Nonsurjective maps between rectangular matrix spaces preserving disjointness, triple products, or norms

TL;DR

This paper characterizes linear maps between rectangular matrix spaces that preserve disjointness, triple products, or norms, revealing their specific structural forms and extending to nonsurjective maps.

Contribution

It provides a complete characterization of linear disjointness-preserving maps and applies these results to maps preserving triple products and matrix norms.

Findings

Disjointness-preserving maps have a specific block-diagonal form.

Characterization of nonsurjective maps preserving JB*-triple products.

Maps preserving Schatten p-norms or Ky Fan k-norms are characterized.

Abstract

Let $M_{m,n}$ be the space of $m\times n$ real or complex rectangular matrices. Two matrices $A, B \in M_{m,n}$ are disjoint if $A^*B = 0_n$ and $AB^* = 0_m$. In this paper, a characterization is given for linear maps $\Phi: M_{m,n} \rightarrow M_{r,s}$ sending disjoint matrix pairs to disjoint matrix pairs, i.e., $A, B \in M_{m,n}$ are disjoint ensures that $\Phi(A), \Phi(B) \in M_{r,s}$ are disjoint. More precisely, it is shown that $\Phi$ preserves disjointness if and only if $\Phi$ is of the form $$\Phi(A) = U\begin{pmatrix} A \otimes Q_1 & 0 & 0 \cr 0 & A^t \otimes Q_2 & 0 \cr 0 & 0 & 0 \cr\end{pmatrix}V$$ for some unitary matrices $U \in M_{r,r}$ and $V\in M_{s,s}$, and positive diagonal matrices $Q_1, Q_2$, where $Q_1$ or $Q_2$ may be vacuous. The result is used to characterize nonsurjective linear maps that preserve the $JB^*$-triple product, or just the zero triple product, on rectangular matrices, defined by $\{A,B,C\} = \frac{1}{2}(AB^*C+CB^*A)$. The result is also applied to characterize linear maps between rectangular matrix spaces of different sizes preserving the Schatten $p$-norms or the Ky Fan $k$-norms.

Equations

$$\Phi (A)=U\left(\begin{array}{ccc}A\otimes Q_1& 0& 0\\ 0& A^t\otimes Q_2& 0\\ 0& 0& 0\end{array}\right)V$$

$$A\perp B,\text{if}{A}^{\ast }B=0_n\text{and}A{B}^{\ast }=0_m.$$

$$\Phi (A)=U\left(\begin{array}{ccc}A\otimes Q_1& 0& 0\\ 0& A^t\otimes Q_2& 0\\ 0& 0& 0\end{array}\right)V$$

$$\Phi (A{B}^{\ast }C+C{B}^{\ast }A)=\Phi (A)\Phi (B{)}^{\ast }\Phi (C)+\Phi (C)\Phi (B{)}^{\ast }\Phi (A)for\; allA,B,C\in {\mathbf{M}}_{m,n},$$

$$A{B}^{\ast }=0_m\text{and}{A}^{\ast }B=0_n⟹\Phi (A)\Phi (B{)}^{\ast }=0_r\text{and}\Phi (A{)}^{\ast }\Phi (B)=0_s,\forall A,B\in {\mathbf{M}}_{m,n},$$

$$\Phi (A)=U\left(\begin{array}{ccc}A\otimes Q_1& 0& 0\\ 0& A^t\otimes Q_2& 0\\ 0& 0& 0\end{array}\right)Vfor\; allA\in {\mathbf{M}}_{m,n}.$$

$$\Phi (E)\perp \Phi (F)\text{whenever}E\perp F$$

$$E_{ij}\mapsto \left(\begin{array}{ccc}{E}_{ij}\otimes Q_1& 0& 0\\ 0& E_{ji}\otimes Q_2& 0\\ 0& 0& 0\end{array}\right)for\; all\; 1\le i\le m,1\le j\le n.$$

$$A\perp B⟺\Psi (A)\perp \Psi (B)for\; allA,B\in {\mathbf{M}}_2.$$

$$\Phi (E_{11})=\left(\begin{array}{ccc}{D}_1& 0& 0\\ 0& 0_{\ell }& 0\\ 0& 0& 0_{r-k-\ell ,s-k-\ell }\end{array}\right)and\Phi (E_{22})=\left(\begin{array}{ccc}{0}_k& 0& 0\\ 0& D_2& 0\\ 0& 0& 0_{r-k-\ell ,s-k-\ell }\end{array}\right),$$

$$\Phi (E_{12}+E_{21})=\left(\begin{array}{ccc}{0}_k& B_{12}& 0\\ B_{12}^{\ast }& 0_k& 0\\ 0& 0& 0_{r-2k,s-2k}\end{array}\right)and\Phi (E_{12}-E_{21})=\left(\begin{array}{ccc}{0}_k& {\hat{C}}_{12}& 0\\ -{{\hat{C}}_{12}}^{\ast }& 0_k& 0\\ 0& 0& 0_{r-2k,s-2k}\end{array}\right),$$

$$X\mapsto (P^{\ast }{R}_2^{\ast }{R}_1^{\ast }\oplus P^{\ast }{R}_2^{\ast }\oplus I_{r-2k})\Phi (X)(R_1{R}_{2}P\oplus R_{2}P\oplus I_{s-2k})$$

$$E_{11}\mapsto \left(\begin{array}{ccc}{Q}_1\oplus Q_2& 0_k& 0\\ 0_k& 0_k& 0\\ 0& 0& 0\end{array}\right),E_{12}\mapsto \left(\begin{array}{ccc}{0}_k& Q_1\oplus 0_{k_2}& 0\\ 0_{k_1}\oplus Q_2& 0_k& 0\\ 0& 0& 0\end{array}\right),$$

$$E_{21}\mapsto \left(\begin{array}{ccc}{0}_k& 0_{k_1}\oplus Q_2& 0\\ Q_1\oplus 0_{k_2}& 0_k& 0\\ 0& 0& 0\end{array}\right),E_{22}\mapsto \left(\begin{array}{ccc}{0}_k& 0_k& 0\\ 0_k& Q_1\oplus Q_2& 0\\ 0& 0& 0\end{array}\right),$$

$$\Phi (E_{12}+E_{21})=\left(\begin{array}{ccc}{B}_{11}& B_{12}& B_{13}\\ B_{21}& B_{22}& B_{23}\\ B_{31}& B_{32}& B_{33}\end{array}\right),$$

$$Z_1=\left(\begin{array}{cc}\gamma & 1\\ 1& \frac{1}{\gamma }\end{array}\right)and{Z}_2=\left(\begin{array}{cc}\frac{1}{\gamma }& -1\\ -1& \gamma \end{array}\right)$$

$$0_k$$

$$0_{k,\ell }$$

$$0_{\ell ,k}$$

$$0_{\ell }$$

$$0_{s-k-\ell }$$

$$0_k$$

$$0_{k,\ell }$$

$$0_{\ell ,k}$$

$$0_{\ell }$$

$$0_{r-k-\ell }$$

$$B_{11}=0_k,B_{22}=0_{\ell },$$

$$B_{12}{B}_{12}^{\ast }=B_{21}^{\ast }{B}_{21}=D_1^2\in {\mathbf{M}}_k,B_{12}^{\ast }{B}_{12}=B_{21}{B}_{21}^{\ast }=D_2^2\in {\mathbf{M}}_{\ell },$$

$$D_1{B}_{12}=B_{21}^{\ast }{D}_2,and{B}_{12}{D}_2=D_1{B}_{21}^{\ast }.$$

$$B_{12}=B_{21}^{\ast }={\alpha }_1{W}_1\oplus \cdots \oplus {\alpha }_v{W}_v$$

$$\Phi (E_{12}-E_{21})=\left(\begin{array}{ccc}{C}_{11}& C_{12}& C_{13}\\ C_{21}& C_{22}& C_{23}\\ C_{31}& C_{32}& C_{33}\end{array}\right),$$

$$Z_3=\left(\begin{array}{cc}\gamma & -1\\ 1& -\frac{1}{\gamma }\end{array}\right)\text{ and }{Z}_4=\left(\begin{array}{cc}\frac{1}{\gamma }& 1\\ -1& -\gamma \end{array}\right)$$

$$0_k$$

$$0_{s-2k}$$

Full text

[TABLE]

for some unitary matrices $U\in{\bf M}_{r,r}$ and $V\in{\bf M}_{s,s}$, and positive diagonal matrices $Q_{1},Q_{2}$, where $Q_{1}$ or $Q_{2}$ may be vacuous. The result is used to characterize nonsurjective linear maps that preserve the $JB^{*}$-triple product, or just the zero triple product, on rectangular matrices, defined by $\{A,B,C\}=\frac{1}{2}(AB^{*}C+CB^{*}A)$. The result is also applied to characterize linear maps between rectangular matrix spaces of different sizes preserving the Schatten $p$-norms or the Ky Fan $k$-norms.

