Boundedness stability properties of linear and affine operators

Let E be a vector space in which some notion of boundedness is defined. The n T : E --> E is said to have the boundedness stability property (BSP) if f or each x is an element of E, the sequence (T-x(n))(n=1)(infinity) is bound ed whenever a subsequence (T(ni)x)(i=1)(infinity) is bounded. It is shown t hat (1) every affine operator on a finite-dimensional Banach space has the (BSP); (2) every affine operator on an infinite-dimensional vector space ha s 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 radiu s less than one and (4) when E is a Hilbert space, every normal or subnorma l bounded linear operator has the (BSP). Some results on affine operators o n a Hilbert space whose linear parts are normal or subnormal are also obtai ned. Finally, some problems are posed.