Rigidity of commutators and elementary operators on Calkin algebras

Abstract

LetA=(A ₁,...,A _n ),B=(B ₁,...,B _n )εL(ℓ^p)ⁿ be arbitraryn-tuples of bounded linear operators on (ℓ^p), with 1<p<∞. The paper establishes strong rigidity properties of the corresponding elementary operators ε _a,b on the Calkin algebraC(ℓ^p)≡L(ℓ^p)/K(ℓ^p);$\varepsilon _{\alpha ,b} (s) = \sum\limits_{i = 1}^n {a_i sb_i }$, where quotient elements are denoted bys=S+K(ℓ^p) forSεL(ℓ^p). It is shown among other results that the kernel Ker(ε _a,b ) is a non-separable subspace ofC(ℓ^p) whenever ε _a,b fails to be one-one, while the quotient$C(\ell ^p )/\overline {\operatorname{Im} \left( {\varepsilon _{\alpha ,b} } \right)}$ is non-separable whenever ε _a,b fails to be onto. These results extend earlier ones in several directions: neither of the subsets {A ₁,...,A _n }, {B ₁,...,B _n } needs to consist of commuting operators, and the results apply to other spaces apart from Hilbert spaces.