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ISRN Algebra
Volume 2012 (2012), Article ID 704960, 6 pages
http://dx.doi.org/10.5402/2012/704960
Research Article

On the Completely Positive and Positive Semidefinite-Preserving Cones—Part III

1Department of Mathematics, Xavier University of Louisiana, New Orleans, LA 70125, USA
2Department of Mathematics, Idaho State University, Pocatello, ID 83209, USA

Received 13 March 2012; Accepted 9 May 2012

Academic Editors: A. Jaballah, A. Kiliçman, F. Uhlig, and P.-H. Zieschang

Copyright © 2012 Andrew J. Klimas and Richard D. Hill. 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.

Abstract

This paper studies the extremals and other faces of the completely positive and positive semidefinite-preserving linear transformations.

1. Introduction

This paper is the third in a sequence giving the related theory of the cone 𝜋(PSD𝑛) of positive semidefinite-preserving linear transformations on the complex vector space of complex matrices of order 𝑛, denoted by 𝑀𝑛, and its self-dual subcone CP𝑛 of the completely positive linear transformations. Following the papers of Barker et al. [1] and Yopp and Hill [2], this third paper studies the extremals and other faces of these cones. The cone of completely copositive linear transformations, coCP𝑛, fits nicely in this work as well.

For a list of all the cone-theoretic definitions used here, see [2]. For a list of characterization theorems for CP𝑛 as well as HP𝑛, the Hermitian preservers, which are an ambient space for all the cones of this paper, see [3].

2. Classes of Extremals of 𝜋(PSD𝑛)

Yopp and Hill [2, 4] have characterized the extremals of CP𝑛 and coCP𝑛 as follows.

Theorem 2.1. Let 𝒯𝑀𝑛𝑀𝑛. Then,(i)𝒯ExtCP𝑛 if and only if there exists 𝐴𝑀𝑛{0} such that 𝒯=𝐴𝐾𝐴; (ii)if 𝒯=𝑟𝑖=1𝐴𝑖𝐾𝐴𝑖, where 𝐴1,,𝐴𝑟𝑀𝑛{0}, then 𝒯ExtCP𝑛 if and only if 𝐴𝑖=𝛼𝑖𝐴1 for all 𝑖=1,,𝑟; (iii)𝒯ExtCP𝑛 if and only if there exist 𝐴1,,𝐴𝑟𝑀𝑛{0} and (𝑑𝑖𝑗)ExtPSD𝑟 such that 𝒯=𝑟𝑖,𝑗=1𝑑𝑖𝑗𝐴𝑖𝐾𝐴𝑗.

Theorem 2.2. Let 𝒯𝑀𝑛𝑀𝑛 be an element of coCP𝑛. Then,(i)𝒯ExtcoCP𝑛 if and only if 𝒯=𝐴𝐾𝐴𝐓 for some 𝐴𝑀𝑛{0};(ii)if 𝒯=𝑟𝑖=1𝐴𝑖𝐾𝐴𝑖𝐓, where 𝐴1,,𝐴𝑟𝑀𝑛{0}, then 𝒯ExtcoCP𝑛 if and only if 𝐴𝑖=𝛼𝑖𝐴1 for all 𝑖=1,,𝑟;(iii)𝒯ExtcoCP𝑛 if and only if there exist 𝐴1,,𝐴𝑟𝑀𝑛{0} and (𝑑𝑖𝑗)ExtPSD𝑟 such that 𝒯=𝑟𝑖,𝑗=1𝑑𝑖𝑗𝐴𝑖𝐾𝐴𝑗𝐓.

The remainder of this section focuses on the extremals of 𝜋(PSD𝑛). The major result of [5] follows.

Theorem 2.3. Let 𝐴𝑀𝑛. If 𝒯=𝐴𝐾𝐴 or 𝒯=𝐴𝐾𝐴𝐓, then 𝒯 is an extremal of 𝜋(PSD𝑛), and when the rank of 𝐴 is 1 or 𝑛, the extremals are exposed.

The following result, originally given in a different setting by Loewy and Schneider [6], yields a (different) large class of extremals, namely, the nonsingular maps. Further, we note that the analogous result does not hold for CP𝑛. We first give an alternative proof for the theorem.

Theorem 2.4. Every nonsingular element of 𝜋(PSD𝑛) is an extremal of 𝜋(PSD𝑛).

