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A note on the Turn $${\check{s}}$$ s ˇ ek theorem

Longfa Sun (), Yinghua Sun (), Yan Guo () and Cuiling Wang ()
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Longfa Sun: North China Electric Power University
Yinghua Sun: North China Electric Power University
Yan Guo: North China Electric Power University
Cuiling Wang: Xiamen University

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 1, 175-179

Abstract: Abstract Let H and K be two real Hilbert spaces and $$f:H\rightarrow K$$ f : H → K be a standard $$\varepsilon $$ ε -phase isometry for some $$\varepsilon \ge 0$$ ε ≥ 0 , i.e., $$f(0)=0$$ f ( 0 ) = 0 and $$\begin{aligned} \min \{ \big |\Vert f(x)-f(y)\Vert -\Vert x-y\Vert \big |,\big |\Vert f(x)-f(y)\Vert -\Vert x+y\Vert \big |\}\le \varepsilon , \end{aligned}$$ min { | ‖ f ( x ) - f ( y ) ‖ - ‖ x - y ‖ | , | ‖ f ( x ) - f ( y ) ‖ - ‖ x + y ‖ | } ≤ ε , for all $$x,y\in H$$ x , y ∈ H . In this note, we show that if f is almost surjective, then there exists a surjective linear isometry $$U:H\rightarrow K$$ U : H → K and a phase function $$\gamma :H\rightarrow \{-1,1\}$$ γ : H → { - 1 , 1 } such that $$\begin{aligned} \Vert \gamma (x)f(x)-U(x)\Vert \le 2\sqrt{2}\varepsilon , \;\mathrm{for\; all}\;x\in H. \end{aligned}$$ ‖ γ ( x ) f ( x ) - U ( x ) ‖ ≤ 2 2 ε , for all x ∈ H .

Keywords: $$\varepsilon $$ ε -Phase-isometries; Linear isometries; Stability; Primary 39B05; 46C50; 47J05 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-023-00468-1

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