1. Introduction

The fruitful history of linear preserver problems starts with a rather surprising result of Frobenius. He showed in [15] that a linear map $\Phi:{\bf M}_{n}({\mathbb{C}})\rightarrow{\bf M}_{n}({\mathbb{C}})$ of $n\times n$ complex matrices preserving determinant, i.e., $\det(A)=\det(\Phi(A))$, must be of the form $A\mapsto MAN$ or $A\mapsto MA^{t}N$ for some matrices $M,N\in{\bf M}_{n}({\mathbb{C}})$ with $\det(MN)=1$. Another seminal work is due to Kadison. In [19], Kadison showed that a unital surjective isometry between two $C^{*}$-algebras ${\mathcal{A}}$ and ${\mathcal{B}}$ must be a $C^{*}$-isomorphism; in particular, a linear map $\Phi:{\bf M}_{n}({\mathbb{C}})\rightarrow{\bf M}_{n}({\mathbb{C}})$ leaving the operator norm invariant must be of the form $A\mapsto UAV$ or $A\mapsto UA^{t}V$ for some unitary matrices $U,V\in{\bf M}_{n}({\mathbb{C}})$.

Researchers have developed many results and techniques in the study of linear preserver problems; see, e.g., [2, 22, 26]. Many of the results have been extended in different directions and applied to other topics such as geometrical structure of Banach spaces, and quantum mechanics; see, e.g., [13, 14, 30]. In spite of these advances, there are some intriguing basic linear preserver problems which remain open. In particular, characterizing linear preservers between different matrix or operator spaces without the surjectivity assumption is very challenging and sometimes intractable; see, for example, [3, 7, 23, 24, 32, 33, 34]. Even for finite dimensional spaces, the problem is highly non-trivial. For instance, there is no easy description of a linear norm preserver $\Phi:M_{n}\rightarrow M_{r}$ if $n\neq r$; see [8].

In this paper, we study nonsurjective linear maps between rectangular matrix spaces preserving disjointness, the Schatten $p$-norms, or the Ky-Fan $k$-norms. The result is used to characterize linear maps that preserve the $JB^{*}$-triple product, or just the zero triple product. Note that there are interesting results on disjointness preserving maps on different kinds of products over general operator spaces or algebras, see, e.g., [21, 27, 28, 16, 17]. However, the basic problem on disjointness preservers from a rectangular matrix space to another rectangular matrix space is unknown, and the existing results do not cover this case. It is our hope that our study will lead to some general techniques for the study of disjointness preservers in a more general context, say, for general $JB^{*}$-triples, to supplement those established in the few literature, e.g., [1].

To better describe the questions addressed in this paper, we introduce some notation. Let ${\bf M}_{m,n}$ be the set of $m\times n$ real or complex rectangular matrices, and let ${\bf M}_{n}={\bf M}_{n,n}$. A pair of matrices $A,B\in{\bf M}_{m,n}$ are disjoint, denoted by

[TABLE]

Here the adjoint $A^{*}$ of a rectangular matrix $A$ is its conjugate transpose $\overline{A^{t}}$. If $A$ is a real matrix, then $A^{*}$ reduces to $A^{t}$, the transpose of $A$. Clearly, $A$ and $B$ are disjoint if and only if they have orthogonal ranges and initial spaces. A rectangular matrix $A$ is called a partial isometry if $AA^{*}A=A$. In this case, $A^{*}A$ is the range projection and $AA^{*}$ is the initial projection of $A$. Two partial isometries are disjoint if and only if they have orthogonal range and initial projections.

We will characterize linear maps $\Phi:{\bf M}_{m,n}\rightarrow{\bf M}_{r,s}$ that preserve disjointness, i.e., $\Phi(A)\perp\Phi(B)$ whenever $A\perp B$, and apply the result to some related topics. In particular, we show in Section 2 that such a map has the form

[TABLE]

for some unitary (orthogonal in the real case) matrices $U\in{\bf M}_{r},V\in{\bf M}_{s}$ and diagonal (square) matrices $Q_{1},Q_{2}$ with positive diagonal entries, where $Q_{1}$ or $Q_{2}$ may be vacuous.

In Section 3, we regard the space of rectangular matrices as JB*-triples carrying the Jordan triple product $\{A,B,C\}=\frac{1}{2}(AB^{*}C+CB^{*}A)$, and use our result in Section 2 to study JB*-triple homomorphisms on rectangular matrices, i.e., linear maps $\Phi:{\bf M}_{m,n}\rightarrow{\bf M}_{r,s}$ satisfying

[TABLE]

and also linear maps preserving matrix triples with zero Jordan triple product.

We also apply our result in Section 2 to study linear maps $\Phi:{\bf M}_{m,n}\rightarrow{\bf M}_{r,s}$ preserving the Schatten $p$-norms and the Ky Fan $k$-norms in Section 4. Open problems and future research possibilities are mentioned in Section 5.

Throughout the paper, we will always assume that $m,n,r,s$ are positive integers, and use the following notation.

1. ${\bf M}_{m,n}={\bf M}_{m,n}(\mathbb{F})$: the vector space of $m\times n$ matrices over ${\mathbb{F}}={\mathbb{R}}$ or ${\mathbb{C}}$.
2. ${\bf M}_{n}={\bf M}_{n}(\mathbb{F})$: the set of $n\times n$ matrices over ${\mathbb{F}}={\mathbb{R}}$ or ${\mathbb{C}}$.
3. ${\bf U}_{n}={\bf U}_{n}(\mathbb{F})=\{A\in{\bf M}_{n}:A^{*}A=I_{n}\}$: the set of real orthogonal or complex unitary matrices depending on ${\mathbb{F}}={\mathbb{R}}$ or ${\mathbb{C}}$.
4. ${\bf H}_{n}={\bf H}_{n}(\mathbb{F})=\{A\in{\bf M}_{n}:A=A^{*}\}$: the set of real symmetric or complex Hermitian matrices depending on ${\mathbb{F}}={\mathbb{R}}$ or ${\mathbb{C}}$.

2. Nonsurjective preservers of disjointness

In this section, we will prove the following.

Theorem 2.1**.**

A linear map $\Phi:{\bf M}_{m,n}\rightarrow{\bf M}_{r,s}$ preserves disjointness, i.e.,

[TABLE]

if and only if there exist $U\in{\bf U}_{r},V\in{\bf U}_{s}$ and diagonal matrices $Q_{1},Q_{2}$ with positive diagonal entries such that

[TABLE]

Here $Q_{1}$ or $Q_{2}$, may be vacuous.

Several remarks are in order concerning Theorem 2.1.

1. (1)

Observing the symmetry and avoiding the triviality, we can assume that $2\leq m\leq n$. 2. (2)

$AB^{*}=0_{m}$ and $A^{*}B=0_{n}$ mean that $A$ and $B$ have orthogonal ranges and orthogonal initial spaces. This amounts to saying that we can obtain their singular value decompositions, $UAV=\sum_{j=1}^{k}a_{j}E_{jj}$ and $UBV=\sum_{j=k+1}^{p}b_{j}E_{jj}$, for some positive scalars $a_{1},..,a_{k}$, $b_{k+1},...,b_{p}$, and unitary matrices $U\in{\bf U}_{m}$ and $V\in{\bf U}_{n}$. 3. (3)

In view of the singular value decompositions, (2.1) in Theorem 2.1 holds if the condition

[TABLE]

is verified just for rank one disjoint partial isometries $E,F$ in ${\bf M}_{m,n}$. 4. (4)

In Theorem 2.1, unless $r\geq m$ and $s\geq n$, or $s\geq m$ and $r\geq n$, $\Phi$ will be the zero map. If $(m,n)=(r,s)$ (resp. $(s,r)$) and $m\neq n$, then $\Phi$ will be the zero map or of the form $A\mapsto UAV$ (resp. $A\mapsto UA^{t}V$) with $U\in{\bf U}_{r},V\in{\bf U}_{s}$. 5. (5)

By relaxing the terminology, the rectangular matrix $A\otimes Q_{1}$ is permutationally similar to $q_{1}A\oplus\cdots\oplus q_{r}A$ if $Q_{1}={\rm diag}\,(q_{1},\dots,q_{r})$. Similarly $A^{t}\otimes Q_{2}$ is permutationally similar to a direct sum of positive multiples of $A^{t}$. So, the theorem asserts that up to a fixed unitary equivalence $\Phi(A)$ is a direct sum of positive multiples of $A$ and $A^{t}$. 6. (6)

In addition to real and complex rectangular matrices, the conclusions in Theorem 2.1 is also valid with the same proof for a real linear map $\Phi:{\bf H}_{n}\rightarrow{\bf M}_{r,s}$ preserving disjointness. We can further assume that the co-domain is ${\bf H}_{r}$, i.e., $\Phi:{\bf H}_{n}\rightarrow{\bf H}_{r}$. In this case, the disjointness assumption on $\Phi$ reduces to that $AB=0$ implies $\Phi(A)\Phi(B)=0$. Adjusting the proof of Theorem 2.1, we can achieve the equality $U=V^{*}$, at the expenses that the diagonal matrices $Q_{1},Q_{2}$ may have negative entries. 7. (7)

If the domain is the set ${\bf M}_{n}(\mathbb{C})$ of $n\times n$ complex matrices or the set ${\bf H}_{n}(\mathbb{C})$ of $n\times n$ complex Hermitian matrices, our results can be deduced from the abstract theorems on $C^{*}$-algebras; e.g., see [28, 21, 20, 4], and also [27, 6]. However, the proofs there do not seem to work for rectangular matrix spaces, or real square matrix spaces. 8. (8)

Our proof is computational and long. It would be nice to have some short and conceptual proofs.

