Operators Admitting a Closed Subspace of Cyclic Vectors

Abstract

For an operator T acting on a complex infinite dimensional Banach space X such that \(T\oplus T\) is cyclic on \(X\oplus X\), we show that T admits a closed infinite dimensional subspace of cyclic vectors (excluding \(\{0\}\)). We give some applications of this result, and we show several examples of cyclic operators T with \(T\oplus T\) non-cyclic admitting a closed infinite dimensional subspace of cyclic vectors.