# Riesz decomposition theorem

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.

How to Cite This Entry:
Riesz decomposition theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_decomposition_theorem&oldid=50379
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article