Generalized Derivation modulo the Ideal of All Compact Operators

The related inequality (1.1) was obtained by Maher [3, Theorem 3.2] who showed that, if A is normal and AT = TA, where T ∈ Cp , then ‖T − (AX −XA)‖p ≥ ‖T‖p for all X ∈ ( ), where Cp is the von Neumann-Schatten class, 1≤ p <∞, and ‖·‖p its norm. Here we show that Maher’s result is also true in the case where Cp is replaced by ( ), the ideal of all compact operators with ‖·‖∞ its norm.Which allows to generalize these results, we prove that if the pair (A,B) has (PF) ( ), the Putnam-Fuglede’s property in ( ), andAT = TB, where T ∈ ( ), then ‖T−(AX−XB)‖∞ ≥ ‖T‖∞ for allX ∈ ( ).