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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 520795, 5 pages
Maps Preserving Schatten -Norms of Convex Combinations
1Department of Applied Mathematics, National Sun Yat-Sen University, Kaohsiung 80424, Taiwan
2School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079, China
Received 18 October 2013; Accepted 30 December 2013; Published 14 January 2014
Academic Editor: Antonio M. Peralta
Copyright © 2014 David Li-Wei Kuo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study maps of positive operators of the Schatten -classes (), which preserve the -norms of convex combinations, that is, . They are exactly those carrying the form for a unitary or antiunitary . In the case , we have the same conclusion whenever it just holds for all the positive Hilbert-Schmidt class operators of norm . Some examples are demonstrated.
The Mazur-Ulam theorem states that every bijective distance preserving map from a Banach space onto another is affine; that is, After translation, we can assume that and is indeed a surjective real linear isometry. Let us consider another version of this statement. Suppose that is a bijective map from a Hilbert space onto and preserves norm of convex combinations: Let us further relax the assumption that (2) holds for just one fixed in . By letting in (2), we see that for all in . Squaring both sides of (2), we will see that the real parts of the inner products coincide; that is, Then the classical Wigner theorem (see, e.g., [1, Theorem 3]) ensures that there is a surjective real linear isometry such that for all in .
Characterizing isometries, linear or not, of spaces of operators under various norms has been a fruitful area of research for a long time. See, for example, [2, 3] for good surveys. In particular, the spaces of the Schatten -class operators on a (complex) Hilbert space () are important objects in both analysis and physics. They are widely used in operator theory and quantum mechanics, for example.
Let be the set of all positive operators in , and let be the set of all positive operators in of -norm one. Recall that an affine automorphism (or S-automorphism in  or Kadison automorphism in ) is a bijective affine map ; that is, It is known (see, e.g., ) that affine automorphisms are exactly those carrying the form for a unitary or antiunitary on .
Recently, Nagy  established a Mazur-Ulam-type result for the Schatten -class operators. Suppose that () is a bijective map preserving the distance induced by the norm . Then is implemented by a unitary or an antiunitary operator such that . In this paper, we will establish a counterpart of Nagy’s result similar to the one demonstrated in the first paragraph. More precisely, we will characterize those maps satisfying We will show that they are implemented by a unitary or an antiunitary operator.
Our main theorem follows.
Theorem 1. Let be a separable complex Hilbert space of finite or infinite dimension. Let . Suppose that is a map from into , which will be assumed to be surjective when . The following conditions are equivalent.(1)preserves the Schatten -norms of convex combinations; that is, (2) preserves the pairings; that is, for all , one has , and (3)There exists a unitary or antiunitary operator on such that
The proof of Theorem 1 is given in Section 2. When , we see in Section 3 that for carrying the expected form stated in Theorem 1(3) it suffices to say that condition (6) held for only one fixed in . Finally, we demonstrate some examples in Section 4.
2. Proof of the Main Theorem
In what follows, we fix some notation and definitions used throughout the paper. Let stand for a separable complex Hilbert space of finite dimension or infinite dimension. Let denote the algebra of all bounded linear operators on . For a compact operator in , let denote the singular values of , that is, the eigenvalues of arranged in their decreasing order (repeating according to multiplicity). A compact operator belongs to the Schatten -classes () if where denotes the trace functional. We call the Schatten -norm of . In particular, is the trace class and is the Hilbert-Schmidt class. The cone of positive operators in is denoted by , and the set of rank one projections in is denoted by .
Recall that the norm of a normed space is Fréchet differentiable at if exists and uniform for all norm one vectors .
Lemma 2 (see [8, Theorem 2.3]). Let and in be nonzero. The norm of is Fréchet differentiable at . For any in , one has
Lemma 3. Suppose . The following conditions are equivalent.(1).(2) for all in and all in .(3) for all in .
Proof. is obvious.
: Differentiating both sides of at , we have by Lemma 2.
: Since and are positive, is Hermitian. There exists an orthonormal basis of such that . Choosing , we have for all . It follows that .
We say that two self-adjoint operators in are orthogonal if , which is equivalent to the property that they have mutually orthogonal ranges.
Lemma 4. Suppose that for . The following conditions are equivalent.(1) are orthogonal; that is, .(2) for any (and thus all) in .(3).(4) for all in ; that is, in Birkhoff's sense.(5).
