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 ∈ ( ).


Introduction.
Let ᏸ(Ᏼ) be the algebra of all bounded operators acting on a complex Hilbert space Ᏼ.For A and B in ᏸ(Ᏼ), let δ A,B denote the operator on ᏸ(Ᏼ) defined by δ A,B (X) = AX − XB.If A = B, then δ A is called the inner derivation induced by A. In [1, Theorem 1.7], Anderson showed that if A is normal and commutes with T then, for all X ∈ ᏸ(Ᏼ), T − (AX − XA) ≥ T . ( In [4], we generalized this inequality, we showed that if the pair (A, B) has the Putnam-Fuglede's property (in particular if A and B are normal operators) and AT = T B, then for all X ∈ ᏸ(Ᏼ), T − (AX − XB) ≥ T . (1. 2) The related inequality (1.1) was obtained by Maher [3,Theorem 3.2] who showed that, if A is normal and , where C p 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 C p is replaced by (Ᏼ), 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) (Ᏼ) , the Putnam-Fuglede's property in (Ᏼ), and AT = T B, where T ∈ (Ᏼ), then T −(AX −XB) ∞ ≥ T ∞ for all X ∈ ᏸ(Ᏼ).

Normal derivations.
In this section, we investigate on the orthogonality of the range and the kernel of a normal derivation modulo the ideal of all compact operators.We recall that the pair (A, B) has the property (PF) (Ᏼ) if AT = T B, where T ∈ (Ᏼ) implies A * T = T B * .Before proving this result we need the following lemmas.
Proof.Let λ 1 ,λ 2 ,...,λ n be eigenvalues of the diagonal operator N.Then, the operator N can be written under the following matrix form: According to the following decomposition of Ᏼ: Let |δ ij | and |X ij | be the matrix representations of S and X according to the above decomposition of Ᏼ.Then Here * stands for some entry.

.14)
Then Theorem 2.3 would imply that T is compact and (2.16) 3. Generalized derivations.In this section, we generalize the above results to a large class of operators.We show that if the pair (A, B) has the property (PF) (Ᏼ) , and AS = SB such that δ N,M (X) + S ∈ (Ᏼ), then S ∈ (Ᏼ) and Before proving this result, we need the following lemma.(2)⇒(1).Let T ∈ (Ᏼ) such that AT = T B. Taking the two decompositions of Ᏼ, ⊥ and Ᏼ 2 = Ᏼ = ker(T ) ⊥ ⊕ ker T .Then we can write A and B on where A 1 and B 1 are normal operators.Also we can write T and X on Ᏼ 2 into Ᏼ 1 Since A 1 and B 1 are normal operators, then, by applying the Fuglede-Putnam's theorem, we obtain (3.4) Proof.Since the pair (A, B) satisfies the property (PF) (Ᏼ) , it follows by Lemma 3.1 that R(S) reduces A, ker(S) ⊥ reduces B, and A| R(S) and B| ker(S) ⊥ are normal operators.

.7)
Since A 1 and B 1 are two normal operators, then it results from Corollary 2.4 that S 1 is compact and in each of the following cases: (1) if A, B ∈ ᏸ(Ᏼ) such that Ax ≥ x ≥ Bx for all x ∈ Ᏼ; (2) if A is invertible and B such that A −1 B ≤ 1.
Proof.(1) The result of Tong [5, Lemma 1] guarantees that the above condition implies that for all T ∈ ker(δ A,B | (Ᏼ)), R(T ) reduces A, ker(T ) ⊥ reduces B, and A| R(T ) and B| ker(T ) ⊥ are unitary operators.Hence, it results from Lemma 3.1 that the pair (A, B) has the property (PF) (Ᏼ) and the result holds by Theorem 3.2.
Inequality (3.10) holds in particular if A = B is isometric; in other words, Ax = x for all x ∈ Ᏼ.
(2) In this case, it suffices to take A 1 = B −1 A and B 1 = B −1 B, then A 1 x ≥ x ≥ B 1 x and the result holds by (1) for all x ∈ Ᏼ.