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

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(PSD_𝑛) of positive semidefinite-preserving linear transformations on the complex vector space of complex matrices of order 𝑛, denoted by 𝑀_𝑛, and its self-dual subcone CP_𝑛 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, coCP_𝑛, 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 CP_𝑛 as well as HP_𝑛, the Hermitian preservers, which are an ambient space for all the cones of this paper, see [3].

2. Classes of Extremals of 𝜋(PSD_𝑛)

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

Theorem 2.1. Let 𝒯∶𝑀_𝑛→𝑀_𝑛. Then,(i)𝒯∈ExtCP_𝑛 if and only if there exists 𝐴∈𝑀_𝑛⧵{0} such that ⟨𝒯⟩=𝐴⊗_𝐾𝐴; (ii)if ⟨𝒯⟩=∑^𝑟𝑖=1𝐴_𝑖⊗_𝐾𝐴_𝑖, where 𝐴₁,…,𝐴_𝑟∈𝑀_𝑛⧵{0}, then 𝒯∈ExtCP_𝑛 if and only if 𝐴_𝑖=𝛼_𝑖𝐴₁ for all 𝑖=1,…,𝑟; (iii)𝒯∈ExtCP_𝑛 if and only if there exist 𝐴₁,…,𝐴_𝑟∈𝑀_𝑛⧵{0} and (𝑑_𝑖𝑗)∈ExtPSD_𝑟 such that ⟨𝒯⟩=∑^{𝑟𝑖,𝑗=1}𝑑_𝑖𝑗𝐴_𝑖⊗_𝐾𝐴_𝑗.

Theorem 2.2. Let 𝒯∶𝑀_𝑛→𝑀_𝑛 be an element of coCP_𝑛. Then,(i)𝒯∈ExtcoCP_𝑛 if and only if ⟨𝒯⟩=𝐴⊗_𝐾𝐴∘𝐓 for some 𝐴∈𝑀_𝑛⧵{0};(ii)if ⟨𝒯⟩=∑^𝑟𝑖=1𝐴_𝑖⊗_𝐾𝐴_𝑖∘𝐓, where 𝐴₁,…,𝐴_𝑟∈𝑀_𝑛⧵{0}, then 𝒯∈ExtcoCP_𝑛 if and only if 𝐴_𝑖=𝛼_𝑖𝐴₁ for all 𝑖=1,…,𝑟;(iii)𝒯∈ExtcoCP_𝑛 if and only if there exist 𝐴₁,…,𝐴_𝑟∈𝑀_𝑛⧵{0} and (𝑑_𝑖𝑗)∈ExtPSD_𝑟 such that ⟨𝒯⟩=∑^{𝑟𝑖,𝑗=1}𝑑_𝑖𝑗𝐴_𝑖⊗_𝐾𝐴_𝑗∘𝐓.

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

Theorem 2.3. Let 𝐴∈𝑀_𝑛. If ⟨𝒯⟩=𝐴⊗_𝐾𝐴 or ⟨𝒯⟩=𝐴⊗_𝐾𝐴∘𝐓, then 𝒯 is an extremal of 𝜋(PSD_𝑛), 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 CP_𝑛. We first give an alternative proof for the theorem.

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

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

The following example shows that this result does not apply to CP_𝑛. Let 𝐴₁=¹⁰⁰², and let 𝐴₂=²⁰⁰¹. Then, the linear transformation 𝒯 with matrix representation ⟨𝒯⟩=²_𝑖=1𝐴_𝑖⊗_𝐾𝐴_𝑖=⎡⎢⎢⎣5000040000400005⎤⎥⎥⎦(2.1) is a nonsingular element of CP₂. However, we observe that there exists no 𝐴∈𝑀₂ such that ⟨𝒯⟩=𝐴⊗_𝐾𝐴. It follows from [2, Theorem 3.4(i)] that 𝒯 is not an extremal of CP₂.

From the subcones CP_𝑛 and coCP_𝑛, we find two more classes of extremals of 𝜋(PSD_𝑛).

Theorem 2.5. Every extremal of CP_𝑛 and of coCP_𝑛 is also an extremal of 𝜋(PSD_𝑛).

Proof. Assume that 𝒯∶𝑀_𝑛→𝑀_𝑛 is an extremal of CP_𝑛. By [2, Theorem 3.4(i)], 𝒯 is an extremal of CP_𝑛 if and only if there exists 𝐴∈𝑀_𝑛 such that ⟨𝒯⟩=𝐴⊗_𝐾𝐴. Then, [5, Lemma 3.2] gives us that 𝒯 is an extremal of 𝜋(PSD_𝑛). An analogous argument implies that every extremal of coCP_𝑛 is also an extremal of 𝜋(PSD_𝑛).