Proof. : From [9, Lemma 2.6], we know that for any two positive operators in , we have
Here, the equality holds if and only if . Setting and , we get
: One direction is obvious. For the other, because are positive, This forces , and thus .
: Since , there exists an orthonormal basis of such that , , , , and for all . Hence,
: Without loss of generality, we can assume that . Define . Then is differentiable and attains its minimum at . From Lemma 2, and assertion (5) follows.
: As in proving , we have . Then, there exists an orthonormal basis of such that , , with , , and for all . Thus, .
Lemma 5. Let . Suppose that is a map from into preserving the Schatten -norms of convex combinations; that is, (6) holds. Then, one has
Proposition 6. Suppose that satisfies Then the following assertions hold.(1) preserves orthogonality in both directions; that is(2)When , maps rank-one projections to rank-one projections. This also holds when and is surjective.(3)When , one hasThis also holds when and is surjective.
Proof. (1) follows from Lemma 4.
(2) First, we assume that . Suppose is a rank-one projection. We can find pairwise orthogonal rank-one projections such that for . From (1), we know that are nonzero and pairwise orthogonal. This forces that has rank one since . By (18), taking , we see that . Therefore, the rank-one positive operator is a projection.
Next, we consider the case and is surjective. Suppose that there exists a rank-one projection in such that has rank greater than one. Then, there are two nonzero orthogonal operators and in such that . Since is surjective and preserves orthogonality in both directions, there are two nonzero orthogonal operators and in such that and . For any in with , we have It forces that and hence , because , , and are all positive. This implies . Therefore, and for some nonzero . This contradicts the fact that .
(3) From (2), we know that , are rank-one projections in . Therefore, , . Using (18) with , , we have
Proof of Theorem 1. follows from Lemma 5.
: From Proposition 6, we obtain that satisfies for all rank-one projections in . From a nonsurjective version of Wigner’s theorem, cf. [6, Theorem ], there exists an isometry or anti-isometry on such that Note that is indeed surjective even when is of infinite dimension, since is assumed to be surjective in this case.
For any rank-one projection in , setting in (7), we have We have by Lemma 3. This gives .
3. Maps Preserving Norms of Just a Special Convex Combination
A careful look at the proof of Lemma 5 tells us that the condition suffices to hold for the members of any sequence in converging to rather than for any point in . Indeed, in order to get some good properties of stated in the previous section, we only need to assume that preserves the Schatten -norm of convex combination with a given system of coefficients.
Proposition 7. Let . Let in be arbitrary but fixed. Suppose The following properties are satisfied. (1) is injective.(2) preserves orthogonality in both directions.(3)When , maps rank-one projections to rank-one projections. This also holds when and is surjective.
Proof. (1) Assume . We have . From (26), we get . Hence,
This forces since the norm is strictly convex for .
(2) Assume . From Lemma 4, we have Together with (26), we have Hence, we have from Lemma 4 again. The other implication follows similarly.
(3) The proof is similar to that of Proposition 6(2).
When , we get an improvement of Theorem 1.
Theorem 8. Let be a separable complex Hilbert space. Suppose that , which needs to be surjective when . The following conditions are equivalent.(1) preserves the Hilbert-Schmidt norms of all convex combinations; that is, (2)For any (and thus all) in one has A special case states that (3) for all in .(4)There exists a unitary or antiunitary operator such that
Proof. We prove only. Observe We have .
We remark that all results in previous sections hold for a map which satisfies instead of (6) the condition The proofs go in exactly the same ways.
The following example shows that a norm preserver of might not be affine.
Example 1. Let be a finite dimensional Hilbert space with an orthonormal basis . Let . Define a map from into itself by where is a rank-one projection for . Obviously, is positive and for all in . However, does not preserve the Schatten -norms of convex combinations, as the eigenvalues of and can be different from each other.
Our theorems are about the Schatten -norms for . Here is an example of a map of which preserves trace norms of convex combinations. However, it is not implemented by a unitary or antiunitary.
Example 2. Consider Example 1 in the case where . In this case, It is easy to see that the map satisfies the condition But there does not exist a unitary or antiunitary such that for all in .
Example 3. Let be a separable Hilbert space of infinite dimension, and let be a basis of . Let be the unilateral shift on defined by for . Let be defined by for in . The map is not surjective, as is not in its range. It is easy to see that holds for all in and in . However, is not implemented by a unitary or antiunitary.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research is supported partially by the Aim for the Top University Plan of NSYSU, the NSC Grant (102-2115-M-110-002-MY2), and the NSFC Grant (no. 11171126).
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