Phase-isometries on real normed spaces☆
Introduction
The study of isometries between two normed spaces dates back to 1930s. The classical Mazur-Ulam theorem ([19]) states that an isometry f of a real normed space X “onto” another real normed space Y with is linear. For strictly convex Y, this result is trivial even for “into” maps (see [2]), but this is false in general. For example, if is defined by for all , then it is an “into” isometry with which is not linear since the function on is not linear. In 1968, T. Figiel proved in [8] the following substitute without the onto assumption for the Mazur-Ulam theorem: For every isometry f from a real normed space X to another real normed space Y with , there is a linear operator F of norm one from to X such that is the identity on X. Recently, a statement [5, Lemma 2.4] equivalent to Figiel's theorem was obtained by Cheng, Dong and Zhang who showed that if is an isometry with , then for every there exists a linear functional with such that for all . Figiel's theorem and its generalizations play an important role in the study of isometric embedding problems (see, for instance [6], [13]). More information and background on linear isometries of normed spaces can be found in the two volume set [9], [10].
Another important result related to linear isometries is the famous Wigner's theorem which plays a fundamental role in quantum mechanics. This result has been formulated in several ways (see [3], [4], [11], [14], [20], [22], [24], [25], [26] to list just some of them). One of them (see [24]) characterizes maps on a Hilbert space preserving the absolute value of the inner product of any pair of vectors. Namely, if H and K are complex Hilbert spaces and is a map with the property then there exists a phase function with such that is a linear isometry or conjugate linear isometry. A real version of this result was presented in [22, Corollary 8(a)] by Rätz. It is worth mentioning that L. Molnár [20] described the form of all bijective maps on the set of all rank-one idempotents of a Banach space which preserve zero products in both directions. This not only extends Uhlhorn's version of Wigner's theorem to indefinite inner product spaces but also significantly generalizes a result of Van den Broek [26].
Wigner's theorem is so important that it worth studying from various points of view. A new general version of this theorem may certainly improve our understanding of it. We shall be trying to present a Wigner-type result for normed spaces over real fields. Let H and K be real inner product spaces. By the polar identity, we can easily see that a map satisfies (1.1) if and only if it satisfies
This inspires us to introduce the following definition.
Definition 1 Let X and Y be normed spaces. A map is called a phase-isometry if it satisfies the functional equation
Let X and Y be real normed spaces. Two maps are called phase equivalent if there exists a phase function such that . We can easily see that a map that is phase equivalent to a linear isometry is definitely a phase-isometry. It is interesting in seeing whether the converse also holds for real normed spaces. Motivated by the Mazur-Ulam theorem, the following natural problem arises. Problem 1 Let X and Y be real normed spaces, and let be a surjective phase-isometry. Is it true that f is phase equivalent to a linear isometry?
By the real version of Wigner's theorem, Problem 1 is solved in the positive for real Hilbert spaces (see [18]). In our paper [15], we provide an affirmative answer to Problem 1 with X and Y being spaces . The first author and Jia presented in [17] a similar result for -type spaces.
A normed space X is said to have the Wigner property if Problem 1 has an affirmative answer for an arbitrary target Y. It should be mentioned that the above results on real Hilbert spaces, spaces and -type spaces are not enough to allow these spaces to have the Wigner property since the target spaces are not arbitrary. We start in the present paper to attack Problem 1 for general normed spaces. Although we feel our paper is interesting in its own right, we hope that it will serve as a stepping stone to show that every surjective phase-isometry between two real normed spaces is phase equivalent to a linear isometry (i.e., all real normed spaces have the Wigner-property).
This paper is organized as follows. In Section 2, we first apply the fundamental theorem of projective geometry to present a sufficient condition such that Problem 1 can be solved in the positive for general normed spaces. Next, we give a Figiel-type result for phase-isometries. The technique of this result that we use draw on the famous Figiel theorem [8] and an idea of [5, lemma 2.4] for a special case . It plays a crucial role and is used frequently in every main theorem of the present paper.
