Jacek Chmieliński

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Linear mappings approximately preserving orthogonality

2004-10-21

Jacek Chmieliński

Approximate orthogonality in normed spaces and its applications

2017-06-07

Jacek Chmieliński , Tomasz Stypuła , Paweł Wójcik

On a ρ-orthogonality

2010-09-01

Jacek Chmieliński , Paweł Wójcik

In a normed space we introduce an exact and approximate orthogonality relation connected with “norm derivatives” $${\rho^{\prime}_{\pm}}$$ . We also consider classes of linear mappings preserving (exactly and approximately) this … In a normed space we introduce an exact and approximate orthogonality relation connected with “norm derivatives” $${\rho^{\prime}_{\pm}}$$ . We also consider classes of linear mappings preserving (exactly and approximately) this kind of orthogonality.

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Remarks on orthogonality preserving mappings in normed spaces and some stability problems

2007-01-01

Jacek Chmieliński

On an -Birkhoff orthogonality.

2005-01-01

Jacek Chmieliński

We define an approximate Birkhoff orthogonality relation in a normed space. We compare it with the one given by S.S. Dragomir and establish some properties of it. In particular, we … We define an approximate Birkhoff orthogonality relation in a normed space. We compare it with the one given by S.S. Dragomir and establish some properties of it. In particular, we show that in smooth spaces it is equivalent to the approximate orthogonality stemming from the semi-inner-product.

Stability of the orthogonality preserving property in finite-dimensional inner product spaces

2005-06-29

Jacek Chmieliński

Isosceles-orthogonality preserving property and its stability

2009-08-27

Jacek Chmieliński , Paweł Wójcik

ρ-Orthogonality and its preservation –- revisited

2013-01-01

Jacek Chmieliński , Paweł Wójcik

The aim of the paper is to present results concerning the $\rho$-orthogonality and its preservation (both accurate and approximate) by linear operators. We survey on the results presented in [11] … The aim of the paper is to present results concerning the $\rho$-orthogonality and its preservation (both accurate and approximate) by linear operators. We survey on the results presented in [11] and [23], as well as give some new and more general ones.

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Approximate symmetry of Birkhoff orthogonality

2018-01-31

Jacek Chmieliński , Paweł Wójcik

On the stability of the orthogonality equation and the orthogonality-preserving property with two unknown functions

2016-09-30

Jacek Chmieliński , Radosław Łukasik , Paweł Wójcik

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The Stability of the Wigner Equation on a Restricted Domain

2001-02-01

Jacek Chmieliński , ‎Soon-Mo Jung

Decomposition of mappings approximately inner product preserving

2005-06-14

Roman Badora , Jacek Chmieliński

Perturbation of the Wigner equation in inner product C*-modules

2008-03-01

Jacek Chmieliński , Dijana Ilišević , Mohammad Sal Moslehian +1 more

Let A be a C*-algebra and B be a von Neumann algebra that both act on a Hilbert space H. Let M and N be inner product modules over A … Let A be a C*-algebra and B be a von Neumann algebra that both act on a Hilbert space H. Let M and N be inner product modules over A and B, respectively. Under certain assumptions, we show that for each mapping f:M→N satisfying ‖∣⟨f(x),f(y)⟩∣−∣⟨x,y⟩∣‖⩽φ(x,y) (x,y∊M), where φ is a control function, there exists a solution I:M→N of the Wigner equation ∣⟨I(x),I(y)⟩∣=∣⟨x,y⟩∣ (x,y∊M) such that ‖f(x)−I(x)‖⩽φ(x,x) (x∊M).

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Orthogonality equation with two unknown functions

2015-06-20

Jacek Chmieliński

The orthogonality equation involving two unknown functions $$\begin{array}{ll}\langle{f(x)}|{g(y)\rangle}=\langle{x}|{y}\rangle\end{array}$$ and, among related problems, the orthogonality preserving property $$\begin{array}{ll}x\bot y \Rightarrow f(x)\bot g(y)\end{array}$$ are considered. The orthogonality equation involving two unknown functions $$\begin{array}{ll}\langle{f(x)}|{g(y)\rangle}=\langle{x}|{y}\rangle\end{array}$$ and, among related problems, the orthogonality preserving property $$\begin{array}{ll}x\bot y \Rightarrow f(x)\bot g(y)\end{array}$$ are considered.

