1. Introduction
Let X and Y be normed spaces over
$\mathbb {F}\in \{\mathbb {R},\mathbb {C}\}$
, where
$\mathbb {R}$
and
$\mathbb {C}$
are the fields of real and complex numbers, respectively. Denote
$\mathbb {T}=\{\alpha \in \mathbb {F}:|\alpha |=1\}$
. A function
$\sigma :X\rightarrow \mathbb {T}$
whose values are of modulus one is called a phase function on X. A mapping
${f: X \rightarrow Y}$
is said to be phase equivalent to another mapping
$g: X \rightarrow Y$
if there exists a phase function
$\sigma :X\rightarrow \mathbb {T}$
such that
$f=\sigma \cdot g$
, that is,
$f(x)=\sigma (x)g(x)$
for
$x\in X$
.
The celebrated Wigner’s unitary–anti-unitary theorem is particularly important in the mathematical foundations of quantum mechanics. It states that for inner product spaces
$(X,\langle \cdot ,\cdot \rangle )$
and
$(Y,\langle \cdot ,\cdot \rangle )$
over
$\mathbb {F}$
, a mapping
$f: X \rightarrow Y$
satisfies
$$ \begin{align} |\langle f(x), f(y) \rangle | = |\langle x, y \rangle|, \quad x, y\in X \end{align} $$
if and only if f is phase equivalent to a linear or anti-linear isometry in the case
$\mathbb {F}=\mathbb {C}$
and to a linear isometry in the case
$\mathbb {F}=\mathbb {R}$
. There are several proofs of this result, see [Reference Almeida and Sharma1, Reference Bargmann2, Reference Gehér4, Reference Győry6, Reference Maksa and Páles13, Reference Rätz18, Reference Turnšek22] to list just some of them. For further generalisations of this fundamental result, we mention the papers [Reference Chevalier3, Reference Gehér5, Reference Molnár15, Reference Qian, Wang, Wu and Yuan17]. Wigner’s theorem is very important and therefore worthy of study from various points of view.
A mapping
$f: X \rightarrow Y$
between normed spaces over
$\mathbb {F}$
is called a phase-isometry if it satisfies the functional equation
$$ \begin{align} \{\|f(x)+\alpha f(y)\|: \alpha\in \mathbb{T}\}=\{\|x+\alpha y\|: \alpha\in \mathbb{T}\}, \quad x,y\in X. \end{align} $$
It is worth noting that if X and Y are inner product spaces, then
$f: X \rightarrow Y$
satisfies (1.1) if and only if f satisfies (1.2). Indeed, with the substitution
$y=x$
, we deduce from either (1.1) or (1.2) that f is norm-preserving. Squaring the norms on both sides of (1.2), it follows that (1.2) holds if and only if
$$ \begin{align*} \{\mbox{Re}(\alpha\langle f(x),f(y) \rangle): \alpha\in \mathbb{T}\}=\{\mbox{Re}(\alpha\langle x, y \rangle): \alpha\in \mathbb{T}\}, \quad x,y\in X, \end{align*} $$
which happens if and only if (1.1) holds. Due to Wigner’s theorem, a mapping between inner product spaces is a phase-isometry if and only if it is phase equivalent to a linear or anti-linear isometry in the case
$\mathbb {F}=\mathbb {C}$
and to a linear isometry in the case
$\mathbb {F}=\mathbb {R}$
.