The rest of the section is devoted to the proof of Theorem 2.1. We describe our proof strategy. Let $\{E_{11},E_{12},\dots,E_{mn}\}$ be the standard basis for ${\bf M}_{m,n}$. We will show that one can apply a series of replacements of $\Phi$ by mappings of the form $X\mapsto\tilde{U}\Phi(X)\tilde{V}$ for some $\tilde{U}\in{\bf U}_{r},\tilde{V}\in{\bf U}_{s}$ so that the resulting map satisfies

[TABLE]

The result will then follow. We carry out the above scheme with an inductive argument, and divide the proofs into several lemmas.

Note that in this section only the linearity and the disjointness structure of the rectangular matrices are concerned. As will be shown below, the (real or complex) matrix space ${\bf M}_{2}=\operatorname{span}\{E_{11},E_{12},E_{21},E_{22}\}$ and the matrix space $\operatorname{span}\{E_{ij},E_{ik},E_{lj},E_{lk}\}$ can be considered as the same object during our discussion.

Lemma 2.2**.**

Let $i\neq l$ and $j\neq k$. The bijective linear map $\Psi:{\bf M}_{2}\to\operatorname{span}\{E_{ij},E_{ik},E_{lj},E_{lk}\}$, sending $E_{11},E_{12},E_{21},E_{22}$ to $E_{ij},E_{ik},E_{lj},E_{lk}\in{\bf M}_{m,n}$ respectively, preserves the disjointness in two directions, i.e.,

[TABLE]

Proof.

The assertion follows from the fact that $\Psi(A)=UAV$, where $U=E_{i1}+E_{l2}\in{\bf M}_{m,2}$ and $V=E_{1j}+E_{2k}\in{\bf M}_{2,n}$ are partial isometries such that $U^{*}U=VV^{*}=I_{2}$, the $2\times 2$ identity matrix.

The technical lemma will be used heavily in the subsequent proofs. Although the statement is stated and proved for the case when the domain is ${\bf M}_{2}$, it is indeed valid for all the rectangular matrix space $\operatorname{span}\{E_{ij},E_{ik},E_{lj},E_{lk}\}$ due to Lemma 2.2. In the future application, the lemma ensures that if $\Phi(E_{ij})$ and $\Phi(E_{lk})$ have some nice structure for a disjointness preserving linear map $\Phi:{\bf M}_{m,n}\rightarrow{\bf M}_{r,s}$, then much can be said about $\Phi(E_{ik}+E_{lj})$ and $\Phi(E_{ik}-E_{lj})$. One can then compose $\Phi$ with some unitaries so that all $\Phi(E_{ij}),\Phi(E_{ik}),\Phi(E_{lj})$ and $\Phi(E_{lk})$ have simple structure.

Lemma 2.3**.**

Let $\Phi:{\bf M}_{2}\rightarrow{\bf M}_{r,s}$ be a nonzero linear map preserving disjointness such that

[TABLE]

where $D_{1}\in{\bf M}_{k},D_{2}\in{\bf M}_{\ell}$ are diagonal matrices with positive diagonal entries arranged in descending order, and $D_{1}=\alpha_{1}I_{u_{1}}\oplus\cdots\oplus\alpha_{v}I_{u_{v}}$ with $\alpha_{1}>\cdots>\alpha_{v}>0$ and $u_{1}+\cdots+u_{v}=k$.

1. (a)

We have $D_{1}=D_{2}$. Moreover,

[TABLE]

where $B_{12}=\alpha_{1}W_{1}\oplus\cdots\oplus\alpha_{v}W_{v}$ and ${\widehat{C}_{12}}=\alpha_{1}W_{1}V_{1}\oplus\cdots\oplus\alpha_{v}W_{v}V_{v}$ with $W_{j},V_{j}\in{\bf U}_{u_{j}}$. 2. (b)

There are unitaries $R_{1},R_{2}\in{\bf U}_{k}$ and a permutation $P\in{\bf M}_{k}$ such that the map

[TABLE]

satisfies

[TABLE]

[TABLE]

where $Q_{1}\in{\bf M}_{k_{1}}$, $Q_{2}\in{\bf M}_{k_{2}}$, $k_{1}+k_{2}=k$, are diagonal matrices with positive diagonal entries from $\{\alpha_{1},\ldots,\alpha_{v}\}$ arranged in descending order.

Proof.

(a) Suppose $\Phi:{\bf M}_{2}\rightarrow{\bf M}_{r,s}$ satisfies the assumption. Let

[TABLE]

where $B_{11}\in{\bf M}_{k},B_{22}\in{\bf M}_{\ell}$. For every nonzero $\gamma\in\mathbb{R}$, the pair of the matrices

[TABLE]

are disjoint, and so are the pair $T_{1}=\Phi(Z_{1})$ and $T_{2}=\Phi(Z_{2})$. Considering the $(1,1)$, $(1,2)$, $(2,1)$, $(2,2)$, $(3,3)$ blocks of the matrix $T_{1}^{*}T_{2}$, we get the following:

[TABLE]

Considering the $(1,1)$, $(1,2)$, $(2,1)$, $(2,2)$, $(3,3)$ blocks of the matrix $T_{1}T_{2}^{*}$, we get the following:

[TABLE]

In view of the $(3,3)$ blocks of $T_{1}^{*}T_{2}$ and $T_{1}T_{2}^{*}$ being zero blocks, we see that $B_{13},B_{23},B_{33},B_{31},B_{32}$ are zero blocks. Since $0\neq\gamma$ is arbitrary and $D_{1}$, $D_{2}$ are invertible, we see that

[TABLE]

Note that $B_{12}B_{12}^{*}$ and $B_{12}^{*}B_{12}$ have the same nonzero eigenvalues (counting multiplicities). Because $D_{1},D_{2}$ have positive diagonal entries arranged in descending order, it follows from (2.2) that $k=\ell$ and $D_{1}=D_{2}$.

We can now assume that $D_{1}=D_{2}=\alpha_{1}I_{u_{1}}\oplus\cdots\oplus\alpha_{v}I_{u_{v}}$ with $\alpha_{1}>\cdots>\alpha_{v}>0$ and $u_{1}+\cdots+u_{v}=k$. Furthermore, from (2.2) the matrices $B_{12},B_{12}^{*},B_{21}$ and $B_{21}^{*}$ have orthogonal columns with Euclidean norms equal to the diagonal entries of $D_{1}$. By (2.3), we see that

[TABLE]

for some $W_{1}\in{\bf U}_{u_{1}},\dots,W_{v}\in{\bf U}_{u_{v}}$.

Let $R_{1}=W_{1}\oplus\cdots\oplus W_{v}$. Replace $\Phi$ by $X\mapsto(R_{1}^{*}\oplus I_{r-k})\Phi(X)(R_{1}\oplus I_{s-k})$. We may assume that $B_{12}=B_{21}^{*}=D_{1}$. Let

[TABLE]

where $C_{11}\in{\bf M}_{k},C_{22}\in{\bf M}_{\ell}$.

Now, the pair of matrices

[TABLE]

are disjoint, and so are the pair of matrices $T_{3}=\Phi(Z_{3})$ and $T_{4}=\Phi(Z_{4})$. Consider the $(1,1)$, $(1,2)$, $(2,1)$, $(2,2)$, $(3,3)$ blocks of the matrix $T_{3}^{*}T_{4}$. By the fact that $k=\ell$ and $D_{1}=D_{2}$, we get the following:

[TABLE]

Consider the $(1,1)$, $(1,2)$, $(2,1)$, $(2,2)$, $(3,3)$ blocks of the matrix $T_{3}T_{4}^{*}$. We get the following:

[TABLE]

By a similar argument for the pair $(T_{1},T_{2})$, we conclude that $C_{11}$, $C_{22}$, $C_{13},C_{23},C_{33},C_{31},C_{32}$ are zero blocks. Furthermore,

$C_{21}^{*}C_{21}=C_{12}C_{12}^{*}=C_{21}C_{21}^{*}=C_{12}^{*}C_{12}=D_{1}^{2}$,  $D_{1}C_{12}=-C_{21}^{*}D_{1}$,  and  $C_{12}D_{1}=-D_{1}C_{21}^{*}$.

