ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 46, 47 Tylli, Hans-Olav, University of Helsinki, Finland Rigidity of commutators and elementary operators on Calkin algebras Let E be a Banach space, C(E) = L(E)/K(E) the Calkin algebra, and denote quotient elements by s = S + K(E). Let E_{a,b} : s ↦ ∑_{i=1}^{n} a_{i}sb_{i} stand for the elementary operator on C(E) induced by the n-tuples A = (A_{1}, …, A_{n}), B = (B_{1}, …, B_{n}) ∈ L(E)^{n}. The following strong rigidity results are presented for arbitrary n-tuples A and B: (i) if E has an unconditional Schauder basis, then dist(E_{a,b}, K(C(E)) is uniformly comparable with ∥E_{a,b}∥ [1], (ii) E_{a,b} is injective on C(ℓ^{p}) iff it is bounded below [2], (iii) E_{a,b} has dense range on C(ℓ^{p}) iff it is surjective [2]. These results are joint work with E. Saksman. References: [1] E. Saksman, and H. -O. Tylli, The Apostol-Fialkow formula for elementary operators on Banach spaces, (preprint, University of Helsinki, 1995). [2] E. Saksman, and H. -O. Tylli, Rigidity of commutators and elementary operators on Calkin algebras, Israel J. Math. (to appear).