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 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 , .
F. Riesz himself, to whom the original result is due, called it the Zerlegungssatz.
|[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|
Riesz decomposition theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_decomposition_theorem&oldid=16314