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

Authors: Andrew J. Klimas; Richard D. Hill
Journal: ISRN Algebra
 (2012)

---

## Abstract

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


---

## Body

## 1. Introduction

This paper is the third in a sequence giving the related theory of the coneπ(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 ofπ(PSDn)

Yopp and Hill [2, 4] have characterized the extremals of CPn and coCPn as follows.Theorem 2.1.
Let𝒯:Mn→Mn. Then,(i)
𝒯∈ExtCPn if and only if there exists A∈Mn∖{0} such that 〈𝒯〉=A¯⊗KA;(ii)
if〈𝒯〉=∑i=1r‍A¯i⊗KAi, where A1,…,Ar∈Mn∖{0}, then 𝒯∈ExtCPn if and only if Ai=αiA1 for all i=1,…,r;(iii)
𝒯∈ExtCPn if and only if there exist A1,…,Ar∈Mn∖{0} and (dij)∈ExtPSDr such that 〈𝒯〉=∑i,j=1r‍dijAi¯⊗KAj.Theorem 2.2.
Let𝒯:Mn→Mn be an element of coCPn. Then,(i)
𝒯∈ExtcoCPn if and only if 〈𝒯〉=A¯⊗KA∘T for some A∈Mn∖{0};(ii)
if〈𝒯〉=∑i=1r‍A¯i⊗KAi∘T, where A1,…,Ar∈Mn∖{0}, then 𝒯∈ExtcoCPn if and only if Ai=αiA1 for all i=1,…,r;(iii)
𝒯∈ExtcoCPn if and only if there exist A1,…,Ar∈Mn∖{0} and (dij)∈ExtPSDr such that 〈𝒯〉=∑i,j=1r‍dijAi¯⊗KAj∘T.The remainder of this section focuses on the extremals ofπ(PSDn). The major result of [5] follows.Theorem 2.3.
LetA∈Mn. If 〈𝒯〉=A¯⊗KA or 〈𝒯〉=A¯⊗KA∘T, 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 ofπ(PSDn) is an extremal of π(PSDn).Proof.
LetA 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 toCPn. Let A1=[1002], and let A2=[2001]. Then, the linear transformation 𝒯 with matrix representation
(2.1)〈𝒯〉=∑i=12‍Ai¯⊗KAi=[5000040000400005]
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 subconesCPn and coCPn, we find two more classes of extremals of π(PSDn).Theorem 2.5.
Every extremal ofCPn and of coCPn is also an extremal of π(PSDn).Proof.
Assume that𝒯:Mn→Mn is an extremal of CPn. By [2, Theorem 3.4(i)], 𝒯 is an extremal of CPn if and only if there exists A∈Mn such that 〈𝒯〉=A¯⊗KA. 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 ofCPn 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.
Let𝒯:Mn→Mn. Then,(i)
if〈𝒯〉=∑i=1r‍Ai¯⊗KAi or 〈𝒯〉=∑i=1r‍Ai¯⊗KAi∘T, where A1,…,Ar∈Mn∖{0}, and if Ai=αiA1 for all i=1,…,r, then 𝒯∈Extπ(PSDn);(ii)
if there existA1,…,Ar∈Mn∖{0} and (dij)∈ExtPSDr such that 〈𝒯〉=∑i=1r‍dijAi¯⊗KAj or 〈𝒯〉=∑i=1r‍dijAi¯⊗KAj∘T, then 𝒯∈Extπ(PSDn).


