0, ∞) is a one-parameter semigroup but fails to be strongly continuous.

The Infinitesimal Generator of a one-parameter semigroup {''T''_''t''}_{''t'' ∈ [0,∞)} is an operator ''A'' defined on a possibly proper subspace of ''E'' as follows:

The domain of ''A'' is the set of ψ ∈ ''E'' such that

h^{-1}\bigg(T_h \psi - \psi\bigg)

:approaches a limit as ''h'' aproaches 0.

The value of ''A'' ψ is the value of the above limit. In other words ''A'' ψ is the derivative at 0 of the function

t \mapsto T_t \psi.

Theorem. The infinitesimal generator of a strongly continuous one-parameter semigroup is a closed linear operator defined on a dense subspace of ''E''.

The Hille-Yosida theorem provides a necessary and sufficient condition for a closed linear operator ''A'' on a Banach space to be the infinitesimal generator of a strongly continuous one-parameter semigroup.

STATEMENT OF THE THEOREM

We first state the theorem for the special case of the contraction semigroup {''T''_''t''}_{''t'' ∈ [0,∞)}, that is, the semigroup in which all operators ''T''_''t'' have norm at most 1.

Theorem. Let ''A'' be a (partially defined) operator on the Banach space ''E''. A necessary and sufficient condition ''A'' be the infinitesimal generator of a strongly continuous contraction semigroup on ''E'' is that
# ''A'' is densely defined;
# For all λ > 0, the operator λ ''I'' − ''A'' has an everywhere defined inverse R(λ, ''A'') such that

	CATEGORIES ABOUT HILLE–YOSIDA THEOREM

	functional analysis

	semigroup theory

	mathematical theorems

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Information About

Hille-yosida Theorem