# Riesz operator

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The Riesz operators on a Banach space are, roughly speaking, those bounded linear operators that have a Riesz spectral theory, i.e. that have a spectral theory like that of compact operators, [a8] (see also Spectral theory of compact operators).

The precise definition is as follows ([a2], [a5]). Let be a bounded operator on a Banach space , and let be the spectrum of . A point is isolated if is closed in , i.e. if there is an open subset such that . A point is a Riesz point if it is isolated and is the direct sum of a closed subspace and a finite-dimensional subspace , both invariant under and such that is nilpotent on and a homeomorphism on .

A bounded operator is a Riesz operator if all points are Riesz points. Every compact operator is a Riesz operator (the Riesz theory of compact operators).

A bounded operator on is called quasi-nilpotent if (which is equivalent to ).

A bounded operator is a Riesz operator if and only if, [a3]:

where runs over all compact operators (see Compact operator).

It is a long-standing question (still open as of 2000) whether every Riesz operator splits as the sum of a quasi-nilpotent operator and a compact operator. Such a decomposition is called a West decomposition, after T.T. West, who proved this for the case that is a Hilbert space, [a6]. Further results can be found in [a1], [a7].

There is another, quite different, notion in which the phrase "Riesz operator" occurs, viz. the parametrized family of multiplier operators

called the Bochner–Riesz operator, [a4]. They are important in Bochner–Riesz summability (see also Riesz summation method).

#### References

 [a1] K.R. Davidson, D.A. Herrero, "Decomposition of Banach space operators" Indiana Univ. Math. J. , 35 (1986) pp. 333–343 [a2] J. Dieudonné, "Foundations of modern analysis" , Acad. Press (1960) pp. 323 [a3] A.F. Ruston, "Operators with a Fredholm theory" J. London Math. Soc. , 29 (1954) pp. 318–326 [a4] E.M. Stein, "Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals" , Princeton Univ. Press (1993) pp. 389 [a5] T.T. West, "Riesz operators in Banach spaces" Proc. London Math. Soc. , 16 (1966) pp. 131–140 [a6] T.T. West, "The decomposition of Riesz operators" Proc. London Math. Soc. , 16 (1966) pp. 737–752 [a7] H. Zhong, "On B-convex spaces and West decomposition of Riesz operators on them" Acta Math. Sinica , 37 (1994) pp. 563–569 [a8] H.R. Dowson, "Spectral theory of linear operators" , Acad. Press (1978) pp. 67ff.
How to Cite This Entry:
Riesz operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_operator&oldid=15754
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article