On Stability of Isometries in Banach Spaces

Abstract

We analyze the problem of stability of linear isometries (SLI) of Banach spaces. Stability means the existence of a function σ(ε) such that σ(ε) → 0 as ε → 0 and for any ε-isometry A of the space X (i.e., $\,(1 - \epsilon )\,\|x\|\, \leq \,{\bigl \| Ax\bigr \|}\, \leq \, (1 + \epsilon )\,{\bigl \|x\bigr \|}\,$ for all x ∈ X) there is an isometry T such that $\|A - T\| \leq \sigma (\epsilon )$. It is known that all finite-dimensional spaces, Hilbert space, the spaces C(K) and L _p (μ) possess the SLI property. We construct examples of Banach spaces X, which have an infinitely smooth norm and are arbitrarily close to the Hilbert space, but fail to possess SLI, even for surjective operators. We also show that there are spaces that have SLI only for surjective operators. To obtain this result we find the functions σ(ε) for the spaces l ₁ and l _∞ . Finally we observe some relations between the conditional number of operators and their approximation by operators of similarity.

Mathematics Subject Classification (2000): Primary 39A30, 41A65, 46B20