Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 23, pp. 356-374, April 2012
ELA
http://math.technion.ac.il/iic/ela
ADDITIVE PRESERVERS OF TENSOR PRODUCT OF
RANK ONE HERMITIAN MATRICES∗
MING-HUAT LIM†
Abstract. Let K be a field of characteristic not two or three with an involution and F be
its fixed field. Let Hm be the F -vector space of all m-square Hermitian matrices over K. Let
Nk
ρm denote the set of all rank-one matrices in Hm . In the tensor product space
i=1 Hmi , let
Nk
Nk
i=1 Ai such that Ai ∈ ρmi , i = 1, . . . , k.
i=1 ρmi denote the set of all decomposable elements
In this paper, additive maps T from Hm ⊗ Hn to Hs ⊗ Ht such that T (ρm ⊗ ρn ) ⊆ (ρs ⊗ ρt ) ∪ {0}
are characterized. From this, a characterization of linear maps is found between tensor products of
two real vector spaces of complex Hermitian matrices that send separable pure states to separable
Nk
pure states. Also classified in this paper are almost surjective additive maps L from
i=1 Hmi to
Nl
Nl
k
i=1 ρni where 2 ≤ k ≤ l. When K is algebraically closed
i=1 Hni such that L ⊗i=1 ρmi ⊆
Nk
Nk
and K = F , it is shown that every linear map on
i=1 ρmi is induced by
i=1 Hmi that preserves
k bijective linear rank-one preservers on Hmi , i = 1, . . . , k.
Key words. Hermitian matrix, Rank-one preserver, Rank-one non-increasing map, Tensor
product, Additive map, Separable pure state.
AMS subject classifications. 15A03, 15A04, 15A69.
1. Introduction. A map from a vector space of matrices over a field to another
is called a rank-one preserver if it sends rank-one matrices to rank-one matrices and
is called rank-one non-increasing if it sends rank-one matrices to matrices of rank less
than or equal to one. Many authors have studied the structures of linear and additive
rank-one preservers and rank-one non-increasing maps between spaces of Hermitian
matrices [2], [3], [7]–[11], [16]–[18]. This paper is concerned with linear and additive
maps on tensor products of spaces of Hermitian matrices that carry the set of tensor
product of rank-one matrices into itself.
¯ = a for any a ∈ K. Let
Let K be a field with an automorphism − such that ā
F = {a ∈ K : ā = a} be its fixed field. For each positive integer m ≥ 2, let Hm
denote the F -vector space of all m-square Hermitian matrices over K. Consider the
Nk
tensor product space i=1 Hmi over F . When the automorphism − of K is of order
Nk
product of
2,
i=1 Hmi can be identified naturally with Hm1 m2 ...mk via kronecker
Nk
matrices. Similarly, when the automorphism − of K is the identity,
i=1 Hmi can
∗ Received
by the editors on June 20, 2011. Accepted for publication on March 10, 2012. Handling
Editor: Shmuel Friedland.
† Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia
([email protected]). This research was supported by a Research Grant of the University of Malaya.
356
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 23, pp. 356-374, April 2012
ELA
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Additive Preservers of Tensor Product of Rank One Hermitian Matrices
357
be identified naturally with a proper subspace of Hm1 m2 ...mk . Let ρm be the set of
Nk
all rank-one matrices in Hm and let
i=1 ρmi denote the set of all decomposable
Nk
,
i
= 1, . . . , k. We first characterize additive
elements
A
such
that
A
∈
ρ
i
mi
i=1 i
maps ϕ from Hm ⊗ Hn to Hs ⊗ Ht such that ϕ(ρm ⊗ ρn ) ⊆ (ρs ⊗ ρt ) ∪ {0} where F
has characteristic not two or three by using the structure of additive maps between
tensor products of two vector spaces that preserve decomposable elements and also
rank-one non-increasing additive maps between spaces of Hermitian matrices. An
additive mapping η from a vector space to another vector space V is said to be
almost surjective if V is linearly spanned by Imη. In Section 3, we show that if
char F 6= 2 and (K, F ) 6= (3, 3), and
N surjective additive map from
Nθ is an almost
Nl
Nk
l
k
i=1 ρni where 2 ≤ k ≤ l, then
i=1 ρmi ⊆
i=1 Hni such that θ
i=1 Hmi to
k = l, θ is quasilinear and induced by k quasilinear rank-one preservers between
spaces of Hermitian matrices with respect to the same endomorphism of F . When
K is algebraically closed and K = F , we show in Section 4 that every linear map on
Nk
Nk
i=1 ρmi is induced by k bijective linear rank-one preservers
i=1 Hmi that preserves
on Hmi , i = 1, . . . , k.
Suppose now that K is the complex field and F is the real field. Then Hm is the
real vector space of complex Hermitian matrices. A positive semi-definite matrix of
Nk
trace one in Hm is called a density matrix. A mixed state in
i=1 Hmi , k ≥ 2, is
called separable if it is a convex combination of elements of the form A1 ⊗ · · · ⊗ Ak
where Ai is a density matrix in Hmi , i = 1, . . . , k. In [5], Friedland et al. determined
Nk
the group G(m1 , . . . , mk ) of all bijective linear maps on i=1 Hmi that leave invariant
the set S(m1 , . . . , mk ) of all separable states. The bipartite case of this result was first
obtained in [1] by a different method. Let Pm denote the set of all rank-one matrices
Nk
in Hm with trace 1. Let
i=1 Pmi be the set of all separable pure states, i.e., all
decomposable elements of the form B1 ⊗ · · · ⊗ Bk where Bi ∈ Pmi , i = 1, . . . , k.
Nk
Since
, . . . , mk ), it follows that
i=1 Pmi is the set of all extreme points of S(m
Nk 1
G(m1 , . . . , mk ) is the set of all bijective linear maps on i=1 Hmi that leave invariant
Nk
the set i=1 Pmi (see [1], [5]). Thus, our work concerning linear and additive maps
N
N
on ki=1 Hmi that preserve ki=1 ρmi is a generalization the above mentioned results.
Nk
We remark that Westwick [19] showed that every linear map on i=1 C mi that
preserves the product states, i.e., the non-zero decomposable elements, is induced by
k bijective linear maps on C mi , i = 1, . . . , k. Here C is the complex field. The case
for k = 2 was proved in [14]. Additive preservers of non-zero decomposable tensors
have been studied by several authors [6, 12, 13, 15, 20].
2. Linear preservers of bipartite separable pure states. For each positive
integer m, let Hm denote the F -vector space of all m-square Hermitian matrices over
K and K m be the set of all m × 1 column vectors over K. For each m × n matrix
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
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358
M.H. Lim
A = (aij ) over K and each endomorphism σ of K, let Aσ = (σ(aij )) and A∗ denote
the transpose of the matrix (a¯ij ).
