# Hille–Yosida theorem

Hille–Yosida theorem In functional analysis, the Hille–Yosida theorem characterizes the generators of strongly continuous one-parameter semigroups of linear operators on Banach spaces. 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, Isao Miyadera, and Ralph Phillips). The contraction semigroup case is widely used in the theory of Markov processes. 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 mathematicians Einar Hille and Kōsaku Yosida who independently discovered the result around 1948.

Contents 1 Formal definitions 2 Statement of the theorem 3 Hille-Yosida theorem for contraction semigroups 4 See also 5 Notes 6 References Formal definitions Main article: C0-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)} t ∈ [0, ∞) such that {displaystyle T(0)=Iquad } {displaystyle T(s+t)=T(s)circ T(t),quad forall t,sgeq 0.} The semigroup is said to be strongly continuous, also called a (C0) semigroup, if and only if the mapping {displaystyle tmapsto T(t)x} 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 {displaystyle h^{-1}{bigg (}T(h)x-x{bigg )}} has a limit as h approaches 0 from the right. The value of A x is the value of the above limit. In other words, A x is the right-derivative at 0 of the function {displaystyle tmapsto T(t)x.} The infinitesimal generator of a strongly continuous one-parameter semigroup is a closed linear operator defined on a 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, ω a real number, and M > 0. Then A generates a strongly continuous semigroup T that satisfies {displaystyle |T(t)|leq M{rm {e}}^{omega t}} if and only if[1] A is closed and D(A) is dense in X, every real λ > ω belongs to the resolvent set of A and for such λ and for all positive integers n, {displaystyle |(lambda I-A)^{-n}|leq {frac {M}{(lambda -omega )^{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 X. Then A generates a contraction semigroup if and only if[2] A is closed and D(A) is dense in X, every real λ > 0 belongs to the resolvent set of A and for such λ, {displaystyle |(lambda I-A)^{-1}|leq {frac {1}{lambda }}.} See also C0 semigroup Lumer–Phillips theorem Stone's theorem on one-parameter unitary groups Notes ^ Engel and Nagel Theorem II.3.8, Arendt et al. Theorem 3.3.4, Staffans Theorem 3.4.1 ^ Engel and Nagel Theorem II.3.5, Arendt et al. Corollary 3.3.5, Staffans Corollary 3.4.5 References Riesz, F.; Sz.-Nagy, B. (1995), Functional analysis. Reprint of the 1955 original, Dover Books on Advanced Mathematics, Dover, ISBN 0-486-66289-6 Reed, Michael; Simon, Barry (1975), Methods of modern mathematical physics. II. Fourier analysis, self-adjointness., Academic Press, ISBN 0125850506 Engel, Klaus-Jochen; Nagel, Rainer (2000), One-parameter semigroups for linear evolution equations, Springer Arendt, Wolfgang; Batty, Charles; Hieber, Matthias; Neubrander, Frank (2001), Vector-valued Laplace Transforms and Cauchy Problems, Birkhauser Staffans, Olof (2005), Well-posed linear systems, Cambridge University Press Feller, William (1971), An introduction to probability theory and its applications. Vol. II. Second edition, John Wiley & Sons, New York Vrabie, Ioan I. (2003), C0-semigroups and applications. North-Holland Mathematics Studies, 191., North-Holland Publishing Co., Amsterdam hide vte Functional analysis (topics – glossary) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraic/topological)locally convexreflexiveseparable Theorems Hahn–BanachRiesz representationclosed graphuniform boundedness principleKakutani fixed-pointKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operators adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebras Banach algebraC*-algebraspectrum of a C*-algebraoperator algebragroup algebra of a locally compact groupvon Neumann algebra Open problems invariant subspace problemMahler's conjecture Applications Hardy spacespectral theory of ordinary differential equationsheat kernelindex theoremcalculus of variationsfunctional calculusintegral operatorJones polynomialtopological quantum field theorynoncommutative geometryRiemann hypothesisdistribution (or generalized functions) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory Categories: Semigroup theoryTheorems in functional analysis

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