When X and Y are normed spaces, one can easily see that if
$f:X \rightarrow Y$
is phase equivalent to a linear or anti-linear isometry, then f is a phase-isometry. For instance, if
$f=\sigma \cdot U$
, where U is a linear isometry and
$\sigma :X\rightarrow \mathbb {T}$
is a phase function, then for
$x,y\in X$
and
$\alpha \in \mathbb {T}$
,
$$ \begin{align*} \|f(x)+\alpha f(y)\|&=\|\sigma(x)U(x)+\alpha\sigma(y)U(y)\|=\|U(\sigma(x)x+\alpha\sigma(y)y)\|\\ &=\|\sigma(x)x+\alpha\sigma(y)y\|=\|x+\alpha\overline{\sigma(x)}\sigma(y)y\| \end{align*} $$
and then
$$ \begin{align*} \|x+\alpha y\|=\|x+(\alpha\sigma(x)\overline{\sigma(y)})\overline{\sigma(x)}\sigma(y)y\|=\|f(x)+\alpha\sigma(x)\overline{\sigma(y)} f(y)\|. \end{align*} $$
Similar reasoning applies when U is an anti-linear isometry. Therefore, a natural problem posed by Maksa and Páles [Reference Maksa and Páles13] (the case
$\mathbb {F}=\mathbb {R}$
), and Wang and Bugajewski [Reference Wang and Bugajewski23] (the case
$\mathbb {F}=\mathbb {C}$
), can be restated as the following problem.
Problem 1.1. Under what conditions is every phase-isometry between two normed spaces over
$\mathbb {F}$
phase equivalent to a linear or anti-linear isometry in the case
$\mathbb {F}=\mathbb {C}$
and to a linear isometry in the case
$\mathbb {F}=\mathbb {R}$
?
A normed space X over
$\mathbb {F}$
is said to have the Wigner property if for any normed space Y over
$\mathbb {F}$
, every surjective phase-isometry
$f: X \rightarrow Y$
is phase equivalent to a linear or anti-linear isometry in the case
$\mathbb {F}=\mathbb {C}$
and to a linear isometry in the case
$\mathbb {F}=\mathbb {R}$
.
There have been several recent papers considering Problem 1.1 or the Wigner property in the case
$\mathbb {F}=\mathbb {R}$
. For relevant results, please refer to [Reference Huang and Tan7–Reference Ilišević, Omladič and Turnšek9, Reference Ilišević and Turnšek11–Reference Maksa and Páles13, Reference Tan and Huang19–Reference Tan and Zhang21, Reference Wang and Bugajewski23]. In particular, Tan and Huang [Reference Tan and Huang19] proved that smooth real normed spaces have the Wigner property. Further, Ilišević et al. [Reference Ilišević, Omladič and Turnšek9] proved that any real normed spaces have the Wigner property. However, to the best of our knowledge, apart from the case where X and Y are inner product spaces, there has been no progress in addressing Problem 1.1 in the case
$\mathbb {F}=\mathbb {C}$
. The aim of this paper is to give a partial solution for the case
$\mathbb {F}=\mathbb {C}$
. Specifically, we show that every smooth complex normed space has the Wigner property. As a by-product, we give a Figiel-type result for phase-isometries. Although our paper is interesting in its own right, we hope that it will serve as a stepping stone to show that all complex normed spaces have the Wigner property.
2. Results
In the remainder of this paper, unless otherwise specified, all the normed spaces are over
$\mathbb {F}\in \{\mathbb {R},\mathbb {C}\}$
. Although the real case has been solved, for the sake of brevity and universality, we will present our lemmas, theorems and proofs in the united form
$\mathbb {F}$
rather than the single form
$\mathbb {C}$
. For a normed space X, we use the notation
$S_X, B_X$
and
$X^*$
to represent the unit sphere, closed unit ball and dual space of X, respectively. The set of positive integers is denoted by
$\mathbb {N}$
.
We start this section with a simple and frequently-used property of phase-isometries between two normed spaces.
Lemma 2.1. Let X and Y be normed spaces and
$f: X \rightarrow Y$
a phase-isometry. Then f is a norm-preserving map. Moreover, if f is surjective, then
$$ \begin{align*} \{f(\alpha x): \alpha\in \mathbb{T}\}=\{\alpha f(x): \alpha\in \mathbb{T}\}, \quad x\in X. \end{align*} $$
Proof. With the substitution
$y=x$
, it follows from (1.2) that
$$ \begin{align*} 2\|f(x)\|=\max\{\|f(x)+\alpha f(x)\|: \alpha\in \mathbb{T}\} =\max\{\|x+\alpha x\|: \alpha\in \mathbb{T}\}=2\|x\|, \end{align*} $$
which shows that f is norm-preserving.