Throughout this paper, all Hermitian matrices we consider are of size at least 2.
In this section, we study additive map from Hm ⊗ Hn to Hs ⊗ Ht such that
ϕ(ρm ⊗ ρn ) ⊆ (ρs ⊗ ρt ) ∪ {0}.
We need the following result proved in [7, 10] for the spaces of symmetric matrices
and in [16] for the spaces of Hermitian matrices where K 6= F .
Lemma 2.1. Let η : Hm → Hn be a rank-one non-increasing additive map. If
(K, F ) ∈
/ {(4, 2), (3, 3), (2, 2)}, then one of the following holds:
(i) η(X) = ζ(X)B for some additive functional ζ : Hm → F and some rank-one
matrix B ∈ Hn ,
(ii) η(X) = λP X σ P ∗ for some n × m matrix P over K, some nonzero endomorphism σ of K commuting with the automorphism −, and some nonzero
λ ∈ F.
An additive subgroup of the tensor product of several vector spaces over a field
is called decomposable (pure) if it consists of decomposable (pure) elements.
Lemma 2.2. Suppose that F is of the characteristic not two or three. Let ϕ be an
additive map from Hm ⊗ Hn to Hs ⊗ Ht such that ϕ(ρm ⊗ ρn ) ⊆ (ρs ⊗ ρt ) ∪ {0}. Let
A be a fixed nonzero element in Hm . Suppose that for each B ∈ ρn , either ϕ(A ⊗ B)
is the zero element or a non-zero decomposable element of the form fB ⊗ gB where
either fB ∈ ρs or gB ∈ ρt . Then ϕ(A ⊗ Hn ) is a decomposable subgroup of Hs ⊗ Ht .
Proof. Since ϕ is additive and the additive span of ρn is Hn , it suffices to show
that ϕ(A ⊗ ρn ) is contained in a decomposable subgroup of Hs ⊗ Ht . Suppose the
contrary. Then there nonzero vectors u, v ∈ K n and nonzero scalars α, β ∈ F such
that
ϕ(A ⊗ αuu∗ ) = U and ϕ(A ⊗ βvv ∗ ) = V
where U − V is not decomposable. Since 4α = (1 + α)2 − (1 − α)2 and ϕ is additive,
it follows that there exists δ ∈ {1 + α, 1 − α} such that ϕ(A ⊗ δ 2 uu∗ ) 6= 0. Let y = δu.
Since the map θ : Hm → Hst defined by θ(X) = ϕ(X ⊗ uu∗ ) is an additive rank-one
non-increasing map, it follows from Lemma 2.1 that ϕ(A⊗yy ∗ ) ∈ hU i\{0}. Similarly,
ϕ(A ⊗ zz ∗ ) ∈ hV i \ {0} for some z ∈ hvi \ {0}. Hence,
ϕ(A ⊗ yy ∗ ) = f1 ⊗ g1 , ϕ(A ⊗ zz ∗ ) = f2 ⊗ g2
for some linearly independent vectors f1 , f2 ∈ Hs and some linearly independent
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
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Additive Preservers of Tensor Product of Rank One Hermitian Matrices
359
vectors g1 , g2 ∈ Ht . Let
ϕ(A ⊗ (y + z)(y + z)∗ ) = h ⊗ k.
Let G be the prime subfield of K. Since for any λ ∈ G \ {0, 1},
(2.1)
=
=
ϕ(A ⊗ (y + λz)(y + λz)∗ )
(1 − λ)f1 ⊗ g1 + (λ2 − λ)f2 ⊗ g2 + λh ⊗ k
hλ ⊗ kλ ,
where either hλ or kλ has rank less than 2. It follows that
(1 − λ)f1 ⊗ g1 + (λ2 − λ)f2 ⊗ g2 = −λh ⊗ k + hλ ⊗ kλ .
Since the left hand side is a rank 2 tensor in Hs ⊗ Ht , it follows that
hf1 , f2 i = hh, hλ i, hg1 , g2 i = hk, kλ i.
This shows that
h ⊗ k = af1 ⊗ g1 + bf1 ⊗ g2 + cf2 ⊗ g1 + df2 ⊗ g2
for some a, b, c, d in F . From (2.1), we have for any λ ∈ G \ {0, 1},
=
ϕ(A ⊗ (y + λz)(y + λz)∗ )
((a − 1)λ + 1)f1 ⊗ g1 + bλf1 ⊗ g2 + cλf2 ⊗ g1 + (λd − λ + λ2 )f2 ⊗ g2 .
Hence,
λ(a − 1) + 1
λb
cλ
λd − λ + λ2
= (d − 1)λ + ((a − 1)(d − 1) − bc + 1)λ2 + (a − 1)λ3 = 0
for any λ ∈ G \ {0, 1}. Since |G| ≥ 5, it follows that d = a = 1, bc = 1. Hence,
=
=
=
ϕ(A ⊗ (y + λz)(y + λz)∗ )
f1 ⊗ g1 + bλf1 ⊗ g2 + cλf2 ⊗ g1 + λ2 f2 ⊗ g2
(f1 + cλf2 ) ⊗ (g1 + bλg2 )
hλ ⊗ kλ .
Since both f1 , f2 and g1 , g2 are linearly independent Hermitian matrices, either
f1 or g1 is of rank-one, either g1 or g2 is of rank-one, and |G| ≥ 5, it is not hard to
see that there exists λ ∈ G \ {0, 1} such that both f1 + cλf2 and g1 + bλg2 are of rank
at least 2. This is a contradiction since one of hλ , kλ has rank less than 2. Hence,
ϕ(A ⊗ Hn ) is a decomposable subgroup of Hs ⊗ Ht .
The following result was essentially proved in [6] (see also [4, 12]).
Lemma 2.3. Let U1 , U2 , V1 , V2 be vectors spaces over a field. Let θ be an additive
map from U1 ⊗ U2 to V1 ⊗ V2 that sends decomposable elements to decomposable
elements. Then one of the following holds:
Electronic Journal of Linear Algebra ISSN 1081-3810
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M.H. Lim
1. Im θ is a decomposable subgroup of V1 ⊗ V2 ,
2. There exist a permutation π on {1, 2} and quasilinear maps θi : Uπ(i) → Vi ,
i = 1, 2, with respect to a endomorphism σ on the underlying field such that
θ(u1 ⊗ u2 ) = θ1 (uπ(1) ) ⊗ θ2 (uπ(2) )
for any u1 ∈ U1 , u2 ∈ U2 .
