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 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. |
Riesz operator. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riesz_operator&oldid=49930