# Normal Categories of Normed Algebra of Finite Rank Bounded Operators

P. G. Romeo1, A. Anju2 1 Department of Mathematics, Cochin University of
Science And Technology, Kerala, India.  romeo_<EMAIL_ADDRESS>2
Department of Mathematics, Cochin University of Science And Technology,
Kerala, India<EMAIL_ADDRESS>
(Date:
2 Corresponding author)

###### Abstract.

In this article, we introduce the normal category $\mathscr{L}(\mathcal{S})$
[$\mathscr{R}(\mathcal{S})$] of principal left [right] ideals of the normed
algebra $\mathcal{S}$ of all finite rank bounded operators on a Hilbert space
$H$ and is shown that they are isomorphic, using Hilbert space duality. We
also described the semigroup of all normal cones in $\mathscr{L}(\mathcal{S})$
which is isomorphic to the semigroup of all finite rank operators on $H$.
Further, we construct bounded normal cones in $\mathscr{L}(\mathcal{S})$ such
that the set of all bounded normal cones in $\mathscr{L}(\mathcal{S})$ is a
normed algebra isomorphic to the normed algebra $\mathcal{S}$.

###### Key words and phrases:

Regular semigroup, Semigroup of finite rank bounded operators on a Hilbert
space, Normal category, Normal cone.

###### 2020 Mathematics Subject Classification:

20M17, 20M50, 47B02.

##  1\. Introduction

In 1974, P. A. Grillet and K. S. S. Nambooripad [6] introduced the cross-
connections of regular partially ordered sets to study the structure of
fundamental regular semigroups. Subsequently, K. S. S. Nambooripad [10]
extended this cross-connection to describe the structure of arbitrary regular
semigroups using normal categories rather than regular partially ordered sets.

In this paper, we consider the regular semigroup $\mathcal{S}$ of all finite
rank bounded operators on a Hilbert space $H$ as a normed algebra and
construct bounded normal cones in the category of principal left ideals of
$\mathcal{S}$ in such a way that the cones forms a normed algebra isomorphic
to the normed algebra $\mathcal{S}$. For this, consider the normal category
$\mathscr{L}(\mathcal{S})[\mathscr{R}(\mathcal{S})]$ of principal left [right]
ideals of the normed algebra $\mathcal{S}$ whose morphisms are bounded linear
maps and each hom-set of $\mathscr{L}(\mathcal{S})[\mathscr{R}(\mathcal{S})]$
is a normed space. Note that, the normal category $\mathscr{L}(\mathcal{S})$
is isomorphic to the normal category $\mathscr{F}(H)$ of finite dimensional
subspaces of $H$ with linear maps as morphisms, and this isomorphism preserves
the normed space structure of hom-sets. Similarly, the normal category
$\mathscr{R}(\mathcal{S})$ is isomorphic to the normal category
$\mathscr{F}(H^{\prime})$ of finite dimensional subspaces of the dual of $H$
with linear maps as morphisms. Further, using Hilbert space duality it is
shown that the normal categories $\mathscr{L}(\mathcal{S})$ and
$\mathscr{R}(\mathcal{S})$ are isomorphic, and restriction of this isomorphism
to hom-sets is a surjective conjugate-linear isometry. Finally, it is shown
that the regular semigroup of all normal cones in $\mathscr{L}(\mathcal{S})$
is isomorphic to the semigroup of all finite rank operators on $H$ and we
described the bounded normal cone in $\mathscr{L}(\mathcal{S})$ such that the
set of all bounded normal cones in $\mathscr{L}(\mathcal{S})$ produces a
normed algebra isomorphic to the normed algebra $\mathcal{S}$. Similarly, the
bounded normal cones in $\mathscr{R}(\mathcal{S})$ is also a normed algebra
which is conjugate-linear isomorphic to the normed algebra $\mathcal{S}$.

##  2\. Preliminaries

The following we recall provide definitions and results regarding cross-
connections needed in the sequel, for more details see [9], [11], [10].
Let $\mathscr{C}$ to be a category. Then $\textit{{v}}\mathscr{C}$ denotes the
object class of $\mathscr{C}$ and $\mathscr{C}$ itself denotes the morphism
class of $\mathscr{C}$. For objects $a,b\in\textit{{v}}\mathscr{C}$, the
collection of all morphisms from $a$ to $b$ is $\mathscr{C}(a,b)$, called a
hom-set and for each $a\in\textit{{v}}\mathscr{C},\,1_{a}$ is the identity
morphism in $\mathscr{C}(a,a)$. Let the identity functor on the category
$\mathscr{C}$ be $1_{\mathscr{C}}$. A category $\mathscr{C}$ is termed as a
small category if the class of all morphisms in $\mathscr{C}$ is a set.

A category $\mathscr{P}$ is a preorder if for any
$p,p^{\prime}\in\textit{{v}}\mathscr{P}$, the hom-set
$\mathscr{P}(p,p^{\prime})$ contains at most one morphism. Let $\mathscr{P}$
be a preorder, then the relation $\subseteq$ on $\textit{{v}}\mathscr{P}$
defined by $p\subseteq p^{\prime}$ if and only if
$\mathscr{P}(p,p^{\prime})\neq\phi$ is a quasiorder. A preorder $\mathscr{P}$
is said to be a strict preorder if $\subseteq$ on $\textit{{v}}\mathscr{P}$ is
a partial order.

######  Definition 2.1.

_[10]_ A small category $\mathscr{C}$ is called a category with subobjects if
$\mathscr{C}$ has a strict preorder $\mathscr{P}$ as a subcategory with
$\textit{{v}}\mathscr{P}=\textit{{v}}\mathscr{C}$ such that every morphism in
$\mathscr{P}$ is a monomorphism in $\mathscr{C}$ and if $f=hg$ for
$f,g\in\mathscr{P}$, $h\in\mathscr{C}$, then $h\in\mathscr{P}$.

In this case, the morphisms in $\mathscr{P}$ are called inclusions in
$\mathscr{C}$, and a morphism $q$ in $\mathscr{C}$ is a retraction if $q$ is a
right inverse of an inclusion in $\mathscr{C}$. A normal factorization of a
morphism $f$ in a category $\mathscr{C}$ with subobjects is a factorization of
the form $f=quj$ where $q$ is a retraction, $u$ is an isomorphism, and $j$ is
an inclusion. If $f=quj$ is a normal factorization of $f$, then $f^{\circ}=qu$
is said to be the epimorphic component of $f$.

Let $\mathscr{C}$ be a category with subobjects. A cone $\gamma$ in
$\mathscr{C}$ with vertex $d\in\textit{{v}}\mathscr{C}$ is a map from
$\textit{{v}}\mathscr{C}$ to $\mathscr{C}$ such that
$\gamma(c)\in\mathscr{C}(c,d)$ $\forall c\in\textit{{v}}\mathscr{C}$, and if
$j\in\mathscr{C}(c^{\prime},c)$ is an inclusion in $\mathscr{C}$, then
$j\gamma(c)=\gamma(c^{\prime})$. For a cone $\gamma$ in
$\mathscr{C},\,c_{\gamma}$ denotes the vertex of $\gamma$ and for each
$c\in\textit{{v}}\mathscr{C}$, the morphism $\gamma(c)$ is called the
component of $\gamma$ at $c$. A cone $\gamma$ with vertex $d$ is said to be a
normal cone if at least one $c\in\textit{{v}}\mathscr{C}$ exists such that
$\gamma(c)$ is an isomorphism from $c$ to $d$ _[3]_.

######  Definition 2.2.

_([3],[2])_ A normal category $\mathscr{C}$ is a category with subobjects such
that every morphism in $\mathscr{C}$ has a normal factorization, every
inclusion in $\mathscr{C}$ splits, and for each $d\in\textit{{v}}\mathscr{C}$,
there is a normal cone $\gamma$ with vertex $d$ and $\gamma(d)=1_{d}$.

Two normal categories $\mathscr{C}$ and $\mathscr{D}$ are isomorphic if there
exists inclusion preserving functors $F:\mathscr{C}\rightarrow\mathscr{D}$ and
$G:\mathscr{D}\rightarrow\mathscr{C}$ such that $FG=1_{\mathscr{C}}$ and
$GF=1_{\mathscr{D}}$ (_[10],[13]_). If $\mathscr{C}$ is a normal category,
then the set $T\mathscr{C}$ of all normal cones in $\mathscr{C}$ is a regular
semigroup under the product defined by
$(\gamma\cdot\sigma)(c)=\gamma(c)(\sigma(c_{\gamma}))^{\circ}$ for
$\gamma,\sigma\in T\mathscr{C}$ and $c\in\textit{{v}}\mathscr{C}$, where
$(\sigma(c_{\gamma}))^{\circ}$ is the epimorphic component of
$\sigma(c_{\gamma})$ (_[10],[2]_).

######  Proposition 2.1.

[2] If two normal categories $\mathscr{C}$ and $\mathscr{D}$ are isomorphic,
then the semigroups $T\mathscr{C}$ and $T\mathscr{D}$ are isomorphic.

Let $S$ be a regular semigroup and $E(S)$ be the set of all idempotents in
$S$. The category $\mathscr{L}(S)$ of principal left ideals of $S$ is a
category whose objects are principal left ideals $Se$ of $S$ for $e\in E(S)$,
and for objects $Se$ and $Sf$, the hom-set $\mathscr{L}(S)(Se,Sf)$ consists of
partial right translations $\rho(e,u,f)$ for each $u\in eSf$, which maps $x\in
Se$ to $xu\in Sf$ [10]. Let $S^{op}$ be the left-right dual of $S$. Then the
category $\mathscr{R}(S)$ of principal right ideals of $S$ is
$\mathscr{L}(S^{op})$ (_[9],[10]_).

######  Lemma 2.1.

[10] Let $\mathscr{L}(S)$ be the category of principal left ideals of a
regular semigroup $S$. Then,

  1. (1)

$\mathscr{L}(S)$ is a category with subobjects whose inclusions are the usual
set inclusions.

  2. (2)

For every $u,f\in E(S)$, the map $u\mapsto\rho(e,u,f)$ from $eSf$ to
$\mathscr{L}(S)(Se,Sf)$ is a bijection.

  3. (3)

$\rho(e,u,f)=\rho(e^{\prime},v,f^{\prime})$ if and only if
$e\mathscr{L}e^{\prime}$, $f\mathscr{L}f^{\prime}$, $u\in eSf$, $v\in
e^{\prime}Sf^{\prime}$ and $v=e^{\prime}u$.

  4. (4)

$\rho(e,u,f)$ is an isomorphism in $\mathscr{L}(S)$ if and only if
$e\,\mathscr{R}\,u\,\mathscr{L}\,f$.

  5. (5)

If $Se\subseteq Sf$, then $j=\rho(e,e,f)$ is the inclusion from $Se$ to $Sf$,
and $\rho$ is a retraction of $j$ if and only if $\rho=\rho(f,g,e)$ for some
$g\in E(L_{e})\cap\omega(f)$.

Let $a\in S$ and $f\in E(L_{a})$. The map $\rho^{a}$ from
$\textit{{v}}\mathscr{L}(S)$ to $\mathscr{L}(S)$ defined by
$\rho^{a}(Se)=\rho(e,ea,f)$ for $Se\in\textit{{v}}\mathscr{L}(S)$ is a normal
cone in $\mathscr{L}(S)$ with vertex $Sa$. The normal cones of the form
$\rho^{a}$ in $\mathscr{L}(S)$ are called principal cones [10].

######  Proposition 2.2.

[10] Let $S$ be a regular semigroup. Then, the map $a\mapsto\rho^{a}$ is a
homomorphism from $S$ to $T\mathscr{L}(S)$.

##  3\. Ideal categories of semigroup of operators on a Hilbet space

It is well known that finite rank bounded operators on a Hilbert space $H$
over a field K, where K is either $\mathbb{R}$ or $\mathbb{C}$ is a regular
semigroup under the function composition $(T_{1}T_{2})(x)=T_{2}(T_{1}(x))$ for
$T_{1},T_{2}\in\mathcal{S}$, and $x\in H$ and denote it by $S$. In the
following, we discuss properties of $\mathscr{L}(\mathcal{S})$ and
$\mathscr{R}(\mathcal{S})$ when $\mathcal{S}$ is regarded as a normed algebra.

The object set $\textit{{v}}\mathscr{L}(\mathcal{S})$ of the category
$\mathscr{L}(\mathcal{S})$ of principal left ideals of $\mathcal{S}$ is

$\textit{{v}}\mathscr{L}(\mathcal{S})=\\{\mathcal{S}P:P\in E(\mathcal{S})\\},$

and for $P_{1},P_{2}\in E(\mathcal{S})$, the hom-set
$hom\,(\mathcal{S}P_{1},\mathcal{S}P_{2})$ is

$\mathscr{L}(\mathcal{S})(\mathcal{S}P_{1},\mathcal{S}P_{2})=\\{\rho(P_{1},T,P_{2}):T\in
P_{1}\mathcal{S}P_{2}\\},$

where the morphism $\rho(P_{1},T,P_{2})$ maps $A\in\mathcal{S}P_{1}$ to
$AT\in\mathcal{S}P_{2}$. It is clear that each object $\mathcal{S}P$ is a left
ideal of the normed algebra $\mathcal{S}$, and for $T\in
P_{1}\mathcal{S}P_{2}$, the morphism $\rho(P_{1},T,P_{2})$ is the composition
operator determined by $T$ and $\rho(P_{1},T,P_{2})$ is a bounded linear map.
Dually of principal right ideals of $\mathcal{S}$, with morphisms of the form
$\lambda(P_{2},T,P_{1})$ maps $A\in\mathcal{P}_{1}S$ to
$TA\in\mathcal{P}_{2}S$ is the category $\mathscr{R}(\mathcal{S})$. For
$T\in\mathcal{S}$, $R(T)$ and $Z(T)$ denote the range space and the zero space
of $T$, respectively. We often use the following lemma without indicating it.