Theorem 2.4. Suppose that F is of the characteristic not two or three. Let ϕ
be an additive map from Hm ⊗ Hn to Hs ⊗ Ht . Then ϕ(ρm ⊗ ρn ) ⊆ (ρs ⊗ ρt ) ∪ {0} if
and only if one of the following holds:
(i) ϕ(Z) = C ⊗ψ(Z) for some C ∈ ρs and some additive map ψ : Hm ⊗Hn → Ht
such that ψ(ρm ⊗ ρn ) ⊆ ρt ∪ {0},
(ii) ϕ(Z) = π(Z)⊗D for some D ∈ ρt and some additive map π : Hm ⊗Hn → Hs
such that π(ρm ⊗ ρn ) ⊆ ρs ∪ {0},
(iii) ϕ(A ⊗ B) = ϕ1 (A) ⊗ ϕ2 (B), A ∈ Hm , B ∈ Hn , for some rank-one nonincreasing additive maps ϕ1 : Hm → Hs and ϕ2 : Hn → Ht ,
(iv) ϕ(A ⊗ B) = ϕ2 (B) ⊗ ϕ1 (A), A ∈ Hm , B ∈ Hn , for some rank-one nonincreasing additive maps ϕ1 : Hm → Ht and ϕ2 : Hn → Hs .
Proof. The sufficiency part is clear. We now consider the necessity part. Let
A ∈ ρm . Then for any B ∈ ρn , ϕ(A ⊗ B) ∈ (ρs ⊗ ρt ) ∪ {0}. Hence, it follows from
Lemma 2.2 that ϕ(A ⊗ Hn ) is a decomposable subgroup of Hs ⊗ Ht . Hence,
ϕ(A ⊗ Hn ) ⊆ A1 ⊗ Ht for some A1 ∈ ρs ,
or
ϕ(A ⊗ Hn ) ⊆ Hs ⊗ A2 for some A2 ∈ ρt .
Let C ∈ Hn be any nonzero matrix. Since for any A ∈ ρm ,
ϕ(A ⊗ C) = 0 or ϕ(A ⊗ C) = fA ⊗ gA
where either fA ∈ ρs or gA ∈ ρt , it follows from Lemma 2.2 that ϕ(Hm ⊗ C) is
a decomposable subgroup of Hs ⊗ Ht . Hence, ϕ sends decomposable elements to
decomposable elements. The result now follows from Lemma 2.3 and the hypothesis
that ϕ(ρm ⊗ ρn ) ⊆ (ρs ⊗ ρt ) ∪ {0}.
Remark 2.5.
(a) Theorem 2.4 is true for any field F with at least 5 elements if we replace
“additive” by “linear”.
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(b) The following result follows from Lemma 2.1. Let η : Hm → Hn be a rank-one
non-increasing linear map where (K, F ) ∈
/ {(4, 2), (3, 3), (2, 2)}. Then one of
the following holds:
(i) η(X) = ζ(X)B for some linear functional ζ : Hm → F and some rankone matrix B ∈ Hn ,
(ii) there exist an n × m matrix P over K and a nonzero λ ∈ F such that η
has the form A 7→ λP AP ∗ or A 7→ λP At P ∗ .
For the rest of this section, Hi is the real vector space of i-square complex Hermitian matrices. Let Pi denote the set of all rank-one matrices in Hi with trace 1.
Then Pm ⊗ Pn is the set of all bipartite separable pure states in Hm ⊗ Hn .
Lemma 2.6. [5] Let η : Hm → Hn be a linear map such that η(Pm ) ⊆ Pn . Then
one of the following holds:
(i) η(X) = (Tr X)R for some R ∈ Pn ,
(ii) m ≤ n, there exists an n × m complex matrix U with U ∗ U = Im such that η
has the form A 7→ U AU ∗ or A 7→ U At U ∗ .
From Theorem 2.4, we obtain following characterization of linear maps that send
bipartite separable pure states to bipartite separable pure states.
Corollary 2.7. Let ϕ be a linear map from Hm ⊗ Hn to Hs ⊗ Ht . Then
ϕ(Pm ⊗ Pn ) ⊆ Ps ⊗ Pt if and only if one of the following holds:
(i) ϕ(Z) = C ⊗ ψ(Z) for some C ∈ Ps and some linear map ψ : Hm ⊗ Hn → Ht
such that ψ(Pm ⊗ Pn ) ⊆ Pt ,
(ii) ϕ(Z) = π(Z) ⊗ D for some D ∈ Pt and some linear map π : Hm ⊗ Hn → Hs
such that π(Pm ⊗ Pn ) ⊆ Ps ,
(iii) ϕ(A ⊗ B) = ϕ1 (A) ⊗ ϕ2 (B) for A ∈ Hm , B ∈ Hn where ϕ1 : Hm → Hs is a
linear map such that ϕ1 (Pm ) ⊆ Ps , and ϕ2 : Hn → Ht is a linear map such
that ϕ2 (Pn ) ⊆ Pt ,
(iv) ϕ(A ⊗ B) = ϕ2 (B) ⊗ ϕ1 (A) for A ∈ Hm , B ∈ Hn where ϕ1 : Hm → Ht is a
linear map such that ϕ1 (Pm ) ⊆ Pt , and ϕ2 : Hn → Hs is a linear map such
that ϕ2 (Pn ) ⊆ Ps .
Characterization of linear maps ϕ on Hm ⊗ Hn such that ϕ(Pm ⊗ Pn ) = Pm ⊗ Pn
was obtained in [1]. The following follows immediately from Corollary 2.7 and Lemma
2.6.
Corollary 2.8. Let ϕ be a linear map from Hm ⊗ Hn to Hs ⊗ Ht such that
ϕ(Pm ⊗ Pn ) ⊆ Ps ⊗ Pt and the rank of ϕ is larger than max {dim Hs , dim Ht }. Then
one of the following holds:
(i) m ≤ s, n ≤ t, ϕ(A ⊗ B) = ϕ1 (A) ⊗ ϕ2 (B) for A ∈ Hm , B ∈ Hn , where
Electronic Journal of Linear Algebra ISSN 1081-3810
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M.H. Lim
ϕ1 : Hm → Hs has the form A 7→ U AU ∗ or A 7→ U At U ∗ with U ∗ U = Im and
ϕ2 : Hn → Ht has the form B 7→ V BV ∗ or B 7→ V B t V ∗ with V ∗ V = In ,
(ii) m ≤ t, n ≤ s, ϕ(A ⊗ B) = ϕ2 (B) ⊗ ϕ1 (A) for A ∈ Hm , B ∈ Hn , where
ϕ1 : Hm → Ht has the form A 7→ U AU ∗ or A 7→ U At U ∗ with U ∗ U = Im and
ϕ2 : Hn → Hs has the form B 7→ V BV ∗ or B 7→ V B t V ∗ with V ∗ V = In .
