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

Longfa Sun (), Yinghua Sun (), Yan Guo () and Cuiling Wang ()
Additional contact information
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
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s13226-023-00468-1 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:indpam:v:56:y:2025:i:1:d:10.1007_s13226-023-00468-1

Ordering information: This journal article can be ordered from
https://www.springer.com/journal/13226

DOI: 10.1007/s13226-023-00468-1

Access Statistics for this article

Indian Journal of Pure and Applied Mathematics is currently edited by Nidhi Chandhoke

More articles in Indian Journal of Pure and Applied Mathematics from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().