Now suppose that f is surjective. Let us take a nonzero
$x\in X$
and
$\alpha \in \mathbb {T}$
. The surjectivity guarantees that there exists some
$y\in X$
such that
$f(y)=\alpha f(x)$
. Then
$$ \begin{align*} \min\{\|y+\beta x\|: \beta\in \mathbb{T}\}=\min\{\|f(y)+\beta f(x)\|: \beta\in \mathbb{T}\}=0, \end{align*} $$
which implies that
$$ \begin{align*} \{\alpha f(x): \alpha\in \mathbb{T}\}\subset\{f(\alpha x): \alpha\in \mathbb{T}\}. \end{align*} $$
Moreover, we conclude from (1.2) that
$$ \begin{align*} \min\{\|f(\alpha x)+\beta f(x)\|: \beta\in \mathbb{T}\}=\min\{\|\alpha x+\beta x\|: \beta\in \mathbb{T}\}=0, \end{align*} $$
which shows that
$$ \begin{align*} \{f(\alpha x): \alpha\in \mathbb{T}\}\subset\{\alpha f(x): \alpha\in \mathbb{T}\}. \end{align*} $$
This competes the proof.
From [Reference Tan and Huang19, Lemma 2], it follows that every surjective phase-isometry between two real normed spaces is injective. The following example shows that a surjective phase-isometry between two complex normed spaces may not be injective.
Example 2.2. Let X be a complex normed space and
$x_0\in X\backslash \{0\}$
. Define
${f:X\rightarrow X}$
by
$f(\alpha x_0)=\alpha ^2x_0$
for all
$\alpha \in \mathbb {T}$
and
$f(x)=x$
otherwise. Then f is a surjective phase-isometry, but it is not injective since
$f(-x_0)=x_0=f(x_0)$
.
In Example 2.2, f is phase equivalent to the identity mapping, letting the phase function
$\sigma $
be
$\sigma (\alpha x_0)=\alpha $
for all
$\alpha \in \mathbb {T}$
and
$\sigma (x)=1$
otherwise.
Recall that a support functional
$\phi $
at
$x\in X\backslash \{0\}$
is a norm-one linear functional in
$X^*$
such that
$\phi (x)=\|x\|$
. Denote by
$D(x)$
the set of all support functionals at
$x\neq 0$
, that is,
$$ \begin{align*} D(x)=\{\phi\in S_{X^*}: \phi(x)=\|x\|\}. \end{align*} $$
The Hahn–Banach theorem implies that
$D(x)\neq \emptyset $
for every
$x\in X\backslash \{0\}$
. A normed space X is said to be smooth at
$x\neq 0$
if there exists a unique supporting functional at x, that is,
$D(x)$
consists of only one element. If X is smooth at every
$x\neq 0$
, then X is said to be smooth. It follows from [Reference Megginson14, Proposition 5.4.20] that each subspace of a smooth normed space is smooth.
Recall also the concept of Gateaux differentiability. Let X be a normed space,
${x, y\in X}$
. Define
$$ \begin{align*} G_+(x,y):=\lim_{t\to0^+, t\in\mathbb{R}}\frac{\|x+ty\|-\|x\|}{t}=\lim_{t\to+\infty, t\in\mathbb{R}}(\|tx+y\|-\|tx\|) \end{align*} $$
and
$$ \begin{align*} G_-(x,y):=\lim_{t\to0^-, t\in\mathbb{R}}\frac{\|x+ty\|-\|x\|}{t}=\lim_{t\to+\infty, t\in\mathbb{R}}(\|tx\|-\|tx-y\|). \end{align*} $$
It is known [Reference Megginson14, Reference Phelps16] that both
$G_+(x,y)$
and
$G_-(x,y)$
exist for each
$x,y\in X$
and
$$ \begin{align*} G_+(x,y)=\max\{\mbox{Re}\,\phi(y):\phi\in D(x)\},\quad G_-(x,y)=\min\{\mbox{Re}\,\phi(y):\phi\in D(x)\}. \end{align*} $$
We say that the norm of X is Gateaux differentiable at
$x\neq 0$
whenever
$G_+(x,y)=G_-(x,y)$
for all
$y\in X$
, in which case the common value of
$G_+(x,y)$
and
$G_-(x,y)$
is denoted by
$G(x,y)$
. It is easy to see that a normed space X is smooth at x if and only if the norm is Gateaux differentiable at x.
