Difference between revisions of "Semi-group of operators"
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− | A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s0841401.png" /> of operators on a [[Banach space|Banach space]] or [[Topological vector space|topological vector space]] with the property that the composite of any two operators in the family is again a member of the family. If the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s0841402.png" /> are | + | A family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s0841401.png" /> of operators on a [[Banach space|Banach space]] or [[Topological vector space|topological vector space]] with the property that the composite of any two operators in the family is again a member of the family. If the operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s0841402.png" /> are "indexed" by elements of some abstract [[Semi-group|semi-group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s0841403.png" /> and the binary operation of the latter is compatible with the composition of operators, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s0841404.png" /> is known as a representation of the semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s0841405.png" />. The most detailed attention has been given to one-parameter semi-groups (cf. [[One-parameter semi-group|One-parameter semi-group]]) of bounded linear operators on a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s0841406.png" />, which yield a representation of the additive semi-group of all positive real numbers, i.e. families <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s0841407.png" /> with the property |
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s0841408.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s0841408.png" /></td> </tr></table> | ||
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140341.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140341.png" /></td> </tr></table> | ||
− | In a non-normed locally convex space, the infinitesimal generator of a semi-group of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140342.png" /> may have no resolvent at any point. An example is: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140343.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140344.png" /> of infinitely-differentiable functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140345.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140346.png" />. As a substitute for the resolvent one can take a continuous operator whose product with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140347.png" />, from the right and the left, differs by a | + | In a non-normed locally convex space, the infinitesimal generator of a semi-group of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140342.png" /> may have no resolvent at any point. An example is: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140343.png" /> in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140344.png" /> of infinitely-differentiable functions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140345.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140346.png" />. As a substitute for the resolvent one can take a continuous operator whose product with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140347.png" />, from the right and the left, differs by a "small amount" from the identity operator. |
A continuous operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140348.png" /> defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140349.png" /> in a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140350.png" /> is called an asymptotic resolvent for a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140351.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140352.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140353.png" />, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140354.png" /> can be extended from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140355.png" /> to a continuous operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140356.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140357.png" />, and if there exists a limit point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140358.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140359.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140360.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140361.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140362.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140363.png" />, where | A continuous operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140348.png" /> defined for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140349.png" /> in a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140350.png" /> is called an asymptotic resolvent for a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140351.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140352.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140353.png" />, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140354.png" /> can be extended from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140355.png" /> to a continuous operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140356.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140357.png" />, and if there exists a limit point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140358.png" /> of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140359.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140360.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140361.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140362.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140363.png" />, where | ||
