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

✍ Scribed by B.P Duggal

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 114 KB
• Volume: 263
• Category: Article
• ISSN: 0022-247X
• DOI: 10.1006/jmaa.2001.7601

✦ Synopsis

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 .