######  Lemma 3.1.

[12] If $T_{1},T_{2}\in\mathcal{S}$, then

  1. _(i)_

$\mathcal{S}T_{1}\subseteq\mathcal{S}T_{2}$ if and only if $R(T_{1})\subseteq
R(T_{2})$.

  2. _(ii)_

$T_{1}\mathcal{S}\subseteq T_{2}\mathcal{S}$ if and only if $Z(T_{2})\subseteq
Z(T_{1})$.

Hence, for principal left ideals $\mathcal{S}T_{1}$ and $\mathcal{S}T_{2}$,
$\mathcal{S}T_{1}=\mathcal{S}T_{2}$ if and only if $R(T_{1})=R(T_{2})$, and
for principal right ideals $T_{1}\mathcal{S}$ and $T_{2}\mathcal{S}$,
$T_{1}\mathcal{S}=T_{2}\mathcal{S}$ if and only if $Z(T_{1})=Z(T_{2})$.

It is known that $\mathcal{S}$ is an involution semigroup under the involution
$T\mapsto T^{*}$, where $T^{*}$ is the adjoint of $T$ and an idempotent $P$ in
$\mathcal{S}$ is a projection if and only if $Z(P)=R(P)^{\perp}$. Thus the
objects of $\mathscr{L}(\mathcal{S})$ and $\mathscr{R}(\mathcal{S})$ can
uniquely be identified by the projections in $\mathcal{S}$. For every
$\mathcal{S}P\in\textit{{v}}\mathscr{L}(\mathcal{S})$, $R(P)$ is a finite-
dimensional subspace of $H$, call it $M$, then
$\mathcal{S}P=\mathcal{S}P_{M}$, where $P_{M}$ is the projection of $H$ onto
$M$ and

$\textit{{v}}\mathscr{L}(\mathcal{S})=\\{\mathcal{S}P_{M}:M\text{ is a finite-
dimensional subspace of }H\\}.$

Dually for the category $\mathscr{R}(\mathcal{S})$ of principal right ideals
of $\mathcal{S}$, if $P\in E(\mathcal{S})$, then
$P\mathcal{S}=P_{M}\mathcal{S}$, where $M=Z(P)^{\perp}$. For any finite-
dimensional subspace $M$ of $H$, if $T\in\mathcal{S}P_{M}$, then
$R(T)\subseteq M$. Conversely, if $T\in\mathcal{S}$ and $R(T)\subseteq M$,
then $\mathcal{S}T\subseteq\mathcal{S}P_{M}$ implies $T\in\mathcal{S}P_{M}$.
Hence, the operators in $\mathcal{S}P_{M}$ can be characterized as
$\mathcal{S}P_{M}=\\{T\in\mathcal{S}:R(T)\subseteq M\\}.$

$\displaystyle\text{But, }T\in P_{M}\mathcal{S}$
$\displaystyle\Longleftrightarrow T^{*}\in\mathcal{S}P_{M}$
$\displaystyle\Longleftrightarrow R(T^{*})\subseteq M$
$\displaystyle\Longleftrightarrow M^{\perp}\subseteq Z(T).$

Thus, for finite-dimensional subspaces $M$ and $N$ of $H$, $T\in
P_{M}\mathcal{S}P_{N}$ if and only if $T\in\mathcal{S},\,M^{\perp}\subseteq
Z(T)$, and $R(T)\subseteq N$. Hence, morphisms in the hom-sets of the category
$\mathscr{L}(\mathcal{S})$ of principal left ideals of $\mathcal{S}$ can be
identified as

$\mathscr{L}(\mathcal{S})(\mathcal{S}P_{M},\mathcal{S}P_{N})=\\{\rho(P_{M},T,P_{N}):T\in\mathcal{S},\,M^{\perp}\subseteq
Z(T),\,R(T)\subseteq N\\}.$

Summarizing the above results, we can state the following lemma.

######  Lemma 3.2.

Let $\mathscr{L}(\mathcal{S})$ be the category of principal left ideals of
$\mathcal{S}$. Then, the following holds.

  1. (1)

$\textit{{v}}\mathscr{L}(\mathcal{S})=\\{\mathcal{S}P_{M}:M\text{ is a finite-
dimensional subspace of }H\\}$.

  2. (2)

If $M$ is a finite-dimensional subspace of $H$, then
$\mathcal{S}P_{M}=\\{T\in\mathcal{S}:R(T)\subseteq M\\}.$

  3. (3)

If $M$ and $N$ are finite-dimensional subspaces of $H$, then the hom-set

$\mathscr{L}(\mathcal{S})(\mathcal{S}P_{M},\mathcal{S}P_{N})=\\{\rho(P_{M},T,P_{N}):T\in\mathcal{S},\,M^{\perp}\subseteq
Z(T),\,R(T)\subseteq N\\}.$

######  Example 3.1.

Let $H=\mathbb{R}^{2}$ and let $L$ be a line passing through the origin in
$\mathbb{R}^{2}$. If $\theta$ is the angle between the line $L$ and the
$X$-axis, then the projection of $H$ onto the line $L$ is given by
$P_{\theta}=\begin{bmatrix}\cos^{2}\theta&\cos\theta\sin\theta\\\
\cos\theta\sin\theta&\sin^{2}\theta\end{bmatrix}.$ Thus, the objects of
$\mathscr{L}(\mathcal{S})$ are $\mathcal{S}P_{H},\mathcal{S}P_{\\{0\\}},$ and
$\mathcal{S}P_{\theta}$, where $0\leq\theta<\pi.$

######  Example 3.2.

Let $H=\mathbb{R}^{3},M=X$-axis and $N=YZ$-plane. Then,

  * •

$\mathcal{S}P_{M}=\left\\{\begin{bmatrix}a&0&0\\\ b&0&0\\\
c&0&0\end{bmatrix}:a,b,c\in\mathbb{R}\right\\},$

  * •

$\mathscr{L}(\mathcal{S})(\mathcal{S}P_{M},\mathcal{S}P_{N})=\left\\{\rho(P_{M},T,P_{N}):T=\begin{bmatrix}0&a&b\\\
0&0&0\\\ 0&0&0\end{bmatrix};a,b\in\mathbb{R}\right\\}.$

We now state the dual of the Lemma 3.2.

######  Lemma 3.3.

Let $\mathscr{R}(\mathcal{S})$ be the category of principal right ideals of
$\mathcal{S}$. Then we have the following.

  1. (1)

$\textit{{v}}\mathscr{R}(\mathcal{S})=\\{P_{M}\mathcal{S}:M\text{ is a finite-
dimensional subspace of }H\\}$.

  2. (2)

If $M$ is a finite-dimensional subspace of $H$, then
$P_{M}\mathcal{S}=\\{T\in\mathcal{S}:M^{\perp}\subseteq Z(T)\\}$.

  3. (3)

If $M$ and $N$ are finite-dimensional subspaces of $H$, then the hom-set

$\mathscr{R}(\mathcal{S})(P_{M}\mathcal{S},P_{N}\mathcal{S})=\\{\lambda(P_{M},T,P_{N}):T\in\mathcal{S},\,N^{\perp}\subseteq
Z(T),\,R(T)\subseteq M\\}.$

Next, we proceed to establish that the hom-sets of $\mathscr{L}(\mathcal{S})$
are normed spaces. Let $M$ and $N$ be finite-dimensional subspaces of $H$. If
$T_{1},T_{2}\in P_{M}\mathcal{S}P_{N}$, then $T_{1}+T_{2},\,kT_{1}\in
P_{M}\mathcal{S}P_{N}$, for all $k\in\textbf{K}$. Hence,

$\displaystyle\rho(P_{M},T_{1},P_{N})+\rho(P_{M},T_{2},P_{N})$
$\displaystyle=\rho(P_{M},T_{1}+T_{2},P_{N}),\text{ and }$ $\displaystyle
k\rho(P_{M},T_{1},P_{N})$ $\displaystyle=\rho(P_{M},kT_{1},P_{N}).$

define addition and scalar multiplication on
$\mathscr{L}(\mathcal{S})(\mathcal{S}P_{M},\mathcal{S}P_{N})$. Clearly
$\mathscr{L}(\mathcal{S})(\mathcal{S}P_{M},\mathcal{S}P_{N})$ is a linear
space and for $T\in P_{M}\mathcal{S}P_{N}$, $\rho=\rho(P_{M},T,P_{N})$ is a
bounded linear map from the normed space $\mathcal{S}P_{M}$ to the normed
space $\mathcal{S}P_{N}$ and the operator norm is a norm on
$\mathscr{L}(\mathcal{S})(\mathcal{S}P_{M},\mathcal{S}P_{N})$. Clearly,

$\Arrowvert\rho\Arrowvert\leq\Arrowvert T\Arrowvert.$ $\text{ But,
}\;\;\frac{\Arrowvert(P_{M})\rho\Arrowvert}{\Arrowvert
P_{M}\Arrowvert}=\Arrowvert P_{M}T\Arrowvert=\Arrowvert T\Arrowvert,$

therefore, $\mathscr{L}(\mathcal{S})(\mathcal{S}P_{M},\mathcal{S}P_{N})$ is a
normed space with respect to the norm
$\Arrowvert\rho(P_{M},T,P_{N})\Arrowvert=\Arrowvert T\Arrowvert$. Thus, we
have the following proposition.

######  Proposition 3.1.

Let $M$ and $N$ be finite-dimensional subspaces of $H$. Then the hom-set
$\mathscr{L}(\mathcal{S})(\mathcal{S}P_{M},\mathcal{S}P_{N})$ of
$\mathscr{L}(\mathcal{S})$ is a normed space.

The dual of this proposition is:

######  Proposition 3.2.

If $M$ and $N$ are finite-dimensional subspaces of $H$, then the hom-set
$\mathscr{R}(\mathcal{S})(P_{M}\mathcal{S},P_{N}\mathcal{S})$ of
$\mathscr{R}(\mathcal{S})$ is a normed space under the functions

$\displaystyle\lambda(P_{M},T_{1},P_{N})+\lambda(P_{M},T_{2},P_{N})$
$\displaystyle=\lambda(P_{M},T_{1}+T_{2},P_{N}),$ $\displaystyle
k\lambda(P_{M},T,P_{N})$ $\displaystyle=\lambda(P_{M},kT,P_{N}),\text{ and }$
$\displaystyle\Arrowvert\lambda(P_{M},T,P_{N})\Arrowvert$
$\displaystyle=\Arrowvert T\Arrowvert,$

for $T,T_{1},T_{2}\in P_{N}\mathcal{S}P_{M}$ and $k\in\textbf{K}.$

It is known that for finite-dimensional subspaces $M$ and $N$ of $H$,
$\mathcal{S}P_{M}\subseteq\mathcal{S}P_{N}$ if and only if $M\subseteq N$ and
the norm of any inclusion is $1$. Next we show, for each inclusion in
$\mathscr{L}(\mathcal{S})$, there exists a unique retraction with norm $1$.

######  Proposition 3.3.

Let $M$ and $N$ be finite-dimensional subspaces of $H$ with $M\subseteq N$.
Then $\rho(P_{N},P_{M},P_{M})$ is the unique retraction to the inclusion
$\rho(P_{M},P_{M},P_{N})$ with norm $1$.

###### Proof.

Let $P_{1},P_{2}\in E(\mathcal{S})$ with $R(P_{1})=N$ and $R(P_{2})=M$. If
$\rho(P_{1},P,P_{2})$ is a retraction to $\rho(P_{M},P_{M},P_{N})$ with norm
$1$, then $P\in E(\mathcal{S})$ and $\Arrowvert P\Arrowvert=1$, thus, $P$ is a
projection. By Lemma 2.1, $\rho(P_{1},P,P_{2})=\rho(P_{N},P_{N}P,P_{M})$. But,

$\displaystyle\rho(P_{M},P_{M},P_{M})$
$\displaystyle=\rho(P_{M},P_{M},P_{N})\rho(P_{1},P,P_{2})=\rho(P_{M},P_{M},P_{N})\rho(P_{N},P_{N}P,P_{M})$
$\displaystyle=\rho(P_{M},P_{M}P_{N}P,P_{M})=\rho(P_{M},P_{M}P,P_{N})$

hence, $P_{M}=P_{M}P$ by Lemma 2.1. So, $P=P_{M}$ and
$\rho(P_{1},P,P_{2})=\rho(P_{N},P_{N}P_{M},P_{M})=\rho(P_{N},P_{M},P_{M})$. ∎

For $T\in P_{M}\mathcal{S}P_{N}$, $\rho(P_{M},T,P_{N})$ is an isomorphism from
$\mathcal{S}P_{M}$ to $\mathcal{S}P_{N}$ if and only if
$P_{M}\,\mathscr{R}\,T\,\mathscr{L}\,P_{N}$ by Lemma 2.1. But,
$P_{M}\,\mathscr{R}\,T\,\mathscr{L}\,P_{N}$ if and only if $Z(T)=M^{\perp}$
and $R(T)=N$. Thus, the following proposition.