Now, $C_{21},C_{12}^{*},C_{21}^{*},C_{12}$ have orthogonal columns with Euclidean norms equal to the diagonal entries of $D_{1}$, and together with the fact that $D_{1}C_{12}=-C_{21}^{*}D_{1}$, and $C_{12}D_{1}=-D_{1}C_{21}^{*}$, we see that

[TABLE]

where $V=D_{1}^{-1}C_{12}=V_{1}\oplus\cdots\oplus V_{v}$ is unitary. Thus in its original form, we see that

[TABLE]

(b) Continue the arguments in (a), and in particular assume that $B_{12}=B_{21}^{*}=D_{1}$ and $C_{12}=-C_{21}^{*}=\alpha_{1}V_{1}\oplus\cdots\oplus\alpha_{v}V_{v}=D_{1}V$. There is a unitary matrix $R_{2}=U_{1}\oplus\cdots\oplus U_{v}\in{\bf U}_{k}$ with $U_{1}\in{\bf M}_{u_{1}},\dots,U_{v}\in{\bf M}_{u_{v}}$ such that $R_{2}^{*}VR_{2}={\rm diag}\,(g_{1},\dots,g_{k})=G\in{\bf U}_{k}$. Now, we may replace $\Phi$ by the map $X\mapsto(R_{2}^{*}\oplus R_{2}^{*}\oplus I_{r-2k})\Phi(X)(R_{2}\oplus R_{2}\oplus I_{s-2k})$ and assume that $C_{12}=-C_{21}^{*}=D_{1}G$. In particular,

[TABLE]

We claim that $G$ is permutationally similar to $I_{k_{1}}\oplus-I_{k_{2}}$ with $k_{1}+k_{2}=k$. To see this, consider the pair

[TABLE]

One readily checks that the pair are disjoint if and only if $(I_{k}+G)(I_{k}-G^{*})=0_{k}$, equivalently, $G$ is a real diagonal unitary matrix. Thus, there is a permutation matrix $P\in{\bf M}_{k}$ such that $P^{t}GP=I_{k_{1}}\oplus-I_{k_{2}}$ with $k_{1}+k_{2}=k$. With a further permutation, we can assume $P^{t}D_{1}GP=Q_{1}\oplus-Q_{2}\in{\bf M}_{k}$ so that $Q_{1},Q_{2}$ are diagonal matrices with descending positive diagonal entries.

We may replace $\Phi$ by a map

[TABLE]

so that

[TABLE]

[TABLE]

Adding and subtracting the matrices $\Phi(E_{12}+E_{21})$ and $\Phi(E_{12}-E_{21})$, we get the desired forms of $\Phi(E_{12})$ and $\Phi(E_{21})$. The result follows.

Lemma 2.4**.**

Theorem 2.1 holds if $m=n\geq 2$.

Proof.

We prove the result by induction on $m=n\geq 2$. Suppose $m=n=2$. We may choose $V_{1}\in{\bf U}_{r},V_{2}\in{\bf U}_{s}$ such that

[TABLE]

where $D_{1}\in{\bf M}_{k},D_{2}\in{\bf M}_{\ell}$ are diagonal matrices with positive diagonal entries arranged in descending order. We may replace $\Phi$ by the map $X\mapsto V_{1}\Phi(X)V_{2}$ so that the resulting map will preserve disjointness and send $E_{jj}$ to $Y_{j}$ for $j=1,2$. By Lemma 2.3, we can modify $V_{1}$ and $V_{2}$ so that the resulting map satisfies

[TABLE]

for some diagonal matrices $Q_{1},Q_{2}$ with descending positive diagonal entries.

Now, we can find a permutation matrix $\hat{P}\in{\bf M}_{2k}$ satisfying $[X_{1}|X_{2}|X_{3}|X_{4}]\hat{P}=[X_{1}|X_{3}|X_{2}|X_{4}]$ whenever $X_{1},X_{3}\in{\bf M}_{2k,k_{1}},X_{2},X_{4}\in{\bf M}_{2k,k_{2}}$. Then the map $X\mapsto(\hat{P}\oplus I_{r-2k})^{t}\Phi(X)(\hat{P}\oplus I_{s-2k})$ will satisfy

[TABLE]

This establishes the assertion for the case when $m=n=2$.

Now, suppose the result holds for square matrices of size smaller than $n$ with $n>2$. Then the restriction of $\Phi$ on matrices $A\in{\bf M}_{n}$ with the last row and last column equal to zero verifies the conclusion. So, there exist $U\in{\bf U}_{r}$ and $V\in{\bf U}_{s}$ such that

[TABLE]

Here $\{E_{ij}:1\leq i,j\leq n\}$ is the standard basis for ${\bf M}_{n}$, and $\{\hat{E}_{ij}:1\leq i,j\leq n-1\}$ is the standard basis for ${\bf M}_{n-1}$, $Q_{1}\in{\bf M}_{k_{1}},Q_{2}\in{\bf M}_{k_{2}}$ are diagonal matrices with positive diagonal entries, and $k=k_{1}+k_{2}$.

Note that $E_{nn}$ and $E_{ij}$ are disjoint for all $1\leq i,j<n$. So, we may assume that

[TABLE]

for some matrix $Y\in{\bf M}_{r-(n-1)k,s-(n-1)k}$. There exist $U_{1}\in{\bf U}_{r-(n-1)k},V_{1}\in{\bf U}_{s-(n-1)k}$ such that

[TABLE]

where $D$ is a diagonal matrix with positive diagonal entries arranged in descending order. We may replace $\Phi$ by the map

[TABLE]

and assume that $U_{1}=I_{r-(n-1)k}$ and $V_{1}=I_{s-(n-1)k}.$

Consider the restriction of the map on the $\operatorname{span}\{E_{11},E_{1n},E_{n1},E_{nn}\}$. Applying the proof of Lemma 2.3 to the restriction map, we see that there is a permutation matrix $P$ such that $D=P^{t}(Q_{1}\oplus Q_{2})P$. Now, replace $\Phi$ by the map

[TABLE]

After a further permutation, we can replace $\hat{E}_{ij}$ with $E_{ij}$ for $1\leq i,j<n$, and the resulting map $\Phi$ satisfies

[TABLE]

[TABLE]

[TABLE]

where $\hat{D}=P^{t}(Q_{1}\oplus-Q_{2})P$.

For $j=1,2,\dots,n-1$, apply Lemma 2.3(a) to the restriction map on the rectangular matrix space $\operatorname{span}\{E_{jj},E_{jn},E_{nj},E_{nn}\}$. We see that

[TABLE]

Here $B_{jn},C_{jn}\in{\bf M}_{k}$ and $D^{-1}B_{jn},D^{-1}C_{jn}\in{\bf U}_{k}$ commute with $D$.

Because every matrix in the range of the map $\Phi$ has its last $r-nk$ rows and last $s-nk$ columns equal to zero, we will assume that $r=nk$ and $s=nk$ for simplicity (by removing the last $r-nk$ rows and $s-nk$ columns from every matrix in the range space). Let $\{e_{1},\dots,e_{n}\}$ be the standard basis for ${\mathbb{C}}^{n}$. For $j=2,\dots,n-1$, consider the disjoint pair

[TABLE]

Then $\Phi(X_{1})$ and $\Phi(X_{2})$ are disjoint. If we partition $\Phi(X_{1}),\Phi(X_{2})$ as $n\times n$ block matrices $Z=(Z_{ij})_{1\leq i,j\leq n}$ such that each block is in ${\bf M}_{k}$, then all the blocks are zero except for the $(p,q)$ blocks with $p,q\in\{1,j,n\}$. Deleting all the zero blocks, we get the following two $3\times 3$ block matrices.

[TABLE]

Both the $(1,1)$ and $(1,2)$ blocks of $Z_{1}Z_{2}^{*}$ equal $0_{k}$, i.e.,

[TABLE]

We see that $B_{1n}B_{1n}^{*}=D^{2}=B_{1n}B_{jn}^{*}$. Since $B_{1n}$ is the product of $D$ and a unitary matrix, it is invertible. So, $B_{1n}=B_{jn}$ for $j=2,\dots,n-1$.

Similarly, we can consider the disjoint pair

[TABLE]

Then removing the zero blocks of $\Phi(X_{3})$ and $\Phi(X_{4})$, we get

[TABLE]

Both the $(1,1)$ and $(1,2)$ blocks of $Z_{3}Z_{4}^{*}$ equal $0_{k}$, i.e.,

[TABLE]

We see that $C_{1n}C_{1n}^{*}=D^{2}=C_{1n}C_{jn}^{*}$. Since $C_{1n}$ is the product of $D$ and a unitary (real orthogonal) matrix, it is invertible. Thus, $C_{1n}=C_{jn}$ for $j=2,\dots,n-1$.

