Stability of basic sequences via nonlinear ε-isometries

Duanxu Dai ()
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Duanxu Dai: Quanzhou Normal University

Indian Journal of Pure and Applied Mathematics, 2018, vol. 49, issue 3, 571-579

Abstract: Abstract In this paper, Let X, Y be two real Banach spaces and ε ≥ 0. A mapping f: X → Y is said to be a standard ε-isometry provided f(0) = 0 and (1) $$\parallel f\left( x \right) - f\left( y \right)\parallel - \parallel x - y\parallel | \leqslant \varepsilon $$ ‖ f ( x ) − f ( y ) ‖ − ‖ x − y ‖ | ≤ ε for all x, y ∈ X. If ε = 0, then it is simply called a standard isometry. We prove a sufficient and necessary condition for which {f(xn)}n≥1 is a basic sequence of Y equivalent to {xn}n≥1 whenever {xn}n≥1 is a basic sequence in X and f: X → Y is a nonlinear standard isometry. As a corollary we obtain the stability of basic sequences under the perturbation by nonlinear and non-surjective standard ε-isometries.

Keywords: ε-isometry; basic sequence; stability; Banach space (search for similar items in EconPapers)
Date: 2018
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s13226-018-0286-3

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