######  Proposition 3.4.

Let $T\in P_{M}\mathcal{S}P_{N}$. Then, $\rho(P_{M},T,P_{N})$ is an
isomorphism from $\mathcal{S}P_{M}$ to $\mathcal{S}P_{N}$ if and only if
$T|_{M}:M\rightarrow N$ is an isomorphism.

Thus it is seen that normal factorization for each morphism
$\rho(P_{M},T,P_{N})$ in $\mathscr{L}(\mathcal{S})$ as
$\rho(P_{M},T,P_{N})=\rho(P_{M},P_{U},P_{U})\rho(P_{U},T,P_{V})\rho(P_{V},P_{V},P_{N})$,
where $U=Z(T)^{\perp}$ and $V=R(T)$.

As a left /right dual the proposition below follows

######  Proposition 3.5.

Let $M$ and $N$ be finite-dimensional subspaces of $H$. Then we have the
following.

  1. (1)

$P_{M}\mathcal{S}\subseteq P_{N}\mathcal{S}$ if and only if $M\subseteq N$.

  2. (2)

If $M\subseteq N$, then $\lambda(P_{N},P_{M},P_{M})$ is the unique retraction
of the inclusion $\lambda(P_{M},P_{M},P_{N})$ with norm $1$.

  3. (3)

Let $T\in P_{N}\mathcal{S}P_{M}$. Then, $\lambda(P_{M},T,P_{N})$ is an
isomorphism from $P_{M}\mathcal{S}$ to $P_{N}\mathcal{S}$ if and only if
$T|_{N}$ is an isomorphism from $N$ onto $M$.

  4. (4)

For $T\in P_{N}\mathcal{S}P_{M}$,
$\lambda(P_{M},T,P_{N})=\lambda(P_{M},P_{V},P_{V})\lambda(P_{V},T,P_{U})\lambda(P_{U},P_{U},P_{N})$,
where $U=Z(T)^{\perp}$ and $V=R(T)$ is a normal factorization of
$\lambda(P_{M},T,P_{N})$.

##  4\. Isomorphism between $\mathscr{L}(\mathcal{S})$ and
$\mathscr{R}(\mathcal{S})$

Let $\mathscr{F}(H)$ be the category whose vertices are finite-dimensional
subspaces of a Hilbert space $H$. For finite-dimensional subspaces $M$ and $N$
of $H$, the hom-set $\mathscr{F}(H)(M,N)$ is the set of all linear maps from
$M$ to $N$. Then $\mathscr{F}(H)$ is a category with subobjects in which
inclusions are the usual subspace inclusion. For $M\subseteq N$, denote the
inclusion map from $M$ to $N$ by $J_{M}^{N}$. Obviously, every inclusion in
$\mathscr{F}(H)$ splits and for any linear map $T:M\longrightarrow N$,
$T=PT_{0}J$ is a normal factorization of $T$, where $P$ is the projection of
$M$ onto $Z(T)^{\perp}$, $T_{0}=T|_{Z(T)^{\perp}}$ is the isomorphism from
$Z(T)^{\perp}$ onto $R(T)$ and $J$ is the inclusion map from $R(T)$ to $N$.
Moreover, for each $M\in\textit{{v}}\mathscr{F}(H)$, the map
$\gamma:\textit{{v}}\mathscr{F}(H)\longrightarrow\mathscr{F}(H)$ defined by

$\gamma(N)=(P_{M})|_{N}:N\longrightarrow M,$

for $N\in\textit{{v}}\mathscr{F}(H)$ is a normal cone in $\mathscr{F}(H)$ with
vertex $M$ and $\gamma(M)=1_{M}$,that is., $\mathscr{F}(H)$ is a normal
category. Further, for each finite-dimensional subspaces $M$ and $N$ of $H$,
the linear space $\mathscr{F}(H)(M,N)$ is a normed space with respect to the
operator norm.

The following theorem says that the normal categories
$\mathscr{L}(\mathcal{S})$ and $\mathscr{F}(H)$ are isomorphic and the
isomorphism preserves normed space structure of hom-sets.

######  Theorem 4.1.

The normal category $\mathscr{L}(\mathcal{S})$ is isomorphic to the normal
category $\mathscr{F}(H)$, and for finite-dimensional subspaces $M$ and $N$ of
$H$, this isomorphism is a normed space isomorphism from the hom-set
$\mathscr{L}(\mathcal{S})(\mathcal{S}P_{M},\mathcal{S}P_{N})$ of
$\mathscr{L}(\mathcal{S})$ onto the hom-set $\mathscr{F}(H)(M,N)$ of
$\mathscr{F}(H)$.

###### Proof.

To prove $\mathscr{L}(\mathcal{S})$ is isomorphic to $\mathscr{F}(H)$ as
normal categories, it will suffices to show that there exists order-preserving
functors $F:\mathscr{L}(\mathcal{S})\rightarrow\mathscr{F}(H)$ and
$G:\mathscr{F}(H)\rightarrow\mathscr{L}(\mathcal{S})$ such that
$FG=1_{\mathscr{L}(\mathcal{S})}$ and $GF=1_{\mathscr{F}(H)}$. Define
$F:\mathscr{L}(\mathcal{S})\rightarrow\mathscr{F}(H)$ by,

$\displaystyle\text{ for
}\mathcal{S}P_{M}\in\textit{{v}}\mathscr{L}(\mathcal{S}),$
$\displaystyle\;F(\mathcal{S}P_{M})=M,$ $\displaystyle\text{ and for
}\rho(P_{M},T,P_{N}):\mathcal{S}P_{M}\rightarrow\mathcal{S}P_{N},$
$\displaystyle\;F(\rho(P_{M},T,P_{N}))=T|_{M}:M\rightarrow N.$

If $T_{1}\in P_{M}\mathcal{S}P_{N}$ and $T_{2}\in P_{N}\mathcal{S}P_{U}$, then

$\displaystyle F(\rho(P_{M},T_{1},P_{N})\rho(P_{N},T_{2},P_{U}))$
$\displaystyle=F(\rho(P_{M},T_{1}T_{2},P_{U}))=(T_{1}T_{2})|_{M}=(T_{1}|_{M})(T_{2}|_{N})$
$\displaystyle=F(\rho(P_{M},T_{1},P_{N}))F(\rho(P_{N},T_{2},P_{U})),$

and $F(1_{\mathcal{S}P_{M}})$ $=F(\rho(P_{M},P_{M},P_{M}))=P_{M}|_{M}=1_{M}$.
Therefore, $F$ is a functor from $\mathscr{L}(\mathcal{S})$ to
$\mathscr{F}(H)$.

Similarly, define $G$ from $\mathscr{F}(H)$ to $\mathscr{L}(\mathcal{S})$ by
$G(M)=\mathcal{S}P_{M}$ where $M\in\textit{{v}}\mathscr{F}(H)$, and for
$T:M\rightarrow
N,\;G(T)=\rho(P_{M},P_{M}TP_{N},P_{N}):\mathcal{S}P_{M}\rightarrow\mathcal{S}P_{N}.$
For $T_{1}:M\rightarrow N$ and $T_{2}:N\rightarrow U$,

$\displaystyle G(T_{1}T_{2})$
$\displaystyle=\rho(P_{M},P_{M}T_{1}T_{2}P_{U},P_{U})=\rho(P_{M},P_{M}T_{1}P_{N}T_{2}P_{U},P_{U})$
$\displaystyle=\rho(P_{M},P_{M}T_{1}P_{N},P_{N})\rho(P_{N},P_{N}T_{2}P_{U},P_{U})=G(T_{1})G(T_{2}),$

and
$G(1_{M})=\rho(P_{M},P_{M}1_{M}P_{M},P_{M})=\rho(P_{M},P_{M},P_{M})=1_{\mathcal{S}P_{M}}$,
ie., $G$ is a functor from $\mathscr{F}(H)$ to $\mathscr{L}(\mathcal{S})$.

Since

$\mathcal{S}P_{M}\subseteq\mathcal{S}P_{N}\Longleftrightarrow M\subseteq N,$

$F$ and $G$ are inclusion preserving functors with
$FG=1_{\mathscr{L}(\mathcal{S})}$ and $GF=1_{\mathscr{F}(H)}$.

Finally, for finite-dimensional subspaces $M$ and $N$ of $H$, $F$ is a normed
space isomorphism from
$\mathscr{L}(\mathcal{S})(\mathcal{S}P_{M},\mathcal{S}P_{N})$ onto
$\mathscr{F}(H)(M,N)$. For, let $T,T_{1},T_{2}\in P_{M}\mathcal{S}P_{N}$ and
$k\in\textbf{K}$. Then,

$\displaystyle F(\rho(P_{M},T_{1},P_{N})+\rho(P_{M},T_{2},P_{N}))$
$\displaystyle=F(\rho(P_{M},T_{1}+T_{2},P_{N}))=(T_{1}+T_{2})|_{M}=(T_{1}|_{M})+(T_{2}|_{M})$
$\displaystyle=F(\rho(P_{M},T_{1},P_{N}))+F(\rho(P_{M},T_{2},P_{N})),and$
$\displaystyle F(k\rho(P_{M},T,P_{N}))$
$\displaystyle=F(\rho(P_{M},kT,P_{N}))=(kT)|_{M}=k(T|_{M})=kF(\rho(P_{M},T,P_{N})),$
$\displaystyle\text{and }\Arrowvert F(\rho(P_{M},T,P_{N}))\Arrowvert$
$\displaystyle=\Arrowvert T|_{M}\Arrowvert=\Arrowvert
P_{M}T\Arrowvert=\Arrowvert
T\Arrowvert=\Arrowvert\rho(P_{M},T,P_{N})\Arrowvert.$

Since $F$ is full and faithful, $F$ is a normed space isomorphism from the
hom-set $\mathscr{L}(\mathcal{S})(\mathcal{S}P_{M},\mathcal{S}P_{N})$ of
$\mathscr{L}(\mathcal{S})$ onto the hom-set $\mathscr{F}(H)(M,N)$ of
$\mathscr{F}(H)$. ∎

Let $H^{\prime}$ denote the dual of the Hilbert space $H$. The category
$\mathscr{F}(H^{\prime})$ of all finite-dimensional subspaces of $H^{\prime}$.
is a normal category. $M$ be a finite-dimensional subspace of $H$, for each
$m\in M$, define the map $f_{m}:H\rightarrow\textbf{K}$ by $f_{m}(x)=\langle
x,m\rangle$ for $x\in H$. For $M^{\prime}=\\{f_{m}\in H^{\prime}:m\in M\\}$,
by Riesz representation theorem,

$\textit{{v}}\mathscr{F}(H^{\prime})=\\{M^{\prime}:M\in\textit{{v}}\mathscr{F}(H)\\}.$

Moreover, for finite-dimensional subspaces $M$ and $N$ of $H$,
$\phi:M^{\prime}\rightarrow N^{\prime}$ is a linear map if and only if there
exists a unique linear map $T:M\rightarrow N$ such that
$\phi=(P_{N}T^{*}P_{M})^{\prime}|_{M^{\prime}}$, where $T^{*}$ is the adjoint
of $T$ and $(P_{N}T^{*}P_{M})^{\prime}$ is the transpose of $P_{N}T^{*}P_{M}$.
Then, $F:\mathscr{F}(H)\to\mathscr{F}(H^{\prime})$ defined by

$\displaystyle F(M)$
$\displaystyle=M^{\prime},\,\text{(}{for}\,M\in\textit{{v}}\mathscr{F}(H)$
$\displaystyle F(T)$
$\displaystyle=(P_{N}T^{*}P_{M})^{\prime}|_{M^{\prime}},\,\text{for
every}\,\,T:M\to N$

is a normal category isomorphism, for $T,T_{1},T_{2}:M\rightarrow N$ and
$k\in\textbf{K}$,

$\displaystyle F(T_{1}+T_{2})$
$\displaystyle=(P_{N}(T_{1}+T_{2})^{*}P_{M})^{\prime}|_{M^{\prime}}=(P_{N}T_{1}^{*}P_{M})^{\prime}|_{M^{\prime}}+(P_{N}T_{2}^{*}P_{M})^{\prime}|_{M^{\prime}}=F(T_{1})+F(T_{2}),$
$\displaystyle F(kT)$
$\displaystyle=(P_{N}(kT)^{*}P_{M})^{\prime}|_{M^{\prime}}=\bar{k}(P_{N}T^{*}P_{M})^{\prime}|_{M^{\prime}}=\bar{k}F(T),$
$\displaystyle\text{ and }\Arrowvert F(T)\Arrowvert$
$\displaystyle=\Arrowvert(P_{N}T^{*}P_{M})^{\prime}|_{M^{\prime}}\Arrowvert=\Arrowvert(P_{N}T^{*}P_{M})^{\prime}\Arrowvert=\Arrowvert
P_{N}T^{*}P_{M}\Arrowvert=\Arrowvert T^{*}\Arrowvert=\Arrowvert T\Arrowvert.$

Therefore, $F$ is a conjugate linear isometry from the hom-set
$\mathscr{F}(H)(M,N)$ of $\mathscr{F}(H)$ onto the hom-set
$\mathscr{F}(H^{\prime})(M^{\prime},N^{\prime})$ of $\mathscr{F}(H^{\prime})$.
Thus, we obtain the following theorem using the Hilbert space duality.