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A generalized function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140431.png" /> with support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140432.png" />, possessing the properties (7), is naturally called the fundamental function of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140434.png" />. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140435.png" /> is the infinitesimal operator of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140436.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140437.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140438.png" /> is the fundamental function of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140439.png" />. The converse statement is true under certain additional assumptions about the order of singularity of the fundamental function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140440.png" /> (or, more precisely, of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140441.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140442.png" />). | A generalized function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140431.png" /> with support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140432.png" />, possessing the properties (7), is naturally called the fundamental function of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140434.png" />. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140435.png" /> is the infinitesimal operator of a semi-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140436.png" /> of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140437.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140438.png" /> is the fundamental function of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140439.png" />. The converse statement is true under certain additional assumptions about the order of singularity of the fundamental function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140440.png" /> (or, more precisely, of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140441.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140442.png" />). | ||
− | A useful notion for the characterization of semi-groups in a locally convex space is that of the generalized resolvent. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140443.png" /> denote the Laplace transform of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140444.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140445.png" /> be the space of all such transforms. A topology is induced in this space, via the Laplace transform, from the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140446.png" />. The Laplace transform of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140447.png" />-valued generalized function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140448.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140449.png" />. Under these conditions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140450.png" /> is a continuous mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140451.png" /> into the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140452.png" /> of continuous linear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140453.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140454.png" /> be the space of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140455.png" /> obtained from functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140456.png" /> with support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140457.png" />, with the natural topology. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140458.png" /> is a linear operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140459.png" />, it can be | + | A useful notion for the characterization of semi-groups in a locally convex space is that of the generalized resolvent. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140443.png" /> denote the Laplace transform of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140444.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140445.png" /> be the space of all such transforms. A topology is induced in this space, via the Laplace transform, from the topology of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140446.png" />. The Laplace transform of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140447.png" />-valued generalized function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140448.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140449.png" />. Under these conditions, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140450.png" /> is a continuous mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140451.png" /> into the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140452.png" /> of continuous linear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140453.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140454.png" /> be the space of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140455.png" /> obtained from functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140456.png" /> with support in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140457.png" />, with the natural topology. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140458.png" /> is a linear operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140459.png" />, it can be "lifted" to an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140460.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140461.png" /> via the equality |
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140462.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140462.png" /></td> </tr></table> | ||