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

Theorem 2.6. Let 𝒯∶𝑀_𝑛→𝑀_𝑛. Then,(i)if ⟨𝒯⟩=∑^𝑟𝑖=1𝐴_𝑖⊗_𝐾𝐴_𝑖 or ⟨𝒯⟩=∑^𝑟𝑖=1𝐴_𝑖⊗_𝐾𝐴_𝑖∘𝐓, where 𝐴₁,…,𝐴_𝑟∈𝑀_𝑛⧵{0}, and if 𝐴_𝑖=𝛼_𝑖𝐴₁ for all 𝑖=1,…,𝑟, then 𝒯∈Ext𝜋(PSD_𝑛); (ii) if there exist 𝐴₁,…,𝐴_𝑟∈𝑀_𝑛⧵{0} and (𝑑_𝑖𝑗)∈ExtPSD_𝑟 such that ⟨𝒯⟩=∑^𝑟𝑖=1𝑑_𝑖𝑗𝐴_𝑖⊗_𝐾𝐴_𝑗 or ⟨𝒯⟩=∑^𝑟𝑖=1𝑑_𝑖𝑗𝐴_𝑖⊗_𝐾𝐴_𝑗∘𝐓, then 𝒯∈Ext𝜋(PSD_𝑛).

3. Other Faces of CP_𝑛 and 𝜋(PSD_𝑛)

It is well known that the transpose map 𝐓 is an element of 𝜋(PSD_𝑛) but not CP_𝑛. Since ⟨𝐓⟩=𝐼_𝑛⊗_𝐾𝐼_𝑛∘𝐓, it follows that 𝐓∈Ext(𝜋(PSD_𝑛)). Thus, 𝐹=Φ(𝐓)⊴𝜋(PSD_𝑛) but 𝐹⋬CP_𝑛. While it is well known that CP_𝑛 is a subcone but not a face of 𝜋(PSD_𝑛), it is an open question whether every proper face of CP_𝑛 (in the sense of a proper subset) is a face of 𝜋(PSD_𝑛). Examples of such faces do exist. The Siler cone 𝐾₁=𝒯∈HP_𝑛∶𝐴^∗𝒯(𝐴)∈PSD_𝑛∀𝐴∈𝑀_𝑛(ℂ)(3.1) is a one-dimensional face of both CP_𝑛 and 𝜋(PSD_𝑛) [8, page 33].

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

Theorem 3.1. 𝐹⊴CP_𝑛 if and only if there exist 𝐵₁,…,𝐵_𝑟∈𝑀_𝑛 such that 𝐹=𝒯∈CP_𝑛∶⟨𝒯⟩=^𝑠_𝑖=1𝐴_𝑖⊗_𝐾𝐴_𝑖𝑤ℎ𝑒𝑟𝑒vec𝐴^∗𝑖^∗𝑥=0𝑤ℎ𝑒𝑛𝑒𝑣𝑒𝑟𝑥∈^𝑟_𝑖=1vec𝐵^∗𝑖^⟂.(3.2)

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

Theorem 3.2. If 𝐹⊴𝐾 and 𝐹≠𝐾, then 𝐹⊂bd𝐾.

In [13], Kye gives us a criterion by which we can determine whether a given linear transformation is an element of int 𝜋(PSD_𝑛) as follows. Let 𝑣∈ℂ^𝑛 be nonzero, and let 𝑃_𝑣 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 PSD_𝑛, 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 𝒯∈𝜋(PSD_𝑛) is an interior point of 𝜋(PSD_𝑛) if and only if 𝒯(𝑃_𝑣) is nonsingular for all extremals 𝑃_𝑣.

A more common characterization of the interior of a positive cone of linear maps when specialized to 𝜋(PSD_𝑛) gives the following: int𝜋PSD_𝑛=𝒯∈𝜋PSD_𝑛∶𝒯PSD_𝑛⧵{0}⊆PD_𝑛.(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(PSD_𝑛). For nonzero 𝑣∈ℂ^𝑛, we have that tr𝑃_𝑣=𝑣₁𝑣₁+𝑣₂𝑣₂+⋯+𝑣_𝑛𝑣_𝑛𝐼_𝑛,(3.4) which is nonsingular, giving us that the trace map is an element of int 𝜋(PSD_𝑛).

Kye's second example is the linear map tran_𝐾∶𝐴↦𝐾^1/2𝐴^𝑇𝐾^1/2,(3.5) where 𝐾^1/2 is the standard square root of a positive definite matrix (cf. [10]). We know that 𝐴∈PSD_𝑛⇒𝐴^𝑇∈PSD_𝑛. Since 𝐾 is positive definite, 𝐾^1/2 is also positive definite, and Sylvester's law of inertia gives us that 𝐴^𝑇 and 𝐾^1/2𝐴^𝑇𝐾^1/2 have the same inertias, so that 𝐾^1/2𝐴^𝑇𝐾^1/2∈PSD_𝑛. Thus, tran_𝐾∈𝜋(PSD_𝑛) for all positive definite matrices 𝐾. Kye claims, for any 𝐾∈PD_𝑛, tran_𝐾∈int𝜋(PSD_𝑛). Actually, it is not (making it an element of bd 𝜋(PSD_𝑛)). To see this, let 𝐾=𝐼_𝑛, which is positive definite. Then, 𝐾^1/2=𝐼_𝑛. In this case, tran_𝐼𝑛∶𝐴↦𝐼^𝑛1/2𝐴^𝑇𝐼^𝑛1/2=𝐴^𝑇(3.6) for 𝐴∈𝑀_𝑛, that is, tran_𝐼𝑛 is simply the transpose map, an extremal of 𝜋(PSD_𝑛), as we observed at the beginning of this section, and thus not an interior point of 𝜋(PSD_𝑛).

