Editing Hille–Yosida theorem

{{Short description|Theorem}}
In [[functional analysis]], the '''Hille–Yosida theorem''' characterizes the generators of [[C0-semigroup|strongly continuous one-parameter semigroup]]s of [[linear operator]]s on [[Banach space]]s. It is sometimes stated for the special case of [[contraction semigroup]]s, with the general case being called the '''Feller–Miyadera–Phillips theorem''' (after [[William Feller]], Isao Miyadera, and Ralph Phillips). The contraction semigroup case is widely used in the theory of [[Markov process]]es. In other scenarios, the closely related [[Lumer–Phillips theorem]] is often more useful in determining whether a given operator generates a [[C0-semigroup|strongly continuous contraction semigroup]]. The theorem is named after the [[mathematician]]s [[Einar Hille]] and [[Kōsaku Yosida]] who independently discovered the result around 1948.

== Formal definitions ==
{{Main article| C0-semigroup|l1=C<sub>0</sub>-semigroup}}
If ''X'' is a Banach space, a [[one-parameter semigroup]] of operators on ''X'' is a family of operators indexed on the non-negative real numbers
{''T''(''t'')}<sub> ''t ∈ [0, &infin;)''</sub> such that 
*<math> T(0)= I \quad </math>
*<math> T(s+t)= T(s) \circ T(t), \quad \forall t,s \geq 0. </math>
The semigroup is said to be '''strongly continuous''', also called a (''C''<sub>0</sub>) semigroup, if and only if the mapping
:<math> t \mapsto T(t) x </math>
is continuous for all ''x ∈ X'', where ''[0, &infin;)'' has the usual topology and ''X'' has the norm topology.

The infinitesimal generator of a one-parameter semigroup ''T'' is an operator ''A'' defined on a possibly proper subspace of ''X'' as follows:
*The domain of ''A'' is the set of ''x ∈ X'' such that
:: <math> h^{-1}\bigg(T(h) x - x\bigg) </math>
:has a limit as ''h'' approaches ''0'' from the right.
* The value of ''Ax'' is the value of the above limit. In other words, ''Ax'' is the right-derivative at ''0'' of the function
::<math> t \mapsto T(t)x. </math>

The infinitesimal generator of a strongly continuous one-parameter semigroup is a [[closed linear operator]] defined on a [[dense (topology)|dense]] [[linear subspace]] of ''X''.

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 ==
Let ''A'' be a linear operator defined on a linear subspace ''D''(''A'') of the Banach space ''X'', ''&omega;'' a real number, and ''M''&nbsp;>&nbsp;0. Then ''A'' generates a [[C0 semigroup|strongly continuous semigroup]] ''T'' that satisfies <math>\|T(t)\|\leq M{\rm e}^{\omega t}</math> if and only if<ref>Engel and Nagel Theorem II.3.8, Arendt et al. Theorem 3.3.4, Staffans Theorem 3.4.1</ref>
# ''A'' is [[Unbounded operator#Closed linear operators|closed]] and ''D''(''A'') is [[dense (topology)|dense]] in ''X'',
# every real ''&lambda;''&nbsp;>&nbsp;''&omega;'' belongs to the [[resolvent set]] of ''A'' and for such λ and for all positive [[integers]] ''n'',
:::<math>\|(\lambda I-A)^{-n}\|\leq\frac{M}{(\lambda-\omega)^n}.</math>

==Hille-Yosida theorem for contraction semigroups==

In the general case the Hille–Yosida theorem is mainly of theoretical importance since the estimates on the powers of the [[resolvent operator]] that appear in the statement of the theorem can usually not be checked in concrete examples. In the special case of [[contraction semigroup]]s (''M''&nbsp;=&nbsp;1 and ''&omega;''&nbsp;=&nbsp;0 in the above theorem) only the case ''n''&nbsp;=&nbsp;1 has to be checked and the theorem also becomes of some practical importance. The explicit statement of the Hille–Yosida theorem for contraction semigroups is:

Let ''A'' be a linear operator defined on a linear subspace ''D''(''A'') of the [[Banach space]] ''X''. Then ''A'' generates a [[contraction semigroup]] if and only if<ref>Engel and Nagel Theorem II.3.5, Arendt et al. Corollary 3.3.5, Staffans Corollary 3.4.5</ref>
# ''A'' is [[Unbounded operator#Closed linear operators|closed]] and ''D''(''A'') is [[dense (topology)|dense]] in ''X'',
# every real ''&lambda;''&nbsp;>&nbsp;0 belongs to the resolvent set of ''A'' and for such ''&lambda;'',
:::<math>\|(\lambda I-A)^{-1}\|\leq\frac{1}{\lambda}.</math>

== See also ==
* [[Stone's theorem on one-parameter unitary groups]]

==Notes==

{{reflist|2}}

==References==
*{{citation|last=Riesz|first= F.|authorlink1=Frigyes Riesz|last2=Sz.-Nagy|first2= B.|title=Functional analysis. Reprint of the 1955 original|series= Dover Books on Advanced Mathematics|publisher=Dover|year= 1995|isbn= 0-486-66289-6}}
*{{citation|last=Reed|first= Michael|last2= Simon|first2= Barry|title=Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. |publisher=Academic Press|year=1975|isbn=0-12-585050-6 }}
*{{citation | last1=Engel| first1=Klaus-Jochen| last2=Nagel| first2=Rainer | title=One-parameter semigroups for linear evolution equations | year=2000| publisher=Springer |isbn=0-387-98463-1 }}
*{{citation | last1=Arendt| first1=Wolfgang| last2=Batty| first2=Charles | last3=Hieber| first3=Matthias| last4=Neubrander| first4=Frank |title=Vector-valued Laplace Transforms and Cauchy Problems | year=2001| publisher=Birkhauser |isbn=0-8176-6549-8 }}
*{{citation | last1=Staffans| first1=Olof| title=Well-posed linear systems | year=2005| publisher=Cambridge University Press |isbn=0-521-82584-9 }}
*{{citation | last1=Feller |first1= William| title=An introduction to probability theory and its applications |volume=II |edition=Second |year=1971|publisher=John Wiley & Sons |location=New York |isbn=0-471-25709-5 }}
*{{citation | last1=Vrabie |first1= Ioan I.| title=C<sub>0</sub>-semigroups and applications |series=North-Holland Mathematics Studies |volume=191 |year=2003|publisher=North-Holland Publishing |location=Amsterdam |isbn=0-444-51288-8 }}

{{Functional analysis}}

{{DEFAULTSORT:Hille-Yosida theorem}}
[[Category:Semigroup theory]]
[[Category:Theorems in functional analysis]]