Stability of symmetric $$\varepsilon $$ ε -isometries on wedges

Longfa Sun ()
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Longfa Sun: North China Electric Power University

Indian Journal of Pure and Applied Mathematics, 2021, vol. 52, issue 1, 11-21

Abstract: Abstract Let X, Y be Banach spaces, W be a closed wedge of X, and $$f:W\cup -W\rightarrow Y$$ f : W ∪ - W → Y be a symmetric $$\varepsilon $$ ε -isometry. We firstly establish a weak stability formula about the symmetric isometry f. Making use of it, we prove a series of new stability theorems for the symmetric isometries defined on the positive cones W of C(K)-spaces. For example, if $$f(W\cup -W)$$ f ( W ∪ - W ) contains a reproducing wedge of Y, then there exists a linear surjective isometry $$U:C(K)\rightarrow Y$$ U : C ( K ) → Y such that $$f-U$$ f - U is uniformly bounded by $$\frac{3}{2}\varepsilon $$ 3 2 ε on $$W\cup -W$$ W ∪ - W ; and if $$\overline{\mathrm{co}}f(W\cup -W)$$ co ¯ f ( W ∪ - W ) contains a reproducing wedge P of Y, then there exists a bounded linear operator $$T:Y\rightarrow C(K)$$ T : Y → C ( K ) with $$\Vert T\Vert =1$$ ‖ T ‖ = 1 such that $$\begin{aligned} \Vert Tf(x)-x\Vert \le \frac{3}{2}\varepsilon ,~~{\mathrm{for\;\; all\;}}~ x\in W\cup -W. \end{aligned}$$ ‖ T f ( x ) - x ‖ ≤ 3 2 ε , for all x ∈ W ∪ - W .

Keywords: Symmetric $$\varepsilon $$ ε -isometry; Stability; Weak stability; Banach space; Primary 46B04; 46B20; 47A58 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s13226-021-00126-4

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