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There are two different theorems that go by this name.
 
There are two different theorems that go by this name.
  
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Roughly speaking, this asserts that a super- or subharmonic function is the sum of a potential and a harmonic function. For precise statements, see [[Subharmonic function|Subharmonic function]] (where it is called the Riesz local representation theorem), and [[Riesz theorem(2)|Riesz theorem]] (where it is simply called the Riesz theorem), [[#References|[a12]]], [[#References|[a20]]]. See also [[#References|[a8]]], 1.IV.8–9, 1.IX.11, 1.XIV.9, and [[#References|[a8]]], 1.XV.7, 1.XVII.7, for a corresponding result for superparabolic functions. In [[#References|[a4]]] the decomposition formula is called the Riesz integral representation and Riesz representation of a superharmonic function
 
Roughly speaking, this asserts that a super- or subharmonic function is the sum of a potential and a harmonic function. For precise statements, see [[Subharmonic function|Subharmonic function]] (where it is called the Riesz local representation theorem), and [[Riesz theorem(2)|Riesz theorem]] (where it is simply called the Riesz theorem), [[#References|[a12]]], [[#References|[a20]]]. See also [[#References|[a8]]], 1.IV.8–9, 1.IX.11, 1.XIV.9, and [[#References|[a8]]], 1.XV.7, 1.XVII.7, for a corresponding result for superparabolic functions. In [[#References|[a4]]] the decomposition formula is called the Riesz integral representation and Riesz representation of a superharmonic function
  
There is also an abstract version (see also [[Potential theory, abstract|Potential theory, abstract]]), dealing with harmonic spaces, which states (see [[#References|[a5]]], Thm. 2.2.2, p. 38) that every superharmonic function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r1301301.png" /> on a harmonic space which has a subharmonic minorant may be written uniquely as the sum of a potential and a harmonic function. This harmonic function is the greatest hypo-harmonic minorant of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r1301302.png" /> and is the infimum of any Perron set generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r1301303.png" />.
+
There is also an abstract version (see also [[Potential theory, abstract|Potential theory, abstract]]), dealing with harmonic spaces, which states (see [[#References|[a5]]], Thm. 2.2.2, p. 38) that every superharmonic function $u$ on a harmonic space which has a subharmonic minorant may be written uniquely as the sum of a potential and a harmonic function. This harmonic function is the greatest hypo-harmonic minorant of $u$ and is the infimum of any Perron set generated by $u$.
  
 
An immediate consequence is the Brelot–Bauer theorem ([[#References|[a5]]], Corol. 2.2.1, p. 38) that the real vector space of differences of positive harmonic functions on a harmonic space is a conditionally complete [[Vector lattice|vector lattice]] ([[Riesz space|Riesz space]]) with respect to the natural order (i.e., pointwise comparison). This gives a link with the [[Riesz decomposition property|Riesz decomposition property]].
 
An immediate consequence is the Brelot–Bauer theorem ([[#References|[a5]]], Corol. 2.2.1, p. 38) that the real vector space of differences of positive harmonic functions on a harmonic space is a conditionally complete [[Vector lattice|vector lattice]] ([[Riesz space|Riesz space]]) with respect to the natural order (i.e., pointwise comparison). This gives a link with the [[Riesz decomposition property|Riesz decomposition property]].
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==Riesz decomposition theorem for operators.==
 
==Riesz decomposition theorem for operators.==
This theorem is also called the Riesz splitting theorem and deals with splitting the spectrum of an operator. Following [[#References|[a10]]], p. 9ff, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r1301304.png" /> be a bounded [[Linear operator|linear operator]] on a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r1301305.png" /> with spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r1301306.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r1301307.png" /> be an isolated part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r1301308.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r1301309.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013010.png" /> are both closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013011.png" />. Let
+
This theorem is also called the Riesz splitting theorem and deals with splitting the spectrum of an operator. Following [[#References|[a10]]], p. 9ff, let $A$ be a bounded [[Linear operator|linear operator]] on a [[Banach space|Banach space]] $X$ with spectrum $\sigma ( A )$. Let $\sigma \subset \sigma ( A )$ be an isolated part of $\sigma ( A )$, i.e. $\sigma$ and $\tau = \sigma ( A ) \backslash \sigma$ are both closed in $\sigma ( A )$. Let
  
