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Báo cáo hóa học: "ORTHOGONALITY PRESERVING PROPERTY, WIGNER EQUATION, AND STABILITY"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: ORTHOGONALITY PRESERVING PROPERTY, WIGNER EQUATION, AND STABILITY | ORTHOGONALITY PRESERVING PROPERTY WIGNER EQUATION AND STABILITY JACEK CHMIELINSKI Received 3 November 2005 Accepted 2 July 2006 We deal with the stability of the orthogonality preserving property in the class of mappings phase-equivalent to linear or conjugate-linear ones. We give a characterization of approximately orthogonality preserving mappings in this class and we show some connections between the considered stability and the stability of the Wigner equation. Copyright 2006 Jacek Chmielihski. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Let X and Y be real or complex inner-product spaces by K we denote the scalar field by and II II the inner product and the corresponding norm and by the standard orthogonality relation . A mapping f X Y is called an isometry if and only if II f x -f y II l x - y II for all x y e X and is called inner-product preserving if it is a solution of the orthogonality equation f x f y x y for x y e X. 1.1 One can show that f satisfies 1.1 if and only if it is a linear isometry. Similarly f X Y is a solution of the functional equation f x f y ylx for x y e X 1.2 if and only if f is a conjugate-linear isometry where conjugate-linear means that f Ax py Af x pf y for x y e X and A p e K. Functions f g X Y are called phaseequivalent if and only if there exists a mapping ơ X K such that g x Ơ x f x and l Ơ x l 1 for each x e X. Let us denote by ẩ J X Y the class of all mappings which are phase-equivalent to linear or conjugate-linear ones. A mapping f X Y which Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2006 Article ID 76489 Pages 1-9 DOI 10.1155 JIA 2006 76489 2 Orthogonality preserving property and its stability satisfies the condition Vx y e X x y f x f y 1.3 will be called orthogonality preserving o.p. . These mappings maybe