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Orthogonality Preserving Property and its Ulam Stability

2011-01-01

Jacek Chmieliński

On a singular case in the Hyers–Ulam–Rassias stability of the Wigner equation

2003-11-22

Jacek Chmieliński

On approximate solutions of the Pexider equation

1993-08-01

Jacek Chmieliński , Jacek Tabor

Normed spaces equivalent to inner product spaces and stability of functional equations

2013-03-22

Jacek Chmieliński

Let $${(X,\| \cdot \|)}$$ be a normed space. If $${\| \cdot \|_i}$$ is an equivalent norm coming from an inner product, then the original norm satisfies an approximate parallelogram law. … Let $${(X,\| \cdot \|)}$$ be a normed space. If $${\| \cdot \|_i}$$ is an equivalent norm coming from an inner product, then the original norm satisfies an approximate parallelogram law. Applying methods and results from the theory of stability of functional equations we study the reverse implication.

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On approximate solutions of the Pexider equation

1993-02-01

Jacek Chmieliński , Jacek Tabor

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Operators reversing orthogonality in normed spaces

2016-12-01

Jacek Chmieliński

APPROXIMATELY C*-INNER PRODUCT PRESERVING MAPPINGS

2008-02-29

Jacek Chmieliński , Mohammad Sal Moslehian

A mapping f : <TEX>$M{\rightarrow}N$</TEX> between Hilbert <TEX>$C^*$</TEX>-modules approximately preserves the inner product if <TEX>$$\parallel</TEX><TEX><</TEX><TEX>f(x),\;f(y)</TEX><TEX>></TEX><TEX>-</TEX><TEX><</TEX><TEX>x,y</TEX><TEX>></TEX><TEX>\parallel\leq\varphi(x,y)$$</TEX> for an appropriate control function <TEX>$\varphi(x,y)$</TEX> and all x, y <TEX>$\in$</TEX> M. In this paper, … A mapping f : <TEX>$M{\rightarrow}N$</TEX> between Hilbert <TEX>$C^*$</TEX>-modules approximately preserves the inner product if <TEX>$$\parallel</TEX><TEX><</TEX><TEX>f(x),\;f(y)</TEX><TEX>></TEX><TEX>-</TEX><TEX><</TEX><TEX>x,y</TEX><TEX>></TEX><TEX>\parallel\leq\varphi(x,y)$$</TEX> for an appropriate control function <TEX>$\varphi(x,y)$</TEX> and all x, y <TEX>$\in$</TEX> M. In this paper, we extend some results concerning the stability of the orthogonality equation to the framework of Hilbert <TEX>$C^*$</TEX>-modules on more general restricted domains. In particular, we investigate some asymptotic behavior and the Hyers-Ulam-Rassias stability of the orthogonality equation.

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Stability of angle-preserving mappings on the plane

2005-01-01

Jacek Chmieliński

We prove that for the mappings of the plane the property of preserving the angle between vectors is stable.We apply this result to prove some kind of stability of the … We prove that for the mappings of the plane the property of preserving the angle between vectors is stable.We apply this result to prove some kind of stability of the Wigner equation on the plane.

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Continuity and discontinuity of seminorms on infinite-dimensional vector spaces

2019-05-09

Jacek Chmieliński , Moshe Goldberg

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Local Approximate Symmetry of Birkhoff–James Orthogonality in Normed Linear Spaces

2021-06-17

Jacek Chmieliński , Divya Khurana , ‎Debmalya Sain

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Approximate smoothness in normed linear spaces

2023-04-27

Jacek Chmieliński , Divya Khurana , ‎Debmalya Sain

Abstract We introduce the notion of approximate smoothness in a normed linear space. We characterize this property and show the connections between smoothness and approximate smoothness for some spaces. As … Abstract We introduce the notion of approximate smoothness in a normed linear space. We characterize this property and show the connections between smoothness and approximate smoothness for some spaces. As an application, we consider in particular the Birkhoff–James orthogonality and its right-additivity under the assumption of approximate smoothness.

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Almost approximately inner product preserving mappings

2000-05-01

Jacek Chmieliński

On a $$\rho $$-Orthogonally Additive Mappings

2020-07-01

Jacek Chmieliński , Justyna Sikorska , Paweł Wójcik

Abstract We show that a real normed linear space endowed with the $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ρ</mml:mi></mml:math> -orthogonality relation, in general need not be an orthogonality space in the sense of … Abstract We show that a real normed linear space endowed with the $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ρ</mml:mi></mml:math> -orthogonality relation, in general need not be an orthogonality space in the sense of Rätz. However, we prove that $$\rho $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ρ</mml:mi></mml:math> -orthogonally additive mappings defined on some classical Banach spaces have to be additive. Moreover, additivity (and approximate additivity) under the condition of an approximate orthogonality is considered.