######  Theorem 4.2.

The normal categories $\mathscr{F}(H)$ and $\mathscr{F}(H^{\prime})$ are
isomorphic, and this isomorphism is a conjugate linear isometry from the hom-
set $\mathscr{F}(H)(M,N)$ of $\mathscr{F}(H)$ onto the hom-set
$\mathscr{F}(H^{\prime})(M^{\prime},N^{\prime})$ of $\mathscr{F}(H^{\prime})$
for any finite-dimensional subspaces $M$ and $N$ of $H$.

Analogous Theorem 4.1, we have the following theorem.

######  Theorem 4.3.

The normal category $\mathscr{R}(\mathcal{S})$ is isomorphic to the normal
category $\mathscr{F}(H^{\prime})$. For each finite-dimensional subspace $M$
and $N$ of $H$, this isomorphism is a normed space isomorphism of hom-set
$\mathscr{R}(\mathcal{S})(P_{M}\mathcal{S},P_{N}\mathcal{S})$ of
$\mathscr{R}(\mathcal{S})$ onto the hom-set
$\mathscr{F}(H^{\prime})(M^{\prime},N^{\prime})$ of $\mathscr{F}(H^{\prime})$.

###### Proof.

Define $F:\mathscr{R}(\mathcal{S})\rightarrow\mathscr{F}(H^{\prime})$ by,

$\displaystyle\;F(P_{M}\mathcal{S})=M\,$ $\displaystyle\text{ for
}P_{M}\mathcal{S}\in\textit{{v}}\mathscr{R}(\mathcal{S})$ $\displaystyle\text{
and for }\lambda(P_{M},T,P_{N}):P_{M}\mathcal{S}\rightarrow P_{N}\mathcal{S},$
$\displaystyle\;F(\lambda(P_{M},T,P_{N}))=T^{\prime}|_{M^{\prime}}:M^{\prime}\rightarrow
N^{\prime}.$

As in the proof of Theorem 4.1, it is seen that $F$ is a normal category
isomorphism from $\mathscr{R}(\mathcal{S})$ to $\mathscr{F}(H^{\prime})$, and
also a normed space isomorphism from the hom-set
$\mathscr{R}(\mathcal{S})(P_{M}\mathcal{S},P_{N}\mathcal{S})$ of
$\mathscr{R}(\mathcal{S})$ onto the hom-set
$\mathscr{F}(H^{\prime})(M^{\prime},N^{\prime})$ of $\mathscr{F}(H^{\prime})$
of finite-dimensional subspaces $M$ and $N$. ∎

From the above discussions we can conclude that the functor $F$ from
$\mathscr{L}(\mathcal{S})$ to $\mathscr{R}(\mathcal{S})$ given by,

$\displaystyle F(\mathcal{S}P_{M})=P_{M}\mathcal{S},\,$
$\displaystyle\,\text{for\;}\mathcal{S}P_{M}\in\textit{{v}}\mathscr{L}(\mathcal{S}),$
$\displaystyle\text{and
for\;}\rho(P_{M},T,P_{N}):\mathcal{S}P_{M}\rightarrow\mathcal{S}P_{N},$
$\displaystyle\;F(\rho(P_{M},T,P_{N}))=\lambda(P_{M},T^{*},P_{N})$

is a normal category isomorphism from $\mathscr{L}(\mathcal{S})$ to
$\mathscr{R}(\mathcal{S})$ and for finite-dimensional subspaces $M$ and $N$ of
$H$, $F$ is a conjugate-linear isometry from the hom-set
$\mathscr{L}(\mathcal{S})(\mathcal{S}P_{M},\mathcal{S}P_{N})$ of
$\mathscr{L}(\mathcal{S})$ onto the hom-set
$\mathscr{R}(\mathcal{S})(P_{M}\mathcal{S},P_{N}\mathcal{S})$ of
$\mathscr{R}(\mathcal{S})$. Thus we have the following:

######  Theorem 4.4.

The normal category $\mathscr{L}(\mathcal{S})$ is isomorphic to the normal
category $\mathscr{R}(\mathcal{S})$. Moreover, this isomorphism is a
conjugate-linear isometry from the hom-set
$\mathscr{L}(\mathcal{S})(\mathcal{S}P_{M},\mathcal{S}P_{N})$ of
$\mathscr{L}(\mathcal{S})$ onto the hom-set
$\mathscr{R}(\mathcal{S})(P_{M}\mathcal{S},P_{N}\mathcal{S})$ of
$\mathscr{R}(\mathcal{S})$ for finite-dimensional subspaces $M$ and $N$ of
$H$.

##  5\. Bounded normal cones

It is well known that isomorphic normal categories have isomorphic semigroups
of normal cones (see Proposition 2.1). Since $\mathscr{F}(H)$,
$\mathscr{F}(H^{\prime})$, $\mathscr{L}(\mathcal{S})$ and
$\mathscr{R}(\mathcal{S})$ are isomorphic as normal categories, it will
sufficies to find the semigroup $T\mathscr{F}(H)$ of all normal cones in
$\mathscr{F}(H)$.

######  Theorem 5.1.

The semigroup $T\mathscr{F}(H)$ of all normal cones in $\mathscr{F}(H)$ is
isomorphic to the semigroup of all finite rank operators on the Hilbert space
$H$.

###### Proof.

Let $\mathcal{S}^{\prime}$ be the semigroup of all finite rank operators on
$H$. For $T\in\mathcal{S}^{\prime}$, define the map $\gamma^{T}$ from
$\textit{{v}}\mathscr{F}(H)$ to $\mathscr{F}(H)$ by,

$\gamma^{T}(M)=T|_{M}:M\rightarrow R(T),$

for $M\in\textit{{v}}\mathscr{F}(H)$. If $M_{1}\subseteq M_{2}$, then
$J_{M_{1}}^{M_{2}}\,\gamma^{T}(M_{2})=J_{M_{1}}^{M_{2}}\,(T|_{M_{2}})=T|_{M_{1}}=\gamma^{T}(M_{1}),$
therefore, $\gamma^{T}$ is a cone in $\mathscr{F}(H)$. For any complement $M$
of $Z(T)$, $\,T|_{M}$ is an isomorphism from $M$ onto $R(T)$, hence,
$\gamma^{T}$ is a normal cone with vertex $R(T)$. Thus, for any finite rank
operator $T$on $H$, there is a normal cone $\gamma^{T}$ in $\mathscr{F}(H)$
with vertex $R(T)$. Define $\phi:\mathcal{S}^{\prime}\rightarrow
T\mathscr{F}(H)$ by

$\phi(T)=\gamma^{T}.$

Let $T_{1},T_{2}\in\mathcal{S}^{\prime}$ and $N=R(T_{1})$. Then for every
$M\in\textit{{v}}\mathscr{F}(H)$,

$\displaystyle\phi(T_{1}T_{2})(M)$
$\displaystyle=\gamma^{T_{1}T_{2}}(M)=(T_{1}T_{2})|_{M}=(T_{1}|_{M})(T_{2}|_{N})^{\circ}$
$\displaystyle=\gamma^{T_{1}}(M)(\gamma^{T_{2}}(N))^{\circ}=\gamma^{T_{1}}\cdot\gamma^{T_{2}}(M)$
$\displaystyle=(\phi(T_{1})\cdot\phi(T_{2}))(M),$

that is, $\phi(T_{1}T_{2})=\phi(T_{1})\cdot\phi(T_{2})$ and so $\phi$ is a
homomorphism from $\mathcal{S}^{\prime}$ to $T\mathscr{F}(H)$.

Next, we have to show that for any normal cone $\gamma$ in $\mathscr{F}(H)$
with vertex $N$, there is a unique finite rank operator $T$ on $H$ such that
$R(T)=N$ and $\gamma^{T}=\gamma$. Let $\gamma$ be a normal cone in
$\mathscr{F}(H)$ with vertex $N$ and $\mathscr{B}$ be a basis of $H$. Define
the map $T$ on $H$ by for $b\in\mathscr{B}$,

$T(b)=\gamma(\langle b\rangle)(b),$

where $\langle b\rangle=\text{Span}\\{b\\}$, and extend $T$ linearly to all of
$H$. That is, if $T_{b}=\gamma(\langle b\rangle)$ for $b\in\mathscr{B}$, then
$T(b)=T_{b}(b)$. To prove that $\gamma(M)=T|_{M}$ for any finite-dimensional
subspace $M$ of $H$, it is enough to prove that $\gamma(\langle
x\rangle)=T|_{\langle x\rangle}$ for any $x\in H$. Let
$x=c_{1}b_{1}+c_{2}b_{2}+\cdots+c_{n}b_{n}$ for $b_{i}\in\mathscr{B}$ and
$c_{i}\in\textbf{K}$, $i=1,2,\cdots,n$. Let
$U=\text{Span}\\{b_{1},b_{2},\cdots,b_{n}\\}$, and $T_{b_{i}}=\gamma(\langle
b_{i}\rangle)$ for $i=1,2,\cdots,n$, then for $i=1,2,\cdots,n$, since $\langle
b_{i}\rangle\subseteq U$, $T(b_{i})=T_{b_{i}}(b_{i})=\gamma(\langle
b_{i}\rangle)(b_{i})=\gamma(U)(b_{i})$, therefore, $\gamma(U)=T|_{U}$, and
$\gamma(\langle x\rangle)=J_{\langle x\rangle}^{U}\,\gamma(U)=J_{\langle
x\rangle}^{U}\,T|_{U}=T|_{\langle x\rangle}$ for any $x\in H$. Since, $\gamma$
is a normal cone, there is a finite-dimensional subspace $M$ of $H$ such that
$\gamma(M)=T|_{M}$ is an isomorphism from $M$ to $N$. Thus, $R(T)=N$ and
$\gamma^{T}=\gamma$.

Finally to prove the uniqueness of $T$, suppose that $\gamma=\gamma^{A}$ for
some $A\in\mathcal{S}^{\prime}$. Then for each $b\in\mathscr{B},(T|_{\langle
b\rangle})(b)=\gamma^{T}(\langle b\rangle)(b)=\gamma^{A}(\langle
b\rangle)(b)=(A|_{\langle b\rangle})(b)$. Thus, $T(b)=A(b)$ for each
$b\in\mathscr{B}$. Therefore, $T=A$. Hence, $\phi$ is an isomorphism from the
semigroup $\mathcal{S}^{\prime}$ of all finite rank operators on $H$ to the
semigroup $T\mathscr{F}(H)$ of all normal cones in $\mathscr{F}(H)$. ∎

Since the semigroup $T\mathscr{F}(H)$ of all normal cones in $\mathscr{F}(H)$
is a regular semigroup, we have the following.

######  Corollary 5.1.

The semigroup of all finite rank operators on $H$ is a regular semigroup.

######  Example 5.1.

Let $H=l^{2}$ and for $n=1,2,3,\ldots$, $e_{n}=(0,\ldots,0,1,0,\ldots),$ where
$1$ occurs only in the $nth$ entry. Also let $\mathscr{B}$ be a basis of
$l^{2}$ containing $e_{1},e_{2},e_{3},\ldots$. Define $T:l^{2}\rightarrow
l^{2}$ by,

$T(x)=\begin{cases}ne_{1}&\text{ if }x=e_{n}\\\ 0&\text{ if
}x\in\mathscr{B},x\neq e_{n}\text{ for any }n.\end{cases}$

By extending $T$ linearly to $l^{2}$, $T$ is a finite rank operator on
$l^{2}$, which is not bounded. Then one can define a normal cone $\gamma^{T}$
in $\mathscr{F}(H)$ with vertex $\langle e_{1}\rangle=Span\\{e_{1}\\}$ by for
$M\in\textit{{v}}\mathscr{F}(H)$,

$\gamma^{T}(M)=T|_{M}:M\rightarrow\langle e_{1}\rangle.$

Similarly it is easy to see that one can define normal cone $\rho^{T}$ in
$\mathscr{L}(\mathcal{S})$ with vertex $\mathcal{S}P_{\langle e_{1}\rangle}$
by, for $\mathcal{S}P_{M}\in\textit{{v}}\mathscr{L}(\mathcal{S})$,

$\rho^{T}(\mathcal{S}P_{M})=\rho(P_{M},P_{M}T,P_{\langle e_{1}\rangle}),$

and normal cone $\lambda^{T}$ in $\mathscr{R}(\mathcal{S})$ with vertex
$P_{\langle e_{1}\rangle}\mathcal{S}$ by, for
$P_{M}\mathcal{S}\in\textit{{v}}\mathscr{R}(\mathcal{S})$,

$\lambda^{T}(P_{M}\mathcal{S})=\lambda(P_{M},(P_{M}T)^{*},P_{\langle
e_{1}\rangle}).$

Next we proceed to descibe bounded normal cone in $\mathscr{F}(H)$ such that
the set of all bounded normal cones in $\mathscr{F}(H)$ forms a normed algebra
isomorphic to the normed algebra $\mathcal{S}$.