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957) {{MR|0089373}} {{ZBL|0392.46001}} {{ZBL|0033.06501}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> Yu.M. Vuvunikyan, "Evolutionary representations of algebras of generalized functions" , ''Theory of operators in function spaces'' , Novosibirsk (1977) pp. 99–120 (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> P.P. Zabreiko, A.V. Zafievskii, "On a certain class of semigroups" ''Soviet Math. Dokl.'' , '''10''' : 6 (1969) pp. 1523–1526 ''Dokl. Akad. Nauk SSSR'' , '''189''' : 5 (1969) pp. 934–937 {{MR|264459}} {{ZBL|}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.V. Zafievskii, ''Trudy Mat. Inst. Voronezh. Univ.'' , '''1''' (1970) pp. 206–210</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, §1 {{MR|0617913}} {{ZBL|0435.46002}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> S.G. Krein, "Linear differential equations in Banach space" , ''Transl. Math. Monogr.'' , '''29''' , Amer. Math. Soc. (1971) (Translated from Russian) {{MR|0342804}} {{ZBL|0179.20701}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> Yu.T. Sil'chenko, "An evolutionary equation with an operator generating a nonlinear semigroup" ''Differential Equations'' , '''15''' : 2 (1979) pp. 255–258 ''Differentsial'nye Uravneniya'' , '''15''' : 2 (1979) pp. 363–366 {{MR|}} {{ZBL|0505.34046}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> J. Chazarain, "Problèmes de Cauchy abstracts et applications à quelques problèmes mixtes" ''J. Funct. Anal.'' , '''7''' : 3 (1971) pp. 386–446</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> I. Ciorânescu, "La caracterisation spectrale d'opérateur, générateurs des semi-groupes distributions d'ordre fini de croissance" ''J. Math. Anal. Appl.'' , '''34''' (1971) pp. 34–41</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> I. Ciorânescu, "A characterization of distribution semigroups of finite growth order" ''Rev. Roum. Math. Pures Appl.'' , '''22''' : 8 (1977) pp. 1053–1068 {{MR|500280}} {{ZBL|0374.46032}} </TD></TR><TR><TD valign="top">[11]</TD> <TD valign="top"> T. Kato, "A characterization of holomorphic semigroups" ''Proc. Amer. Math. Soc.'' , '''25''' : 3 (1970) pp. 495–498 {{MR|0264456}} {{ZBL|0199.45604}} </TD></TR><TR><TD valign="top">[12]</TD> <TD valign="top"> J. Lions, "Les semigroupes distributions" ''Portugal. Math.'' , '''19''' (1960) pp. 141–164</TD></TR><TR><TD valign="top">[13]</TD> <TD valign="top"> A. Pazy, "On the differentiability and compactness of semi-groups of linear operators" ''J. Math. Mech.'' , '''17''' : 12 (1968) pp. 1131–1141 {{MR|231242}} {{ZBL|0162.45903}} </TD></TR><TR><TD valign="top">[14]</TD> <TD valign="top"> A. Pazy, "Approximations of the identity operator by semigroups of linear operators" ''Proc. Amer. Math. Soc.'' , '''30''' : 1 (1971) pp. 147–150 {{MR|0287362}} {{ZBL|}} </TD></TR><TR><TD valign="top">[15]</TD> <TD valign="top"> T. Ushijima, "On the abstract Cauchy problems and semi-groups of linear operators in locally convex spaces" ''Sci. Papers College Gen. Educ. Univ. Tokyo'' , '''21''' (1971) pp. 93–122 {{MR|0312324}} {{ZBL|0239.47031}} </TD></TR><TR><TD valign="top">[16]</TD> <TD valign="top"> T. Ushijima, "On the generation and smoothness of semi-groups of linear operators" ''J. Fac. Sci. Univ. Tokyo, Sec. 1A'' , '''19''' : 1 (1972) pp. 65–127 {{MR|0308854}} {{ZBL|0239.47032}} </TD></TR><TR><TD valign="top">[17]</TD> <TD valign="top"> C. Wild, "Semi-groupes de croissance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140481.png" /> holomorphes" ''C.R. Acad. Sci. Paris Sér. A'' , '''285''' (1977) pp. 437–440 (English abstract) {{MR|448159}} {{ZBL|0359.47024}} </TD></TR><TR><TD valign="top">[18]</TD> <TD valign="top"> J.A. Gol'dstein, "Semigroups of linear operators and application" , Oxford Univ. Press (1985) (Translated from Russian)</TD></TR><TR><TD valign="top">[19]</TD> <TD valign="top"> A. Pazy, "Semigroups of linear operators and application to partial differential equations" , Springer (1983) {{MR|0710486}} {{ZBL|}} </TD></TR><TR><TD valign="top">[20]</TD> <TD valign="top"> Ph. Clément, H.J.A.M. Heijmans, S. Angenent, C.J. van Duijn, B. de Pagter, "One-parameter semigroups" , ''CWI Monographs'' , '''5''' , North-Holland (1987) {{MR|0915552}} {{ZBL|0636.47051}} </TD></TR></table> |
Line 265: | Line 265: | ||
====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> P. Butzer, H. Berens, "Semigroups of operators and approximation" , Springer (1967) {{MR|230022}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> H. Kellermann, M. Hieber, "Integrated semigroups" ''J. Funct. Anal.'' , '''84''' (1989) pp. 160–180 {{MR|0999494}} {{ZBL|0604.47025}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> I. Miyadera, N. Tanaka, "Exponentially bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084140/s084140482.png" />-semigroups and integrated semigroups" ''Tokyo J. Math.'' , '''12''' (1989) pp. 99–115 {{MR|1001735}} {{ZBL|}} </TD></TR></table> |
Revision as of 17:01, 15 April 2012
A family of operators on a Banach space or topological vector space with the property that the composite of any two operators in the family is again a member of the family. If the operators
are "indexed" by elements of some abstract semi-group
and the binary operation of the latter is compatible with the composition of operators,
is known as a representation of the semi-group
. The most detailed attention has been given to one-parameter semi-groups (cf. One-parameter semi-group) of bounded linear operators on a Banach space
, which yield a representation of the additive semi-group of all positive real numbers, i.e. families
with the property
![]() |
If is strongly measurable,
, then
is a strongly-continuous semi-group; this will be assumed in the sequel.
The limit
![]() |
exists; it is known as the type of the semi-group. The functions increase at most exponentially.
An important characteristic is the infinitesimal operator (infinitesimal generator) of the semi-group:
![]() |
defined on the linear set of all elements
for which the limit exists; the closure,
, of this operator (if it exists) is known as the generating operator, or generator, of the semi-group. Let
be the subspace defined as the closure of the union of all values
; then
is dense in
. If there are no non-zero elements in
such that
, then the generating operator
exists. In the sequel it will be assumed that
and that
implies
.
The simplest class of semi-groups, denoted by , is defined by the condition:
as
for any
. This is equivalent to the condition: The function
is bounded on any interval
. In that case
has a generating operator
whose resolvent
satisfies the inequalities
![]() | (1) |
where is the type of the semi-group. Conversely, if
is a closed operator with domain of definition dense in
and with a resolvent satisfying (1), then it is the generating operator of some semi-group
of class
such that
. Condition (1) is satisfied if
![]() |
(the Hill–Yosida condition). If, moreover, , then
is a contraction semi-group:
.