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Approximate Birkhoff-James Orthogonality in Normed Linear Spaces and Related Topics

2022-01-01

Jacek Chmieliński

Approximately $C^*$-inner product preserving mappings

2006-01-01

Jacek Chmieliński , Mohammad Sal Moslehian

A mapping $f: {\mathcal M} \to {\mathcal N}$ between Hilbert $C^*$-modules approximately preserves the inner product if \[\|< f(x), f(y)> - < x, y> \| \leq ϕ(x, y),\] for an … A mapping $f: {\mathcal M} \to {\mathcal N}$ between Hilbert $C^*$-modules approximately preserves the inner product if \[\|< f(x), f(y)> - < x, y> \| \leq ϕ(x, y),\] for an appropriate control function $ϕ(x,y)$ and all $x, y \in {\mathcal M}$. In this paper, we extend some results concerning the stability of the orthogonality equation to the framework of Hilbert $C^*$-modules on more general restricted domains. In particular, we investigate some asymptotic behavior and the Hyers--Ulam--Rassias stability of the orthogonality equation.

Orthogonality preserving property, Wigner equation, and stability

2006-01-01

Jacek Chmieliński

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 … 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.

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Convex functions and approximate Birkhoff–James orthogonality

2023-10-09

Jacek Chmieliński , Karol Gryszka , Paweł Wójcik

Abstract For the well developed notion of approximate Birkhoff-James orthogonality, in a real or complex normed linear space, we formulate a new characterization. It can be derived from other, already … Abstract For the well developed notion of approximate Birkhoff-James orthogonality, in a real or complex normed linear space, we formulate a new characterization. It can be derived from other, already known, characterizations as well as obtained in a more elementary and direct way, on the basis of some simple inequalities for real convex functions.

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On a conditional Wigner equation

1998-08-01

Jacek Chmieliński

On functional equations related to additive mappings and isometries

2014-07-14

Jacek Chmieliński

Certain functional equations, related to the problem of characterization of metrics generated by norms, are considered. The solutions of these equations are strongly connected with additive and isometric mappings. Certain functional equations, related to the problem of characterization of metrics generated by norms, are considered. The solutions of these equations are strongly connected with additive and isometric mappings.

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A note on a characterization of metrics generated by norms

2015-12-01

Jacek Chmieliński

With reference to papers of Oikhberg and Rosenthal [3] and Šemrl [4, 5], we make a contribution to the problem of characterization of metrics generated by norms. With reference to papers of Oikhberg and Rosenthal [3] and Šemrl [4, 5], we make a contribution to the problem of characterization of metrics generated by norms.

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Continuity and discontinuity of seminorms on infinite-dimensional vector spaces. II

2020-02-22

Jacek Chmieliński , Moshe Goldberg

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A comparison of some conditional functional equations

2005-01-01

Jacek Chmieliński

Approximately $C^*$-inner product preserving mappings

2006-01-12

Jacek Chmieliński , Mohammad Sal Moslehian

A mapping $f: {\mathcal M} \to {\mathcal N}$ between Hilbert $C^*$-modules approximately preserves the inner product if \[\| - \| \leq \phi(x, y),\] for an appropriate control function $\phi(x,y)$ and … A mapping $f: {\mathcal M} \to {\mathcal N}$ between Hilbert $C^*$-modules approximately preserves the inner product if \[\| - \| \leq \phi(x, y),\] for an appropriate control function $\phi(x,y)$ and all $x, y \in {\mathcal M}$. In this paper, we extend some results concerning the stability of the orthogonality equation to the framework of Hilbert $C^*$-modules on more general restricted domains. In particular, we investigate some asymptotic behavior and the Hyers--Ulam--Rassias stability of the orthogonality equation.

Birkhoff–James Orthogonality Reversing Property and Its Stability

2019-01-01

Jacek Chmieliński , Paweł Wójcik

Local approximate symmetry of Birkhoff-James orthogonality in normed linear spaces

2020-12-15

Jacek Chmieliński , Divya Khurana , ‎Debmalya Sain

Two different notions of approximate Birkhoff-James orthogonality in nor\-med linear spaces have been introduced by Dragomir and Chmie\-lin\-ski. In the present paper we consider a global and a local approximate … Two different notions of approximate Birkhoff-James orthogonality in nor\-med linear spaces have been introduced by Dragomir and Chmie\-lin\-ski. In the present paper we consider a global and a local approximate symmetry of the Birkhoff-James orthogonality related to each of the two definitions. We prove that the considered orthogonality is approximately symmetric in the sense of Dragomir in all finite-dimensional Banach spaces. For the other case, we prove that for finite-dimensional polyhedral Banach spaces, the approximate symmetry of the orthogonality is equivalent to some newly introduced geometric property. Our investigations complement and extend the scope of some recent results on a global approximate symmetry of the Birkhoff-James orthogonality.

