Stability of Riesz bases

Abstract:

The Kato Theorem on similarity for sequences of projections in a Hilbert space is extended to the case when both sequences consist of nonselfadjoint projections. Passing to subspaces, this leads to stability theorems for Riesz bases of subspaces, at least one of which is finite dimensional, and for arbitrary vector Riesz bases. The following is proved as an application. If $\left \{\phi _n\right \}_{n=1}^{\infty }$ is a Riesz basis and $|\theta _n|\leq C$ for large $n$, where the constant $C$ depends only on $\left \{\phi _n\right \}_{n=1}^{\infty }$, then $\left \{\phi _n+\theta _n \phi _{n+1}\right \}_{n=1}^{\infty }$ also forms a Riesz basis.