In
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics ...
, the Hille–Yosida theorem characterizes the generators of
strongly continuous one-parameter semigroups of
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
s on
Banach space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
s. It is sometimes stated for the special case of
contraction semigroups, with the general case being called the Feller–Miyadera–Phillips theorem (after
William Feller
William "Vilim" Feller (July 7, 1906 – January 14, 1970), born Vilibald Srećko Feller, was a Croatian–American mathematician specializing in probability theory.
Early life and education
Feller was born in Zagreb to Ida Oemichen-Perc, a Cro ...
, Isao Miyadera, and Ralph Phillips). The contraction semigroup case is widely used in the theory of
Markov process
In probability theory and statistics, a Markov chain or Markov process is a stochastic process describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, ...
es. In other scenarios, the closely related
Lumer–Phillips theorem is often more useful in determining whether a given operator generates a
strongly continuous contraction semigroup. The theorem is named after the
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...
s
Einar Hille
Carl Einar Hille (28 June 1894 – 12 February 1980) was an American mathematics professor and scholar. Hille authored or coauthored twelve mathematical books and a number of mathematical papers.
Early life and education
Hille was born in New Y ...
and
Kōsaku Yosida
was a Japanese mathematician who worked in the field of functional analysis. He is known for the Hille-Yosida theorem concerning ''C0''-semigroups. Yosida studied mathematics at the University of Tokyo, and held posts at Osaka and Nagoya ...
who independently discovered the result around 1948.
Formal definitions
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 ∈ [0, ∞)'' such that
*
*
The semigroup is said to be strongly continuous, also called a (''C''
0) semigroup, if and only if the mapping
:
is continuous for all ''x ∈ X'', where ''[0, ∞)'' 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
::
: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
::
The infinitesimal generator of a strongly continuous one-parameter semigroup is a closed linear operator defined on a
dense
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ...
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'', ''ω'' a real number, and ''M'' > 0. Then ''A'' generates a
strongly continuous semigroup
In mathematical analysis, a ''C''0-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions of scalar linear constant coefficient or ...
''T'' that satisfies
if and only if
# ''A'' is
closed and ''D''(''A'') is
dense
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ...
in ''X'',
# every real ''λ'' > ''ω'' belongs to the
resolvent set In linear algebra and operator theory, the resolvent set of a linear operator is a set of complex numbers for which the operator is in some sense "well-behaved". The resolvent set plays an important role in the resolvent formalism.
Definitions
L ...
of ''A'' and for such λ and for all positive
integers
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
''n'',
:::
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 semigroups (''M'' = 1 and ''ω'' = 0 in the above theorem) only the case ''n'' = 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
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and ...
''X''. Then ''A'' generates a
contraction semigroup if and only if
[Engel and Nagel Theorem II.3.5, Arendt et al. Corollary 3.3.5, Staffans Corollary 3.4.5]
# ''A'' is
closed and ''D''(''A'') is
dense
Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ...
in ''X'',
# every real ''λ'' > 0 belongs to the resolvent set of ''A'' and for such ''λ'',
:::
See also
*
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 \mathcal and one-parameter families
:(U_)_
o ...
Notes
References
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{{DEFAULTSORT:Hille-Yosida theorem
Semigroup theory
Theorems in functional analysis