Local approximate symmetry of Birkhoff-James orthogonality in normed linear spaces

2020-01-01

Jacek Chmieliński , Divya Khurana , ‎Debmalya Sain

Two different notions of approximate Birkhoff-James orthogonality in nor\-med linear spaces have been introduced by Dragomir and Chmie\-li\'n\-ski. In the present paper we consider a global and a local approximate … Two different notions of approximate Birkhoff-James orthogonality in nor\-med linear spaces have been introduced by Dragomir and Chmie\-li\'n\-ski. In the present paper we consider a global and a local approximate symmetry of the Birkhoff-James orthogonality related to each of the two definitions. We prove that the considered orthogonality is approximately symmetric in the sense of Dragomir in all finite-dimensional Banach spaces. For the other case, we prove that for finite-dimensional polyhedral Banach spaces, the approximate symmetry of the orthogonality is equivalent to some newly introduced geometric property. Our investigations complement and extend the scope of some recent results on a global approximate symmetry of the Birkhoff-James orthogonality.

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Continuity and Discontinuity of Seminorms on Infinite-Dimensional Vector Spaces. II

2020-01-01

Jacek Chmieliński , Moshe Goldberg

In this paper we extend our findings in [3] and answer further questions regarding continuity and discontinuity of seminorms on infinite-dimensional vector spaces. In this paper we extend our findings in [3] and answer further questions regarding continuity and discontinuity of seminorms on infinite-dimensional vector spaces.

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Continuity and Discontinuity of Seminorms on Infinite-Dimensional Vector Spaces

2019-01-01

Jacek Chmieliński , Moshe Goldberg

Let $S$ be a seminorm on an infinite-dimensional real or complex vector space $X$. Our purpose in this note is to study the continuity and discontinuity properties of $S$ with … Let $S$ be a seminorm on an infinite-dimensional real or complex vector space $X$. Our purpose in this note is to study the continuity and discontinuity properties of $S$ with respect to certain norm-topologies on $X$.

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Approximate smoothness in normed linear spaces

2021-01-01

Jacek Chmieliński , Divya Khurana , ‎Debmalya Sain

We introduce the notion of approximate smoothness in a normed linear space. We characterize this property and show the connections between smoothness and approximate smoothness for some spaces. As an … We introduce the notion of approximate smoothness in a normed linear space. We characterize this property and show the connections between smoothness and approximate smoothness for some spaces. As an application, we consider in particular the Birkhoff-James orthogonality and its right-additivity under the assumption of approximate smoothness.

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Smoothness and approximate smoothness in normed linear spaces and operator spaces

2025-02-28

Jacek Chmieliński , Souvik Ghosh , Kallol Paul +1 more

Approximate Orthogonality and Its Applications to Specific Classes of Linear Operators

2025-05-01

Najla Altwaijry , Jacek Chmieliński , Cristian Conde +1 more

Approximate Orthogonality and Its Applications to Specific Classes of Linear Operators

2025-05-01

Najla Altwaijry , Jacek Chmieliński , Cristian Conde +1 more

Smoothness and approximate smoothness in normed linear spaces and operator spaces

2025-02-28

Jacek Chmieliński , Souvik Ghosh , Kallol Paul +1 more

Convex functions and approximate Birkhoff–James orthogonality

2023-10-09

Jacek Chmieliński , Karol Gryszka , Paweł Wójcik

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Approximate smoothness in normed linear spaces

2023-04-27

Jacek Chmieliński , Divya Khurana , ‎Debmalya Sain

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Approximate Birkhoff-James Orthogonality in Normed Linear Spaces and Related Topics

2022-01-01

Jacek Chmieliński

Local Approximate Symmetry of Birkhoff–James Orthogonality in Normed Linear Spaces

2021-06-17

Jacek Chmieliński , Divya Khurana , ‎Debmalya Sain

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Approximate smoothness in normed linear spaces

2021-01-01

Jacek Chmieliński , Divya Khurana , ‎Debmalya Sain

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Local approximate symmetry of Birkhoff-James orthogonality in normed linear spaces