Remark 2.9. The corresponding results of Corollaries 2.7 and 2.8 are true for
the spaces of real symmetric matrices.
3. Extension to multipartite systems. In this section, we shall study alNk
Nl
Nk
most surjective additive maps from i=1 Hmi to i=1 Hni that send
i=1 ρmi to
Nl
i=1 ρni where 2 ≤ k ≤ l and mi , nj ≥ 2, i = 1, . . . , k, j = 1, . . . , l.
Let X = x1 ⊗ · · · ⊗ xm and Y = y1 ⊗ · · · ⊗ ym be two nonzero decomposable
Nm
elements in the tensor product
i=1 Wi of m vector spaces W1 , . . . , Wm over F .
Then each xi is called a factor of X. If xi and yi are linearly independent for at most
one i, then we write X ∼ Y . It is known that X − Y is decomposable if and only if
X ∼ Y . If X ∼ Y and X, Y are linearly independent, then we say that X and Y are
Nm
adjacent. It is easily seen that an additive subgroup S of i=1 Wi is decomposable
if and only if X ∼ Y for any non-zero elements X, Y ∈ S, and every decomposable
N
subgroup of m
i=1 Wi is of the form
(3.1)
w1 ⊗ · · · ⊗ wi−1 ⊗ G ⊗ wi+1 ⊗ · · · ⊗ wm
for some i, some nonzero vectors wj ∈ Wj , j 6= i and some subgroup G of Wi . The
nonzero vectors w1 , . . . , wi−1 , wi+1 , . . . , wm are called factors of the decomposable
N
subgroup. A decomposable subgroup of m
i=1 Wi is said to be of type-i if it is of the
form (3.1). If G = Wi , then it is called a maximal decomposable subgroup of type-i.
A rank-one non-increasing map is said to be degenerate if its image consists of
matrices of rank less than or equal to one.
For the following five Lemmas, we assume that char F 6= 2 and (K, F ) 6= (3, 3).
Nk
Nl
Lemma 3.1. Let θ be an additive map from
i=1 Hmi to
i=1 Hni such that
N
Nk
Nl
k
θ
i=1 ρmi and A ∼ B, then θ(A) ∼ θ(B).
i=1 ρni . If A, B ∈
i=1 ρmi ⊆
Proof. Without loss of generality, we may assume that
A = A1 ⊗ · · · ⊗ Ak−1 ⊗ αuu∗ , B = A1 ⊗ · · · ⊗ Ak−1 ⊗ βvv ∗
Nk−1
for some A1 ⊗ · · · ⊗ Ak−1 in i=1 ρmi , some nonzero vectors u, v ∈ K mk and some
nonzero scalars α, β ∈ F . Suppose the contrary that θ(A)−θ(B) is not decomposable.
Using the same argument as in the first paragraph of the proof of Lemma 2.2, we see
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that there exist
C = A1 ⊗ · · · ⊗ Ak−1 ⊗ yy ∗ where y ∈ hui \ {0}
and
D = A1 ⊗ · · · ⊗ Ak−1 ⊗ zz ∗ where z ∈ hvi \ {0}
such that θ(C) − θ(D) is not decomposable. We have
θ(C) = αu1 u∗1 ⊗ · · · ⊗ ul u∗l , θ(D) = bv1 v1∗ ⊗ · · · ⊗ vl vl∗
for some nonzero vectors ui , vi ∈ K ni , i = 1, . . . , l, some nonzero a, b ∈ F . Since
θ(C) − θ(D) is not decomposable, we may assume without loss of generality that
both u1 u∗1 , v1 v1∗ and u2 u∗2 , v2 v2∗ are linearly independent. Let π be the natural
Nl
additive map from Hmk to i=1 Hni that is induced by the restriction map of θ to
A1 ⊗ · · · ⊗ Ak−1 ⊗ Hmk . Then π is a rank-one preserver from Hmk to Hn1 ...nl which
is non-degenerate. By Lemma 2.1, there exist an n × mk matrix U over K where
n = n1 . . . nl , a non-zero endomorphism σ of K commuting with − and a nonzero
scalar c ∈ F such that
π(X) = cU X σ U ∗
for all X ∈ Hmk . Let
H = A1 ⊗ · · · ⊗ Ak−1 ⊗ (y + z)(y + z)∗ .
Then θ(H) = dw1 w1∗ ⊗ · · · ⊗ wl wl∗ for some nonzero d ∈ F and for some nonzero
vectors wi ∈ K ni , i = 1, . . . , l. Since
θ(C) = cU (yy ∗ )σ U ∗ , θ(D) = cU (zz ∗)σ U ∗
and
θ(H) = cU ((y + z)(y + z)∗ )σ U ∗ ,
it follows that
hU y σ i = hu1 ⊗ · · · ⊗ ul i, hU z σ i = hv1 ⊗ · · · ⊗ vl i
and
hU (y + z)σ i = hw1 ⊗ · · · ⊗ wl i.
Hence,
w1 ⊗ · · · ⊗ wl = λu1 ⊗ · · · ⊗ ul + µv1 ⊗ · · · ⊗ vl
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M.H. Lim
for some nonzero scalars λ, µ ∈ F . This yields a contradiction since both u1 , v1 and
u2 , v2 are linearly independent vectors. Hence, θ(A) ∼ θ(B).
Nk
Lemma 3.2. Letθ be an almost
surjective additive map from
i=1 Hmi to
Nl
Nl
Nk
⊆
i=1 Hni such that θ
i=1 ρni . Let M be any maximal decomposi=1 ρmi
Nk
Then θ(M ) is a
able subgroup of
i whose factors are rank-one matrices.
i=1 HmN
l
decomposable subgroup of
i=1 Hni whose factors are rank-one matrices such that
dimhθ(M )i ≥ 2.
Nl
Proof. By Lemma 3.1, we see that θ(M ) is a decomposable subgroup of i=1 Hni
whose factors are rank-one matrices. Suppose that
θ(M ) ⊆ hJ1 ⊗ · · · ⊗ Jl i
for some Ji ∈ ρni , i = 1, . . . , l. Without loss of generality, we may assume that M =
Nk−1
A1 ⊗ · · · ⊗ Ak−1 ⊗ Hmk for some A1 ⊗ · · · ⊗ Ak−1 in i=1
ρmi . Let Ω := X1 ⊗ · · · ⊗ Xk
Nk
.