######  Definition 5.1.

A normal cone $\gamma$ in $\mathscr{F}(H)$ with vertex $N$ is said to be a
bounded normal cone if there exists some $\alpha_{\gamma}\in\mathbb{R}$ such
that $\Arrowvert\gamma(M)\Arrowvert\leq\alpha_{\gamma}$, for all
$M\in\textit{{v}}\mathscr{F}(H)$.

Analogous to this, we can define bounded normal cones in
$\mathscr{L}(\mathcal{S})$ and $\mathscr{R}(\mathcal{S})$. The following
theorem revels that the set $B\mathscr{L}(\mathcal{S})$ of all bounded normal
cones in $\mathscr{L}(\mathcal{S})$ is a normed algebra isomorphic to the
normed algebra $\mathcal{S}$.

######  Theorem 5.2.

The set $B\mathscr{L}(\mathcal{S})$ of all bounded normal cones in
$\mathscr{L}(\mathcal{S})$ is a normed algebra, which is isomorphic to the
normed algebra $\mathcal{S}$.

###### Proof.

For $T\in\mathcal{S}$ the principal cone $\rho^{T}$ is a normal cone in
$\mathscr{L}(\mathcal{S})$ with vertex $\mathcal{S}P_{N}$, where $N=R(T)$ and
for any $\mathcal{S}P_{M}\in\textit{{v}}\mathscr{L}(\mathcal{S})$,

$\Arrowvert\rho^{T}(\mathcal{S}P_{M})\Arrowvert=\Arrowvert\rho(P_{M},P_{M}T,P_{N})\Arrowvert=\Arrowvert
P_{M}T\Arrowvert\leq\Arrowvert T\Arrowvert,$

therefore, $\rho^{T}$ is a bounded normal cone in $\mathscr{L}(\mathcal{S})$.
Define the function $\phi:\mathcal{S}\rightarrow B\mathscr{L}(\mathcal{S})$
by,

$\phi(T)=\rho^{T},$

clearly $\phi$ is one-one. To prove that $\phi$ is onto, suppose that $\gamma$
is a bounded normal cone in $\mathscr{L}(\mathcal{S})$ with vertex
$\mathcal{S}P_{N}$, by Theorem 5.1, there exists a finite rank operator $T$ on
$H$ such that $\gamma=\rho^{T}$ and $N=R(T)$, ie., for any
$\mathcal{S}P_{M}\in\textit{{v}}\mathscr{L}(\mathcal{S})$,
$\gamma(\mathcal{S}P_{M})=\rho^{T}(\mathcal{S}P_{M})=\rho(P_{M},P_{M}T,P_{N})$.

Since $\rho^{T}$ is a bounded normal cone, there exists some
$\alpha\in\mathbb{R}$ such that
$\Arrowvert\rho^{T}(\mathcal{S}P_{M})\Arrowvert\leq\alpha$, for all
$\mathcal{S}P_{M}\in\textit{{v}}\mathscr{L}(\mathcal{S})$, so for any
$\mathcal{S}P_{M}\in\textit{{v}}\mathscr{L}(\mathcal{S})$, $\Arrowvert
P_{M}T\Arrowvert=\Arrowvert\rho(P_{M},P_{M}T,P_{N})\Arrowvert=\Arrowvert\rho^{T}(\mathcal{S}P_{M})\Arrowvert\leq\alpha$.
Hence, $\Arrowvert P_{\langle x\rangle}T\Arrowvert\leq\alpha$ for any $x\in
H$. Thus, for any $x\in H$,

$\Arrowvert T(x)\Arrowvert=\Arrowvert(P_{\langle
x\rangle}T)(x)\Arrowvert\leq\alpha\Arrowvert x\Arrowvert,$

therefore, $T$ is a finite rank bounded operator on $H$ and $\gamma=\phi(T)$,
and $B\mathscr{L}(\mathcal{S})$ is a normed space. Further for
$\rho^{T_{1}},\rho^{T_{2}}$ in $B\mathscr{L}(\mathcal{S})$,

$\rho^{T_{1}}+\rho^{T_{2}}=\rho^{T_{1}+T_{2}},$ $k\rho^{T_{1}}=\rho^{kT_{1}},$
$\text{ and }\Arrowvert\rho^{T_{1}}\Arrowvert=\Arrowvert T_{1}\Arrowvert.$

But,

$\phi(T_{1}T_{2})=\rho^{T_{1}T_{2}}=\rho^{T_{1}}\cdot\rho^{T_{2}}=\phi(T_{1})\cdot\phi(T_{2}),$
$\phi(T_{1}+T_{2})=\rho^{T_{1}+T_{2}}=\rho^{T_{1}}+\rho^{T_{2}}=\phi(T_{1})+\phi(T_{2}),$
$\phi(kT)=\rho^{kT}=k\rho^{T}=k\phi(T),$ $\text{ and
}\Arrowvert\phi(T)\Arrowvert=\Arrowvert\rho^{T}\Arrowvert=\Arrowvert
T\Arrowvert.$

Thus, the set $B\mathscr{L}(\mathcal{S})$ of all bounded normal cones in
$\mathscr{L}(\mathcal{S})$ is a normed algebra that is isomorphic to the
normed algebra $\mathcal{S}$. ∎

Consier the normal cone $\gamma^{\prime}$ in $\mathscr{R}(\mathcal{S})$ with
vertex $P_{N}\mathcal{S}$ corresponding to the normal cone $\gamma$ in
$\mathscr{L}(\mathcal{S})$ with vertex $\mathcal{S}P_{N}$ given by for
$P_{M}\mathcal{S}\in\textit{{v}}\mathscr{R}(\mathcal{S})$,

$\gamma^{\prime}(P_{M}\mathcal{S})=F(\gamma(F^{-1}(P_{M}\mathcal{S}))),$

where $F$ is the isomorphism from $\mathscr{L}(\mathcal{S})$ to
$\mathscr{R}(\mathcal{S})$ in Theorem 4.4. Since $F$ is an isometry on each
hom-set, there is a bijection between the set $B\mathscr{R}(\mathcal{S})$ of
all bounded normal cones in $\mathscr{R}(\mathcal{S})$, and the set
$B\mathscr{L}(\mathcal{S})$ of all bounded normal cones in
$\mathscr{L}(\mathcal{S})$. Furthermore, for $T\in\mathcal{S}$, the normal
cone corresponding to the normal cone $\rho^{T}$ in $\mathscr{L}(\mathcal{S})$
is $\lambda^{T^{*}}$. For
$P_{M}\mathcal{S}\in\textit{{v}}\mathscr{R}(\mathcal{S})$,

$\gamma^{\prime}(P_{M}\mathcal{S})=F(\rho^{T}(\mathcal{S}P_{M}))=F(\rho(P_{M},P_{M}T,P_{N}))=\lambda(P_{M},(P_{M}T)^{*},P_{N})=\lambda^{T^{*}}(P_{M}\mathcal{S}),$

hence, for $\lambda^{T_{1}},\lambda^{T_{2}}$ in $B\mathscr{R}(\mathcal{S})$.
Also

$\lambda^{T_{1}}+\lambda^{T_{2}}=\lambda^{T_{1}+T_{2}},$
$k\lambda^{T_{1}}=\lambda^{kT_{1}},$ $\text{ and
}\Arrowvert\lambda^{T_{1}}\Arrowvert=\Arrowvert T_{1}\Arrowvert.$

Thus we have following theorem.

######  Theorem 5.3.

The set $B\mathscr{R}(\mathcal{S})$ of all bounded normal cones in
$\mathscr{R}(\mathcal{S})$ is a normed algebra, and this normed algebra
$B\mathscr{R}(\mathcal{S})$ is conjugate linear isomorphic to the normed
algebra $\mathcal{S}$.

###### Proof.

Define the map $\phi:\mathcal{S}\rightarrow B\mathscr{R}(\mathcal{S})$ by,
$\phi(T)=\lambda^{T^{*}}$. Then,

$\phi(T_{1}T_{2})=\lambda^{(T_{1}T_{2})^{*}}=\lambda^{T_{2}^{*}T_{1}^{*}}=\lambda^{T_{1}^{*}}\cdot\lambda^{T_{2}^{*}}=\phi(T_{1})\cdot\phi(T_{2}),$
$\phi(T_{1}+T_{2})=\lambda^{(T_{1}+T_{2})^{*}}=\lambda^{(T_{1})^{*}+(T_{2})^{*}}=\lambda^{T_{1}^{*}}+\lambda^{T_{2}^{*}}=\phi(T_{1})+\phi(T_{2}),$
$\phi(kT)=\lambda^{(kT)^{*}}=\lambda^{\bar{k}T^{*}}=\bar{k}\lambda^{T^{*}}=\bar{k}\phi(T),$
$\text{and
}\Arrowvert\phi(T)\Arrowvert=\Arrowvert\lambda^{T^{*}}\Arrowvert=\Arrowvert
T^{*}\Arrowvert=\Arrowvert T\Arrowvert,$

therefore, $B\mathscr{R}(\mathcal{S})$ is a normed algebra, and it is
conjugate linear isomorphic to the normed algebra $\mathcal{S}$. ∎

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# Positive Definite Tensors to Nonlinear Complementarity Problems

Maolin Che E-mail<EMAIL_ADDRESS>and<EMAIL_ADDRESS>School
of Mathematical Sciences, Fudan University, Shanghai, 200433, P. R. of China.
This author is supported by the National Natural Science Foundation of China
under grant 11271084. Liqun Qi  E-mail<EMAIL_ADDRESS>Department of
Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon,
Hong Kong. This author’s work was supported by the Hong Kong Research Grant
Council (Grant No. PolyU 502510, 502111, 501212, 501913). Yimin Wei
Corresponding author (Y. Wei). E-mail<EMAIL_ADDRESS>and
<EMAIL_ADDRESS>School of Mathematical Sciences and Shanghai Key
Laboratory of Contemporary Applied Mathematics, Fudan University, Shanghai,
200433, P. R. of China. This author is supported by the National Natural
Science Foundation of China under grant 11271084.

###### Abstract

The main purpose of this note is to investigate some kinds of nonlinear
complementarity problems (NCP). For the structured tensors, such as, symmetric
positive definite tensors and copositive tensors, we derive the existence
theorems on a solution of these kinds of nonlinear complementarity problems.
We prove that a unique solution of the NCP exists under the condition of
diagonalizable tensors.

Keywords: Copositive tensor; Symmetric tensor; Positive definite tensor;
Diagonalizable tensors; Nonlinear complementarity problem.

AMS subject classifications: 15A18, 15A69, 65F15, 65F10

##  1 Introduction

Let $F$ be a mapping from $\mathds{R}^{n}$ into itself. The nonlinear
complementarity problem, denoted by $\mathrm{NCP}(F)$, is to find a vector
$\mathbf{x}^{*}\in\mathds{R}_{+}^{n}$ such that

$F(\mathbf{x}^{*})\in\mathds{R}_{+}^{n},\quad
F(\mathbf{x}^{*})^{\top}\mathbf{x}=0.$

When $F(\mathbf{x})$ is an affine function of $\mathbf{x}$, say
$F(\mathbf{x})=\mathbf{q}+M\mathbf{x}$ for some given vectors
$\mathbf{q}\in\mathds{R}^{n}$ and matrix $M\in\mathds{R}^{n\times n}$, the
problem $\mathrm{NCP}(F)$ reduces to the linear complementarity problem, which
is denoted by $\mathrm{LCP}(\mathbf{q},M)$. The results of the linear
complementarity problem can be found in the references ([1, 3]).

A further generalization of the $\mathrm{NCP}(F)$ is the variational
inequality problem: given a mapping
$F:\mathds{R}^{n}\rightarrow\mathds{R}^{n}$, and
$\emptyset\neq\mathfrak{K}\subseteq\mathds{R}^{n}$, find a
$\mathbf{x}^{*}\in\mathfrak{K}$ satisfying

$(\mathbf{y}-\mathbf{x}^{*})^{\top}F(\mathbf{x}^{*})\geq 0,\quad\text{for all
}\mathbf{y}\in\mathfrak{K},$

denoted by $\mathrm{VI}(\mathfrak{K},F)$.

If $\mathfrak{K}=\\{\mathbf{x}:\mathbf{x}\geq{\bf 0}\\}$, then a
$\mathbf{x}^{*}$ is a solution of $\mathrm{VI}(\mathfrak{K},F)$, solves the
$\mathrm{NCP}(F)$.

Over the past decade, the research of finite-dimensional variational
inequality and complementarity problems ([4, 5, 7, 8, 11, 2, 13]) has been
rapidly developed in the theory of existence, uniqueness and sensitivity of
solutions, theory of algorithms, and the application of these techniques to
transportation planning, regional science, socio-economic analysis, energy
modeling and game theory.

Qi [14] defined two kinds of eigenvalues and described some relative results
similar to the matrix eigenvalues. Lim [10] proposed another definition of
eigenvalues, eigenvectors, singular values, and singular vectors for tensors
based on a constrained variational approach, much like the Rayleigh quotient
for symmetric matrix eigenvalues, independently.

