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

Abstract

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 δ A B B H → B H denote the generalized derivation δ AB 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 δ A m B m X = 0 for some X ∈ B H and if either of A and B * is injective, then a necessary and sufficient condition for δ AB X = 0 is that A r XB n−r − A n−r XB r = 0 for (any) two consecutive values of r 1 ≤ r < n. (ii) If δ A m B m X and δ A n B n X ∈ p for some X ∈ B H , and if m = 2 or 3, then either δ n AB X or δ n+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 δ AB X ∈ 2 tn p . (iii) If δ A m B m X and δ A n B n X ∈ p for some X ∈ B H , and if either A is semi-Fredholm with ind A ≤ 0 or 1 − A * A ∈ p , then δ AB X ∈ p .

Some Intertwining Relations Modulo Operator Ideals

“…By a slight modification of the proof of Theorem C given in [10], Duggal [5,Theorem 5] has extended this result to relatively prime powers other than 2 and 3.…”