2020-12-15

Jacek Chmieliński , Divya Khurana , ‎Debmalya Sain

On a $$\rho $$-Orthogonally Additive Mappings

2020-07-01

Jacek Chmieliński , Justyna Sikorska , Paweł Wójcik

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Continuity and discontinuity of seminorms on infinite-dimensional vector spaces. II

2020-02-22

Jacek Chmieliński , Moshe Goldberg

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Local approximate symmetry of Birkhoff-James orthogonality in normed linear spaces

2020-01-01

Jacek Chmieliński , Divya Khurana , ‎Debmalya Sain

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Continuity and Discontinuity of Seminorms on Infinite-Dimensional Vector Spaces. II

2020-01-01

Jacek Chmieliński , Moshe Goldberg

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Continuity and discontinuity of seminorms on infinite-dimensional vector spaces

2019-05-09

Jacek Chmieliński , Moshe Goldberg

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Birkhoff–James Orthogonality Reversing Property and Its Stability

2019-01-01

Jacek Chmieliński , Paweł Wójcik

Continuity and Discontinuity of Seminorms on Infinite-Dimensional Vector Spaces

2019-01-01

Jacek Chmieliński , Moshe Goldberg

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Approximate symmetry of Birkhoff orthogonality

2018-01-31

Jacek Chmieliński , Paweł Wójcik

Approximate orthogonality in normed spaces and its applications

2017-06-07

Jacek Chmieliński , Tomasz Stypuła , Paweł Wójcik

Operators reversing orthogonality in normed spaces

2016-12-01

Jacek Chmieliński

On the stability of the orthogonality equation and the orthogonality-preserving property with two unknown functions

2016-09-30

Jacek Chmieliński , Radosław Łukasik , Paweł Wójcik

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A note on a characterization of metrics generated by norms

2015-12-01

Jacek Chmieliński

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Orthogonality equation with two unknown functions

2015-06-20

Jacek Chmieliński

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On functional equations related to additive mappings and isometries

2014-07-14

Jacek Chmieliński

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Normed spaces equivalent to inner product spaces and stability of functional equations

2013-03-22

Jacek Chmieliński

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ρ-Orthogonality and its preservation –- revisited

2013-01-01

Jacek Chmieliński , Paweł Wójcik

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Orthogonality Preserving Property and its Ulam Stability

2011-01-01

Jacek Chmieliński

On a ρ-orthogonality

2010-09-01

Jacek Chmieliński , Paweł Wójcik

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Isosceles-orthogonality preserving property and its stability

2009-08-27

Jacek Chmieliński , Paweł Wójcik

Perturbation of the Wigner equation in inner product C*-modules

2008-03-01

Jacek Chmieliński , Dijana Ilišević , Mohammad Sal Moslehian +1 more

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APPROXIMATELY C*-INNER PRODUCT PRESERVING MAPPINGS

2008-02-29

Jacek Chmieliński , Mohammad Sal Moslehian

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Remarks on orthogonality preserving mappings in normed spaces and some stability problems

2007-01-01

Jacek Chmieliński

Approximately $C^*$-inner product preserving mappings

2006-01-12

Jacek Chmieliński , Mohammad Sal Moslehian

Approximately $C^*$-inner product preserving mappings

2006-01-01

Jacek Chmieliński , Mohammad Sal Moslehian

Orthogonality preserving property, Wigner equation, and stability

2006-01-01

Jacek Chmieliński

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Stability of the orthogonality preserving property in finite-dimensional inner product spaces

2005-06-29

Jacek Chmieliński

Decomposition of mappings approximately inner product preserving

2005-06-14

Roman Badora , Jacek Chmieliński

A comparison of some conditional functional equations

2005-01-01

Jacek Chmieliński

On an -Birkhoff orthogonality.

2005-01-01

Jacek Chmieliński

Stability of angle-preserving mappings on the plane

2005-01-01

Jacek Chmieliński

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Linear mappings approximately preserving orthogonality