It is well-known that $A$ is a $P$ matrix ([1]), if and only if the linear
complementarity problem,

$\rm{find}~{}~{}z\in\mathds{R}^{n}~{}~{}\rm{such~{}that}~{}~{}z\geq\mathbf{0},\quad
q+Az\geq\mathbf{0},~{}~{}\rm{and}~{}~{}z^{\top}(q+Az)=0.$

has a unique solution for all $q\in\mathds{R}^{n}$. Then for a $P$ tensor
([16]) $\mathcal{A}\in T_{m,n},(m>2)$, does a similar property hold for the
following nonlinear complementarity problem,

$\rm{find}~{}~{}x\in\mathds{R}^{n}~{}~{}\rm{such~{}that}~{}~{}x\geq\mathbf{0},\quad
q+\mathcal{A}x^{m-1}\geq\mathbf{0},~{}~{}\rm{and}~{}~{}x^{\top}(q+\mathcal{A}x^{m-1})=0?$

In this note, we will consider a special kind of $\mathrm{NCP}(F)$, that is,
$F_{i}(\mathbf{x})$ is a multivariate polynomial and the degree of
$F_{i}(\mathbf{x})$ is $k_{i}$, then $F(\mathbf{x})$ can be expressed by,

$F(\mathbf{x})=\sum\limits_{i=1}^{k}\mathcal{A}_{i}\mathbf{x}^{i-1},$

where $\mathcal{A}_{i}\in T_{i-1,n}$, $\mathcal{A}_{i}\mathbf{x}^{i-1}$ means
the tensor-vector product given in Section 2 and $k=\max\limits_{1\leq i\leq
n}k_{i}$. Specially, $\mathcal{A}_{1}$ is a vector and $\mathcal{A}_{2}$ is a
matrix.

Throughout this paper, we assume that $m,\ n\ (\geq 2)$ are positive integers
and $m$ is even. We use small letters $x,u,v,\dots,$ for scalers, small bold
letters $\mathbf{x},\mathbf{u},\mathbf{v},\dots,$ for vectors, capital letters
$A,B,C,\dots,$ for matrices, calligraphic letters
$\mathcal{A},\mathcal{B},\mathcal{C},\dots,$ for tensors, and
$\mathfrak{A},\mathfrak{B},\mathfrak{C},\dots,$ for the subsets in
$\mathds{R}^{n}$. Denote $[n]=\\{1,2,\dots,n\\}$. $\mathbf{0}$ means a column
vector in $\mathds{R}^{n}$, where its all entries are zeros.
$\mathds{R}_{+}^{n}$ denotes the nonnegative orthant of $\mathds{R}^{n}$.

Let $\mathds{R}$ be a real field. An order $m$ dimension $n$ tensor
$\mathcal{A}$ consists of $n^{m}$ real entries. Let $T_{m,n}$ denote all order
$m$ dimension $n$ tensors where every entry of each tensor in $T_{m,n}$ is in
$\mathds{R}$. $\mathcal{D}\in T_{m,n}$ is diagonal if all off-diagonal entries
are zero. Particularly, when the diagonal entries of $\mathcal{D}$ are 1, then
$\mathcal{D}$ is called the identity tensor and denote it as $\mathcal{I}$.

The rest of this note is organized as follows. Section 2 introduces
definitions of basic preliminaries and some results of the nonlinear
complementarity problems. The main result in Section 3 is to consider the
solvability of Questions 1 and 2. We conclude our paper in Section 4.

##  2 Preliminaries

In this section, we present five definitions about tensors and nonlinear
mappings, four lemmas associated with these definitions and the solutions of
the nonlinear complementarity problems, and two questions proposed as follows.

###  2.1 Definitions

The mode-$k$ tensor-matrix product and mode-$k$ tensor-vector product of
$\mathcal{A}\in T_{m,n}$ are defined as follows.

######  Definition 1.

([9]) The mode-$k$ product of a tensor $\mathcal{A}\in T_{m,n}$ and a matrix
$B\in\mathds{R}^{n\times n}$, denoted by $\mathcal{A}\times_{k}B$, is a tensor
$\mathcal{C}\in T_{m,n}$ of which are given by

$\mathcal{C}_{i_{1}\dots i_{k-1}ji_{k+1}\dots
i_{m}}=\sum_{i_{k}=1}^{n}\mathcal{A}_{i_{1}\dots i_{k-1}i_{k}i_{k+1}\dots
i_{m}}b_{ji_{k}},\quad k\in[m].$

Particularly, the mode-$k$ multiplication of a tensor $\mathcal{A}\in T_{m,n}$
and a vector $\mathbf{x}\in\mathds{R}^{n}$ is denoted by
$\mathcal{A}\bar{\times}_{k}\mathbf{x}$. Set
$\mathcal{C}=\mathcal{A}\bar{\times}_{k}\mathbf{x}$, then, element-wise, we
have

$\mathcal{C}_{i_{1}\dots i_{k-1}i_{k+1}\dots
i_{m}}=\sum\limits_{i_{k}=1}^{n}\mathcal{A}_{i_{1}\dots
i_{k-1}i_{k}i_{k+1}\dots i_{m}}x_{i_{k}}.$

According to Definition 1, let $m$ vectors $\mathbf{x}_{k}\in\mathds{R}^{n}$,
the formula
$\mathcal{A}\bar{\times}_{1}\mathbf{x}_{1}\dots\bar{\times}_{m}\mathbf{x}_{m}$
is easy to define. If these $m$ vectors are the same vector, denoted by
$\mathbf{x}$, then
$\mathcal{A}\bar{\times}_{1}\mathbf{x}\dots\bar{\times}_{m}\mathbf{x}$ can be
simplified as $\mathcal{A}\mathbf{x}^{m}$.

Our next definition is motivated by the class of copositive matrices ([1]),
which in turn generalizes that of nonnegative matrices.

######  Definition 2.

([7]) A mapping $F:\mathfrak{X}\rightarrow\mathds{R}^{n}$ is said to be

  1. (a)

Copositive with respect to $\mathfrak{X}$, if

$[F(\mathbf{x})-F(\mathbf{0})]^{\top}\mathbf{x}\geq
0,\quad\forall\mathbf{x}\in\mathfrak{X}.$

  2. (b)

Strictly copositive with respect to $\mathfrak{X}$, if

$[F(\mathbf{x})-F(\mathbf{0})]^{\top}\mathbf{x}>0,\quad\forall\mathbf{x}\in\mathfrak{X},\
\mathbf{x}\neq\mathbf{0}.$

  3. (c)

Strongly copositive with respect to $\mathfrak{X}$, if there exists a positive
scalar $\alpha$ such that

$[F(\mathbf{x})-F(\mathbf{0})]^{\top}\mathbf{x}\geq\alpha\|\mathbf{x}\|_{2}^{2},\quad\forall\mathbf{x}\in\mathfrak{X}.$

The definition of a symmetric tensor ([10, 14]) is stated as follows.

######  Definition 3.

Suppose that $\mathcal{A}\in T_{m,n}$. $\mathcal{A}$ is called symmetric if
$\mathcal{A}_{i_{1}i_{2}\dots i_{m}}$ is invariant by any permutation $\pi$,
that is, $\mathcal{A}_{i_{1}i_{2}\dots
i_{m}}=\mathcal{A}_{\pi(i_{1},i_{2},\dots,i_{m})}$ where all $i_{k}\in[n]$
with $k\in[m]$. We denote all symmetric tensors by $ST_{m,n}$.

When $m$ is even and $\mathcal{A}$ is symmetric, we say that

  1. (a)

$\mathcal{A}$ is positive definite ([14]), if $\mathcal{A}\mathbf{x}^{m}>0$
for all $\mathbf{x}\in\mathds{R}^{n}$ and $\mathbf{x}\neq\mathbf{0}$,

  2. (b)

$\mathcal{A}$ is a copositive tensor ([15]), if for any
$x\in\mathds{R}_{+}^{n}$, then $\mathcal{A}\mathbf{x}^{m}\geq 0$,

  3. (c)

$\mathcal{A}$ is a strictly copositive tensor ([15]), if for any
$x\in\mathds{R}_{+}^{n}$, then $\mathcal{A}\mathbf{x}^{m}>0$.

The set of all positive definite tensors is denoted by $SPT_{m,n}$.

The mapping

$G(\mathbf{x})=F(\mathbf{x})-F(\mathbf{0})$ (1)

plays an important role in the nonlinear complementarity problem ([7]), this
again is motivated by the linear complementarity problem.

The strict copositivity of a mapping can be relaxed through the introduction
of the class of d-regular mappings.

######  Definition 4.

([7]) For any vector $\mathbf{x}\in\mathds{R}_{+}^{n}$, we define the index
sets,

$I_{+}(\mathbf{x})=\\{i:x_{i}>0\\}\quad\text{and}\quad
I_{0}(\mathbf{x})=\\{i:x_{i}=0\\}.$

Let $\mathbf{d}>\mathbf{0}$ be an arbitrary vector in $\mathds{R}^{n}$. A
mapping $G:\mathds{R}^{n}\rightarrow\mathds{R}^{n}$ is said to be d-regular,
if the following system of equations has no solution in
$(\mathbf{x},t)\in\mathds{R}_{+}^{n}\times\mathds{R}_{+}$ with
$\mathbf{z}\neq\mathbf{0}$,

$\begin{split}G_{i}(\mathbf{x})+td_{i}&=0,\quad i\in I_{+}(\mathbf{x}),\\\
G_{i}(\mathbf{x})+td_{i}&\geq 0,\quad i\in I_{0}(\mathbf{x}).\end{split}$ (2)

Equivalently, $G$ is d-regular if, for any scalar $r>0$, the augmented
nonlinear complementarity problem $NCP(H)$ defined by
$H:\mathds{R}^{n+1}\rightarrow\mathds{R}^{n+1}$,

$H\begin{pmatrix}\mathbf{x}\\\
t\end{pmatrix}=\begin{pmatrix}G(\mathbf{x})+t\mathbf{d}\\\
r-\mathbf{d}^{\top}\mathbf{x}\end{pmatrix},$

has no solution $(\mathbf{x},t)$ with $\mathbf{x}\neq\mathbf{0}$.

Similar to the diagonalizable matrices ([6]), the definition of the
diagonalizable tensors is presented as follows.

######  Definition 5.

Suppose that $\mathcal{A}\in ST_{m,n}$. $\mathcal{A}$ is called diagonalizable
if $\mathcal{A}$ can be represented as

$\\{\mathcal{A}\in
T_{m,n}|\mathcal{A}=\mathcal{D}\times_{1}B\times_{2}B\dots\times_{m}B\\},$

where $B\in\mathds{R}^{n\times n}$ with $\det(B)\neq 0$ and $\mathcal{D}$ is a
diagonal tensor. Denote all diagonalizable tensors by $D_{m,n}$.

###  2.2 Lemmas

The following lemma is an existence and uniqueness theorem by Cottle ([2]). It
involves the notion of positively bounded Jacobians, and the original proof
was constructive, in the sense that an algorithm was employed to actually
compute the unique solution.

######  Lemma 1.

([2, 7]) Let $F:\mathds{R}_{+}^{n}\rightarrow\mathds{R}^{n}$ be continuously
differentiable and suppose that there exists one $\delta\in(0,1)$, such that
all principal minors of the Jacobian matrix $\nabla F(\mathbf{x})$ are bounded
between $\delta$ and $\delta^{-1}$, for all $\mathbf{x}\in\mathds{R}_{+}^{n}$.
Then the $\mathrm{NCP}(F)$ has a unique solution.

If mapping $F$ is strictly copositive, then the following result holds.

######  Lemma 2.

([11]) Let $F:\mathds{R}_{+}^{n}\rightarrow\mathds{R}^{n}$ be continuous and
strictly copositive with respect to $\mathds{R}_{+}^{n}$. If there exists a
mapping $c:\mathds{R}_{+}\rightarrow\mathds{R}$ such that
$c(\lambda)\rightarrow\infty$ as $\lambda\rightarrow\infty$, and for all
$\lambda\geq 1$, $\mathbf{x}\geq\mathbf{0}$,

$[F(\lambda\mathbf{x})-F(\mathbf{0})]^{\top}\mathbf{x}\geq
c(\lambda)[F(\mathbf{x})-F(\mathbf{0})]^{\top}\mathbf{x},$ (3)

then the problem $\mathrm{NCP}(F)$ has a nonempty, compact solution set.

For the linear complementarity problem, the mapping $G$, given in formula (1),
is obviously linear and thus, condition (3) is satisfied with
$c(\lambda)=\lambda$. More generally, the same condition will hold with
$c(\lambda)=\lambda^{\alpha}$, if $G$ is positively homogeneous of degree
$\alpha>0$; i.e., if $G(\lambda\mathbf{x})=\lambda^{\alpha}G(\mathbf{x})$ for
$\lambda>0$.

If $F$ is strictly copositive with respect to $\mathds{R}_{+}^{n}$, then the
mapping $G$ in (1) is d-regular for any $\mathbf{d}>\mathbf{0}$. The following
lemma presents an existence result for the nonlinear complementarity problem
with d-regular mapping.