<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/r/r130/r130130/r13013012.png" /></td> </tr></table>
+
\begin{equation*} P _ { \sigma } = \frac { 1 } { 2 \pi i } \int _ { \Gamma } ( \lambda - A ) ^ { - 1 } d \lambda \end{equation*}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013013.png" /> is a contour in the resolvent set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013014.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013015.png" /> in its interior and separating <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013016.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013017.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013018.png" /> is a projection (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013019.png" />), called the Riesz projection or Riesz projector (cf. also [[Spectral synthesis|Spectral synthesis]] (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013020.png" /> a single point) and [[Krein space|Krein space]]). Put <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013022.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013023.png" />, both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013025.png" /> are invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013026.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013028.png" />.
+
where $\Gamma$ is a contour in the resolvent set of $A$ with $\sigma$ in its interior and separating $\sigma$ from $\tau$. Then $P _ { \sigma }$ is a projection (i.e. $P _ { \sigma } ^ { 2 } = P _ { \sigma }$), called the Riesz projection or Riesz projector (cf. also [[Spectral synthesis|Spectral synthesis]] (for $\sigma$ a single point) and [[Krein space|Krein space]]). Put $M = \operatorname { Im } ( P _ { \sigma } )$, $L = \operatorname { Ker } ( P _ { \sigma } )$. Then $X = M \oplus L$, both $M$ and $L$ are invariant under $A$, and $\sigma ( A | _ { M } ) = \sigma$, $\sigma ( A | _ { L } ) = \tau$.
  
If, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013029.png" /> is the disjoint union of two closed subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013031.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130130/r13013033.png" />.
+
If, moreover, $\sigma ( A )$ is the disjoint union of two closed subsets $\sigma$ and $\tau$, then $P _ { \sigma } + P _ { \tau } =\operatorname {id}$, $P _ { \sigma } P _ { \tau } = 0 = P _ { \tau } P _ { \sigma }$.
  
 
For more general results (for closed linear operators), see [[#References|[a10]]], p. 326ff. See also [[Functional calculus|Functional calculus]] (particularly the part dealing with the Riesz–Dunford functional calculus) and, e.g., [[#References|[a13]]].
 
