# Riesz operator

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).

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