A point
$\phi \in S_{X^*}$
is said to be a
$w^*$
-exposed point of
$B_{X^*}$
provided that
$\phi $
is the only supporting functional for some smooth point
$u\in S_X$
. Recently, Tan and Huang [Reference Tan and Huang19] showed that for every phase-isometry f of a real normed space X into another real normed space Y and every
$w^*$
-exposed point
$\phi $
of
$B_{X^*}$
, there exists
$\varphi \in S_{Y^*}$
such that
$\phi (x)=\pm \varphi (f(x))$
for all
$x\in X$
. This result can be viewed as an extension of Figiel’s theorem, which plays an important role in the study of isometric embedding. We will present a similar result for a phase-isometry between two normed spaces over
${\mathbb {F}\in \{\mathbb {R},\mathbb {C}\}}$
.
Lemma 2.3. Let X and Y be normed spaces and
$f:X\rightarrow Y$
a phase-isometry. Then for every
$w^*$
-exposed point
$\phi $
of
$B_{X^*}$
, there exists
$\varphi \in S_{Y^*}$
such that
$$ \begin{align*} |\phi(x)|=|\varphi(f(x))|, \quad x\in X. \end{align*} $$
Proof. Let
$u\in S_X$
be a smooth point such that
$\phi (u)=1$
. For every
$n\in \mathbb {N}$
, the Hahn–Banach theorem guarantees the existence of
$\varphi _n\in S_{Y^*}$
such that
$$ \begin{align*}\varphi_n(f(nu))=\|f(nu)\|= \|nu\|=n.\end{align*} $$
For
$t\in [0,n]$
, there exists some
$\alpha _{t,n}\in \mathbb {T}$
such that
$$ \begin{align*}\|f(nu)-\alpha_{t,n}f(tu)\|=\|nu-tu\|=n-t.\end{align*} $$
Consequently, we deduce that
$$ \begin{align*} 2n&=|\varphi_n (f(nu)-\alpha_{t,n}f(tu))+\varphi_n(f(nu)+\alpha_{t,n}f(tu))|\\ &\leq|\varphi_n (f(nu)-\alpha_{t,n}f(tu))|+|\varphi_n(f(nu)+\alpha_{t,n}f(tu))|\\ &\leq\|f(nu)-\alpha_{t,n}f(tu)\|+\|f(nu)+\alpha_{t,n}f(tu)\|\\&\leq (n-t)+(n+t)=2n, \end{align*} $$
which implies that
$\varphi _n(\alpha _{t,n}f(tu))=t$
. This means that for each
$t\in (0, n]$
, there exists a unique
$\alpha _{t,n}\in \mathbb {T}$
such that
$\varphi _n(f(tu))=\overline {\alpha _{t,n}}t$
. By Alaoglu’s theorem, the sequence
$\{\varphi _n\}$
has a cluster point
$\varphi \in S_{Y^*}$
in the
$w^*$
topology. It follows that for each
$t>0$
, there exists
$\alpha _t\in \mathbb {T}$
depending only on t such that
$\varphi (f(tu))=\alpha _t t$
.
