A note on generalized derivation ranges (Q1064531)

Let X,Y be infinite dimensional Banach spaces over complex field C. For \(A\in B(X)\), \(B\in B(Y)\) we define the generalized derivation \({\mathcal J}_{AB}\) by the equation \({\mathcal J}_{AB}S=AS-SB,\quad (\forall)S\in B(Y,X).\) If A,B are compact, then we have that \({\mathcal J}_{AB}\) has closed range if and only if A and B have closed range. The proof employs a new concept, namely ''a sequence of approximately linearly independent vectors''. A sequence of vectors \(\{u_ k\}_{k\in {\mathbb{N}}}\) is called approximately linearly independent, if for any \(n\in N\), there exists \(m\in N\) such that if \(k_ n>k_{n-1}>...>k_ 1\geq m\), then any n vectors \(u_{k_ 1},...,u_{k_ n}\) are linearly independent.