LetA=(A1,...,An),B=(B1,...,Bn)eL(lp)n be arbitraryn-tuples of bounded linear operators on (lp), with 1<p<∞. The paper establishes strong rigidity properties of the corresponding elementary operators ea,b on the Calkin algebraC(lp)≡L(lp)/K(lp);\(\varepsilon _{\alpha ,b} (s) = \sum\limits_{i = 1}^n {a_i sb_i } \), where quotient elements are denoted bys=S+K(lp) forSeL(lp). It is shown among other results that the kernel Ker(ea,b) is a non-separable subspace ofC(lp) whenever ea,b fails to be one-one, while the quotient\(C(\ell ^p )/\overline {\operatorname{Im} \left( {\varepsilon _{\alpha ,b} } \right)} \) is non-separable whenever ea,b fails to be onto. These results extend earlier ones in several directions: neither of the subsets {A1,...,An}, {B1,...,Bn} needs to consist of commuting operators, and the results apply to other spaces apart from Hilbert spaces.