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5 results

  • THE WIGNER PROPERTY OF SMOOTH NORMED SPACES

  • Part of
  • XUJIAN HUANG, JIABIN LIU, SHUMING WANG
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 110 / Issue 3 / December 2024
    Published online by Cambridge University Press:
    09 May 2024, pp. 545-553
    Print publication:
    December 2024
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  • We prove that every smooth complex normed space X has the Wigner property. That is, for any complex normed space Y and every surjective mapping $f: X\rightarrow Y$ satisfying

    $$ \begin{align*} \{\|f(x)+\alpha f(y)\|: \alpha\in \mathbb{T}\}=\{\|x+\alpha y\|: \alpha\in \mathbb{T}\}, \quad x,y\in X, \end{align*} $$

    where $\mathbb {T}$ is the unit circle of the complex plane, there exists a function $\sigma : X\rightarrow \mathbb {T}$ such that $\sigma \cdot f$ is a linear or anti-linear isometry. This is a variant of Wigner’s theorem for complex normed spaces.

  • MIN-PHASE-ISOMETRIES IN STRICTLY CONVEX NORMED SPACES

  • Part of
  • DONGNI TAN, FAN ZHANG
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 109 / Issue 1 / February 2024
    Published online by Cambridge University Press:
    22 March 2023, pp. 152-160
    Print publication:
    February 2024
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  • Suppose that X and Y are two real normed spaces. A map $f:X\rightarrow Y$ is called a min-phase-isometry if it satisfies

    $$ \begin{align*} \min\{\|f(x)+f(y)\|,\|f(x)-f(y)\|\}=\min\{\|x+y\|,\|x-y\|\} \quad (x,y\in X). \end{align*} $$

    We present properties of min-phase-isometries in the case that Y is strictly convex and show that if a min-phase-isometry f (not necessarily surjective) fixes the origin, then it is phase-equivalent to a linear isometry, that is, $f(x)=\varepsilon (x)g(x)$ for $x\in X$, where $g:X\rightarrow Y$ is a linear isometry and $\varepsilon $ is a map from X to $\{-1,1\}$.

Wigner-Type Theorems for Hilbert Grassmannians
  • Mark Pankov
  • Published online:
    06 January 2020
    Print publication:
    16 January 2020
  • Wigner's theorem is a fundamental part of the mathematical formulation of quantum mechanics. The theorem characterizes unitary and anti-unitary operators as symmetries of quantum mechanical systems, and is a key result when relating preserver problems to quantum mechanics. At the heart of this book is a geometric approach to Wigner-type theorems, unifying both classical and more recent results. Readers are initiated in a wide range of topics from geometric transformations of Grassmannians to lattices of closed subspaces, before moving on to a discussion of applications. An introduction to all the key aspects of the basic theory is included as are plenty of examples, making this book a useful resource for beginning graduate students and non-experts, as well as a helpful reference for specialist researchers.

  • WIGNER’S THEOREM IN ${\mathcal{L}}^{\infty }(\unicode[STIX]{x1D6E4})$-TYPE SPACES

  • Part of
  • WEIKE JIA, DONGNI TAN
  • Journal:
    Bulletin of the Australian Mathematical Society / Volume 97 / Issue 2 / April 2018
    Published online by Cambridge University Press:
    04 December 2017, pp. 279-284
    Print publication:
    April 2018
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  • We investigate surjective solutions of the functional equation

    $$\begin{eqnarray}\displaystyle \{\Vert f(x)+f(y)\Vert ,\Vert f(x)-f(y)\Vert \}=\{\Vert x+y\Vert ,\Vert x-y\Vert \}\quad (x,y\in X), & & \displaystyle \nonumber\end{eqnarray}$$
    where $f:X\rightarrow Y$ is a map between two real ${\mathcal{L}}^{\infty }(\unicode[STIX]{x1D6E4})$-type spaces. We show that all such solutions are phase equivalent to real linear isometries. This can be considered as an extension of Wigner’s theorem on symmetry for real ${\mathcal{L}}^{\infty }(\unicode[STIX]{x1D6E4})$-type spaces.