Wigner’s Theorem in -Type Spaces

Abstract

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.

The wigner property for CL-spaces and finite-dimensional polyhedral Banach spaces

“…In our paper [8], we provide an affirmative answer to Question 1.1 with X and Y being l p (Γ) spaces (0 < p < ∞). The first author and Jia presented in [9] a similar result for L ∞ (Γ)-type spaces. Recall that a Banach space X is said to have the Wigner property if Question 1.1 has an affirmative answer for an arbitrary target Y .…”

Wigner’s Theorem in &lt;i&gt;s&lt;/i&gt;* and &lt;i&gt;s&lt;sub&gt;n&lt;/sub&gt;(H)&lt;/i&gt; Spaces

“…If the two spaces are not inner product spaces, Huang and Tan [8] gave a partial answer about the real atomic p L spaces with 0 p > . Jia and Tan [9] get the conclusion about the  -type spaces. In [6], xiaohong Fu proved the problem of isometry extension in the s space detailedly.…”

On Wigner's theorem in strictly convex normed spaces

“…So the main question is if the converse is true. For results in this direction, see [10]- [12]. However, the question in full generality remains open.…”