A spectral characterization of operators having rank k (Q2640849)

The following result is proved. Let \(A\in B(X)\), X a Banach space. Then \(rank A\leq k\quad \Rightarrow \cap^{k+1}_{m=1}\sigma (T+c_ mA)\subset \sigma (T)\) for each \(T\in B(X)\) and arbitrary, distinct scalars \(c_ 1,...,c_{k+1}\); in the reverse direction, \(\cap^{k+1}_{m=1}\sigma (T+c_ mA)\subset \sigma (T)\) for each T of rank at most \((k+1)\) and some nonzero scalars \(c_ 1,...,c_{k+1}\Rightarrow\) rank \(A\leq k\).