BOUNDEDNESS STABILITY PROPERTIES OF LINEAR AND AFFINE OPERATORS

Abstract

Let $E$ be a vector space in which some notion of boundedness is defined. Then $T:E\to E$ is said to have the boundedness stability property (BSP) if for each $x\in E$, the sequence $(T^n_x)^\infty_{n=1}$ is bounded whenever a subsequence $(T^{n_i}x)^\infty_{i=1}$ is bounded. It is shown that (1) every affine operator on a finite-dimensional Banach space has the (BSP); (2) every affine operator on an infinite-dimensional vector space has the functional (BSP); (3) when $E$ is an infinite-dimensional Banach space, an affine operator $T$ on $E$ has the (BSP) if its linear part $A_T=T-T(0)$ is a compact perturbation of a bounded linear operator with spectral radius less than one and (4) when $E$ is a Hilbert space, every normal or subnormal bounded linear operator has the (BSP). Some results on affine operators on a Hilbert space whose linear parts are normal or subnormal are also obtained. Finally, some problems are posed.