Let $W$ be the unitary matrix $D^{-1}B_{1n}\in{\bf M}_{n}.$ Replace $\Phi$ by the map $X\mapsto(I_{(n-1)k}\oplus W)\Phi(X)(I_{(n-1)k}\oplus W^{*})$. Then with $\hat{C}=C_{jn}W^{*}$ for $j=1,\dots,n-1$, we have

[TABLE]

[TABLE]

[TABLE]

Recall that $P$ is a permutation matrix such that $D=P^{t}(Q_{1}\oplus Q_{2})P$. Now replace $\Phi$ by $X\mapsto(I_{n}\otimes P)\Phi(X)(I_{n}\otimes P^{t})$. Then

[TABLE]

[TABLE]

[TABLE]

where $G=P\hat{C}P^{t}$.

It remains to show that $G=Q_{1}\oplus-Q_{2}$ so that $E_{jn}\otimes G-E_{nj}\otimes G^{*}=(E_{jn}-E_{nj})\otimes(Q_{1}\oplus-Q_{2})$. To this end, consider the disjoint pair $X_{5}=E_{22}+E_{nn}-E_{2n}-E_{n2}$ and $X_{6}=E_{12}+E_{1n}-E_{21}-E_{n1}$. Then $Z_{5}=\Phi(X_{5})$ and $Z_{6}=\Phi(X_{6})$ are disjoint. If we partition $\Phi(X_{5}),\Phi(X_{6})$ as $n\times n$ block matrices $Z=(Z_{ij})_{1\leq i,j\leq n}$ such that each block is in ${\bf M}_{k}$, then all the blocks are zero except for the $(p,q)$ blocks with $p,q\in\{1,2,n\}$. Let $Q=Q_{1}\oplus Q_{2}$ and $C_{12}=Q_{1}\oplus-Q_{2}$. Deleting all the zero blocks, we get the following two matrices.

[TABLE]

Now, the $(1,2)$ block of $Z_{6}Z_{5}^{*}$ is zero, i.e., $C_{12}Q=GQ$. It follows that $G=C_{12}=Q_{1}\oplus-Q_{2}$. Thus, the desired result follows.

To prove the theorem when the domain is ${\bf M}_{m,n}$ with $m<n$, we can apply the result for the restriction of $\Phi$ to the subspace spanned by $\{E_{ij}:1\leq i,j\leq m\}$ and assume the restriction map has nice structure. Then we have to show that $\Phi(E_{il})$ also has a nice form for $l>m$. To do that we need another technical lemma showing that if $\Phi(E_{ij})$ and $\Phi(E_{kj})$ have nice forms, then $\Phi(E_{il})$ and $\Phi(E_{kl})$ also have nice forms. We state and prove the results for a special case in the following, in view of Lemma 2.2.

Lemma 2.5**.**

Let $Q_{1}\in{\bf M}_{k_{1}},Q_{2}\in{\bf M}_{k_{2}}$ with $k_{1}+k_{2}=k$ be diagonal matrices with positive diagonal entries arranged in descending order. Let $\Phi:{\bf M}_{2}\rightarrow{\bf M}_{r,s}$ be a nonzero linear map preserving disjointness.

• (a)

Assume

[TABLE]

where $(r_{1},s_{1})=(r-2k_{1}-k_{2},s-k_{1}-2k_{2})$. Then there exist $R=R_{1}\oplus R_{2}\in{\bf U}_{k_{1}}\oplus{\bf U}_{k_{2}}$, $U\in{\bf U}_{r-k},V\in{\bf U}_{s-k}$ such that

[TABLE]

moreover, if $U=\begin{pmatrix}U_{11}&U_{12}\cr U_{21}&U_{22}\cr\end{pmatrix}$ with $U_{11}\in{\bf M}_{k_{1}}$, then the modified map $\Psi$ defined by

[TABLE]

satisfies

[TABLE]

Consequently, before the modification we have

[TABLE]

where $\hat{Y}_{1},\hat{Z}_{1}$ have singular values equal to the diagonal entries of $Q_{1}$, and $\hat{Y}_{2},\hat{Z}_{2}$ have singular values equal to the diagonal entries of $Q_{2}$.

• (b)

Suppose

[TABLE]

and $(r_{2},s_{2})=(r-2k,s-2k)$. Then $\Phi(E_{12})$ also satisfies (2.5).

Proof.

(a) By Lemma 2.3, we know that the disjoint matrices $\Phi(E_{22})$ and $\Phi(E_{11})$ have the same rank. So, $r,s\geq 2k$. Let $P_{1}\in{\bf M}_{2k}$ be a permutation matrix such that $[X_{1}|X_{2}|X_{3}|X_{4}]P_{1}=[X_{1}|X_{3}|X_{2}|X_{4}]$ whenever $X_{1},X_{2}\in{\bf M}_{2k,k_{1}}$ and $X_{3},X_{4}\in{\bf M}_{2k,k_{2}}$. Then the map $\hat{\Phi}$ defined by $\hat{\Phi}(X)=(P_{1}^{t}\oplus I_{r-2k})\Phi(X)$ will still preserve disjointness such that $\hat{\Phi}(E_{11})$ and $\hat{\Phi}(E_{21})$ equal

[TABLE]

Suppose $P_{2}\in{\bf M}_{k}$ is a permutation matrix such that $D_{1}=P_{2}^{t}(Q_{1}\oplus Q_{2})P_{2}$ has diagonal entries arranged in descending order. We can then find $U_{1}\in{\bf U}_{r-k}$ and $V_{1}\in{\bf U}_{s-k}$ such that

[TABLE]

where $D_{2}$ is a diagonal matrix with positive diagonal entries arranged in descending order.

Applying Lemma 2.3, we can find $S_{2}\in{\bf U}_{k},U_{2}\in{\bf U}_{r-k},V_{2}\in{\bf U}_{s-k}$ such that the map $\Psi_{1}$ defined by

[TABLE]

satisfies

[TABLE]

where $\hat{Q}_{1}\in{\bf M}_{\ell_{1}}$ and $\hat{Q}_{2}\in{\bf M}_{\ell_{2}}$ are diagonal matrices with positive diagonal entries arranged in descending order. Let $\Psi$ be defined by $\Psi(X)=\Psi_{1}(X)(I_{k}\oplus P_{3}\oplus I_{s-2k})$, where $P_{3}\in{\bf M}_{k}$ is a permutation matrix such that $[X_{1}|X_{2}]P_{3}=[X_{2}|X_{1}]$ whenever $X_{1}\in{\bf M}_{k,k_{1}}$ and $X_{2}\in{\bf M}_{k,k_{2}}$. Then the map $\Psi$ satisfies

[TABLE]

Let $R=P_{2}S_{2}\in{\bf M}_{k}$, $V=V_{1}V_{2}(P_{3}\oplus I_{s-2k})\in{\bf M}_{s-k}$, and $U=U_{2}U_{1}\in{\bf M}_{r-k}$. Then

[TABLE]

If we partition $\Psi(X)$ into a $2\times 2$ block matrix such that the $(1,1)$ block lies in ${\bf M}_{k}$, then the diagonal entries of $\hat{Q}_{1}$ are the singular values of the $(2,1)$ block of $\hat{\Phi}(E_{21})$ (using the same partition). So, $\hat{Q}_{1}=Q_{1}$ and $\hat{Q}_{2}=Q_{2}$. Hence, $\hat{\Phi}(E_{21})=\Psi(E_{21})$. It follows that

[TABLE]

As a result,

[TABLE]

Thus, $R=R_{1}\oplus R_{2}$ with $R_{1}\in{\bf M}_{k_{1}},R_{2}\in{\bf M}_{k_{2}}$. Since $Q_{1}$ and $Q_{2}$ are diagonal matrices with positive diagonal entries, we see that $R_{1}^{*}Q_{1}R_{1}=Q_{1}$ and $R_{2}^{*}Q_{2}R_{2}=Q_{2}$. Moreover, by (2.6) we have

[TABLE]

One can then check that the modified map $\hat{\Psi}(X)=(P_{1}^{t}\oplus I_{r-2k})\Psi(X)$ has the desired property.