For each
$x\in X$
, there exist
$\alpha _x, \beta _x\in \mathbb {T}$
such that
$\alpha _x\phi (x)=|\phi (x)|$
and
$\beta _x\varphi (f(x))=|\varphi (f(x))|$
. For each
$n\in \mathbb {N}$
, there exists
$\alpha _{x,n}, \beta _{x,n}\in \mathbb {T}$
such that
$$ \begin{align*} \|nu-\alpha_xx\|=\|f(nu)-\alpha_{x,n}\alpha_nf(x)\| & \geq |\varphi(f(nu))-\alpha_{x,n}\alpha_n\varphi(f(x))|\\ &=|\alpha_n n-\alpha_{x,n}\alpha_n\varphi(f(x))|=|n-\alpha_{x,n}\varphi(f(x))| \end{align*} $$
and
$$ \begin{align*} |n+\beta_x\varphi(f(x))|=|\alpha_n n+\alpha_n\beta_x\varphi(f(x))| & =|\varphi(f(nu))+\alpha_n\beta_x \varphi(f(x))|\\ &\leq \|f(nu)+\alpha_n\beta_xf(x)\|=\| nu+\beta_{x,n}x\|. \end{align*} $$
Given that
$\mathbb {T}$
is compact, there must be a strictly increasing sequence
$\{n_j:j\in \mathbb {N}\}$
in
$\mathbb {N}$
and
$\alpha ^{\prime }_x,\beta ^{\prime }_x\in \mathbb {T}$
such that
$\lim _{j\to \infty }\alpha _{x,n_j}=\alpha ^{\prime }_x$
and
$\lim _{j\to \infty }\beta _{x,n_j}=\beta ^{\prime }_x$
. Since
$\phi $
is the only supporting functional at u,
$$ \begin{align*} |\phi(x)|&=\mbox{Re}\,\phi(\alpha_xx)=\lim_{j\to\infty}(\|n_ju\|-\|n_ju-\alpha_xx\|)\\ &\leq\lim_{j\to\infty}(n_j-| n_j-\alpha_{x,n_j}\varphi(f(x))|)=\lim_{j\to\infty}(n_j-| n_j-\alpha^{\prime}_x\varphi(f(x))|)\\ &=\mbox{Re}\,(\alpha^{\prime}_x\varphi(f(x)))\leq|\varphi(f(x))| \end{align*} $$
and
$$ \begin{align*} |\varphi(f(x))|&=\mbox{Re}\,(\beta_x\varphi(f(x)))=\lim_{j\to\infty}(|n_j+\beta_x\varphi(f(x))|-n_j)\\ &\leq\lim_{j\to\infty} (\| n_ju+\beta_{x,n_j}x\|-\|n_ju\|)=\lim_{j\to\infty} (\|n_ju+\beta^{\prime}_xx\|-\|n_ju\|)\\ &=\mbox{Re}\,\phi(\beta^{\prime}_xx)\leq|\phi(x)|. \end{align*} $$
This completes the proof.
Let V be a vector space. For
$M\subset V$
,
$[M]$
denotes the subspace generated by M. If
$x,y\in V$
, then we write
$[x]:=[\{x\}]$
and
$[x,y]:=[\{x,y\}]$
for simplicity.
Lemma 2.4. Let X and Y be normed spaces with X being smooth. Suppose that
${f: X \rightarrow Y}$
is a surjective phase-isometry. Then for every
$x\in X$
,
$$ \begin{align*} f([x])=[f(x)]. \end{align*} $$
Proof. We first prove that
$[f(x)]\subset f([x])$
for each
$x\in X$
. Assume, for a contradiction, that
$tf(x)\notin f([x])$
for some nonzero
$x\in X$
and
$t\in \mathbb {F}$
. Since f is surjective, there exists
$y\in X$
such that
$f(y)=tf(x)$
. The function
$s\mapsto \|y-sx\|$
is continuous and its value tends to infinity when
$|s|$
tends to infinity. Hence, there is at least one point
$s_0\in \mathbb {F}$
such that
$$ \begin{align*} d:=d(y,[x])=\min\{\|y-sx\|: s\in \mathbb{F}\}=\|y-s_0x\|>0. \end{align*} $$
Set
$E:=[x,y]$
. By the Hahn–Banach theorem, there exists
$\phi \in S_{E^*}$
which satisfies
$\phi (y)=d$
and
$\phi (x)=0$
. Note that E being a two-dimensional subspace of X is reflexive. This guarantees the existence of some
$z\in S_E$
such that
$\phi (z)=1$
. Since X is smooth, so is its subspace E. Therefore,
$\phi $
is the only supporting functional at
$z\in S_E$
. We apply Lemma 2.3 to
$f|_E: E\rightarrow Y$
to obtain
$\varphi \in S_{Y^*}$
such that
$|\phi |=|\varphi \circ f|$
on E. Then
$$ \begin{align*} 0<d=|\phi(y)|=|\varphi(f(y))|=|\varphi(tf(x))|=|t||\varphi(f(x))|=|t||\phi(x)|=0, \end{align*} $$
which is a contradiction. This proves
$[f(x)]\subset f([x])$
.