For more general results (for closed linear operators), see [[#References|[a10]]], p. 326ff. See also [[Functional calculus|Functional calculus]] (particularly the part dealing with the Riesz–Dunford functional calculus) and, e.g., [[#References|[a13]]].
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Banalescu,  "On the Riesz decomposition property"  ''Rev. Roum. Math. Pures Appl.'' , '''36'''  (1991)  pp. 107–114</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Ch. Berg,  G. Frost,  "Potential theory on locally compact Abelian groups" , Springer  (1975)  pp. 148</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.M. Blumenthal,  R.K. Getoor,  "Markov processes and potential theory" , Acad. Press  (1968)  pp. 272, Thm. 2.11</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Brelot,  "On topologies and boundaries in potential theory" , Springer  (1971)  pp. 93; 45</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  C. Constantinescu,  A. Cornea,  "Potential theory on harmonic spaces" , Springer  (1972)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J. Deny,  "Noyaux de convolution de Hunt et noyaux associes à une famille fondamentale"  ''Ann. Inst. Fourier (Grenoble)'' , '''12'''  (1962)  pp. 643–667</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  J.L. Doob,  "Semimartingales and subharmonic functions"  ''Trans. Amer. Math. Soc.'' , '''77'''  (1954)  pp. 86–121</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J.L. Doob,  "Classical potential theory and its probabilistic counterpart" , Springer  (1984)</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  R.K. Getoor,  J. Glover,  "Riesz decompositions in Markov process theory"  ''Trans. Amer. Math. Soc.'' , '''285'''  (1984)  pp. 107–132</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  I. Gohberg,  S. Goldberg,  M.A. Kaashoek,  "Classes of linear operators" , '''I''' , Birkhäuser  (1990)</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  M. Goldstein,  W.H. Ow,  "A converse of the Riesz decomposition theorem for harmonic spaces"  ''Math. Z.'' , '''173'''  (1980)  pp. 105–109</TD></TR><TR><TD valign="top">[a12]</TD> <TD valign="top">  W.K. Hayman,  P.B. Kennedy,  "Subharmonic functions" , '''I''' , Acad. Press  (1976)  pp. Sect. 3.5</TD></TR><TR><TD valign="top">[a13]</TD> <TD valign="top">  E. Hille,  "Methods in classical and functional analysis" , Addison-Wesley  (1972)  pp. 349–350</TD></TR><TR><TD valign="top">[a14]</TD> <TD valign="top">  G.A. Hunt,  "Markoff processes and potentials I"  ''Illinois J. Math.'' , '''1'''  (1957)  pp. 44–93</TD></TR><TR><TD valign="top">[a15]</TD> <TD valign="top">  G.A. Hunt,  "Markoff processes and potentials II"  ''Illinois J. Math.'' , '''1'''  (1957)  pp. 316–369</TD></TR><TR><TD valign="top">[a16]</TD> <TD valign="top">  G.A. Hunt,  "Markoff processes and potentials III"  ''Illinois J. Math.'' , '''2'''  (1958)  pp. 151–213</TD></TR><TR><TD valign="top">[a17]</TD> <TD valign="top">  F. Riesz,  "Sur les fonctions subharmoniques et leur rapport à la theorie du potentiel I"  ''Acta Math.'' , '''48'''  (1926)  pp. 329–343</TD></TR><TR><TD valign="top">[a18]</TD> <TD valign="top">  F. Riesz,  "Sur les fonctions subharmoniques et leur rapport à la theorie du potentiel II"  ''Acta Math.'' , '''54'''  (1930)  pp. 321–360</TD></TR><TR><TD valign="top">[a19]</TD> <TD valign="top">  F. Riesz,  "Über die linearen Transformationen des komplexen Hilbertschen Raumes"  ''Acta Sci. Math. (Szeged)'' , '''5'''  (1930/32)  pp. 23–54</TD></TR><TR><TD valign="top">[a20]</TD> <TD valign="top">  E.B. Saff,  V. Totik,  "Logarithmic potentials and external fields" , Springer  (1997)  pp. 100</TD></TR></table>
+
<table><tr><td valign="top">[a1]</td> <td valign="top">  M. Banalescu,  "On the Riesz decomposition property"  ''Rev. Roum. Math. Pures Appl.'' , '''36'''  (1991)  pp. 107–114</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  Ch. Berg,  G. Frost,  "Potential theory on locally compact Abelian groups" , Springer  (1975)  pp. 148</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  R.M. Blumenthal,  R.K. Getoor,  "Markov processes and potential theory" , Acad. Press  (1968)  pp. 272, Thm. 2.11</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  M. Brelot,  "On topologies and boundaries in potential theory" , Springer  (1971)  pp. 93; 45</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  C. Constantinescu,  A. Cornea,  "Potential theory on harmonic spaces" , Springer  (1972)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  J. Deny,  "Noyaux de convolution de Hunt et noyaux associes à une famille fondamentale"  ''Ann. Inst. Fourier (Grenoble)'' , '''12'''  (1962)  pp. 643–667</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  J.L. Doob,  "Semimartingales and subharmonic functions"  ''Trans. Amer. Math. Soc.'' , '''77'''  (1954)  pp. 86–121</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  J.L. Doob,  "Classical potential theory and its probabilistic counterpart" , Springer  (1984)</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  R.K. Getoor,  J. Glover,  "Riesz decompositions in Markov process theory"  ''Trans. Amer. Math. Soc.'' , '''285'''  (1984)  pp. 107–132</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  I. Gohberg,  S. Goldberg,  M.A. Kaashoek,  "Classes of linear operators" , '''I''' , Birkhäuser  (1990)</td></tr><tr><td valign="top">[a11]</td> <td valign="top">  M. Goldstein,  W.H. Ow,  "A converse of the Riesz decomposition theorem for harmonic spaces"  ''Math. Z.'' , '''173'''  (1980)  pp. 105–109</td></tr><tr><td valign="top">[a12]</td> <td valign="top">  W.K. Hayman,  P.B. Kennedy,  "Subharmonic functions" , '''I''' , Acad. Press  (1976)  pp. Sect. 3.5</td></tr><tr><td valign="top">[a13]</td> <td valign="top">  E. Hille,  "Methods in classical and functional analysis" , Addison-Wesley  (1972)  pp. 349–350</td></tr><tr><td valign="top">[a14]</td> <td valign="top">  G.A. Hunt,  "Markoff processes and potentials I"  ''Illinois J. Math.'' , '''1'''  (1957)  pp. 44–93</td></tr><tr><td valign="top">[a15]</td> <td valign="top">  G.A. Hunt,  "Markoff processes and potentials II"  ''Illinois J. Math.'' , '''1'''  (1957)  pp. 316–369</td></tr><tr><td valign="top">[a16]</td> <td valign="top">  G.A. Hunt,  "Markoff processes and potentials III"  ''Illinois J. Math.'' , '''2'''  (1958)  pp. 151–213</td></tr><tr><td valign="top">[a17]</td> <td valign="top">  F. Riesz,  "Sur les fonctions subharmoniques et leur rapport à la theorie du potentiel I"  ''Acta Math.'' , '''48'''  (1926)  pp. 329–343</td></tr><tr><td valign="top">[a18]</td> <td valign="top">  F. Riesz,  "Sur les fonctions subharmoniques et leur rapport à la theorie du potentiel II"  ''Acta Math.'' , '''54'''  (1930)  pp. 321–360</td></tr><tr><td valign="top">[a19]</td> <td valign="top">  F. Riesz,  "Über die linearen Transformationen des komplexen Hilbertschen Raumes"  ''Acta Sci. Math. (Szeged)'' , '''5'''  (1930/32)  pp. 23–54</td></tr><tr><td valign="top">[a20]</td> <td valign="top">  E.B. Saff,  V. Totik,  "Logarithmic potentials and external fields" , Springer  (1997)  pp. 100</td></tr></table>