Lemma 1 Main lemma Let X and Y be real normed spaces, and let be a phase-isometry (not necessarily surjective). Then for every -exposed point of , there exists a linear functional of norm one such that for all .
As a result, our first result reads as follows. Theorem 1 Let X be a real smooth normed space. Then X has the Wigner property.
The following result is an immediate consequence of Theorem 1.
Corollary 1 All real spaces with for an arbitrary measure μ have the Wigner property. In particular, real Hilbert spaces have the Wigner property.
In Section 3, by the properties of Birkhoff orthogonality, we continue to present another sufficient condition such that Problem 1 can be solved in the positive for some special normed spaces. As its application, we present one more result as follows.
Theorem 2 -type spaces and -spaces have the Wigner property.
Section snippets
Main lemma and phase-isometries for smooth normed spaces
In this note, the letters are used to denote real normed spaces and are their dual spaces respectively. For a real normed space X, we denote by and the unit sphere and the closed unit ball of X, respectively. Throughout what follows, we shall freely use without explicit mention notation “±” to mean that either “+” or “−” holds.
We start this section with an interesting result showing that every surjective phase-isometry is an injective odd norm-preserving map.
Lemma 2 Let X and Y be real
Phase-isometries on -type and -type spaces
In this section we consider phase-isometries from -type or -type spaces onto any normed spaces. We shall show that all such maps are phase equivalent to real linear isometries.
To show the following results of this section, we recall some notations and results about Birkhoff orthogonality. Let X be a real normed space. For all , let us denote by the Birkhoff orthogonality relation on X as: This relation is clearly homogeneous, but neither symmetric nor
Acknowledgments
The authors wish to express their appreciation to Guanggui Ding for many very helpful comments regarding isometric theory in Banach spaces and the referees whose kind remarks helped them to make the presentation of the paper clearer.
References (26)
- et al.
On stability of nonsurjective ε-isometries of Banach spaces
J. Funct. Anal.
(2013) - et al.
Isometric embeddings of compact spaces into Banach spaces
J. Funct. Anal.
(2008) An elementary proof for the non-bijective version of Wigner's theorem
Phys. Lett. A
(2014)A new proof of Wigner's theorem
Rep. Math. Phys.
(2004)Orthogonality preserving transformations on indefinite inner product spaces: generalization of Uhlhorn's version of Wigner's theorem
J. Funct. Anal.
(2002)- et al.
A direct proof of Wigner's theorem on maps which preserve transition probabilities between pure states of quantum systems
Ann. Phys.
(1990) Symmetry transformations in indefinite metric spaces: a generalization of Wigner's theorem
Physica A
(1984)- et al.
On Birkhoff orthogonality and isosceles orthogonality in normed linear spaces
Aequ. Math.
(2012) Isometries in normed spaces
Am. Math. Mon.
(1971)Note on Wigner's theorem on symmetry operations
J. Math. Phys.
(1964)
Wigner's theorem on symmetries in indefinite metric spaces
Commun. Math. Phys.
An elementary proof of the fundamental theorem of projective geometry
Geom. Dedic.
On nonlinear isometric embeddings of normed linear spaces
Bull. Acad. Pol. Sci. Math. Astron. Phys.
Cited by (20)
Phase-isometries on the unit sphere of CL-spaces
2023, Journal of Mathematical Analysis and ApplicationsPhase-isometries between normed spaces
2021, Linear Algebra and Its ApplicationsMIN-PHASE-ISOMETRIES IN STRICTLY CONVEX NORMED SPACES
2023, Bulletin of the Australian Mathematical SocietyThe wigner property for CL-spaces and finite-dimensional polyhedral Banach spaces
2021, Proceedings of the Edinburgh Mathematical SocietyOn a universal inequality for approximate phase isometries
2024, Acta Mathematica ScientiaA functional equation related to Wigner’s theorem
2024, Aequationes Mathematicae