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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 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" />.
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The precise definition is as follows ([[#References|[a2]]], [[#References|[a5]]]). Let $R$ be a bounded operator on a [[Banach space|Banach space]] $E$, and let $\sigma ( R )$ be the spectrum of $R$. A point $\lambda \in \sigma ( R )$ is isolated if $\sigma ( R ) \backslash \lambda$ is closed in $\sigma ( R )$, i.e. if there is an open subset $U \subset \mathbf{C}$ such that $U \cap \sigma ( R ) = \{ \lambda \}$. A point $\lambda \in \sigma ( R )$ is a Riesz point if it is isolated and $E$ is the direct sum of a closed subspace $F ( \lambda )$ and a finite-dimensional subspace $N ( \lambda )$, both invariant under $R$ and such that $R - \lambda$ is nilpotent on $N ( \lambda )$ and a homeomorphism on $F ( \lambda )$.
  
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).
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A bounded operator $R$ is a Riesz operator if all points $\lambda \in \sigma ( R ) \backslash \{ 0 \}$ 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" />).
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A bounded operator $T$ on $E$ is called quasi-nilpotent if $\operatorname { lim } _ { n \rightarrow \infty } \| T ^ { n } \| ^ { 1 / n } = 0$ (which is equivalent to $\sigma ( T ) = \{ 0 \}$).
  
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]]]:
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A bounded operator $R$ 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>
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\begin{equation*} \operatorname { lim } _ { n \rightarrow \infty } \{ \operatorname { inf } _ { C } \| R ^ { n } - C \| ^ { 1 / n } \} = 0, \end{equation*}
  
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]]).
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where $C$ 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]]].
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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 $E$ 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
 
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>
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\begin{equation*} f ( x ) \mapsto ( S ^ { \alpha } f ) ( x ) = \int _ { | \xi | \leq 1 } \hat { f} ( \xi )  ( 1 - | \xi | ^ { 2 } ) ^ { \alpha } e ^ { 2 \pi i x . \xi } d \xi,  \end{equation*}
  
 
called the Bochner–Riesz operator, [[#References|[a4]]]. They are important in Bochner–Riesz summability (see also [[Riesz summation method|Riesz summation method]]).
 
called the Bochner–Riesz operator, [[#References|[a4]]]. They are important in Bochner–Riesz summability (see also [[Riesz summation method|Riesz summation method]]).
  
 
====References====
 
====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>
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<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>

Latest revision as of 15:30, 1 July 2020

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 $R$ be a bounded operator on a Banach space $E$, and let $\sigma ( R )$ be the spectrum of $R$. A point $\lambda \in \sigma ( R )$ is isolated if $\sigma ( R ) \backslash \lambda$ is closed in $\sigma ( R )$, i.e. if there is an open subset $U \subset \mathbf{C}$ such that $U \cap \sigma ( R ) = \{ \lambda \}$. A point $\lambda \in \sigma ( R )$ is a Riesz point if it is isolated and $E$ is the direct sum of a closed subspace $F ( \lambda )$ and a finite-dimensional subspace $N ( \lambda )$, both invariant under $R$ and such that $R - \lambda$ is nilpotent on $N ( \lambda )$ and a homeomorphism on $F ( \lambda )$.

A bounded operator $R$ is a Riesz operator if all points $\lambda \in \sigma ( R ) \backslash \{ 0 \}$ are Riesz points. Every compact operator is a Riesz operator (the Riesz theory of compact operators).

A bounded operator $T$ on $E$ is called quasi-nilpotent if $\operatorname { lim } _ { n \rightarrow \infty } \| T ^ { n } \| ^ { 1 / n } = 0$ (which is equivalent to $\sigma ( T ) = \{ 0 \}$).

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

\begin{equation*} \operatorname { lim } _ { n \rightarrow \infty } \{ \operatorname { inf } _ { C } \| R ^ { n } - C \| ^ { 1 / n } \} = 0, \end{equation*}

where $C$ 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 $E$ 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

\begin{equation*} f ( x ) \mapsto ( S ^ { \alpha } f ) ( x ) = \int _ { | \xi | \leq 1 } \hat { f} ( \xi ) ( 1 - | \xi | ^ { 2 } ) ^ { \alpha } e ^ { 2 \pi i x . \xi } d \xi, \end{equation*}

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=49930
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article