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).

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.