A summable semi-group is a semi-group for which the functions are summable on any finite interval for all
. A summable semi-group has a generating operator
. The operator
is closed if and only if, for every
,
![]() |
For one can define the Laplace transform of a summable semi-group,
![]() | (2) |
giving a bounded linear operator which has many properties of a resolvent operator.
A closed operator with domain of definition dense in
is the generating operator of a summable semi-group
if and only if, for some
, the resolvent
exists for
and the following conditions hold: a)
,
; b) there exist a non-negative function
,
,
, jointly continuous in all its variables, and a non-negative function
, bounded on any interval
, such that, for
,
![]() |
![]() |
![]() |
Under these conditions
![]() |
If one requires in addition that the function be summable on finite intervals, a necessary and sufficient condition is the existence of a continuous function
such that, for
,
![]() | (3) |
![]() | (4) |
![]() |
Under these conditions, . By choosing different functions satisfying (3), one can define different subclasses of summable semi-groups. If
, the result is the class
and (1) follows from (4). If
,
, condition (4) implies the condition
![]() |
Semi-groups with power singularities.
If in the previous example , then the integrals in (4) are divergent for
. Hence the generating operator for the corresponding semi-group may not have a resolvent for any
, i.e. it may have a spectrum equal to the entire complex plane. However, for
large enough one can define for such operators functions
which coincide with the functions
in the previous cases. The operator function
is called a resolvent of order
if it is analytic in some domain
and if for
,
![]() |
![]() |
and if for all
implies
. If
, the operator may have a unique resolvent of order
, for which there is a maximal domain of analyticity, known as the resolvent set of order
.
Let be a strongly-continuous semi-group such that the inequality
![]() |
holds for . Then its generating operator
has a resolvent of order
for
, and, moreover,
![]() |
![]() | (5) |
![]() |
Conversely, suppose that for the operator
has a resolvent
of order
satisfying (5) with
. Then there exists a unique semi-group
such that
![]() |
and the generating operator of this semi-group is such that
.
Smooth semi-groups.
If , the function
is continuously differentiable and
![]() |
There exist semi-groups of class such that, if
, the functions
are non-differentiable for all
. However, there are important classes of semi-groups for which the degree of smoothness increases with increasing
. If the functions
,
, are differentiable for any
, then it follows from the semi-group property that the
are twice differentiable if
, three times differentiable if
, etc. Therefore, if these functions are differentiable at any
for
, then
is infinitely differentiable.
Given a semi-group of class , a necessary and sufficient condition for the functions
to be differentiable for all
and
, where
, is that there exist numbers
such that the resolvent
is defined in the domain
![]() |
while in this domain
![]() |
A necessary and sufficient condition for to be infinitely differentiable for all
and
is that, for every
, there exist
such that the resolvent
is defined in the domain
![]() |
and such that
![]() |
Sufficient conditions are: If there exists a for which
![]() |
then the are differentiable for
and
; if
, then the
are infinitely differentiable for all
and
.
The degree of smoothness of a semi-group may sometimes be inferred from its behaviour at zero; for example, suppose that for every there exists a
such that, for
,
![]() |
then the are infinitely differentiable for all
,
.
There are smoothness conditions for summable semi-groups and semi-groups of polynomial growth. If a semi-group has polynomial growth of degree and is infinitely differentiable for
, then the function
![]() |
also has polynomial growth:
![]() |
In the general case there is no rigorous relationship between the numbers and
, and
can be utilized for a more detailed classification of infinitely-differentiable semi-groups of polynomial growth.
Analytic semi-groups.
An important class of semi-groups, related to partial differential equations of parabolic type, comprises those semi-groups which admit an analytic continuation to some sector of the complex plane containing the positive real axis. A semi-group of class
has this property if and only if its resolvent satisfies the following inequality in some right half-plane
:
![]() |
Another necessary and sufficient conditions is: The semi-group is strongly differentiable and its derivative satisfies the estimate
![]() |
Finally, the inequality
![]() |
is also a sufficient condition for to be analytic.