Then
there
exists
a
chain
of elements Ω1 , . . . , Ωs in
be any element of
ρ
m
i
i=1
Nk
such
that
Ω
=
A
⊗
·
·
·⊗
A
⊗
X
,
Ω
=
Ω
and
Ω1 ∼ Ω2 , . . . , Ωs−1 ∼ Ωs .
ρ
1
1
k−1
k
s
i=1 mi
In view of Lemma 3.1 and the hypothesis that k ≤ l, we see that Ji is a factor of
T (Ω) for some i. Let
V = J1 ⊗ Hn2 ⊗ · · · ⊗ Hnl + Hn1 ⊗ J2 ⊗ Hn3 ⊗ · · · ⊗ Hnl + · · · + Hn1 ⊗ · · · ⊗ Hnl−1 ⊗ Jl
N
k
Then θ
i=1 ρmi ⊆ V and this implies that Im(θ) ⊆ V . Any decomposable element
of the form D1 ⊗ · · · ⊗ Dk where Di , Ji are linearly independent does not belong to
V and hence θ is not almost surjective, a contradiction. Hence, dimhθ(M )i ≥ 2.
Nk
Lemma 3.3. Let θ be an almost
surjective additive map from
i=1 Hmi to
Nl
Nl
Nk
such
that
θ
H
where
k
≤
l.
If
M
and
M
⊆
ρ
ρ
ni
1
2 are maxi=1
i=1 ni
i=1 mi
Nk
H , then θ(M1 ) and θ(M2 )
imal decomposable subgroups of the same type in
N i=1 mi
are decomposable subgroups of the same type in li=1 Hni .
Proof. Without loss of generality, we may assume that
M1 = B ⊗ Hmk and M2 = C ⊗ Hmk
Nk−1
for some B, C ∈ i=1 ρmi . Suppose that k ≥ 3. Since there is a chain of decomposNk−1
able elements N1 , . . . , Nt in i=1 ρmi such that B ∼ N1 , . . . , Ni ∼ Ni+1 , . . . , Nt ∼ C,
we may assume without loss of generality that B ∼ C. We may also assume that
B = A ⊗ Y, C = A ⊗ Z, A ∈
k−2
O
ρmi , Y, Z ∈ ρmk−1 .
i=1
Suppose that θ(M1 ) and θ(M2 ) are decomposable subgroups of different types. We
may assume that
θ(M1 ) ⊆ J1 ⊗ Hn2 ⊗ D and θ(M2 ) ⊆ Hn1 ⊗ J2 ⊗ E
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for some J1 ∈ ρn1 , J2 ∈ ρn2 and for some D, E ∈
Nl
i=3
365
ρni . By Lemma 3.2,
dimhθ(Mi )i ≥ 2, i = 1, 2.
Hence, there exist vectors K1 , K2 ∈ ρmk such that
θ(B ⊗ K1 ) = J1 ⊗ W2 ⊗ D and θ(C ⊗ K2 ) = W1 ⊗ J2 ⊗ E,
where Ji , Wi are linearly independent, i = 1, 2. Since
C ⊗ K1 ∼ B ⊗ K1 and C ⊗ K1 ∈ M2 ,
it follows from Lemma 3.1 that D, E are linearly dependent. Let D = J3 ⊗ · · · ⊗ Jl .
For any X ∈ ρmk , let MX = A ⊗ Hmk−1 ⊗ X. Since
θ(MX ) ∩ θ(Mi ) 6= {0}, i = 1, 2,
and θ(MX ) is a decomposable subgroup, it follows that
(3.2)
θ(MX ) ⊆ J1 ⊗ · · · ⊗ Ji−1 ⊗ Hni ⊗ Ji+1 ⊗ · · · ⊗ Jl
Nk
for some i = 1, . . . , l. Let Y = Y1 ⊗· · ·⊗Yk ∈ i=1 ρmi . Since there exist Z1 , . . . , Zs ∈
Nk
i=1 ρmi for some positive integer s ≤ k − 2 such that
Y ∼ Z1 , . . . , Zi ∼ Zi+1 , . . . , Zs ∼ A ⊗ Yk−1 ⊗ Yk ,
it follows that Y and A ⊗ Yk−1 ⊗ Yk have at least 2 common factors. Since
A ⊗ Yk−1 ⊗ Yk ∈ MYk
it follows from that (3.2) that Ji is a factor of θ(Y ) for some i. This shows that θ
is not almost surjective, a contradiction. Hence, θ(M1 ) and θ(M2 ) are decomposable
subgroups of the same type. If k = 2, we see from the above proof that the result is
also true.
For each positive integer k, let [k] denote the set {1, . . . , k}.
Nk
Let θ be an almost
surjective additive map from
i=1 Hmi to
Nl
Nl
Nk
⊆
i=1 Hni such that θ
i=1 ρni where k ≤ l. Then k = l and θ
i=1 ρmi
Nk
sends maximal decomposable subgroups of distinct types in
i=1 Hmi to decomposNl
able subgroups of distinct types in i=1 Hni .
Lemma 3.4.
Proof. Let J be the subset of [l] consisting of all positive integers j such that θ(M )
Nk
is of type-j for some maximal decomposable subgroup M of
i=1 Hmi . Without
loss of generality, we may assume that J = {1, . . . , t}. Suppose that t < l. Let
N
X1 ⊗ · · · ⊗ Xk ∈ ki=1 ρmi and
θ(X1 ⊗ · · · ⊗ Xk ) = Y1 ⊗ · · · ⊗ Yl .
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Consider any decomposable element A1 ⊗ · · · ⊗ Ak ∈ ρmi . In view of Lemma 3.3, we
have
ξ(Hm1 ⊗ X2 ⊗ · · · ⊗ Xk ) is of type ≤ t.
Since
A1 ⊗ X2 ⊗ · · · ⊗ Xk ∈ Hm1 ⊗ X2 ⊗ · · · ⊗ Xk ,
it follows that
(3.3)
where W =
θ(A1 ⊗ X2 ⊗ · · · ⊗ Xk ) ∈ W ⊗ Yt+1 ⊗ · · · ⊗ Yl
Nt
i=1
Hni . Similarly since
A1 ⊗ A2 ⊗ X3 ⊗ · · · ⊗ Xk ∈ A1 ⊗ Hm2 ⊗ X3 ⊗ · · · ⊗ Xk ,
and
θ(A1 ⊗ Hm2 ⊗ X3 ⊗ · · · ⊗ Xk ) is of type ≤ t,
it follows from (3.3) that
ξ(A1 ⊗ A2 ⊗ X3 ⊗ · · · ⊗ Xk ) ∈ W ⊗ Yt+1 ⊗ · · · ⊗ Yl .
