Decomposition of Riesz operators

15.4.

DECOMPOSITION OF RIESZ OPERATORS~

If X is a Banach space let B(X), K(X), and Q(X) denote the sets of bounded, compact, and quasinilpotent operators on X (respectively). T, T ~ B(X), is a Riesz operator if it has a Riesz spectral theory associated with the compact operators, i.e., the spectrum of T is an at most countable set whose only possible accumulation point is the origin and all of whose nonzero points are poles of the resolvent of finite rank. The set of Riesz operators is denoted by R(X). Ruston [I] characterized the Riesz operators as TE~CX)~==> the coset T + K(X) is a quasinilpotent element of the Calkin algebra B(X)/K(X). Clearly R(X) ~ K(X) + Q(X). West [2] proved that if X is a Hilbert space then R(X) = K(X) + Q(X). This decomposition has been generalized to a C*-algebra setting by Smyth [3]. The proof is analogous to the superdiagonalization of a matrix which is then written as the sum of a diagonal matrix and a superdiagonal matrix with a zero diagonal. Nothing is known about the decomposition problem in general Banach spaces. It may be that the decomposability of all Riesz operators characterizes Hilbert spaces up to equivaalence among Banach spaces. LITERATURE CITED I 9

2. 3.

A. F. Ruston, "Operators with a Fredholm theory," J. London Math. Soc., 29, 318-326 (1954)~ T. T. West, "The decomposition of Riesz operators," Proc. London Math. Soc. III Series, 16, 737-752 (1966). M. R. F. Smyth, "Riesz theory in Banach algebras," Math. Z., 145, 145-155 (1975).

~M. R. F. SMYTH and T. T. WEST.

3 9 Trinity College, Dublin 2, Ireland.

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