Now, we turn to $\Phi(E_{12})$ and $\Phi(E_{21})$. If $U=(U_{ij})_{1\leq i,j\leq 3}\in{\bf M}_{r-k}$ with $U_{11}\in{\bf M}_{k_{1}},U_{22}\in{\bf M}_{k_{2}}$, and $V=(V_{ij})_{1\leq i,j\leq 3}\in{\bf M}_{s-k}$ with $V_{11}\in{\bf M}_{k_{2}},V_{22}\in{\bf M}_{k_{1}}$, then

[TABLE]

where

[TABLE]

[TABLE]

Note that $\hat{\Phi}(E_{12})$ and $\hat{\Phi}(E_{21})$ are disjoint. So, $U_{21}^{*}Q_{2}R_{2}^{*},R_{1}Q_{1}V_{12}^{*}\in{\bf M}_{k_{1},k_{2}}$ are zero blocks. Since $R_{1}Q_{1}$ and $Q_{2}R_{2}^{*}$ are invertible, we see that

[TABLE]

As a result, $\Phi(E_{12})=(P_{1}\oplus I_{r-2k})\hat{\Phi}(E_{12})$ has the asserted form with

[TABLE]

Also, $\hat{\Phi}(E_{22})=\begin{pmatrix}0_{k}&0\cr 0&G\cr\end{pmatrix}$ with

[TABLE]

by (2.6), where $R^{\prime}=R_{2}\oplus R_{1}$. Thus, by (2.7), we have

[TABLE]

As a result, $\Phi(E_{22})=(P_{1}\oplus I_{r-2k})\hat{\Phi}(E_{22})$ has the asserted form with

[TABLE]

(b) Applying a block permutation, we may assume that $\Phi(E_{11})$, $\Phi(E_{21})$, $\Phi(E_{22})$ equal

[TABLE]

respectively. We need to show that

[TABLE]

Suppose $\hat{P}\in{\bf M}_{k}$ is a permutation matrix such that $\hat{D}=\hat{P}^{t}(Q_{1}\oplus Q_{2})\hat{P}$ is a diagonal matrix with entries in descending order. Applying Lemma 2.3 to the map

[TABLE]

we conclude that there exist a permutation $P\in{\bf M}_{k}$ and $W_{1},W_{2}\in{\bf U}_{k}$ commuting with $\hat{D}$ such that for $W=\hat{P}W_{1}W_{2}P\oplus\hat{P}W_{2}P\in{\bf M}_{2k}$, the map $\Psi$ defined by $X\mapsto(W^{*}\oplus I_{r-2k})\Phi(X)(W\oplus I_{s-2k})$ has the form

[TABLE]

where $\hat{Q}_{1}\in{\bf M}_{\ell_{1}}$ and $\hat{Q}_{2}\in{\bf M}_{\ell_{2}}$ are diagonal matrices with positive diagonal entries arranged in descending order. Note that the diagonal entries of $\hat{Q}_{1}$ are the singular values of the $(1,2)$ block of $\Phi(E_{12})$. So, $\hat{Q}_{1}=Q_{1}$ and $\hat{Q}_{2}=Q_{2}$. Consequently,

[TABLE]

has the asserted form.

Proof of Theorem 2.1.

Without loss of generality, we assume $2\leq m\leq n$. We prove the result by induction on $n-m$. If $n-m=0$, the result follows from Lemma 2.4. Suppose $n-m=\ell\geq 1$ and the result holds for the cases when $n-m<\ell$.

By the induction assumption on the restriction map of $\Phi$ on the span of ${\mathcal{C}}_{n}=\{E_{ij}:1\leq i\leq m,1\leq j<n\}$, there are diagonal matrices $Q_{1}\in{\bf M}_{k_{1}},Q_{2}\in{\bf M}_{k_{2}}$ with positive entries arranged in descending order, and $U_{1}\in{\bf U}_{r},V_{1}\in{\bf U}_{s}$ such that the map $U_{1}\Phi(X)V_{1}$ satisfies

[TABLE]

where $\{E_{ij}:1\leq i\leq m,1\leq j\leq n\}$ is the standard basis for ${\bf M}_{m,n}$, $\{\hat{E}_{ij}:1\leq i\leq m,1\leq j<n\}$ is the standard basis for ${\bf M}_{m,n-1}$, and $(\hat{r},\hat{s})=(r-mk_{1}-(n-1)k_{2},s-(n-1)k_{1}-mk_{2})$. For notational simplicity, we assume that $U_{1}=I_{r},V_{1}=I_{s}$.

Consider the restriction of $\Phi$ on $\operatorname{span}\{E_{ij},E_{in},E_{mj},E_{mn}\}$ for all $1\leq i<m,1\leq j<n$. By Lemma 2.5 (a), we see that

[TABLE]

where only the last $k_{1}$ rows of $Z_{1}$ can be nonzero, and only the last $k_{2}$ columns of $Z_{2}$ can be nonzero.

Similarly,

[TABLE]

where only the first $k_{1}$ rows of $Y_{1}$ can be nonzero, and only the first $k_{2}$ columns of $Y_{2}$ can be nonzero.

Now, consider the restriction of $\Phi$ on $\operatorname{span}\{E_{11},E_{1n},E_{m1},E_{mn}\}.$ By Lemma 2.5 (a), there exist $R=R_{1}\oplus R_{2}\in{\bf U}_{k_{1}}\oplus{\bf U}_{k_{2}}$, $U\in{\bf U}_{\hat{r}}$ and $V\in{\bf U}_{\hat{s}}$ such that

[TABLE]

moreover, if $U=\begin{pmatrix}U_{11}&U_{12}\cr U_{21}&U_{22}\cr\end{pmatrix}$ with $U_{11}\in{\bf M}_{k_{1}}$, then

[TABLE]

[TABLE]

where $(r_{1},s_{1})=(\hat{r}-2k_{1},\hat{s}-2k_{2})$. Consequently, the modified map $\Psi$ defined by

[TABLE]

satisfies $\Psi(E_{ij})=\Phi(E_{ij})$ for all $1\leq i\leq m,1\leq j<n-1$, and $\Psi(E_{mn})$ has the form (2.9) with

[TABLE]

Let $\tilde{P}\in{\bf M}_{s}$ be the permutation matrix satisfying $[X_{1}|X_{2}|X_{3}|X_{4}]\tilde{P}=[X_{1}|X_{3}|X_{2}|X_{4}]$ whenever $X_{1}\in{\bf M}_{r,(n-1)k_{1}}$, $X_{2}\in{\bf M}_{r,mk_{2}}$, $X_{3}\in{\bf M}_{r,k_{1}}$, $X_{4}\in{\bf M}_{r,\hat{s}-k_{1}}$. Then the map $\hat{\Psi}$ defined by $X\mapsto\Psi(X)\tilde{P}$ satisfies

[TABLE]

for $(i,j)\in\{(u,v):1\leq u\leq m,1\leq v<n\}\cup\{(m,n)\}$. For $j=2,\dots,n-1$, consider the restriction of $\Psi$ on $\operatorname{span}\{E_{jj},E_{jn},E_{mj},E_{mn}\}$. Thus, $\hat{\Psi}(E_{jj}),\hat{\Psi}(E_{mj}),\hat{\Psi}(E_{mn})$ have the form (2.11), and so must $\hat{\Psi}(E_{jn})$ by Lemma 2.5 (b). As a result, $\hat{\Psi}(E_{ij})$ has the form in (2.11) for all $1\leq i\leq m,1\leq j\leq n$.

3. Nonsurjective (zero) Triple Product Preservers and

JB*-homomorphisms on rectangular matrices

Notice that the set ${\bf M}_{n}(\mathbb{C})$ of complex square matrices is a C∗-algebra. Let $T:\mathcal{A}\to\mathcal{B}$ be a bounded linear map between C∗-algebras. In [31, Theorem 3.2], it was shown that $T$ is a triple homomorphism with respect to the Jordan triple product,

[TABLE]

if and only if $T$ preserves disjointness and $T^{**}(1)$ is a partial isometry in $B^{**}$. In the case that $T$ is surjective, the condition on $T^{**}(1)$ can be dropped as shown in [20, Theorem 2.2], see also [27]. In [4], on the other hand, it is obtained a characterization of linear maps from $C^{*}$-algebras into JB*-triples that preserve disjointness with some conditions.

In the following, we consider the Jordan triple product $\{A,B,C\}=\frac{1}{2}(AB^{*}C+CB^{*}A)$ of real or complex matrices $A,B,C\in{\bf M}_{m,n}$. A (real or complex) linear map $\Psi:{\bf M}_{m,n}\to{\bf M}_{r,s}$ between rectangular matrices is called a JB*-triple homomorphism if

[TABLE]

We have the polarization identity

[TABLE]

In the complex case, letting the cube $A^{(3)}=AA^{*}A$, we have

[TABLE]

Therefore, a linear map $\Phi$ between rectangular matrices is a JB*-triple homomorphism exactly when $\Phi(AB^{*}A)=\Phi(A)\Phi(B)^{*}\Phi(A)$, and in the complex case exactly when $\Phi(AA^{*}A)=\Phi(A)\Phi(A)^{*}\Phi(A)$, for all $A,B\in{\bf M}_{m,n}$.

We say that the matrix triple $(A,B,C)$ in ${\bf M}_{m,n}$ has zero triple product if $\{A,B,C\}=0_{m,n}$. A linear map $\Phi:{\bf M}_{m,n}\rightarrow{\bf M}_{r,s}$ preserves zero triple products if

[TABLE]

For more information of JB*-triples, see, e.g., [9].

We have the following result concerning the zero triple product preservers and JB*-triple homomorphisms on rectangular matrices.

Theorem 3.1**.**

Let $\Phi:{\bf M}_{m,n}\to{\bf M}_{r,s}$ be a linear map.