Proof. Let 𝐴 be a nonsingular element of 𝜋(PSD𝑛). Since every nonsingular linear transformation is a vector space isomorphism, it follows from [2, Theorem 1.1(iv)] that 𝐴(ExtPSD𝑛)ExtPSD𝑛. By [7, Theorem 2.B.2], it follows that 𝐴 is an extremal of 𝜋(PSD𝑛).

The following example shows that this result does not apply to CP𝑛. Let 𝐴1=1002, and let 𝐴2=2001. Then, the linear transformation 𝒯 with matrix representation 𝒯=2𝑖=1𝐴𝑖𝐾𝐴𝑖=5000040000400005(2.1) is a nonsingular element of CP2. However, we observe that there exists no 𝐴𝑀2 such that 𝒯=𝐴𝐾𝐴. It follows from [2, Theorem 3.4(i)] that 𝒯 is not an extremal of CP2.

From the subcones CP𝑛 and coCP𝑛, we find two more classes of extremals of 𝜋(PSD𝑛).

Theorem 2.5. Every extremal of CP𝑛 and of coCP𝑛 is also an extremal of 𝜋(PSD𝑛).

Proof. Assume that 𝒯𝑀𝑛𝑀𝑛 is an extremal of CP𝑛. By [2, Theorem 3.4(i)], 𝒯 is an extremal of CP𝑛 if and only if there exists 𝐴𝑀𝑛 such that 𝒯=𝐴𝐾𝐴. Then, [5, Lemma 3.2] gives us that 𝒯 is an extremal of 𝜋(PSD𝑛). An analogous argument implies that every extremal of coCP𝑛 is also an extremal of 𝜋(PSD𝑛).

By definition, all extreme rays of CP𝑛 and coCP𝑛 are also extreme rays of 𝜋(PSD𝑛). Also, Theorem 2.5 and [2, Theorem 3.4] give us the following extremals for 𝜋(PSD𝑛).

Theorem 2.6. Let 𝒯𝑀𝑛𝑀𝑛. Then,(i)if 𝒯=𝑟𝑖=1𝐴𝑖𝐾𝐴𝑖 or 𝒯=𝑟𝑖=1𝐴𝑖𝐾𝐴𝑖𝐓, where 𝐴1,,𝐴𝑟𝑀𝑛{0}, and if 𝐴𝑖=𝛼𝑖𝐴1 for all 𝑖=1,,𝑟, then 𝒯Ext𝜋(PSD𝑛); (ii) if there exist 𝐴1,,𝐴𝑟𝑀𝑛{0} and (𝑑𝑖𝑗)ExtPSD𝑟 such that 𝒯=𝑟𝑖=1𝑑𝑖𝑗𝐴𝑖𝐾𝐴𝑗 or 𝒯=𝑟𝑖=1𝑑𝑖𝑗𝐴𝑖𝐾𝐴𝑗𝐓, then 𝒯Ext𝜋(PSD𝑛).

3. Other Faces of CP𝑛 and 𝜋(PSD𝑛)

It is well known that the transpose map 𝐓 is an element of 𝜋(PSD𝑛) but not CP𝑛. Since 𝐓=𝐼𝑛𝐾𝐼𝑛𝐓, it follows that 𝐓Ext(𝜋(PSD𝑛)). Thus, 𝐹=Φ(𝐓)𝜋(PSD𝑛) but 𝐹CP𝑛. While it is well known that CP𝑛 is a subcone but not a face of 𝜋(PSD𝑛), it is an open question whether every proper face of CP𝑛 (in the sense of a proper subset) is a face of 𝜋(PSD𝑛). Examples of such faces do exist. The Siler cone 𝐾1=𝒯HP𝑛𝐴𝒯(𝐴)PSD𝑛𝐴𝑀𝑛()(3.1) is a one-dimensional face of both CP𝑛 and 𝜋(PSD𝑛) [8, page 33].

In [2, Theorem 3.2], Yopp and Hill have characterized all the faces of CP𝑛 as follows.

