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- Title
Stability Characterizations of ∈-isometries on Certain Banach Spaces.
- Authors
Cheng, Li Xin; Sun, Long Fa
- Abstract
Suppose that X, Y are two real Banach Spaces. We know that for a standard ∈-isometry f: X → Y, the weak stability formula holds and by applying the formula we can induce a closed subspace N of Y*. In this paper, by using again the weak stability formula, we further show a sufficient and necessary condition for a standard ∈-isometry to be stable in assuming that N is w*-closed in Y*. Making use of this result, we improve several known results including Figiel's theorem in reflexive spaces. We also prove that if, in addition, the space Y is quasi-reflexive and hereditarily indecomposable, then L(f)≡span¯[f(x)] contains a complemented linear isometric copy of X; Moreover, if X = Y, then for every ∈-isometry f : X → X, there exists a surjective linear isometry S : X → X such that f − S is uniformly bounded by 2∈ on X.
- Subjects
BANACH spaces; SUBSPACES (Mathematics); STABILITY theory; ISOMETRICS (Mathematics); INDECOMPOSABLE modules
- Publication
Acta Mathematica Sinica, 2019, Vol 35, Issue 1, p123
- ISSN
1439-8516
- Publication type
Article
- DOI
10.1007/s10114-018-8038-1