If a semi-group has an analytic continuation
to a sector
and has polynomial growth at zero,
,
, then the resolvent
of order
of its generating operator
has an analytic continuation to the sector
, and satisfies the following estimate in any sector
,
:
![]() |
Conversely, suppose that the resolvent of an operator
is defined in a sector
and that
![]() |
Then there exists a semi-group of growth
, analytic in the sector
, whose generating operator
is such that
.
Distribution semi-groups.
In accordance with the general concept of the theory of distributions (cf. Generalized function), one can drop the requirement that the operator-valued function be defined for every
, demanding only that it be possible to evaluate the integrals
for all
in the space
of infinitely-differentiable functions with compact support. Hence the following definition: A distribution semi-group on a Banach space
is a continuous linear mapping
of
into the space
of all bounded linear operators on
, with the following properties: a)
if
; b) if
are functions in the subspace
of all functions in
with support in
, then
, where the star denotes convolution:
![]() |
(the semi-group property); c) if for all
, then
; d) the linear hull of the set of all values of
,
,
, is dense in
; e) for any
,
, there exists a continuous
on
with values in
, so that
and
![]() |
for all .
The infinitesimal operator of a distribution semi-group is defined as follows. If there exists a delta-sequence
such that
and
as
, then
and
. The infinitesimal operator has a closure
, known as the infinitesimal generator of the distribution semi-group. The set
is dense in
and contains
for any
.
A closed linear operator with a dense domain of definition in
is the infinitesimal generator of a distribution semi-group if and only if there exist numbers
,
and a natural number
such that the resolvent
exists for
and satisfies the inequality
![]() | (6) |
If is a closed linear operator on
, then the set
can be made into a Fréchet space
by introducing the system of norms
![]() |
The restriction of
to
leaves
invariant. If
is the infinitesimal generator of a semi-group, then
is the infinitesimal generator of a semi-group of class
(continuous for
,
) on
. Conversely, if
is dense in
, the operator
has a non-empty resolvent set and
is the infinitesimal generator of a semi-group of class
on
, then
is the infinitesimal generator of a distribution semi-group on
.
A distribution semi-group has exponential growth of order at most ,
, if there exists an
such that
is a continuous mapping in the topology induced on
by the space
of rapidly-decreasing functions. A closed linear operator is the infinitesimal generator of a distribution semi-group with the above property if and only if it has a resolvent
which satisfies (6) in the domain
![]() |
where . In particular, if
the semi-group is said to be exponential and inequality (6) is valid in some half-plane. There exists a characterization of the semi-groups of the above types in terms of the operator
. Questions of smoothness and analyticity have also been investigated for distribution semi-groups.
Semi-groups of operators in a (separable) locally convex space
.
The definition of a strongly-continuous semi-group of operators continuous on
remains the same as for a Banach space. Similarly, the class
is defined by the property
as
for any
. A semi-group is said to be locally equicontinuous (of class
) if the family of operators
is equicontinuous when
ranges over any finite interval in
. In a barrelled space, a semi-group of class
is always equicontinuous (cf. Equicontinuity).
A semi-group is said to be equicontinuous (of class ) if the family
,
, is equicontinuous.
Infinitesimal operators and infinitesimal generators are defined as in the Banach space case.
Assume from now on that the space is sequentially complete. The infinitesimal generator
of a semi-group of class
is identical to the infinitesimal operator; its domain of definition,
, is dense in
and, moreover, the set
is dense in
. The semi-group
leaves
invariant and
![]() |
If is the infinitesimal generator of a semi-group of class
, the resolvent
is defined for
and is the Laplace transform of the semi-group.
A linear operator is the infinitesimal generator of a semi-group of class
if and only if it is closed, has dense domain of definition in
, and if there exists a sequence of positive numbers
such that, for any
, the resolvent
is defined and the family of operators
,
is equicontinuous. In this situation the semi-group can be constructed by the formula
![]() |
![]() |
In a non-normed locally convex space, the infinitesimal generator of a semi-group of class may have no resolvent at any point. An example is:
in the space
of infinitely-differentiable functions of
on
. As a substitute for the resolvent one can take a continuous operator whose product with
, from the right and the left, differs by a "small amount" from the identity operator.