Theorem 3.1. 𝐹CP𝑛 if and only if there exist 𝐵1,,𝐵𝑟𝑀𝑛 such that 𝐹=𝒯CP𝑛𝒯=𝑠𝑖=1𝐴𝑖𝐾𝐴𝑖𝐴𝑤𝑒𝑟𝑒vec𝑖𝑥=0𝑤𝑒𝑛𝑒𝑣𝑒𝑟𝑥𝑟𝑖=1𝐵vec𝑖.(3.2)

One way in which we could show that a face 𝐹 of CP𝑛 is not a face of 𝜋(PSD𝑛) would be to exhibit (at least) one element of 𝐹 that lies in the interior (equivalently, not in the boundary) of 𝜋(PSD𝑛), as we could then apply the following result due to Barker and Schneider [9, Corollary 2.18].

Theorem 3.2. If 𝐹𝐾 and 𝐹𝐾, then 𝐹bd𝐾.

In [13], Kye gives us a criterion by which we can determine whether a given linear transformation is an element of int 𝜋(PSD𝑛) as follows. Let 𝑣𝑛 be nonzero, and let 𝑃𝑣 denote the one-dimensional projection matrix corresponding to the projection onto the subspace spanned by 𝑣. Note that one-dimensional projections are precisely the extremals of PSD𝑛, as characterized by Barker and Carlson [11] and further by Hill and Waters [12, Theorem 3.8]. This leads to Kye's result [13, Proposition 4.1].

Theorem 3.3. The linear map 𝒯𝜋(PSD𝑛) is an interior point of 𝜋(PSD𝑛) if and only if 𝒯(𝑃𝑣) is nonsingular for all extremals 𝑃𝑣.

A more common characterization of the interior of a positive cone of linear maps when specialized to 𝜋(PSD𝑛) gives the following: int𝜋PSD𝑛=𝒯𝜋PSD𝑛𝒯PSD𝑛{0}PD𝑛.(3.3) Thus, instead of requiring that the image of any nonzero positive semidefinite matrix be positive definite, as in the standard characterization, Kye's characterization instead requires that the image of any extremal be nonsingular.

We continue this section with two examples due to Kye [13].

The first example is the trace map, which is easily seen to be an element of 𝜋(PSD𝑛). For nonzero 𝑣𝑛, we have that 𝑃tr𝑣=𝑣1𝑣1+𝑣2𝑣2++𝑣𝑛𝑣𝑛𝐼𝑛,(3.4) which is nonsingular, giving us that the trace map is an element of int 𝜋(PSD𝑛).

Kye's second example is the linear map tran𝐾𝐴𝐾1/2𝐴𝑇𝐾1/2,(3.5) where 𝐾1/2 is the standard square root of a positive definite matrix (cf. [10]). We know that 𝐴PSD𝑛𝐴𝑇PSD𝑛. Since 𝐾 is positive definite, 𝐾1/2 is also positive definite, and Sylvester's law of inertia gives us that 𝐴𝑇 and 𝐾1/2𝐴𝑇𝐾1/2 have the same inertias, so that 𝐾1/2𝐴𝑇𝐾1/2PSD𝑛. Thus, tran𝐾𝜋(PSD𝑛) for all positive definite matrices 𝐾. Kye claims, for any 𝐾PD𝑛, tran𝐾int𝜋(PSD𝑛). Actually, it is not (making it an element of bd 𝜋(PSD𝑛)). To see this, let 𝐾=𝐼𝑛, which is positive definite. Then, 𝐾1/2=𝐼𝑛. In this case, tran𝐼𝑛𝐴𝐼𝑛1/2𝐴𝑇𝐼𝑛1/2=𝐴𝑇(3.6) for 𝐴𝑀𝑛, that is, tran𝐼𝑛 is simply the transpose map, an extremal of 𝜋(PSD𝑛), as we observed at the beginning of this section, and thus not an interior point of 𝜋(PSD𝑛).

4. Toward a Result Concerning Faces in Shared Boundaries

In Section 3, we have encountered a number of linear maps that lie in the intersection of the boundaries of CP𝑛 and 𝜋(PSD𝑛). We further study this area, which leads us to a still-open question: is a face of CP𝑛 which lies in the boundary of 𝜋(PSD𝑛) necessarily also a face of 𝜋(PSD𝑛)?

We do have the following.

Theorem 4.1. Let 𝐹𝐿𝐾 be a chain of (sub)cones. If 𝐹𝐾, then 𝐹𝐿.