2004-10-21

Jacek Chmieliński

On a singular case in the Hyers–Ulam–Rassias stability of the Wigner equation

2003-11-22

Jacek Chmieliński

The Stability of the Wigner Equation on a Restricted Domain

2001-02-01

Jacek Chmieliński , ‎Soon-Mo Jung

Almost approximately inner product preserving mappings

2000-05-01

Jacek Chmieliński

On a conditional Wigner equation

1998-08-01

Jacek Chmieliński

On approximate solutions of the Pexider equation

1993-08-01

Jacek Chmieliński , Jacek Tabor

On approximate solutions of the Pexider equation

1993-02-01

Jacek Chmieliński , Jacek Tabor

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Coauthor	Papers Together
Paweł Wójcik	9
‎Debmalya Sain	6
Divya Khurana	5
Moshe Goldberg	4
Mohammad Sal Moslehian	4
Jacek Tabor	2
Cristian Conde	1
Karol Gryszka	1
Ghadir Sadeghi	1
Justyna Sikorska	1
Najla Altwaijry	1
Tomasz Stypuła	1
Roman Badora	1
‎Soon-Mo Jung	1
Radosław Łukasik	1
Souvik Ghosh	1
Kais Feki	1
Kallol Paul	1
Dijana Ilišević	1

Orthogonality and linear functionals in normed linear spaces

1947-01-01

Robert James

{*) This definition was used by Birkhoff [3] and by Fortet [6 and 7]. {*) This definition was used by Birkhoff [3] and by Fortet [6 and 7].

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Orthogonality in linear metric spaces

1935-06-01

Garrett Birkhoff

Linear mappings approximately preserving orthogonality

2004-10-21

Jacek Chmieliński

Stability of Functional Equations in Several Variables

1998-01-01

D. H. Hyers , G. Isac , Themistocles M. Rassias

Approximately additive and approximately linear mappings stability of the quadratic functional equation generalizations - the method of invariant means approximately multiplicative mappings - superstability stability of functional equations for trigonometric … Approximately additive and approximately linear mappings stability of the quadratic functional equation generalizations - the method of invariant means approximately multiplicative mappings - superstability stability of functional equations for trigonometric and similar functions functions with bounded nth differences approximately convex functions stability of the generalized orthogonality functional equation stability and set-valued mappings stability of stationary and minimum points functional congruences quasi-additive functions and related topics.

Operators preserving orthogonality are isometries

1993-01-01

Alexander Koldobsky

Synopsis We prove that every operator preserving orthogonality in a real Banach space is an isometry multiplied by a constant. Synopsis We prove that every operator preserving orthogonality in a real Banach space is an isometry multiplied by a constant.

Stability of the orthogonality preserving property in finite-dimensional inner product spaces

2005-06-29

Jacek Chmieliński

On maps that preserve orthogonality in normed spaces

2006-08-01

A. Blanco , Aleksej Turnšek

We show that every orthogonality-preserving linear map between normed spaces is a scalar multiple of an isometry. Using this result, we generalize Uhlhorn's version of Wigner's theorem on symmetry transformations … We show that every orthogonality-preserving linear map between normed spaces is a scalar multiple of an isometry. Using this result, we generalize Uhlhorn's version of Wigner's theorem on symmetry transformations to a wide class of Banach spaces.

Orthogonality in normed linear spaces

1945-06-01

Robert James

Approximate orthogonality in normed spaces and its applications

2017-06-07

Jacek Chmieliński , Tomasz Stypuła , Paweł Wójcik

On mappings approximately preserving orthogonality

2007-03-14

Aleksej Turnšek

On an -Birkhoff orthogonality.

2005-01-01

Jacek Chmieliński

Mappings approximately preserving orthogonality in normed spaces

2010-08-15

Blaž Mojškerc , Aleksej Turnšek

Approximately orthogonality preserving mappings on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mi>C</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:math>-modules

2007-10-25

Dijana Ilišević , Aleksej Turnšek

On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces

2011-09-17

Javier Alonso , Horst Martini , Senlin Wu

Semi-inner-product spaces

1961-01-01

G. Lumer

function as a particular Banach space (whose norm satisfies the parallelogram law), but rather as an inner-product space. It is in terms of the innerproduct space structure that most of … function as a particular Banach space (whose norm satisfies the parallelogram law), but rather as an inner-product space. It is in terms of the innerproduct space structure that most of the terminology and techniques are developed. On the other hand, this type of Hilbert space considerations find no real parallel in the general Banach space setting. Some time ago, while trying to carry over a Hilbert space argument to a general Banach space situation, we were led to use a suitable mapping from a Banach space into its dual in order to make up for the lack of an innerproduct. Our procedure suggested the existence of a general theory which it seemed should be useful in the study of operator (normed) algebras by providing better insight on known facts, a more adequate language to classify special types of operators, as well as new techniques. These ideas evolved into a theory of semi-inner-product spaces which is presented in this paper (together with certain applications)(1). We shall consider vector spaces on which instead of a bilinear form there is defined a form [x, y] which is linear in one component only, strictly positive, and satisfies a Schwarz inequality. Such a form induces a norm, by setting |x| = ([x, x])112; and for every normed space one can construct at least one such form (and, in general, infinitely many) consistent with the