## 3.   Other Faces ofCPn and π(PSDn)

It is well known that the transpose mapT is an element of π(PSDn) but not CPn. Since 〈T〉=In⊗KIn∘T, it follows that T∈Ext(π(PSDn)). Thus, F=Φ(T)⊴π(PSDn) but F⋬CPn. 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
(3.1)K1={𝒯∈HPn:A*𝒯(A)∈PSDn∀A∈Mn(ℂ)}
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.
F
⊴
CP
n if and only if there exist B1,…,Br∈Mn such that
(3.2)F={𝒯∈CPn:〈𝒯〉=∑i=1s‍Ai¯⊗KAiwherevec(Ai*)*x=0wheneverx∈⋂i=1r‍vec(Bi*)⊥}.One way in which we could show that a faceF 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.
IfF⊴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(PSDn) as follows. Let v∈ℂn 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 map𝒯∈π(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 toπ(PSDn) gives the following:
(3.3)intπ(PSDn)={𝒯∈π(PSDn):𝒯(PSDn∖{0})⊆PDn}.
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 ofπ(PSDn). For nonzero v∈ℂn, we have that
(3.4)tr(Pv)=(v¯1v1+v¯2v2+⋯+v¯nvn)In,
which is nonsingular, giving us that the trace map is an element of int π(PSDn).Kye's second example is the linear map(3.5)tranK:A↦K1/2ATK1/2,
where K1/2 is the standard square root of a positive definite matrix (cf. [10]). We know that A∈PSDn⇒AT∈PSDn. 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 K1/2ATK1/2∈PSDn. Thus, tranK∈π(PSDn) for all positive definite matrices K. Kye claims, for any K∈PDn, tranK∈intπ(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,
(3.6)tranIn:A↦In1/2ATIn1/2=AT
for A∈Mn, 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 Section3, 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.
LetF≤L≤K be a chain of (sub)cones. If F⊴K, then F⊴L.Proof.
Assume thatF⊴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 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.
Let𝒯∈CPn, and suppose that there exist A1,…,Ar∈Mn such that 〈𝒯〉=∑i=1r‍Ai¯⊗KAi. If r<n2, then 𝒯∈bdCPn.Theorem 4.3.
Let𝒯∈π(PSDn), and suppose that there exist A1,…,Ar∈Mn such that 〈𝒯〉=∑i=1r‍ϵiAi¯⊗KAi where ϵ=±1. If r<n, then 𝒯∈bdπ(PSDn).Combining these results gives the following.Theorem 4.4.
Let𝒯∈CPn, and suppose that there exist A1,…,Ar∈Mn such that 〈𝒯〉=∑i=1r‍Ai¯⊗KAi. If r<n, then 𝒯∈bdCPn∩bdπ(PSDn).Proof.
Assumer<n. Since r<n≤n2 for n=1,2,…,𝒯∈bdCPn. If we let ϵi=1 for i=1,2,…,r, then 𝒯∈bdπ(PSDn) and the conclusion is immediate.Finally, combining Theorems3.1 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,…,Br∈Mn such that
(4.1)F={𝒯∈CPn:〈𝒯〉=∑i=1s‍Ai¯⊗KAiwheres<n,andvec(Ai*)*x=0wheneverx∈⋂i=1rvec(Bi*)⊥}.Results by Yopp and Hill [2] finish the known material of this section.Theorem 4.6.
Let𝒯∈CPn. Then, 𝒯∈intCPn if and only if there exist linearly independent matrices A1,…,An2 such that 〈𝒯〉=∑i=1n2‍Ai¯⊗KAi.Theorem 4.7.
Let𝒯,𝒮∈π(PSDn). If 𝒯∈Φπ(PSDn)(𝒮), then 𝒯(A)∈ΦPSDn(𝒮(A)) for every A∈PSDn.Proof.
By Corollary 2.10 of [9, page 222], 𝒯∈Φπ(PSDn)(𝒮) if and only if there exists λ>0 such that 𝒮-λ𝒯∈π(PSDn). Therefore, for every A∈PSDn, 𝒮(A)-λ𝒯(A)∈π(PSDn). It follows that 𝒯(A)∈ΦPSDn(𝒮(A)) for every A∈PSDn.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.
LetK, L, and F be cones such that FLK. If F⊴L and F⊂bdK, then F⊴K.Conjecture 4.9.
LetF be a proper subcone of CPn. If F⊴CPn and F⊂bdπ(PSDn), then F⊴π(PSDn).



---

*Source: 704960-2012-07-02.xml*