| In this paper, a derivation δAB mapping into a ideal I of B(H) is considered, when A,B ∈B(H) and I is a norm ideal. If Ran(δAB)?I, let δAB:B(H)→I denote the induced operator and let λ be the scalar such that A- λ∈I, B-λ∈I, we estimate the norm of δAB as follows‖A-λ‖+‖B-λ‖≥‖δAB‖≥‖A- λ‖+‖B-λ‖ when WN(A-λ)∩WN(λ - B)≠?, where WN(A- λ) denotes the normalized maximal numerical range and ‖A-λ‖ denotes the norm of A-λ∈I. In particular when I=Cp(lp, we prove that ‖δAB‖p=‖A-λ‖p+‖B-λ‖p if and only if ‖A-λ‖=‖A-λ‖p and WN(A-λ)∩WN(λ-B)≠?. At last, some examples show that the estimate as above is exact. |