Conversely, fix a nonzero
$x\in X$
. For each
$r\in (0,+\infty )$
, by the above inclusion and the norm preserving property of f, there exists some
$\alpha _r\in \mathbb {T}$
such that
$r^{-1}f(rx)=f(\alpha _r x)$
. For each
$\alpha \in \mathbb {T}$
, by Lemma 2.1, there exist
$\beta _{r,\alpha },\alpha ^{\prime }_r\in \mathbb {T}$
such that
$$ \begin{align*} f(r\alpha x)=\beta_{r,\alpha}f(rx)=\beta_{r,\alpha}rf(\alpha_r x)=r\beta_{r,\alpha}\alpha^{\prime}_rf(x), \end{align*} $$
which implies that
$f([x])\subset [f(x)]$
. The proof is complete.
Note that the conclusion of Lemma 2.4 is equivalent to
$$ \begin{align*} \{f(r\alpha x): \alpha\in \mathbb{T}\}=\{r\alpha f(x): \alpha\in \mathbb{T}\}, \quad x\in X,\ r\in [0,+\infty). \end{align*} $$
Lemma 2.5. Let X and Y be normed spaces with X being smooth. Suppose that
${f: X \rightarrow Y}$
is a surjective phase-isometry. Then for every
$x,y\in X$
,
$$ \begin{align*} \{G_+(f(x),\alpha f(y)): \alpha\in \mathbb{T}\}=\{G(x,\alpha y): \alpha\in \mathbb{T}\}=\{G_-(f(x),\alpha f(y)): \alpha\in \mathbb{T}\}. \end{align*} $$
Proof. We only prove the first equality, the second being similar. Let
$x, y\in X$
be nonzero and
$\alpha \in \mathbb {T}$
. For each
$n\in \mathbb {N}$
, Lemma 2.4 and (1.2) imply that there exist
$\alpha _n,\beta _n,\gamma _n\in \mathbb {T}$
such that
$f(nx)=\alpha _nnf(x)$
and
$$ \begin{align*} \|f(nx)+\alpha_n\alpha f(y)\|=\|nx+\beta_n y\|, \quad \|f(nx)+\alpha_n\gamma_nf(y)\|=\|nx+\alpha y\|. \end{align*} $$
By the compactness of
$\mathbb {T}$
, there is a strictly increasing sequence
$\{n_j:j\in \mathbb {N}\}$
in
$\mathbb {N}$
and
$\beta ,\gamma \in \mathbb {T}$
such that
$\lim _{j\to \infty }\beta _{n_j}=\beta $
and
$\lim _{j\to \infty }\gamma _{n_j}=\gamma $
. Then
$$ \begin{align*} G_+(f(x),\alpha f(y)) & =\lim_{j\to\infty}(\|n_jf(x)+\alpha f(y)\|-\|n_jf(x)\|)\\ & = \lim_{j\to\infty}(\|f(n_jx)+\alpha_{n_j}\alpha f(y)\|-\|n_jf(x)\|) \\ &= \lim_{j\to\infty}(\|n_jx+\beta_{n_j}y\|-\|n_jx\|) = \lim_{j\to\infty}(\|n_jx+\beta y\|-\|n_jx\|)=G(x,\beta y) \end{align*} $$
and
$$ \begin{align*} G(x,\alpha y)&=\lim_{j\to\infty}(\|n_jx+\alpha y\|-\|n_jx\|)\\ &=\lim_{j\to\infty}(\|f(n_jx)+\alpha_{n_j}\gamma_{n_j}f(y)\|-\|f(n_jx)\|)\\ &=\lim_{j\to\infty}(\|n_jf(x)+\gamma_{n_j}f(y)\|-\|n_jf(x)\|)\\ &=\lim_{j\to\infty}(\|n_jf(x)+\gamma f(y)\|-\|n_jf(x)\|)=G_+(f(x),\gamma f(y)). \end{align*} $$
The proof is complete.