Latest revision as of 17:00, 1 July 2020

There are two different theorems that go by this name.

Riesz decomposition theorem for super- or subharmonic functions.

Roughly speaking, this asserts that a super- or subharmonic function is the sum of a potential and a harmonic function. For precise statements, see Subharmonic function (where it is called the Riesz local representation theorem), and Riesz theorem (where it is simply called the Riesz theorem), [a12], [a20]. See also [a8], 1.IV.8–9, 1.IX.11, 1.XIV.9, and [a8], 1.XV.7, 1.XVII.7, for a corresponding result for superparabolic functions. In [a4] the decomposition formula is called the Riesz integral representation and Riesz representation of a superharmonic function

There is also an abstract version (see also Potential theory, abstract), dealing with harmonic spaces, which states (see [a5], Thm. 2.2.2, p. 38) that every superharmonic function $u$ on a harmonic space which has a subharmonic minorant may be written uniquely as the sum of a potential and a harmonic function. This harmonic function is the greatest hypo-harmonic minorant of $u$ and is the infimum of any Perron set generated by $u$.

An immediate consequence is the Brelot–Bauer theorem ([a5], Corol. 2.2.1, p. 38) that the real vector space of differences of positive harmonic functions on a harmonic space is a conditionally complete vector lattice (Riesz space) with respect to the natural order (i.e., pointwise comparison). This gives a link with the Riesz decomposition property.

There is also a converse Riesz decomposition theorem, [a11].

In the mid-1950s, the pioneering work of J.L. Doob and G.A. Hunt, [a7], [a14], [a15], [a16], showed a deep connection between potential theory and stochastic processes. Correspondingly, there are probabilistic Riesz decomposition theorems on decompositions of excessive functions, excessive measures and super-martingales. See [a3], [a9], [a8], 2.III.21, for precise formulations. There are also versions of these on commutative and non-commutative groups, [a1], [a2], [a6].

Riesz decomposition theorem for operators.

This theorem is also called the Riesz splitting theorem and deals with splitting the spectrum of an operator. Following [a10], p. 9ff, let $A$ be a bounded linear operator on a Banach space $X$ with spectrum $\sigma ( A )$. Let $\sigma \subset \sigma ( A )$ be an isolated part of $\sigma ( A )$, i.e. $\sigma$ and $\tau = \sigma ( A ) \backslash \sigma$ are both closed in $\sigma ( A )$. Let

\begin{equation*} P _ { \sigma } = \frac { 1 } { 2 \pi i } \int _ { \Gamma } ( \lambda - A ) ^ { - 1 } d \lambda \end{equation*}