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On a singular case in the Hyers–Ulam–Rassias stability of the Wigner equation

2003-11-22

Jacek Chmieliński

Approximate Birkhoff–James orthogonality in the space of bounded linear operators

2017-10-18

Kallol Paul , ‎Debmalya Sain , ‎Arpita Mal

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On Wigner's theorem: Remarks, complements, comments, and corollaries

1996-02-01

Jürg Rätz

Inner products in normed linear spaces

1947-01-01

Robert James

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Decomposition of mappings approximately inner product preserving

2005-06-14

Roman Badora , Jacek Chmieliński

ρ-Orthogonality and its preservation –- revisited

2013-01-01

Jacek Chmieliński , Paweł Wójcik

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The Stability of the Wigner Equation on a Restricted Domain

2001-02-01

Jacek Chmieliński , ‎Soon-Mo Jung

On a ρ-orthogonality

2010-09-01

Jacek Chmieliński , Paweł Wójcik

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Operator norm attainment and inner product spaces

2013-07-22

‎Debmalya Sain , Kallol Paul

Isosceles-orthogonality preserving property and its stability

2009-08-27

Jacek Chmieliński , Paweł Wójcik

On orthogonally additive mappings

1985-12-01

Jürg Rätz

On isometries of normed linear spaces

1970-01-01

Donald O. Koehler , Peter Rosenthal

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Orthogonality Preserving Property and its Ulam Stability

2011-01-01

Jacek Chmieliński

Banach space theory: the basis for linear and nonlinear analysis

2011-07-01

Marián Fabian , Petr Habala , Petr Hájek +2 more

Preface.- Basic Concepts in Banach Spaces.- Hahn-Banach and Banach Open Mapping Theorems.- Weak Topologies and Banach Spaces.- Schauder Bases.- Structure of Banach Spaces.- Finite-Dimensional Spaces.- Optimization.- C^1 Smoothness in Separable … Preface.- Basic Concepts in Banach Spaces.- Hahn-Banach and Banach Open Mapping Theorems.- Weak Topologies and Banach Spaces.- Schauder Bases.- Structure of Banach Spaces.- Finite-Dimensional Spaces.- Optimization.- C^1 Smoothness in Separable Spaces.- Superreflexive Spaces.- Higher Order Smoothness.- Dentability and differentiability.- Basics in Nonlinear Geometric Analysis.- Weakly Compactly Generated Spaces.- Topics in Weak Topologies on Banach Spaces.- Compact Operators on Banach Spaces.- Tensor Products.- Appendix.- References.- Symbol Index.- Subject Index.- Author Index.

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Characterizations of Rotundity and Smoothness by Approximate Orthogonalities

2016-08-10

Tomasz Stypuła , Paweł Wójcik

Abstract In this paper we consider the approximate orthogonalities in real normed spaces. Using the notion of approximate orthogonalities in real normed spaces, we provide some new characterizations of rotundity … Abstract In this paper we consider the approximate orthogonalities in real normed spaces. Using the notion of approximate orthogonalities in real normed spaces, we provide some new characterizations of rotundity and smoothness of dual spaces.

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Remarks on orthogonality preserving mappings in normed spaces and some stability problems

2007-01-01

Jacek Chmieliński

Approximate symmetry of Birkhoff orthogonality

2018-01-31

Jacek Chmieliński , Paweł Wójcik

An Introduction to Banach Space Theory

1998-01-01

Robert E. Megginson

On the Stability of the Linear Functional Equation

1941-04-15

D. H. Hyers

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The Birkhoff–James orthogonality in Hilbert C∗-modules

2012-06-15

Ljiljana Arambašić , Rajna Rajić

APPROXIMATELY C*-INNER PRODUCT PRESERVING MAPPINGS

2008-02-29

Jacek Chmieliński , Mohammad Sal Moslehian

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Approximate Birkhoff–James orthogonality and smoothness in the space of bounded linear operators

2019-04-08

‎Arpita Mal , Kallol Paul , T. S. S. R. K. Rao +1 more

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Stability of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mi>C</mml:mi><mml:mo>∗</mml:mo></mml:msup></mml:math>-inner products

2005-10-12

Maryam Amyari

Antinorms and Radon curves

2006-09-01

Horst Martini , Konrad J. Swanepoel

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Orthogonality of compact operators

