The spectral picture of (LA, RB)

AbstractLet A and B be commuting n-tuples of operators on a Hilbert space H, and let (LA, RB) be the (2n)-tuple of left and right multiplications induced by A and B on L(H). We show that σT((LA,RB), L(H)) = σT(A,H) × σT(B,H), σTe((LA,RB), L(H)) = [σTe(A,H) × σT(B,H)]∪[σT(A,H) × σTe(B,H)], and ind(LA, RB) = (−1)nind(A)ind(B), whenever A and B are Fredholm. Similar results hold for the restrictions of (LA, RB) to norm ideals in L(H), giving in particular the spectral picture of (A ⊗ I, I ⊗ B) on H ⊗ H. A. S. Fainstein, working independently, has announced versions of these results for Banach spaces