Geodesic
On the Range and the Kernel of Derivations
Serdica Mathematical Journal, Tome 32 (2006) no. 1, pp. 31-38.

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Let H be a separable infinite dimensional complex Hilbert space and let L(H) denote the algebra of all bounded linear operators on H into itself. Given A ∈ L(H), the derivation δA : L(H)→ L(H) is defined by δA(X) = AX-XA. In this paper we prove that if A is an n-multicyclic hyponormal operator and T is hyponormal such that AT = TA, then || δA(X)+T|| ≥ ||T|| for all X ∈ L(H). We establish the same inequality if A is a finite operator and commutes with normal operator T. Some related results are also given.
Keywords: Finite Operator, n-multicyclic hyponormal operator 
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Bouali, Said; Bouhafsi, Youssef. On the Range and the Kernel of Derivations. Serdica Mathematical Journal, Tome 32 (2006) no. 1, pp. 31-38. http://geodesic.mathdoc.fr/item/SMJ2_2006_32_1_a2/