2016-06-14

Paweł Wójcik

On the numerical index of polyhedral Banach spaces

2019-04-26

‎Debmalya Sain , Kallol Paul , Pintu Bhunia +1 more

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Orthogonality of matrices and some distance problems

1999-01-01

Rajendra Bhatia , Peter Šemrl

A note on compact operators which attain their norm

1979-06-01

John M. Baker

For Banach spaces X having the unit cell of X**w*sequentially compact, the compact operators from X into a Banach space Y attain their norm in X**.The same holds for weakly … For Banach spaces X having the unit cell of X**w*sequentially compact, the compact operators from X into a Banach space Y attain their norm in X**.The same holds for weakly compact operators if, in addition, X has the strict Dunford-Pettis property.For Banach spaces X such that the quotient space X**/X is separable and Y the space of absolutely summable sequences, a proper subset P σ of the finite rank operators from X into Y is exhibited.The set P o is shown to consist of operators which attain their norm and to be norm-dense in the operator space.

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Characterizations of Inner Product Spaces

1986-01-01

Dan Amir

Inner product modules over 𝐵*-algebras

1973-01-01

William L. Paschke

This paper is an investigation of right modules over a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>B</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{B^\ast … This paper is an investigation of right modules over a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>B</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{B^\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebra <italic>B</italic> which posses a <italic>B</italic>-valued “inner product” respecting the module action. Elementary properties of these objects, including their normability and a characterization of the bounded module maps between two such, are established at the beginning of the exposition. The case in which <italic>B</italic> is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>W</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{W^\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebra is of especial interest, since in this setting one finds an abundance of inner product modules which satisfy an analog of the self-duality property of Hilbert space. It is shown that such self-dual modules have important properties in common with both Hilbert spaces and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>W</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{W^\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebras. The extension of an inner product module over <italic>B</italic> by a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Superscript asterisk"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>B</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{B^\ast }</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-algebra <italic>A</italic> containing <italic>B</italic> as a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi /> <mml:mo>∗</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">^\ast</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-subalgebra is treated briefly. An application of some of the theory described above to the representation and analysis of completely positive maps is given.

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An Introduction to the Theory of Functional Equations and Inequalities

2009-01-01

Marek Kuczma

Marek Kuczma was born in 1935 in Katowice, Poland, and died there in 1991. After finishing high school in his home town, he studied at the Jagiellonian University in Kraków. … Marek Kuczma was born in 1935 in Katowice, Poland, and died there in 1991. After finishing high school in his home town, he studied at the Jagiellonian University in Kraków. He defended his doctoral d

Orthogonality of matrices

2006-11-29

Carlos Benítez , Manuel Fernández , M. Laura Soriano

Characterizations of smooth spaces by approximate orthogonalities

2014-09-11

Paweł Wójcik

Since the monograph by Amir that appeared in 1986, a lot of attention has been given to the problem of characterizing, by means of properties of the norms, when a … Since the monograph by Amir that appeared in 1986, a lot of attention has been given to the problem of characterizing, by means of properties of the norms, when a Banach space is indeed a Hilbert space, i.e., when the norm derives from an inner product. In this paper, similar investigations will be carried out for smooth spaces instead of inner product spaces. We consider the approximate orthogonalities in real normed spaces. We show that the relations approximate semi-orthogonality and approximate ρ+-orthogonality are generally incomparable (unless the normed space is smooth). As a result, we give a characterization of smooth spaces in terms of those approximate orthogonalities.

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Operator norm attainment and Birkhoff–James orthogonality

2015-03-13

‎Debmalya Sain , Kallol Paul , Sourav Hait

A New Generalization of a Theorem of Jung for the Orthogonality Equation

2002-01-01

Themistocles M. Rassias

The main result of this paper states as follows: Assume that for a closed ball D @ 0 and with center at the origin, a mapping T : D M … The main result of this paper states as follows: Assume that for a closed ball D @ 0 and with center at the origin, a mapping T : D M D satisfies $$ T(0) = 0\ \hbox{and}\ \vert \langle Tx, Ty \rangle - \langle x, y \rangle \vert \leq \varepsilon \eqno (1) $$ for some 0 h l < min { 1/4, d 2 /17} and for all x , y ] D . Then, there exists an isometry I : D M D with $$ \vert Tx - Ix \vert \le \left\{ {\matrix{ {13\sqrt \varepsilon } \hfill &{{\rm for}\; d