A continuous operator defined for
in a set
is called an asymptotic resolvent for a linear operator
if
is continuous on
, the operator
can be extended from
to a continuous operator
on
, and if there exists a limit point
of the set
such that
,
as
for any
, where
![]() |
An asymptotic resolvent possesses various properties resembling those of the ordinary resolvent.
A closed linear operator with a dense domain of definition in
is the infinitesimal generator of a semi-group of class
if and only if there exist numbers
and
such that, for
, there exists an asymptotic resolvent
of
with the properties: the functions
,
,
are strongly infinitely differentiable for
, and the families of operators
![]() |
are equicontinuous.
Generation theorems have also been proved for other classes of semi-groups of operators on a locally convex space.
Adjoint semi-groups.
If is a semi-group of class
on a Banach space
, then the adjoint operators form a semi-group of bounded operators on the adjoint space
. However, the assertion that
as
for any
is valid only in the sense of the weak- topology
. If
is the generating operator, its adjoint
is a weak infinitesimal generator for
, in the sense that
is the set of all
for which the limit of
as
exists in the sense of weak- convergence and is equal to
. The domain of definition
is dense in
— again in the sense of the weak- topology — and the operator
is closed in the weak- topology.
Let be the set of all elements in
such that
as
in the strong sense; then
is a closed subspace of
that is invariant under all
. On
the operators
form a semi-group of class
. The space
is also the strong closure of the set
in
. If the original space is reflexive, then
. Analogous propositions hold for semi-groups of class
in locally convex spaces. Semi-groups of classes
and
generate semi-groups of the same classes in
.
Distribution semi-groups in a (separable) locally convex space.
A distribution semi-group in a sequentially complete locally convex space is defined just as in a Banach space. A semi-group
is said to be locally equicontinuous (of class
) if, for any compact subset
, the family of operators
,
, is equicontinuous. In a barrelled space
, any distribution semi-group is defined by analogy to the Banach case. For semi-groups of class
, the infinitesimal operator is closed
,
is dense in
, and for any
and
,
![]() | (7) |
A generalized function with support in
, possessing the properties (7), is naturally called the fundamental function of the operator
. Thus, if
is the infinitesimal operator of a semi-group
of class
, then
is the fundamental function of the operator
. The converse statement is true under certain additional assumptions about the order of singularity of the fundamental function
(or, more precisely, of the function
, where
).
A useful notion for the characterization of semi-groups in a locally convex space is that of the generalized resolvent. Let denote the Laplace transform of a function
, and let
be the space of all such transforms. A topology is induced in this space, via the Laplace transform, from the topology of
. The Laplace transform of an
-valued generalized function
is defined by
. Under these conditions,
is a continuous mapping of
into the space
of continuous linear operators on
. Let
be the space of all
obtained from functions
with support in
, with the natural topology. If
is a linear operator on
, it can be "lifted" to an operator
on
via the equality
![]() |
Thus, it is defined for all such that the right-hand side of the equality is defined for any
and it extends to a generalized function in
. The continuous operator
on
is defined by
![]() |
If the operator has a continuous inverse
on
, then
is called the generalized resolvent of
.
An operator has a generalized resolvent if and only if the operator
has a locally equicontinuous fundamental function
, constructed by the formula
![]() |
where
![]() |
Subject to certain additional assumptions, is a distribution semi-group. An extension theorem for semi-groups of class
has also been proved in terms of generalized resolvents.
See also Semi-group of non-linear operators.