Continuing the process, we see that
ξ(A1 ⊗ · · · ⊗ Ak ) ∈ W ⊗ Yt+1 ⊗ · · · ⊗ Yl .
This is a contradiction since θ is an almost surjective. Hence, t = l. This shows
that k = l. Hence, θ sends maximal decomposable subgroups of distinct types in
Nl
Nk
i=1 Hni .
i=1 Hmi to decomposable subgroups of the distinct types in
Lemma 3.5. Let ϕ1 , ϕ2 be two non-degenerate additive rank-one preservers from
Hm to Hn such that ϕ1 (A), ϕ2 (A) are linearly dependent for any rank-one matrix A
in Hm . Then ϕ1 , ϕ2 are linearly dependent.
Proof. By Lemmas 2.1, there exist n × m matrices U, V over K, non-zero endomorphisms σ, τ of K commuting with − and non-zero scalars a, b ∈ F such that
ϕ1 (X) = aU X σ U ∗ , ϕ2 (X) = bV X τ V ∗
for all X ∈ Hm . Since for any non-zero vector x ∈ K n , U (xx∗ )σ U ∗ , V (xx∗ )τ V ∗ are
linearly dependent, it follows that U xσ = cx V xτ for some nonzero cx ∈ K. Since ϕ2
is non-degenerate, there exist y, z ∈ K n such that V y τ , V z τ are linearly independent
vectors. Since
U y σ = cy V y τ , U z σ = cz V z τ
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367
and
U (y σ + z σ ) = cy+z V (y + z)τ ,
we have
cy+z V (y + z)τ = cy V y τ + cz V z τ .
Hence, cy = cz = cz+y . Now for any nonzero w ∈ K n , we have either V y τ , V wτ
or V z τ , V wτ are linearly independent. Hence, cw = cy . This implies that there is a
fixed element c ∈ K \ {0} such that U xσ = cV xτ for any non-zero x ∈ K n . For any
non-zero scalar d ∈ K and any non-zero x ∈ K n , we have
U (dx)σ = cV (dx)τ .
Hence, σ(d) = τ (d). Thus, ϕ1 (X) = ab−1 cc∗ ϕ2 (X) for any rank-one matrix X ∈ Hm .
This shows that ϕ1 , ϕ2 are linearly dependent.
Let σ be an endomorphism of F . An additive map T from a vector space V over
F to a vector space W over F is said to be σ-quisilinear if T (cv) = σ(c)T (v) for any
c ∈ F , v ∈ V . If in addition σ is an automorphism of F , then T is called semilinear.
Theorem 3.6. Suppose that char F 6= 2 and (K, F ) 6= (3, 3). Let
Nθ be analNl
Nk
k
most surjective additive map from i=1 Hmi to i=1 Hni such that θ
i=1 ρmi ⊆
Nl
i=1 ρni where k ≤ l. Then k = l, and there exist a permutation τ on [k] and almost
surjective σ-quasilinear rank-one preservers θi from Hmτ (i) to Hni such that
θ(A1 ⊗ · · · ⊗ Ak ) = θ1 (Aτ (1) ) ⊗ · · · ⊗ θk (Aτ (k) )
for any Ai ∈ Hmi , i = 1, . . . , k.
Proof. We see from Lemma 3.3 that for each i ∈ [k], maximal decomposable
Nk
subgroups of i=1 Hmi of type-i are mapped to decomposable subgroups of type-ji
for some ji . By Lemma 3.4, [l] = [k] = {j1 , . . . , jk }. Let α be the permutation on
[k] such that α(ji ) = i, i = 1, . . . , k. Let Sα be the canonical isomorphism from
Nk
Nk
i=1 Hnα(i) such that
i=1 Hni to
Sα (A1 ⊗ · · · ⊗ Ak ) = Aα(1) ⊗ · · · ⊗ Aα(k)
for any Ai ∈ Hni , i = 1, . . . , k. Let ξ = Sα ◦ θ. It follows that ξ maps every maximal
decomposable subgroups of type-j to decomposable subgroups of type-j, j = 1, . . . , k.
Nk−1
From Lemma 3.3, we see that for any C ∈ i=1
ρmi , ξ (C ⊗ Hmk ) ⊆ C̃ ⊗ Hnk for
Nk−1
some C̃ ∈ i=1 ρni and hence there is a additive rank-one preserver πC from Hmk
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to Hnk such that ξ(C ⊗ X) = C̃ ⊗ πC (X) for any X ∈ Hmk . By Lemma 3.2, πC is
Nk−1
non-degenerate. Let D, E ∈ i=1
ρmi be linearly independent. Then we have
ξ(D ⊗ X) = D̃ ⊗ πD (X),
ξ(E ⊗ X) = Ẽ ⊗ πE (X)
Nk−1
for some D̃, Ẽ ∈ i=1 ρni and for any X ∈ ρmk . Suppose that D ∼ E. Since D ⊗ X,
E ⊗ X belong to a maximal decomposable subgroup M of type-s for some s < k,
it follows that ξ(M ) a decomposable subgroup of type-s. This shows that πD (X),
πE (X) are linearly dependent for any X ∈ ρmk . Now, suppose that D, E are not
adjacent decomposable elements. Then there exists a chain of elements Ω1 , . . . , Ωs in
Nk−1
i=1 ρmi such that Ω1 = D, E = Ωs , and Ωi−1 , Ωi are adjacent for i = 1, . . . , s − 1.
We conclude from the previous case that πD (X), πE (X) must be linearly dependent
for any X ∈ ρmk . In view of Lemma 3.5, πD , πE are linearly dependent. This shows
that there exist a non-degenerate additive rank-one preserver π from Hmk to Hnk and
Nk−1
Nk−1
Hni such that
an additive map η from i=1
Hmi to i=1
ξ(C ⊗ X) = η(C) ⊗ π(X)
Nk−1
Note that both π and η are almost
ρmi and any
NX ∈ Hmk . N
k−1
k−1
surjective. Also we see that η
i=1 ρni . If k > 2, by repeating the
i=1 ρmi ⊆
process we see that there are σi -quasilinear rank-one preservers πi from Hmi to Hni ,
such that
for any C ∈
i=1
ξ(A1 ⊗ · · · ⊗ Ak ) = π1 (A1 ) ⊗ · · · ⊗ πk (Ak )
for any Ai ∈ Hmi , i = 1, . . . , k where π = πk . For any λ ∈ F \ {0} and any Ai ∈ Hmi ,
i = 1, . . . , k,
ξ((λA1 ) ⊗ · · · ⊗ Ak ) = π1 (λA1 ) ⊗ · · · ⊗ πk (Ak )
= ξ(A1 ⊗ λA2 ⊗ · · · ⊗ Ak )
= π1 (A1 ) ⊗ π2 (λA2 ) ⊗ · · · ⊗ πk (Ak ).
