# Riesz decomposition theorem

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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 (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 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 and is the infimum of any Perron set generated by .

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 be a bounded linear operator on a Banach space with spectrum . Let be an isolated part of , i.e. and are both closed in . Let where is a contour in the resolvent set of with in its interior and separating from . Then is a projection (i.e. ), called the Riesz projection or Riesz projector (cf. also Spectral synthesis (for a single point) and Krein space). Put , . Then , both and are invariant under , and , .

If, moreover, is the disjoint union of two closed subsets and , then , .

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=16314
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article