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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|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 $u$ 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]].

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

In the mid-1950s, the pioneering work of J.L. Doob and G.A. Hunt, [[#References|[a7]]], [[#References|[a14]]], [[#References|[a15]]], [[#References|[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 [[#References|[a3]]], [[#References|[a9]]], [[#References|[a8]]], 2.III.21, for precise formulations. There are also versions of these on commutative and non-commutative groups, [[#References|[a1]]], [[#References|[a2]]], [[#References|[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 [[#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

\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|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, $\sigma ( A )$ 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]]].

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

====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>