The Commutant Modulo Cp of Co-prime Powers of Operators on a Hilbert Space

Let H be a separable infinite-dimensional complex Hilbert space and let A B ∈ B H , where B H is the algebra of operators on H into itself. Let δA B B H → B H denote the generalized derivation δAB X = AX − XB. This note considers the relationship between the commutant of an operator and the commutant of coprime powers of the operator. Let m n be some co-prime natural numbers and let p denote the Schatten p-class, 1 ≤ p < ∞. We prove (i) If δAmBm X = 0 for some X ∈ B H and if either of A and B∗ is injective, then a necessary and sufficient condition for δAB X = 0 is that ArXBn−r −An−rXBr = 0 for (any) two consecutive values of r 1 ≤ r < n. (ii) If δAmBm X and δAnBn X ∈ p for some X ∈ B H , and if m = 2 or 3, then either δn AB X or δn+3 AB X ∈ p; for general m and n, if A and B∗ are normal or subnormal, then there exists a natural number t such that δAB X ∈ 2tnp. (iii) If δAmBm X and δAnBn X ∈ p for some X ∈ B H , and if either A is semi-Fredholm with indA ≤ 0 or 1 − A∗A ∈ p, then δAB X ∈ p