1. (a)

$\Phi$* preserves zero triple products if and only if there are $U\in{\bf U}_{r},V\in{\bf U}_{s}$, and diagonal matrices $Q_{1},Q_{2}$ with positive diagonal entries such that*

[TABLE]

Here $Q_{1}$ or $Q_{2}$, may be vacuous. 2. (b)

$\Phi$* is a JB*-triple homomorphism if and only if there exist $U\in{\bf U}_{r},V\in{\bf U}_{s}$, and nonnegative integers $q_{1},q_{2}$ such that*

[TABLE]

where the size of the zero block at the bottom right corner is $(r-(q_{1}m+q_{2}n))\times(s-(q_{1}n+q_{2}m))$.

To prove the above theorem, we need the following lemma, which is valid for both real and complex matrices. See [4, Lemma 1] for the complex case. Recall that $A^{*}=A^{t}$ in the real case.

Lemma 3.2**.**

Let $A,B\in{\bf M}_{m,n}$. The following conditions are equivalent to each other.

1. (a)

$A^{*}B=0_{n}$* and $AB^{*}=0_{m}$.* 2. (b)

$AA^{*}B+BA^{*}A=0_{m,n}$.

Proof.

It suffices to prove (b)$\implies$(a). Observe that from (b) we have

[TABLE]

Taking adjoints of the Hermitian matrices, we have

[TABLE]

Therefore, the positive semi-definite $n\times n$ matrices $A^{*}A$ and $B^{*}B$ commute. By spectral theory, the product $(B^{*}B)(A^{*}A)=-(B^{*}A)(B^{*}A)^{*}$ is also positive semi-definite, and thus $B^{*}A=0$. Similarly, we have $AB^{*}=0$.

Proof of Theorem 3.1.

(a) Suppose $\Phi$ preserves zero triple products. By Lemma 3.2, if $A,B\in{\bf M}_{m,n}$ are disjoint, then $\Phi(A),\Phi(B)\in{\bf M}_{r,s}$ are disjoint. So, $\Phi$ has the asserted form by Theorem 2.1. The converse is clear.

(b) Suppose $\Phi$ is a JB*-triple homomorphism. Then it will preserve zero triple products, and thus by (a), be of the form (3.2). Since $E_{11}^{(3)}=E_{11}$, we have $\Phi(E_{11})^{(3)}=\Phi(E_{11})$. One gets the conclusions $Q_{1}=I_{q_{1}}$ and $Q_{2}=I_{q_{2}}$ as in (3.3). The converse is clear.

Recall that a rectangular matrix $A$ is called a partial isometry if $AA^{*}A=A$. Equivalently, $A$ has singular values from the set $\{1,0\}$. We state our result using the complex notation. Of course, in the real case, we have $X^{*}=X^{t}$, and a unitary matrix is a real orthogonal matrix. It turns out that JB*-triple homomorphisms are closely related to linear preservers of (disjoint) partial isometries. Some assertions in the following might be known to experts, at least in the complex case.

Theorem 3.3**.**

Suppose $\Phi:{\bf M}_{m,n}\to{\bf M}_{r,s}$ is a linear map. The following conditions are equivalent.

1. (a)

$\Phi$* maps partial isometries in ${\bf M}_{m,n}$ to partial isometries in ${\bf M}_{r,s}$.* 2. (b)

$\Phi$* sends disjoint (rank one) partial isometries to disjoint partial isometries.* 3. (c)

$\Phi$* preserves disjointness, and there is a nonzero partial isometry $P\in{\bf M}_{m,n}$ such that $\Phi(P)$ is a partial isometry.* 4. (d)

$\Phi$* preserves matrix triples with zero JB*-triple product, and there is a nonzero partial isometry $P\in{\bf M}_{m,n}$ such that $\Phi(P)$ is a partial isometry.* 5. (e)

$\Phi$* is a JB*-triple homomorphism and has the form (3.3).*

Proof.

The implication (e) $\implies$ (a) is clear.

(a) $\implies$ (b): Let $A\in{\bf M}_{m,n}$ be a rank one partial isometry, and $\Phi(A)=U\begin{pmatrix}I_{k}&0\cr 0&0\cr\end{pmatrix}V$, where $U\in{\bf U}_{r},V\in{\bf U}_{s}$. Suppose $B\in{\bf M}_{m,n}$ is a rank one partial isometry disjoint from $A$ such that $\Phi(B)=U\begin{pmatrix}Y_{11}&Y_{12}\cr Y_{21}&Y_{22}\cr\end{pmatrix}V$ with $Y_{11}\in{\bf M}_{k}$. Because $\Phi(A)\pm\Phi(B)$ are partial isometries, we see that the Euclidean norm of each of the first $k$ columns of $\Phi(A)+\Phi(B)$ and $\Phi(A)-\Phi(B)$ is not larger than one. Thus, $Y_{11},Y_{21}$ are zero matrices. Considering the norms of the first $k$ rows of $\Phi(A)+\Phi(B)$, we see that $Y_{12}$ is the zero matrix as well. Thus, $\Phi(A),\Phi(B)$ are disjoint partial isometries in ${\bf M}_{r,s}$. In general, due to the singular value decomposition, every rectangular matrix can be written as a linear sum of disjoint rank one partial isometries. Thus $\Phi$ sends disjoint partial isometries to disjoint partial isometries.

(b) $\implies(c)$: $\Phi$ preserves disjointness of rank one partial isometries, and hence preserves disjointness due to the singular value decomposition. Evidently, it sends a nonzero partial isometry to a partial isometry.

(c) $\implies$ (e): Because $\Phi$ preserves disjointness, $\Phi$ has the form described in Theorem 2.1. By the fact that $\Phi$ sends a nonzero partial isometry to a partial isometry, we see that $Q_{1},Q_{2}$ are identity matrices. So, conditions (a), (b), (c) and (e) are equivalent.

By Lemma 3.2 we have (d) $\implies$ (c). The implication (e) $\implies$ (d) is also clear.

Several remarks are in order. Theorem 3.1 and Theorem 3.3 are also valid for real linear maps $\Phi:{\bf H}_{n}\rightarrow{\bf M}_{r,s}$. Note that self-adjoint partial isometries are exactly differences $p-q$ of two orthogonal projections. Indeed, we can further assume that the co-domain is ${\bf H}_{r}$, i.e., $\Phi:{\bf H}_{n}\rightarrow{\bf H}_{r}$. Then we can arrange $U=V^{*}$ in (3.2) and (3.3), at the expenses that $Q_{1}$, $Q_{2}$ may have negative diagonal matrices in (3.2), and (3.3) may look like

[TABLE]

where $q_{1}^{+},q_{1}^{-},q_{2}^{+},q_{2}^{-}$ are nonnegative integers and the zero block matrix in the bottom right corner has size $(r-((q_{1}^{+}+q_{1}^{-})m+(q_{2}^{+}+q_{2}^{-})n))\times(r-((q_{1}^{+}+q_{1}^{-})n+(q_{2}^{+}+q_{2}^{-})m))$.

Theorem 3.1 (a) allows us to obtain the following general result on linear preserver of functions of $JB^{*}$-triple product on matrices.

Corollary 3.4**.**

Let $\nu_{1},\nu_{2}$ be scalar functions on $M_{m,n}$ and $M_{r,s}$ such that

[TABLE]

for all $A$ in $M_{m,n}$ or $M_{r,s}$, respectively. Suppose a linear map $\Phi:M_{m,n}\rightarrow M_{r,s}$ satisfies

[TABLE]

Then $\Phi$ has the form $(\ref{s-form-2})$.

This corollary can be used to determine the structure of linear preservers of functions on triple product of matrices easily. We mention a few examples in the following related to the study in [11, 10, 5, 12, 16, 17, 23] and their references.

Suppose a linear map $\Phi:M_{m,n}\rightarrow M_{r,s}$ satisfies (3.4), where $\nu_{1},\nu_{2}$ are norms on matrices. Then $\Phi$ has the form $(\ref{s-form-2})$. From this, one may easily deduce the conditions on $U,V,Q_{1},Q_{2},$ etc. to ensure the converse of the statement. For example, if $\nu_{1},\nu_{2}$ are the operator norms, then $U,V$ can be any unitary matrices and the operator norm of $D_{1}\oplus D_{2}$ has to be one.

Suppose $(m,r)=(n,s)$, ${\mathbb{F}}={\mathbb{C}}$, and $\nu_{1},\nu_{2}$ are the numerical radius. Then $\Phi:M_{n}\rightarrow M_{r}$ satisfies (3.4) if and only if $\Phi$ has the form (3.2) with $V=RU^{*}$ for a diagonal matrix $R$ such that $((I_{n}\otimes Q_{1})\oplus(I_{n}\otimes Q_{2})\oplus 0)R$ has numerical radius $1$. From this, one may further deduce that when $(m,r)=(n,s)$, ${\mathbb{F}}={\mathbb{C}}$, and $\nu_{1},\nu_{2}$ are the numerical range, $\Phi:M_{n}\rightarrow M_{r}$ satisfies (3.4) if and only if $\Phi$ has the form (3.3) with $V=U^{*}$. Similarly, we can treat the linear preservers $\Phi:M_{n}\rightarrow M_{r}$ leaving invariant the pseudo spectral radius, pseudo spectrum, and other types of scalar or non-scalar functions.