######  Lemma 3.

([8]) Let $F$ be a continuous mapping from $\mathds{R}^{n}$ into itself and
$G$ defined by (1). Suppose that $G$ is positively homogeneous of degree
$\alpha>0$ and that $G$ is d-regular for some $\mathbf{d}>\mathbf{0}$. Then
the problem $\mathrm{NCP}(F)$ has a nonempty, compact solution set.

The main characterization theorem for copositive tensors can be summarized as
follows.

######  Lemma 4.

([15, Theorem 5]) Let $\mathcal{A}\in T_{m,n}$ be a symmetric tensor. Then,
$\mathcal{A}$ is copositive of and only if

$\min\\{\mathcal{A}\mathbf{x}^{m}:\mathbf{x}\in\mathds{R}_{+}^{n},\
\sum_{i=1}^{n}x_{i}^{m}=1\\}\geq 0.$

$\mathcal{A}$ is strictly copositive if and only if

$\min\\{\mathcal{A}\mathbf{x}^{m}:\mathbf{x}\in\mathds{R}_{+}^{n},\
\sum\limits_{i=1}^{n}x_{i}^{m}=1\\}>0.$

###  2.3 Problem Description

In this subsection, we propose two questions which we shall discuss in this
note.

######  Question 1.

([16]) Given $\mathcal{A}\in T_{m,n}$ and $\mathbf{q}\in\mathds{R}^{n}$. The
$\mathrm{NCP}(\mathbf{q},\mathcal{A})$ is to find a vector
$\mathbf{x}\in\mathds{R}_{+}^{n}$ such that

$\mathcal{A}\mathbf{x}^{m-1}+\mathbf{q}\in\mathds{R}_{+}^{n},\quad\mathcal{A}\mathbf{x}^{m}+\mathbf{q}^{\top}\mathbf{x}=0.$

######  Question 2.

Given $\mathcal{A}_{k}\in T_{m-(2k-2),n}$ and $\mathbf{q}\in\mathds{R}^{n}$
with $k=1,2,\dots,m/2$. The $\mathrm{GNCP}(\mathbf{q},\\{\mathcal{A}_{k}\\})$
is to find a vector $\mathbf{x}\in\mathds{R}_{+}^{n}$ such that

$\sum_{k=1}^{m/2}\mathcal{A}_{k}\mathbf{x}^{m-(2k-1)}+\mathbf{q}\in\mathds{R}_{+}^{n},\quad\sum_{k=1}^{m/2}\mathcal{A}_{k}\mathbf{x}^{m-2k+2}+\mathbf{q}^{\top}\mathbf{x}=0,$

where $\mathcal{A}_{m/2}$ is a square matrix.

Let
$\mathrm{FEA}(\mathbf{q},\mathcal{A})=\\{\mathbf{x}\in\mathds{R}_{+}^{n}:\mathcal{A}\mathbf{x}^{m-1}+\mathbf{q}\in\mathds{R}_{+}^{n}\\}$.
If $\mathrm{FEA}(\mathbf{q},\mathcal{A})\neq\emptyset$, then we see that
$\mathrm{NCP}(\mathbf{q},\mathcal{A})$ is feasible. It is obvious that
Question 1 is a special case of Question 2. However, for simplicity, we only
consider the solvability of Question 1, and make use of the results obtained
by solving Question 1, we then consider the solvability of Question 2.

##  3 Main results

Without loss of generality, suppose that $\mathbf{q}\in\mathds{R}^{n}$ in
Questions 1 and 2 is nonzero. For example, let $\mathcal{A}\in ST_{m,n}$ be
positive definite. If $\mathbf{q}$ is zero, then the solution of Question 1 is
zero. This situation is extraordinary, in order to avoid this situation, let
$\mathbf{q}\in\mathds{R}^{n}$ in Questions 1 and 2 be nonzero.

###  3.1 Necessary conditions for Solving Question 1

The cornerstone for the necessary conditions to be presented is the nonlinear
programming formulation of the Question 1,

$\begin{split}\min&\qquad\mathcal{A}\mathbf{x}^{m}+\mathbf{q}^{\top}\mathbf{x}\\\
s.t.&\qquad\mathcal{A}\mathbf{x}^{m-1}+\mathbf{q}\in\mathds{R}_{+}^{n},\
\mathbf{x}\in\mathds{R}_{+}^{n}.\end{split}$ (4)

Because $\mathrm{FEA}(\mathbf{q},\mathcal{A})$ is also the feasible set of
(4), if $\mathbf{x}_{*}$ minimizes the nonlinear programming given in (4) and
$\mathcal{A}\mathbf{x}_{*}^{m}+\mathbf{q}^{\top}\mathbf{x}_{*}=0$, then
$\mathbf{x}_{*}$ is a solution of Question 1. According to first-order
necessary conditions given in [12], we obtain the following result.

######  Theorem 1.

If $\mathrm{FEA}(\mathbf{q},\mathcal{A})\neq\emptyset$ and $\mathbf{x}_{*}$ is
a local solution of (4). Then, there exists a vector $\mathbf{u}_{*}$ of
multipliers satisfying the conditions,

$\begin{split}\mathbf{q}+m\mathcal{A}\mathbf{x}_{*}^{m-1}-(m-1)\mathcal{A}\mathbf{x}_{*}^{m-2}\mathbf{u}_{*}&\geq\mathbf{0}\\\
\mathbf{x}_{*}^{\top}(\mathbf{q}+m\mathcal{A}\mathbf{x}_{*}^{m-1}-(m-1)\mathcal{A}\mathbf{x}_{*}^{m-2}\mathbf{u}_{*})&=0\\\
\mathbf{u}_{*}&\geq\mathbf{0}\\\
\mathbf{u}_{*}^{\top}(\mathbf{q}+\mathcal{A}\mathbf{x}_{*}^{m-1})&=0.\end{split}$
(5)

Finally, the vectors $\mathbf{x}_{*}$ and $\mathbf{u}_{*}$ satisfy

$(m-1)(\mathbf{x}_{*}-\mathbf{u}_{*})_{i}(\mathcal{A}\mathbf{x}_{*}^{m-2}(\mathbf{x}_{*}-\mathbf{u}_{*}))_{i}\leq
0,\quad i\in[n].$ (6)

###### Proof.

Since $\mathrm{FEA}(\mathbf{q},\mathcal{A})\neq\emptyset$, the nonlinear
program (4) is feasible. Such an optimal solution $\mathbf{x}_{*}$ and a
suitable vector $\mathbf{u}_{*}$ of multipliers will satisfy the Karush-Kuhn-
Tucker conditions (5). To prove (6), we examine the inner product

$\mathbf{x}_{*}^{\top}(\mathbf{q}+m\mathcal{A}\mathbf{x}_{*}^{m-1}-(m-1)\mathcal{A}\mathbf{x}_{*}^{m-2}\mathbf{u}_{*})=0,$

at the componentwise level and deduce that for all $i\in[n]$,

$(m-1)(\mathbf{x}_{*})_{i}(\mathcal{A}\mathbf{x}_{*}^{m-2}(\mathbf{x}_{*}-\mathbf{u}_{*}))_{i}\leq
0,$ (7)

using the fact that $\mathbf{x}_{*}\in\mathrm{FEA}(\mathbf{q},\mathcal{A})$.
Similarly, multiplying the $i$th component in

$\mathbf{q}+m\mathcal{A}\mathbf{x}_{*}^{m-1}-(m-1)\mathcal{A}\mathbf{x}_{*}^{m-2}\mathbf{u}_{*}\geq\mathbf{0},$

by $\mathbf{u}_{*}$ and then invoking the complementarity condition

$(\mathbf{u}_{*})_{i}(\mathbf{q}+\mathcal{A}\mathbf{x}_{*}^{m-1})_{i}=0,$

which is implied by $\mathbf{u}_{*}\geq\mathbf{0}$,
$\mathbf{u}_{*}^{\top}(\mathbf{q}+\mathcal{A}\mathbf{x}_{*}^{m-1})=0$, and the
feasibility of of $\mathbf{x}_{*}$, we obtain

$-(m-1)(\mathbf{u}_{*})_{i}(\mathcal{A}\mathbf{x}_{*}^{m-2}(\mathbf{x}_{*}-\mathbf{u}_{*}))_{i}\leq
0.$ (8)

Now, (6) follows by adding (7) and (8). ∎

######  Remark 1.

Cottle ([2]) obtained the general matrix case of Theorem 1.

With the help of Theorem 1, we prove the following existence result for the
$\mathrm{NCP(\mathbf{q},\mathcal{A})}$.

######  Theorem 2.

Let nonzero $\mathbf{x}_{*}$ be a local solution of $($4$)$. If
$\mathcal{A}\mathbf{x}^{m-2}$ is positive definite for all nonzero
$\mathbf{x}\in\mathds{R}^{n}$, then $\mathbf{x}_{*}$ solves
$\mathrm{NCP}(\mathbf{q},\mathcal{A})$.

###### Proof.

According to Theorem 1, there exists a nonnegative vector $\mathbf{u}_{*}$
such that

$(m-1)(\mathbf{x}_{*}-\mathbf{u}_{*})_{i}(\mathcal{A}\mathbf{x}_{*}^{m-2}(\mathbf{x}_{*}-\mathbf{u}_{*}))_{i}\leq
0,\quad i\in[n],$

that is,

$(\mathbf{x}_{*}-\mathbf{u}_{*})^{\top}(\mathcal{A}\mathbf{x}_{*}^{m-2}(\mathbf{x}_{*}-\mathbf{u}_{*}))\leq
0.$

According to proposition assumptions, we know that
$\mathbf{x}_{*}=\mathbf{u}_{*}$. Based on (5), then, $\mathbf{x}_{*}$ solves
$\mathrm{NCP}(\mathbf{q},\mathcal{A}))$. ∎

Moreover, we can derive some results about Question 2, similar to Theorems 1
and 2. Here, we do not list them out.

###  3.2 Solving Question 1

In Question 1, let $F(\mathbf{x})=\mathcal{A}\mathbf{x}^{m-1}+\mathbf{q}$. We
first consider some properties of $F(\mathbf{x})$ when $\mathcal{A}$ is
selected from sets of structured tensors.

######  Theorem 3.

Suppose $\mathcal{A}\in ST_{m,n}$ and $\mathbf{x}\in\mathds{R}_{+}^{n}$.

  1. (a)

If $\mathcal{A}$ is (strictly) copositive, then the mapping $F(\mathbf{x})$ is
(strictly) copositive with respect to $\mathds{R}_{+}^{n}$,

  2. (b)

If $\mathcal{A}$ is positive definite, then the mapping $F(\mathbf{x})$ is
strongly copositive with respect to $\mathds{R}_{+}^{n}$ when
$\alpha\leq\lambda_{\min}\
(\leq\lambda_{\min}\frac{\|\mathbf{x}\|_{2}^{2}}{\|\mathbf{x}\|_{m}^{m}})$,
where $\lambda_{\min}$ is the smallest Z-eigenvalue (H-eigenvalue) of
$\mathcal{A}$.

###### Proof.

According to Definition 2, because of
$F(\mathbf{x})=\mathcal{A}\mathbf{x}^{m-1}+\mathbf{q}$, so
$[F(\mathbf{x})-F(\mathbf{0})]^{\top}\mathbf{x}=\mathcal{A}\mathbf{x}^{m}$.
Since $\mathcal{A}$ is (strictly) copositive, $\mathcal{A}\mathbf{x}^{m}\ (>)\
\geq 0$. Then, part $(a)$ is proved. We now prove part $(b)$. Since
$\mathcal{A}$ is positive definite, according to [14, Theorem 5], we know that
the smallest Z-eigenvalue (H-eigenvalue) of $\mathcal{A}$ is greater than
zero. Hence, part $(b)$ is proved. ∎

When $\mathcal{A}\in D_{m,n}$ is positive definite, the following theorem will
give a porperty of the Jacobian matrix $\nabla F(\mathbf{x})$, where
$\mathbf{x}$ is nonzero vector.

######  Theorem 4.

Let $\mathcal{A}\in SD_{m,n}$ be positive definite. Then the Jacobian matrix
$\nabla F(\mathbf{x})$ is positive definite with $\mathbf{x}\neq\mathbf{0}$.

###### Proof.

As $\mathcal{A}$ is diagonalizable, for a vector $\mathbf{x}$, according to
Definition 1, we have

$\begin{split}\mathcal{A}\mathbf{x}^{m}&=(\mathcal{D}\times_{1}B\times_{2}B\dots\times_{m}B)\mathbf{x}^{m}=\mathcal{D}(B^{\top}\mathbf{x})^{m}\\\
&=\mathcal{D}\mathbf{y}^{m}\
(\mathbf{y}\overset{\triangle}{=}B^{\top}\mathbf{x})=\sum_{i=1}^{n}d_{i}y_{i}^{m},\end{split}$

where $d_{i}$ is the $i$th diagonal entry of $\mathcal{D}$. According to the
proposition assumption, $d_{i}>0$, we have $\mathcal{A}\mathbf{x}^{m}>0$ for
all nonzero vectors $\mathbf{x}$.