References
[1] | E. Hille, R.S. Phillips, "Functional analysis and semi-groups" , Amer. Math. Soc. (1957) MR0089373 Zbl 0392.46001 Zbl 0033.06501 |
[2] | Yu.M. Vuvunikyan, "Evolutionary representations of algebras of generalized functions" , Theory of operators in function spaces , Novosibirsk (1977) pp. 99–120 (In Russian) |
[3] | P.P. Zabreiko, A.V. Zafievskii, "On a certain class of semigroups" Soviet Math. Dokl. , 10 : 6 (1969) pp. 1523–1526 Dokl. Akad. Nauk SSSR , 189 : 5 (1969) pp. 934–937 MR264459 |
[4] | A.V. Zafievskii, Trudy Mat. Inst. Voronezh. Univ. , 1 (1970) pp. 206–210 |
[5] | K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, §1 MR0617913 Zbl 0435.46002 |
[6] | S.G. Krein, "Linear differential equations in Banach space" , Transl. Math. Monogr. , 29 , Amer. Math. Soc. (1971) (Translated from Russian) MR0342804 Zbl 0179.20701 |
[7] | Yu.T. Sil'chenko, "An evolutionary equation with an operator generating a nonlinear semigroup" Differential Equations , 15 : 2 (1979) pp. 255–258 Differentsial'nye Uravneniya , 15 : 2 (1979) pp. 363–366 Zbl 0505.34046 |
[8] | J. Chazarain, "Problèmes de Cauchy abstracts et applications à quelques problèmes mixtes" J. Funct. Anal. , 7 : 3 (1971) pp. 386–446 |
[9] | I. Ciorânescu, "La caracterisation spectrale d'opérateur, générateurs des semi-groupes distributions d'ordre fini de croissance" J. Math. Anal. Appl. , 34 (1971) pp. 34–41 |
[10] | I. Ciorânescu, "A characterization of distribution semigroups of finite growth order" Rev. Roum. Math. Pures Appl. , 22 : 8 (1977) pp. 1053–1068 MR500280 Zbl 0374.46032 |
[11] | T. Kato, "A characterization of holomorphic semigroups" Proc. Amer. Math. Soc. , 25 : 3 (1970) pp. 495–498 MR0264456 Zbl 0199.45604 |
[12] | J. Lions, "Les semigroupes distributions" Portugal. Math. , 19 (1960) pp. 141–164 |
[13] | A. Pazy, "On the differentiability and compactness of semi-groups of linear operators" J. Math. Mech. , 17 : 12 (1968) pp. 1131–1141 MR231242 Zbl 0162.45903 |
[14] | A. Pazy, "Approximations of the identity operator by semigroups of linear operators" Proc. Amer. Math. Soc. , 30 : 1 (1971) pp. 147–150 MR0287362 |
[15] | T. Ushijima, "On the abstract Cauchy problems and semi-groups of linear operators in locally convex spaces" Sci. Papers College Gen. Educ. Univ. Tokyo , 21 (1971) pp. 93–122 MR0312324 Zbl 0239.47031 |
[16] | T. Ushijima, "On the generation and smoothness of semi-groups of linear operators" J. Fac. Sci. Univ. Tokyo, Sec. 1A , 19 : 1 (1972) pp. 65–127 MR0308854 Zbl 0239.47032 |
[17] | C. Wild, "Semi-groupes de croissance ![]() |
[18] | J.A. Gol'dstein, "Semigroups of linear operators and application" , Oxford Univ. Press (1985) (Translated from Russian) |
[19] | A. Pazy, "Semigroups of linear operators and application to partial differential equations" , Springer (1983) MR0710486 |
[20] | Ph. Clément, H.J.A.M. Heijmans, S. Angenent, C.J. van Duijn, B. de Pagter, "One-parameter semigroups" , CWI Monographs , 5 , North-Holland (1987) MR0915552 Zbl 0636.47051 |
Comments
References
[a1] | P. Butzer, H. Berens, "Semigroups of operators and approximation" , Springer (1967) MR230022 |
[a2] | H. Kellermann, M. Hieber, "Integrated semigroups" J. Funct. Anal. , 84 (1989) pp. 160–180 MR0999494 Zbl 0604.47025 |
[a3] | I. Miyadera, N. Tanaka, "Exponentially bounded ![]() |
Semi-group of operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-group_of_operators&oldid=24563