where $\Gamma$ is a contour in the resolvent set of $A$ with $\sigma$ in its interior and separating $\sigma$ from $\tau$. Then $P _ { \sigma }$ is a projection (i.e. $P _ { \sigma } ^ { 2 } = P _ { \sigma }$), called the Riesz projection or Riesz projector (cf. also Spectral synthesis (for $\sigma$ a single point) and Krein space). Put $M = \operatorname { Im } ( P _ { \sigma } )$, $L = \operatorname { Ker } ( P _ { \sigma } )$. Then $X = M \oplus L$, both $M$ and $L$ are invariant under $A$, and $\sigma ( A | _ { M } ) = \sigma$, $\sigma ( A | _ { L } ) = \tau$.

If, moreover, $\sigma ( A )$ is the disjoint union of two closed subsets $\sigma$ and $\tau$, then $P _ { \sigma } + P _ { \tau } =\operatorname {id}$, $P _ { \sigma } P _ { \tau } = 0 = P _ { \tau } P _ { \sigma }$.

For more general results (for closed linear operators), see [a10], p. 326ff. See also Functional calculus (particularly the part dealing with the Riesz–Dunford functional calculus) and, e.g., [a13].

F. Riesz himself, to whom the original result is due, called it the Zerlegungssatz.

References

[a1] M. Banalescu, "On the Riesz decomposition property" Rev. Roum. Math. Pures Appl. , 36 (1991) pp. 107–114
[a2] Ch. Berg, G. Frost, "Potential theory on locally compact Abelian groups" , Springer (1975) pp. 148
[a3] R.M. Blumenthal, R.K. Getoor, "Markov processes and potential theory" , Acad. Press (1968) pp. 272, Thm. 2.11
[a4] M. Brelot, "On topologies and boundaries in potential theory" , Springer (1971) pp. 93; 45
[a5] C. Constantinescu, A. Cornea, "Potential theory on harmonic spaces" , Springer (1972)
[a6] J. Deny, "Noyaux de convolution de Hunt et noyaux associes à une famille fondamentale" Ann. Inst. Fourier (Grenoble) , 12 (1962) pp. 643–667
[a7] J.L. Doob, "Semimartingales and subharmonic functions" Trans. Amer. Math. Soc. , 77 (1954) pp. 86–121
[a8] J.L. Doob, "Classical potential theory and its probabilistic counterpart" , Springer (1984)
[a9] R.K. Getoor, J. Glover, "Riesz decompositions in Markov process theory" Trans. Amer. Math. Soc. , 285 (1984) pp. 107–132
[a10] I. Gohberg, S. Goldberg, M.A. Kaashoek, "Classes of linear operators" , I , Birkhäuser (1990)
[a11] M. Goldstein, W.H. Ow, "A converse of the Riesz decomposition theorem for harmonic spaces" Math. Z. , 173 (1980) pp. 105–109
[a12] W.K. Hayman, P.B. Kennedy, "Subharmonic functions" , I , Acad. Press (1976) pp. Sect. 3.5
[a13] E. Hille, "Methods in classical and functional analysis" , Addison-Wesley (1972) pp. 349–350
[a14] G.A. Hunt, "Markoff processes and potentials I" Illinois J. Math. , 1 (1957) pp. 44–93
[a15] G.A. Hunt, "Markoff processes and potentials II" Illinois J. Math. , 1 (1957) pp. 316–369
[a16] G.A. Hunt, "Markoff processes and potentials III" Illinois J. Math. , 2 (1958) pp. 151–213
[a17] F. Riesz, "Sur les fonctions subharmoniques et leur rapport à la theorie du potentiel I" Acta Math. , 48 (1926) pp. 329–343
[a18] F. Riesz, "Sur les fonctions subharmoniques et leur rapport à la theorie du potentiel II" Acta Math. , 54 (1930) pp. 321–360
[a19] F. Riesz, "Über die linearen Transformationen des komplexen Hilbertschen Raumes" Acta Sci. Math. (Szeged) , 5 (1930/32) pp. 23–54
[a20] E.B. Saff, V. Totik, "Logarithmic potentials and external fields" , Springer (1997) pp. 100
How to Cite This Entry:
Riesz decomposition theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_decomposition_theorem&oldid=16314
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article