Hence, σ1 (λ) = σ2 (λ) for any λ ∈ F \ {0}. Thus, σ1 = σ2 . Similarly, σ1 = σi for
2 < i ≤ k. Since ξ = Sα ◦ θ, we obtain our required conclusion.
Corollary 3.7. Suppose that char F 6= 2 and (K, F )6= (3, 3). Let
be an sur θN
Nk
Nk
k
jective additive map from i=1 Hmi onto itself such that θ
i=1 ρmi .
i=1 ρmi ⊆
Then there exist a permutation τ on [k] with mτ (i) = mi , i = 1, . . . , k, and bijective
σ-semilinear rank-one preservers θi from Hmτ (i) to Hmi such that
θ(A1 ⊗ · · · ⊗ Ak ) = θ1 (Aτ (1) ) ⊗ · · · ⊗ θk (Aτ (k) )
for any Ai ∈ Hmi , i = 1, . . . , k.
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Proof. By Theorem 3.6, there exist a permutation τ on [k] and almost surjective
σ-quasilinear rank-one preservers θi from Hmτ (i) to Hmi such that
θ(A1 ⊗ · · · ⊗ Ak ) = θ1 (Aτ (1) ) ⊗ · · · ⊗ θk (Aτ (k) )
for any Ai ∈ Hmi , i = 1, . . . , k. Hence, θ is σ-quasilinear. Since every surjective
quasilinear mapping from a finite dimensional vector space onto itself is semilinear,
it follows that θ is semilinear. This shows that σ is surjective, and hence, each θi is
bijective σ-semilinear. This completes the proof.
Nk
Corollary 3.8. Let θ be analmost surjective additive map from i=1 Hmi to
Nl
Nk
Nl
⊆ i=1 ρni where k ≤ l. Suppose that F is the
i=1 ρmi
i=1 Hni such that θ
real field. Then k = l, and there exist a permutation τ on [k] and bijective linear
rank-one preservers θi from Hmτ (i) to Hni such that
θ(A1 ⊗ · · · ⊗ Ak ) = θ1 (Aτ (1) ) ⊗ · · · ⊗ θk (Aτ (k) )
for any Ai ∈ Hmi , i = 1, . . . , k.
Proof. The result follows from Theorem 3.6 and the fact that identity map is the
only non-zero endomorphism of the real field.
Since Lemma 3.1 to Lemma 3.5 are true if we replace “additive” by “linear” and
the hypothesis on the fields by (K, F ) ∈
/ {(4, 2), (3, 3), (2, 2)}, we see from the proof
of Theorem 3.6 that the following result is valid.
Theorem 3.9. Suppose that (K, F ) ∈
/ {(4, 2), (3, 3), (2, 2)}. Let θ be a surjective
Nl
Nk
Nl
Nk
linear map from i=1 Hmi to i=1 Hni such that θ
i=1 ρni where
i=1 ρmi ⊆
k ≤ l. Then k = l, and there exist a permutation τ on [k] and bijective linear rank-one
preservers θi from Hmτ (i) to Hni such that
θ(A1 ⊗ · · · ⊗ Ak ) = θ1 (Aτ (1) ) ⊗ · · · ⊗ θk (Aτ (k) )
for any Ai ∈ Hmi , i = 1, . . . , k.
4. Symmetric matrices over algebraically closed fields. For each positive
integer m ≥ 2, let Sm be the vector space of all m-square symmetric matrices over
the field K. In this section, we shall show that if K is algebraically closed, then
Nk
Nk
Nk
every linear map θ on i=1 Smi such that θ
i=1 ρmi is induced by
i=1 ρmi ⊆
k bijective rank-one preservers on spaces of symmetric matrices.
The following result is known for the case where char K 6= 2 and n = m, see [8].
Lemma 4.1. Let η : Sn → Sm be a linear rank-one preserver where K is quadratically closed. Then n ≤ m and there exists an m × n matrix U of rank n such that
η(A) = U AU t
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for all A ∈ Sn .
Proof. Suppose that the rank of η is 1. Let x, y be linearly independent vectors
in K n . Then
η(xxt ) = uut , η(yy t ) = cuut
and
η((x + y)(x + y)t ) = duut
for some nonzero c, d ∈ K and some nonzero vector u ∈ K n . Hence,
η(xy t + yxt ) = (d − c − 1)uut .
For any λ ∈ K,
η((x + λy)(x + λy)t ) = (1 + λ(d − c − 1) + cλ2 )uut .
Since K is quadratically closed, there exists g ∈ K such that
1 + g(d − c − 1) + cg 2 = 0.
Hence,
η((x + gy)(x + gy)t ) = 0,
a contradiction. The result now follows from Lemma 2.1.
Lemma 4.2. Let K be algebraically closed. Then there does not exist a linear
map ϕ from Sm ⊗ Sn to Sm such that ϕ (ρm ⊗ ρn ) ⊆ ρm where m ≥ n.
Proof. Suppose the contrary that such ϕ exists. Let A, B be linearly independent
vectors in ρn . Since the restriction map of ϕ to Sm ⊗ A is a linear rank-one preserver,
it follows from Lemma 4.1 that there exists an invertible m-square matrix P such
that
ϕ(X ⊗ A) = P XP t
for all X ∈ Sm . Similarly, there exists an invertible m-square matrix Q such that
ϕ(X ⊗ B) = QXQt
for all X ∈ Sm . Since K is algebraically closed, it follows that P −1 Q has a non-zero
eigenvalue λ ∈ K. Let u ∈ K m be an eigenvector of P −1 Q corresponding to λ. Then
Q(u) = λP (u). Hence,
ϕ(uut ⊗ (λ2 A − B)) = λ2 P (uut )P t − Q(uut )Qt = 0.
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371
Since the restriction map of ϕ to uut ⊗ Sn is a linear rank-one preserver, it follows
from Lemma 4.1 that this restriction map is injective. Since λ2 A − B 6= 0, we obtain
a contradiction. This completes the proof.
Nk
Lemma 4.3. Let K be
algebraically
closed. Let θ be a linear map from i=1 Smi
Nl
Nl
Nk
to
⊆
i=1 Sni such that θ
i=1 ρni . Let M be any maximal decomi=1 ρmi
Nk
whose factors are rank-one matrices. Then θ(M )
posable subspace of
i=1 Smi N
k
is a decomposable subspace of
i=1 Sni whose factors are rank-one matrices and
dim θ(M ) = dim M .