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 CP_𝑛 and 𝜋(PSD_𝑛). We further study this area, which leads us to a still-open question: is a face of CP_𝑛 which lies in the boundary of 𝜋(PSD_𝑛) necessarily also a face of 𝜋(PSD_𝑛)?

We do have the following.

Theorem 4.1. Let 𝐹≤𝐿≤𝐾 be a chain of (sub)cones. If 𝐹⊴𝐾, then 𝐹⊴𝐿.

Proof. Assume that 𝐹⊴𝐾 and that 𝑥,𝑦−𝑥∈𝐿, and 𝑦∈𝐹. Since 𝐿≤𝐾, it follows that 𝑥,𝑦−𝑥∈𝐾. Since 𝐹⊴𝐾, we have that 𝑥∈𝐹. Thus, 𝐹⊴𝐿.

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

Theorem 4.2. Let 𝒯∈CP_𝑛, and suppose that there exist 𝐴₁,…,𝐴_𝑟∈𝑀_𝑛 such that ⟨𝒯⟩=∑^𝑟𝑖=1𝐴_𝑖⊗_𝐾𝐴_𝑖. If 𝑟<𝑛², then 𝒯∈bdCP_𝑛.

Theorem 4.3. Let 𝒯∈𝜋(PSD_𝑛), and suppose that there exist 𝐴₁,…,𝐴_𝑟∈𝑀_𝑛 such that ⟨𝒯⟩=∑^𝑟𝑖=1𝜖_𝑖𝐴_𝑖⊗_𝐾𝐴_𝑖 where 𝜖=±1. If 𝑟<𝑛, then 𝒯∈bd𝜋(PSD_𝑛).

Combining these results gives the following.

Theorem 4.4. Let 𝒯∈CP_𝑛, and suppose that there exist 𝐴₁,…,𝐴_𝑟∈𝑀_𝑛 such that ⟨𝒯⟩=∑^𝑟𝑖=1𝐴_𝑖⊗_𝐾𝐴_𝑖. If 𝑟<𝑛, then 𝒯∈bdCP_𝑛∩bd𝜋(PSD_𝑛).

Proof. Assume 𝑟<𝑛. Since 𝑟<𝑛≤𝑛² for 𝑛=1,2,…,𝒯∈bdCP_𝑛. If we let 𝜖_𝑖=1 for 𝑖=1,2,…,𝑟, then 𝒯∈bd𝜋(PSD_𝑛) and the conclusion is immediate.

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

Theorem 4.5. 𝐹 is a face of both CP_𝑛 and 𝜋(PSD_𝑛) if and only if there exist 𝐵₁,…,𝐵_𝑟∈𝑀_𝑛 such that 𝐹=𝒯∈CP_𝑛∶⟨𝒯⟩=^𝑠_𝑖=1𝐴_𝑖⊗_𝐾𝐴_𝑖𝑤ℎ𝑒𝑟𝑒𝑠<𝑛,andvec𝐴^∗𝑖^∗𝑥=0𝑤ℎ𝑒𝑛𝑒𝑣𝑒𝑟𝑥∈^𝑟_𝑖=1vec𝐵^∗𝑖^⟂(4.1.)

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

Theorem 4.6. Let 𝒯∈CP_𝑛. Then, 𝒯∈intCP_𝑛 if and only if there exist linearly independent matrices 𝐴₁,…,𝐴_𝑛2 such that ⟨𝒯⟩=∑^𝑛2_𝑖=1𝐴_𝑖⊗_𝐾𝐴_𝑖.

Theorem 4.7. Let 𝒯,𝒮∈𝜋(PSD_𝑛). If 𝒯∈Φ_{𝜋(PSD𝑛)}(𝒮), then 𝒯(𝐴)∈Φ_PSD𝑛(𝒮(𝐴)) for every 𝐴∈PSD_𝑛.

Proof. By Corollary 2.10 of [9, page 222], 𝒯∈Φ_{𝜋(PSD𝑛)}(𝒮) if and only if there exists 𝜆>0 such that 𝒮−𝜆𝒯∈𝜋(PSD_𝑛). Therefore, for every 𝐴∈PSD_𝑛, 𝒮(𝐴)−𝜆𝒯(𝐴)∈𝜋(PSD_𝑛). It follows that 𝒯(𝐴)∈Φ_PSD𝑛(𝒮(𝐴)) for every 𝐴∈PSD_𝑛.

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 𝐹⊴𝐿 and 𝐹⊂bd𝐾, then 𝐹⊴𝐾.

Conjecture 4.9. Let 𝐹 be a proper subcone of CP_𝑛. If 𝐹⊴CP_𝑛 and 𝐹⊂bd𝜋(PSD_𝑛), then 𝐹⊴𝜋(PSD_𝑛).

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.