4. Nonsurjective norm preservers

Denote the singular values of $A\in{\bf M}_{m,n}$ by $s_{1}(A)\geq\cdots\geq s_{h}(A)$ for $h=\min\{m,n\}$. For $p>0$, let

[TABLE]

If $p\geq 1$, then $S_{p}(A)$ is known as the Schatten $p$-norm. In particular, $S_{2}(A)=(\sum_{j=1}^{h}s_{j}(A))^{1/2}=({\rm tr}\,(A^{*}A))^{1/2}$, which is called the Frobenius norm, equips ${\bf M}_{m,n}$ as a Hilbert space. For $1\leq p<+\infty$ but $p\neq 2$, a linear operator $\Psi:{\bf M}_{m,n}\rightarrow{\bf M}_{m,n}$ satisfies $S_{p}(\Psi(A))=S_{p}(A)$ for all $A\in{\bf M}_{m,n}$ if and only if $\Psi$ has the form $A\mapsto UAV$, or $A\mapsto UA^{t}V$ in case $m=n$, for some $U\in{\bf U}_{m},V\in{\bf U}_{n}$ (see, e.g., [5, 25]).

It is more difficult to characterize linear isometries from ${\bf M}_{m,n}$ to ${\bf M}_{r,s}$ for $(m,n)\neq(r,s)$. Only very few results are known; see, for example, [8, 23]. With Theorem 2.1, we get the following result.

Theorem 4.1**.**

Suppose $m,n\geq 2$, $p\in(0,2)\cup(2,+\infty)$, and $\Phi:{\bf M}_{m,n}\rightarrow{\bf M}_{r,s}$ is a linear map. The following conditions are equivalent.

• (a)

$S_{p}(\Phi(A))=S_{p}(A)$* for all $A\in{\bf M}_{m,n}$.*

• (b)

$S_{p}(\Phi(A))=S_{p}(A)$* for all $A\in{\bf M}_{m,n}$ with rank at most 2.*

• (c)

There are $U\in{\bf U}_{r},V\in{\bf U}_{s}$, and diagonal matrices $Q_{1}\in{\bf M}_{q_{1}},Q_{2}\in{\bf M}_{q_{2}}$ with positive diagonal entries such that $S_{p}(Q_{1}\oplus Q_{2})=1$ and

[TABLE]

Here $Q_{1}$ or $Q_{2}$ may be vacuous.

Proof.

The implications (c) $\implies$ (a) $\implies$ (b) are clear. For the implication (b) $\implies$ (c), it follows from a result of McCarthy [29, Theorem 2.7] that $\Phi$ preserves disjointness for rank one matrix pairs. By Theorem 2.1, we get the form of $\Phi$. Applying the fact that $S_{p}(\Phi(E_{11}))=S_{p}(E_{11})$, we easily deduce that $S_{p}(Q_{1}\oplus Q_{2})=1$ .

For $1\leq k\leq\min\{m,n\}$, the Ky Fan $k$-norm of $A$ is defined by

[TABLE]

Linear isometries for the Ky Fan $k$-norm have been studied. Seeing Theorem 4.1, one may think that a similar extension for the Ky Fan $k$-norm can be obtained by similar arguments. It turns out that this can only be done for the complex case because there are real linear isometries for Ky Fan $k$-norms that do not preserve disjointness; see [18, 25]. This reinforces the fact that proof techniques for complex matrices may not apply to real matrices, and it is quite remarkable that a uniform proof of Theorem 2.1 can be used for both real and complex matrices. In any event, we have the following theorem supplementing [23, Theorem 1.1], in which the linear map $\Phi:{\bf M}_{m,n}({\mathbb{C}})\rightarrow{\bf M}_{r,s}({\mathbb{C}})$ is assumed to satisfy that

[TABLE]

Theorem 4.2**.**

Suppose $2\leq k^{\prime}\leq\min\{m,n\}$ and $1\leq k\leq\min\{r,s\}$. The following conditions are equivalent for a linear map $\Phi:{\bf M}_{m,n}({\mathbb{C}})\rightarrow{\bf M}_{r,s}({\mathbb{C}})$.

1. (a)

$F_{k}(\Phi(A))=F_{k^{\prime}}(A)$* for all $A\in{\bf M}_{m,n}({\mathbb{C}})$ with rank at most $2$.* 2. (b)

There are unitary matrices $U\in{\bf M}_{r}({\mathbb{C}})$, $V\in{\bf M}_{s}({\mathbb{C}})$ and positive-definite diagonal matrices $Q_{1},Q_{2}$ (maybe vacuous) of size $q_{1},q_{2}$ such that $k\geq 2(q_{1}+q_{2})$, $Q_{1}\oplus Q_{2}$ has trace $1$, and

[TABLE]

Proof.

The implication (b) $\implies$ (a) is plain.

(a) $\implies$ (b). By [23, Lemma 2.2], $\Phi$ preserves disjoint rank one pairs. By Theorem 2.1, $\Phi$ carries the form (4.1). Consider $A_{\epsilon}=E_{11}+\epsilon E_{22}$ for $0\leq\epsilon<1$. Using (4.1), we can assume

[TABLE]

for some fixed scalars $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{q}>0$ with $q=q_{1}+q_{2}$.

Suppose $k\leq q$ first. Since $k^{\prime}\geq 2$, we have

[TABLE]

This yields a contradiction, because $[0,\lambda_{k}/\lambda_{1}]$ contains infinitely many points $\epsilon$.

Suppose $0<r=k-q<q$. Then we have

[TABLE]

This implies $\lambda_{1}+\lambda_{2}+\cdots+\lambda_{q}=1$, and $1+\epsilon=1+\epsilon\lambda_{1}+\cdots+\epsilon\lambda_{r}$ for all $0<\epsilon\leq\lambda_{q}/\lambda_{r+1}$. This gives us the contradiction that $\lambda_{r+1}=\cdots=\lambda_{q}=0$.

Hence, $k\geq 2q$. In this case, we have

[TABLE]

This gives $1=\lambda_{1}+\lambda_{2}+\cdots+\lambda_{q}$, which equals the trace of $Q_{1}\oplus Q_{2}$.

5. Final remarks and future research

It would be interesting to extend our results in Sections 2 and 3 to the (real or complex) linear space $B(H,K)$ of bounded linear operators between infinite dimensional Banach spaces $H$ and $K$, or to general JB*-triples. Our approach depends on the singular value decomposition of matrices, which is a finite dimensional feature. New techniques will be needed to extend our results.

To conclude the paper, we list several comments and questions concerning the results in Section 4.

1. (1)

As pointed out in [8], the problem for the operator norm, i.e., Ky Fan $1$-norm, is difficult. 2. (2)

Many real linear isometries for Ky Fan $k$-norms also preserve disjointness (although there are exceptions). It would be nice to investigate a version of Theorem 4.2 such that the conclusion also hold for real matrices. 3. (3)

For any linear isometry which preserves disjoint rank one pairs, we can apply Theorem 2.1. It is interesting to characterize such norms other than the Schatten $p$-norms and the Ky Fan $k$-norms. Suggested by the asserted form (4.1), we should put emphasis on unitarily invariant norms. 4. (4)

We have similar results for real symmetric and complex Hermitian matrices. Besides $S_{p}(A)$ and $F_{k}(A)$, can we do it for the $k$-numerical radius on Hermitian matrices ${\bf H}_{n}$ defined by

[TABLE] 5. (5)

In fact, one can also ask for characterizations of $k$-numerical radius preservers $\Phi:{\bf M}_{n}\rightarrow{\bf M}_{r}$. 6. (6)

One may consider linear preservers or non-linear preservers for other types of norms or functions on rectangular matrices, Hermitian, symmetric, or skew-symmetric matrix spaces that are related to disjointness preserving maps.

Acknowledgment

Li is an affiliate member of the Institute for Quantum Computing, University of Waterloo. He is an honorary professor of Shanghai University. His research was supported by USA NSF grant DMS 1331021, Simons Foundation Grant 351047, and NNSF of China Grant 11571220. This research was started when he visited Taiwan in 2018 supported by grants from Taiwan MOST. He would like to express his gratitude to the hospitality of several institutions, including the Academia Sinica, National Taipei University of Science and Technology, National Chung Hsing University, and National Sun Yat-sen University. He would also like to thank Dr. Daniel Puzzuoli for some helpful discussions.

Tsai, Wang and Wong are supported by Taiwan MOST grants 105-2115-M-027-002-MY2, 106-2115-M-005-001-MY2 and 106-2115-M-110-006-MY2, respectively.