Proof. Suppose that M is of type i. Since θ is linear and K is algebraically closed,
we see from the proof of Lemma 3.1 that θ(A) ∼ θ(B) whenever A, B ∈ M . Hence,
Nk
θ(M ) is a decomposable subspace of i=1 Sni of type j for some j whose factors are
rank-one matrices. Let π be the natural linear map from Smi to Snj that is induced
by the restriction map of θ to M . Then π is a rank-one preserver from Smi to Snj .
By Lemma 4.1, π is injective, and hence, dim θ(M ) = dim M . This completes the
proof.
Nk
Lemma 4.4. Let K be algebraically
closed. Let θ be a linear map from i=1 Smi
N
Nl
Nk
to li=1 Sni such that θ
i=1 ρni . If M1 and M2 are maximal decomi=1 ρmi ⊆
Nk
S , then θ(M1 ) and θ(M2 ) are decomposable subspaces of the same type in
Nli=1 mi
posable subspaces of the same type in i=1 Sni .
Proof. We may assume that M1 = B ⊗ Smk and M2 = C ⊗ Smk where B, C ∈
Nk−1
i=1 ρmi are adjacent. Suppose that θ(M1 ) and θ(M2 ) are decomposable subspaces
of different types. We may assume that
θ(M1 ) ⊆ J1 ⊗ Sn2 ⊗ D and θ(M2 ) ⊆ Sn1 ⊗ J2 ⊗ E
Nl
for some J1 ∈ ρn1 , J2 ∈ ρn2 and for some D, E ∈ i=3 ρni if 3 ≤ l. By Lemma 4.1,
there exist injective linear rank-one preservers ξ : Smk → Sn2 and ψ : Smk → Sn1
such that
θ(B ⊗ X) = J1 ⊗ ξ(X) ⊗ D and θ(C ⊗ X) = ψ(X) ⊗ J2 ⊗ E
for any X ∈ ρmk . There exist two linearly independent rank-one matrices U1 , U2 in
Smk such that ξ(Ui ), J2 are linearly independent, i = 1, 2. Since
θ(B ⊗ Ui ) ∼ θ(C ⊗ Ui )
and ξ(Ui ), J2 are linearly independent, it follows that ψ(Ui ), J1 are linearly dependent
for i = 1, 2. This is a contradiction since ψ is an injective linear map. Hence, θ(M1 )
and θ(M2 ) are decomposable subspaces of the same type.
Theorem 4.5.
Let K be algebraically closed.
Let θ be a linear map from
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Nk
i=1 Smi to
M.H. Lim
Nk
i=1 Sni such that θ
N
k
i=1
ρm i
⊆
Nk
i=1
ρni where mi = ni , mi ≥
mi+1 , i = 1, . . . , k − 1 and nk−1 ≥ nk . Then there exist a permutation σ on [k] and
k − 1 bijective linear rank-one preservers θi from Smi to Snσ(i) , i = 1, . . . , k − 1, and
a linear rank-one preserver θk from Smk to Snσ(k) such that
θ(A1 ⊗ · · · ⊗ Ak ) = θπ(1) (Aπ(1) ) ⊗ · · · ⊗ θπ(k) (Aπ(k) )
for any Ai ∈ Smi , i = 1, . . . , k where π = σ −1 .
Proof. We shall show that θ sends maximal decomposable subspaces of distinct
Nk
Nk
types in i=1 Smi to decomposable subspaces of distinct types in i=1 Sni . Suppose
the contrary. Let s be the smallest positive integer such that there exists another
positive integer t with the property that maximal decomposable subspaces of types s
and t are mapped to decomposable subspaces of type h for some positive integer h.
Nk
Let X1 ⊗ · · · ⊗ Xk ∈ i=1 ρmi and
θ(X1 ⊗ · · · ⊗ Xk ) = Y1 ⊗ · · · ⊗ Yk .
N
Let N be the subspace of ki=1 Smi spanned by the set of all non-zero decomposable
elements whose i-factor is Xi for i ∈
/ {s, t}. From the proof of Lemma 3.4, we see that
θ(N ) ⊆ Y1 ⊗ · · · ⊗ Yh−1 ⊗ Snh ⊗ Yh+1 ⊗ · · · ⊗ Yk .
Since mi = ni , mi ≥ mi+1 , i = 1, . . . , k − 1, and nk+1 ≥ nk , it follows from Lemma
4.3 that ms = nh . Hence,
θ(N ) = Y1 ⊗ · · · ⊗ Yh−1 ⊗ Snh ⊗ Yh+1 ⊗ · · · ⊗ Yk .
This shows that there exists a linear map ϕ from Sms ⊗ Smt to Sms such that ϕ(ρms ⊗
ρmt ) = ρms where ms ≥ mt . This is a contradiction to Lemma 4.2. Hence, θ sends
Nk
maximal decomposable subspaces of distinct types in
i=1 Smi to decomposable
Nk
.
Using
this
fact,
Lemmas
4.1, 4.3, 4.4 and
subspaces of distinct types in
S
n
i
i=1
the same argument as in the proof of Theorem 3.6, we obtain the required result.
For each permutation α on [l], let Pα denote the canonical isomorphism on
i=1 Sni such that
Nl
Pα (X1 ⊗ · · · ⊗ Xl ) = Xα(1) ⊗ · · · ⊗ Xα(l)
for any Xi ∈ Sni , i = 1, . . . , l.
Nk
Sm
Corollary 4.6. Let K bealgebraically closed. Let θ be a linear map from
Nl
Nl−k
Nl
Nk
ρm . Then k ≤ l and there exist B ∈
ρm
to
Sm such that θ
ρm ⊆
if k < l, a permutation α on [l], and bijective linear rank-one preservers θ1 , . . . , θk on
Sm such that
(Pα ◦ θ)(A1 ⊗ · · · ⊗ Ak ) = θ1 (A1 ) ⊗ · · · ⊗ θk (Ak ) ⊗ B
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 23, pp. 356-374, April 2012
ELA
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Additive Preservers of Tensor Product of Rank One Hermitian Matrices
373
for any A1 , . . . , Ak ∈ Sm where B is deleted if k = l.
Proof. From the proof of Theorem 4.5, we see that maximal decomposable subNk
spaces of distinct types in
Sm are mapped to decomposable subspaces of distinct
Nl
Sm . This shows that from k ≤ l. If k = l, the result follows from Theorem
types in
N
4.5. If k < l, we see from the proof of Lemma 3.4 that there exist B ∈ l−k ρm , and
a permutation α on [l] such that
Im(Pα ◦ θ) ⊆
k
O
Sm
!
⊗B
and maximal decomposable subspaces of type-i are mapped under Pα ◦ θ to decomposable subspaces of type-i. It is now clear that the result follows from Theorem
4.5.
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