Proof. Assume that 𝐹𝐾 and that 𝑥,𝑦𝑥𝐿, and 𝑦𝐹. Since 𝐿𝐾, it follows that 𝑥,𝑦𝑥𝐾. Since 𝐹𝐾, we have that 𝑥𝐹. Thus, 𝐹𝐿.

In [2], Yopp and Hill give several results concerning the boundary of CP𝑛, one of which, when combined with its analogue for 𝜋(PSD𝑛) (also found in [2]), yields a sufficient condition for a linear map to be in the intersection of their boundaries.

Theorem 4.2. Let 𝒯CP𝑛, and suppose that there exist 𝐴1,,𝐴𝑟𝑀𝑛 such that 𝒯=𝑟𝑖=1𝐴𝑖𝐾𝐴𝑖. If 𝑟<𝑛2, then 𝒯bdCP𝑛.

Theorem 4.3. Let 𝒯𝜋(PSD𝑛), and suppose that there exist 𝐴1,,𝐴𝑟𝑀𝑛 such that 𝒯=𝑟𝑖=1𝜖𝑖𝐴𝑖𝐾𝐴𝑖 where 𝜖=±1. If 𝑟<𝑛, then 𝒯bd𝜋(PSD𝑛).

Combining these results gives the following.

Theorem 4.4. Let 𝒯CP𝑛, and suppose that there exist 𝐴1,,𝐴𝑟𝑀𝑛 such that 𝒯=𝑟𝑖=1𝐴𝑖𝐾𝐴𝑖. If 𝑟<𝑛, then 𝒯bdCP𝑛bd𝜋(PSD𝑛).

Proof. Assume 𝑟<𝑛. Since 𝑟<𝑛𝑛2 for 𝑛=1,2,,𝒯bdCP𝑛. If we let 𝜖𝑖=1 for 𝑖=1,2,,𝑟, then 𝒯bd𝜋(PSD𝑛) and the conclusion is immediate.

Finally, combining Theorems 3.1 and 4.4, we obtain the following sufficient condition for a face of CP𝑛 to also be a face of 𝜋(PSD𝑛):

Theorem 4.5. 𝐹 is a face of both CP𝑛 and 𝜋(PSD𝑛) if and only if there exist 𝐵1,,𝐵𝑟𝑀𝑛 such that 𝐹=𝒯CP𝑛𝒯=𝑠𝑖=1𝐴𝑖𝐾𝐴𝑖𝐴𝑤𝑒𝑟𝑒𝑠<𝑛,andvec𝑖𝑥=0𝑤𝑒𝑛𝑒𝑣𝑒𝑟𝑥𝑟𝑖=1𝐵vec𝑖.(4.1)

Results by Yopp and Hill [2] finish the known material of this section.

Theorem 4.6. Let 𝒯CP𝑛. Then, 𝒯intCP𝑛 if and only if there exist linearly independent matrices 𝐴1,,𝐴𝑛2 such that 𝒯=𝑛2𝑖=1𝐴𝑖𝐾𝐴𝑖.

Theorem 4.7. Let 𝒯,𝒮𝜋(PSD𝑛). If 𝒯Φ𝜋(PSD𝑛)(𝒮), then 𝒯(𝐴)ΦPSD𝑛(𝒮(𝐴)) for every 𝐴PSD𝑛.

Proof. By Corollary 2.10 of [9, page 222], 𝒯Φ𝜋(PSD𝑛)(𝒮) if and only if there exists 𝜆>0 such that 𝒮𝜆𝒯𝜋(PSD𝑛). Therefore, for every 𝐴PSD𝑛, 𝒮(𝐴)𝜆𝒯(𝐴)𝜋(PSD𝑛). It follows that 𝒯(𝐴)ΦPSD𝑛(𝒮(𝐴)) for every 𝐴PSD𝑛.

The material of this paper leaves an open problem that we first state in generality and then in the setting of the paper.

Conjecture 4.8. Let 𝐾, 𝐿, and 𝐹 be cones such that 𝐹/𝐿/𝐾. If 𝐹𝐿 and 𝐹bd𝐾, then 𝐹𝐾.

Conjecture 4.9. Let 𝐹 be a proper subcone of CP𝑛. If 𝐹CP𝑛 and 𝐹bd𝜋(PSD𝑛), then 𝐹𝜋(PSD𝑛).

Acknowledgment

The material of this paper forms a part of the Doctor of Arts thesis written by Klimas under the direction of Hill at Idaho State University.

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