Transitivity for linear operators on a Banach space

Let G be the multiplicative group of invertible elements of E(X), the algebra of all bounded linear operators on a Banach space X. In 1945 Mackey showed that if ${\displaystyle x_1,\dots ,x_n}$ and ${\displaystyle y_1,\dots ,y_n}$ are any two sets of linearly independent elements of X with the same number of items, then there exists T ∈ G so that ${\displaystyle T\left(x_k\right)=y_k}$, ${\displaystyle k=1,\dots ,n}$. We prove that some proper multiplicative subgroups of G have this property.