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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, [[#References|[a8]]] (see also [[Spectral theory of compact operators|Spectral theory of compact operators]]).

The precise definition is as follows ([[#References|[a2]]], [[#References|[a5]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r1301401.png" /> be a bounded operator on a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r1301402.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r1301403.png" /> be the spectrum of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r1301404.png" />. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r1301405.png" /> is isolated if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r1301406.png" /> is closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r1301407.png" />, i.e. if there is an open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r1301408.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r1301409.png" />. A point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r13014010.png" /> is a Riesz point if it is isolated and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r13014011.png" /> is the direct sum of a closed subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r13014012.png" /> and a finite-dimensional subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r13014013.png" />, both invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r13014014.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r13014015.png" /> is nilpotent on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r13014016.png" /> and a homeomorphism on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r13014017.png" />.

A bounded operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r13014018.png" /> is a Riesz operator if all points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r13014019.png" /> are Riesz points. Every compact operator is a Riesz operator (the Riesz theory of compact operators).

A bounded operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r13014020.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r13014021.png" /> is called quasi-nilpotent if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r13014022.png" /> (which is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r13014023.png" />).

A bounded operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r13014024.png" /> is a Riesz operator if and only if, [[#References|[a3]]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r13014025.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r13014026.png" /> runs over all compact operators (see [[Compact operator|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r13014027.png" /> is a [[Hilbert space|Hilbert space]], [[#References|[a6]]]. Further results can be found in [[#References|[a1]]], [[#References|[a7]]].

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

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r130/r130140/r13014028.png" /></td> </tr></table>

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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K.R. Davidson,   D.A. Herrero,   "Decomposition of Banach space operators"  ''Indiana Univ. Math. J.'' , '''35'''  (1986)  pp. 333–343</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Dieudonné,   "Foundations of modern analysis" , Acad. Press  (1960)  pp. 323</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.F. Ruston,   "Operators with a Fredholm theory"  ''J. London Math. Soc.'' , '''29'''  (1954)  pp. 318–326</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E.M. Stein,   "Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals" , Princeton Univ. Press  (1993)  pp. 389</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  T.T. West,   "Riesz operators in Banach spaces"  ''Proc. London Math. Soc.'' , '''16'''  (1966)  pp. 131–140</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  T.T. West,   "The decomposition of Riesz operators"  ''Proc. London Math. Soc.'' , '''16'''  (1966)  pp. 737–752</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  H. Zhong,   "On B-convex spaces and West decomposition of Riesz operators on them"  ''Acta Math. Sinica'' , '''37'''  (1994)  pp. 563–569</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  H.R. Dowson,   "Spectral theory of linear operators" , Acad. Press  (1978)  pp. 67ff.</TD></TR></table>