Lemma 2.6. Let X and Y be normed spaces with X being smooth. Suppose that
${f: X \rightarrow Y}$
is a surjective phase-isometry. Then Y is smooth.
Proof. Let
$x\in X$
be a nonzero element with the unique supporting functional
${\phi _x\in D(x)}$
. It suffices to prove that
$D(f(x))$
is a singleton set. Let
$\varphi ,\psi \in D(f(x))$
and
$f(y)\in \ker \varphi $
. For each
$\alpha \in \mathbb T$
, Lemma 2.5 implies that there exists
$\beta ,\gamma \in \mathbb T$
such that
$$ \begin{align*} \mbox{Re}(\alpha \phi_{x}(y))=\mbox{Re}\phi_{x}(\alpha y)=G(x,\alpha y)=G_+(f(x),\beta f(y))\geq\mbox{Re}\varphi(\beta f(y))=0 \end{align*} $$
and
$$ \begin{align*} \mbox{Re}(\alpha\psi(f(y)))=\mbox{Re}\psi(\alpha f(y))\leq G_+(f(x),\alpha f(y))=G(x,\gamma y)=\mbox{Re} \phi_{x}(\gamma y). \end{align*} $$
Using the arbitrariness of
$\alpha \in \mathbb T$
twice gives
$\phi _{x}(y)=0$
by the first inequality and therefore
$\psi (f(y))=0$
by the second inequality. This shows that
$\ker \varphi \subset \ker \psi $
. Thus,
$\psi =\lambda \varphi $
for some
$\lambda \in \mathbb F$
. Considering that
$\psi ,\varphi \in D(f(x))$
, we find that
$\lambda =1$
. This implies that
$\psi =\varphi $
, which completes the proof.
Recently, Ilišević and Turnšek [Reference Ilišević and Turnšek10, Theorem 2.2 and Remark 2.1] generalised Wigner’s theorem to smooth normed spaces via semi-inner products. This can be translated into the following theorem in the language of supporting functionals.
Theorem 2.7. Let X and Y be smooth normed spaces over
$\mathbb {F}$
and
$f:X\rightarrow Y$
a surjective mapping satisfying, for all nonzero
$x, y\in X$
,
$$ \begin{align*} |\phi_{f(x)}(f(y))| = |\phi_x(y)|. \end{align*} $$
Then f is phase equivalent to a linear or anti-linear surjective isometry in the case
$\mathbb {F}=\mathbb {C}$
and to a linear surjective isometry in the case
$\mathbb {F}=\mathbb {R}$
.
Combining the above results gives our main theorem.
Theorem 2.8. Every smooth normed space has the Wigner property.
Proof. Let X and Y be normed spaces with X being smooth. Suppose that
$f: X \rightarrow Y$
is a surjective phase-isometry. By Lemma 2.6, Y is smooth. Then Lemma 2.5 implies that for all nonzero
$x, y\in X$
,
$$ \begin{align*} \{\mbox{Re}\phi_{f(x)}(\alpha f(y)): \alpha\in \mathbb{T}\}=\{\mbox{Re}\phi_{x}(\alpha y): \alpha\in \mathbb{T}\}. \end{align*} $$
Taking the maximum on both sides, for all nonzero
$x, y\in X$
,
$$ \begin{align*} |\phi_{f(x)}(f(y))|=|\phi_{x}(y)|. \end{align*} $$
By Theorem 2.7, f is phase equivalent to a linear or anti-linear surjective isometry in the case
$\mathbb {F}=\mathbb {C}$
and to a linear surjective isometry in the case
$\mathbb {F}=\mathbb {R}$
. This completes the proof.
It is well known that
$L^p(\mu )$
is a smooth normed space, where
$\mu $
is a measure and
$1<p<\infty $
. The following corollary is immediate.
Corollary 2.9.
$L^{p}(\mu )$
has the Wigner property, where
$\mu $
is a measure and
${1<p<\infty }$
.