Since the Jacobian matrix $\nabla F(\mathbf{x})$ is
$(m-1)\mathcal{A}\mathbf{x}^{m-2}$, for any nonzero vector
$\mathbf{z}\in\mathds{R}^{n}$, $\mathbf{z}^{\top}\nabla
F(\mathbf{x})\mathbf{z}$ can be expressed by

$\mathbf{z}^{\top}\nabla
F(\mathbf{x})\mathbf{z}=(m-1)\sum\limits_{i=1}^{n}d_{i}y_{i}^{m-2}z_{i}^{2}>0.$

Hence, the Jacobian matrix $\nabla F(\mathbf{x})$ is positive definite with
$\mathbf{x}\neq\mathbf{0}$. ∎

######  Theorem 5.

Suppose that $\mathcal{A}\in ST_{m,n}$. For Question 1, the following results
hold.

  1. (a)

If $\mathcal{A}$ is diagonalizable and positive definite, then the
$\mathrm{NCP}(\mathbf{q},\mathcal{A})$ has a unique solution,

  2. (b)

If $\mathcal{A}$ is positive definite, then the
$\mathrm{NCP}(\mathbf{q},\mathcal{A})$ has a nonempty, compact solution set,

  3. (c)

If $\mathcal{A}$ is strictly copositive with respect to $\mathds{R}_{+}^{n}$,
then the $\mathrm{NCP}(\mathbf{q},\mathcal{A})$ has a nonempty, compact
solution set.

###### Proof.

Part $(a)$: it is obvious, according to Lemma 1 and Theorem 4. We will prove
part $(b)$ as follows. Since $\mathcal{A}$ is positive definite, according to
Theorem 3, we have $F(\mathbf{x})=\mathcal{A}\mathbf{x}^{m-1}+\mathbf{q}$ is
strictly copositive. Let $c(\lambda)=\lambda^{\alpha}$ with $0<\alpha\leq m$,
we know that $c(\lambda)\rightarrow\infty$ as $\lambda\rightarrow\infty$,
then, based on Lemma 2, we know that if $\mathcal{A}$ is positive definite,
then the $\mathrm{NCP}(\mathbf{q},\mathcal{A})$ has a nonempty, compact
solution set.

The rest part is to show that part $(c)$. By Theorem 3, we obtain that
$F(\mathbf{x})=\mathcal{A}\mathbf{x}^{m-1}+\mathbf{q}$ is strictly copositive.
In [8], we have that if $F(\mathbf{x})$ is strictly copositive with respect to
$\mathds{R}_{+}^{n}$, then the mapping $G$ in (1) is $d$-regular for any
$d>0$. Then, according to Lemma 3, if $\mathcal{A}$ is strictly copositive
with respect to $\mathds{R}_{+}^{n}$, then the
$\mathrm{NCP}(\mathbf{q},\mathcal{A})$ has a nonempty, compact solution set. ∎

###  3.3 Solving Question 2

In the above subsection, we consider the solvability of Question 1.
Analogously, The following theorems have been described by the solvability of
Question 2.

######  Theorem 6.

Suppose that $\mathcal{A}_{k}\in T_{m-(2k-2),n}$, with $k\in[m/2]$. For
Question 2, the following results hold.

  1. (a)

If $\mathcal{A}_{k}$ are diagonalizable and positive definite, then the
$\mathrm{GNCP}(\mathbf{q},\\{\mathcal{A}_{k}\\})$ has a unique solution,

  2. (b)

If $\mathcal{A}_{k}$ are positive definite, then the
$\mathrm{GNCP}(\mathbf{q},\\{\mathcal{A}_{k}\\})$ has a nonempty, compact
solution set,

  3. (c)

If $\mathcal{A}_{k}$ are strictly copositive with respect to
$\mathds{R}_{+}^{n}$, then the
$\mathrm{GNCP}(\mathbf{q},\\{\mathcal{A}_{k}\\})$ has a nonempty, compact
solution set,

where $\mathcal{A}_{m/2}$ is a square matrix.

It is easy to prove this theorem according to the above description, then, we
omit the proof. Based on Lemma 1, because $\mathcal{A}_{m/2}$ is a square
matrix, so the constraints of matrix $\mathcal{A}_{m/2}$ can be weakened.
Hence, a more general result is given as follows.

######  Theorem 7.

Suppose that $\mathcal{A}_{k}\in T_{m-(2k-2),n}$, with $k\in[m/2-1]$ and
$\mathcal{A}_{m/2}$ is a square matrix. For Question 2, the following results
hold.

  1. (a)

If $\mathcal{A}_{k}$ are diagonalizable and positive definite and there exists
one $\delta\in(0,1)$, such that all principal minors of $\mathcal{A}_{m/2}$
are bounded between $\delta$ and $\delta^{-1}$, then the
$\mathrm{GNCP}(\mathbf{q},\\{\mathcal{A}_{k}\\})$ has a unique solution,

  2. (b)

If $\mathcal{A}_{k}$ are positive definite and $\mathcal{A}_{m/2}$ is strictly
copositive with respect to $\mathds{R}_{+}^{n}$, then the
$\mathrm{GNCP}(\mathbf{q},\\{\mathcal{A}_{k}\\})$ has a nonempty, compact
solution set,

  3. (c)

If $\mathcal{A}_{k}\ (k\in[m/2])$ are strictly copositive with respect to
$\mathds{R}_{+}^{n}$, then the
$\mathrm{GNCP}(\mathbf{q},\\{\mathcal{A}_{k}\\})$ has a nonempty, compact
solution set.

######  Remark 2.

In the above two theorems, the assumptions of $\mathcal{A}_{k}\in
T_{m-(2k-2),n}$, with $k\in[m/2]$ can be appropriately reduced. However, we
don’t consider these situations.

##  4 Conclusion

In this note, by some structured tensors, the main task is to consider the
existence and uniqueness about the solution of Questions 1 and 2. Now we
present two conjectures about the $\mathrm{NCP}(\mathbf{q},\mathcal{A})$.

######  Conjecture 1.

For the part $(\rm a)$ in Theorem 5, if $\mathcal{A}\in ST_{m,n}$ is just
positive definite, then the $NCP(\mathbf{q},\mathcal{A})$ has a unique
solution.

######  Conjecture 2.

If $\mathrm{FEA}(\mathbf{q},\mathcal{A})\neq\emptyset$, then the nonlinear
program (4) has an optimal solution, $\mathbf{x}_{*}$. Moreover, there exists
a vector $\mathbf{u}_{*}$ of multipliers satisfying the conditions,

$\begin{split}\mathbf{q}+m\mathcal{A}\mathbf{x}_{*}^{m-1}-(m-1)\mathcal{A}\mathbf{x}_{*}^{m-2}\mathbf{u}_{*}&\geq\mathbf{0}\\\
\mathbf{x}_{*}^{\top}(\mathbf{q}+m\mathcal{A}\mathbf{x}_{*}^{m-1}-(m-1)\mathcal{A}\mathbf{x}_{*}^{m-2}\mathbf{u}_{*})&=0\\\
\mathbf{u}_{*}&\geq\mathbf{0}\\\
\mathbf{u}_{*}^{\top}(\mathbf{q}+\mathcal{A}\mathbf{x}_{*}^{m-1})&=0.\end{split}$

Finally, the vectors $\mathbf{x}_{*}$ and $\mathbf{u}_{*}$ satisfy

$(m-1)(\mathbf{x}_{*}-\mathbf{u}_{*})_{i}(\mathcal{A}\mathbf{x}_{*}^{m-2}(\mathbf{x}_{*}-\mathbf{u}_{*}))_{i}\leq
0,\quad i\in[n].$

When $m=2$, this is the theorem about the existence result for a solution of
the quadratic program associated with the linear complementarity problem given
in [3]. Unfortunately, Cottle [2] presented some counter examples to explain
that this conjecture is not true for the general nonlinear programming.

Finally, for the existence and uniqueness about the solution of Question 1, we
have an open question given as follows.

######  Question 3.

Suppose $\mathcal{A}\in T_{m,n}$ and nonzero $\mathbf{x}\in\mathds{R}^{n}$.
What conditions of $\mathcal{A}$ will make sure that there exists one
$\delta\in(0,1)$, such that all principal minors of matrix
$\mathcal{A}\mathbf{x}^{m-2}$ are bounded between $\delta$ and $\delta^{-1}$,
for all $\mathbf{x}\in\mathds{R}_{+}^{n}$?

## References

  * [1] A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, vol. 9 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1994.
  * [2] R. W. Cottle, Nonlinear programs with positively bounded Jacobians, SIAM J. Appl. Math., 14 (1966), pp. 147–158.
  * [3] R. W. Cottle, J.-S. Pang, and R. E. Stone, The Linear Complementarity Problem, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1992.
  * [4] F. Facchinei and J.-S. Pang, Finite-dimensional Variational Inequalities and Complementarity Problems. Vol. I, Springer Series in Operations Research, Springer-Verlag, New York, 2003.
  * [5] , Finite-dimensional Variational Inequalities and Complementarity Problems. Vol. II, Springer Series in Operations Research, Springer-Verlag, New York, 2003.
  * [6] G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, fourth ed., 2013.
  * [7] P. T. Harker and J.-S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: a survey of theory, algorithms and applications, Math. Program., 48 (1990), pp. 161–220.
  * [8] S. Karamardian, The complementarity problem, Math. Program., 2 (1972), pp. 107–129.
  * [9] T. G. Kolda and B. W. Bader, Tensor decompositions and applications, SIAM Rev., 51 (2009), pp. 455–500.
  * [10] L. Lim, Singular values and eigenvalues of tensors: A variational approach, in IEEE CAMSAP 2005: First International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, IEEE, 2005, pp. 129–132.
  * [11] J. J. Moré, Classes of functions and feasibility conditions in nonlinear complementarity problems, Math. Program., 6 (1974), pp. 327–338.
  * [12] J. Nocedal and S. J. Wright, Numerical Optimization, Springer-Verlag, 1999.
  * [13] M. A. Noor, On the nonlinear complementarity problem, J. Math. Anal. Appl., 123 (1987), pp. 455–460.
  * [14] L. Qi, Eigenvalues of a real supersymmetric tensor, J. Symbolic Comput., 40 (2005), pp. 1302–1324.
  * [15] , Symmetric nonnegative tensors and copositive tensors, Linear Algebra Appl., 439 (2013), pp. 228–238.
  * [16] Y. Song and L. Qi, Properties of some classes of structured tensors, J Optim. Theory Appl., to appear (2014).


# Orbital-selective behavior in cubanite CuFe2S3

Elizaveta A. Pankrushina  A.N. Zavaritsky Institute of Geology and
Geochemistry, Ural Branch of the Russian Academy of Sciences, Ak. Vonsovskogo
St. 15, 620016, Ekaterinburg, Russia  Alexey V. Ushakov  M.N. Miheev Institute
of Metal Physics of Ural Branch of Russian Academy of Sciences, S.
Kovalevskaya St. 18, 620990 Ekaterinburg, Russia  Mohsen M. Abd-Elmeguid  II.
Physikalisches Institut, Universität zu Koln, Zülpicher Strasse 77, 50937
Köln, Germany  Sergey V. Streltsov<EMAIL_ADDRESS>M.N. Miheev
Institute of Metal Physics of Ural Branch of Russian Academy of Sciences, S.
Kovalevskaya St. 18, 620990 Ekaterinburg, Russia  Department of theoretical
physics and applied mathematics, Ural Federal University, Mira St. 19, 620002
Ekaterinburg, Russia

(August 28, 2024)

###### Abstract

Using ab initio band structure calculations we show that mineral cubanite,
CuFe2S3, demonstrates an orbital-selective behavior with some of the electrons
occupying molecular orbitals of $x^{2}-y^{2}$ symmetry and others localized at
atomic orbitals. This is a rare situation for $3d$ transition metal compounds
explains experimentally observed absence of charge disproportionation,
anomalous Mössbauer data, and ferromagnetic ordering in between nearest
neighbor Fe ions.

###### pacs:

71.27.+a, 71.20.-b, 71.15.Mb

##  I Introduction

In recent years there has been growing interest in investigating the ground
state properties of transition metal (TM) compounds in which the orbital
degrees of freedom play a dominant role Jackeli and Khaliullin (2009); Liu and
Khaliullin (2018); Feiguin et al. (2019); Streltsov and Khomskii (2020); Li et
al. (2021). Particularly interesting is the directional character of the $d$
orbitals that can lead to the formation of various electronic and magnetic
ground states, see e.g. Ref. [6]. Indeed, pronounced effects have been
recently reported for many $4d$ and $5d$ metal cluster compounds revealing
that different orbitals can behave in different ways, i.e. demonstrate
orbital-selective behavior: orbitals directed toward neighbors in a TM dimer
(or trimer) behave as delocalized and can be described by molecular orbitals,
whereas the electrons in other orbitals are localized. This can result in a
suppression of the effective magnetic moment and strongly affects the
mechanism of exchange interaction. Thus, it turns out that the ground state
properties of such cluster compounds are determined by their orbital structure
and the related orbital-selective behavior.

However, in general such a situation is more typical for $4d-5d$ systems,
since a direct overlap between more spatially extended $4d-5d$ orbitals is
much larger than in case of $3d$. In this respect, ternary chalcogenide
compound CuFe2S3 (mineral cubanite) is a promising candidate for such study,
since its electronic and magnetic ground state properties are far from being
clear. CuFe2S3 crystallizes in the orthorhombic structure with a single class
of FeSzymański (1974); Fleet (1970); Goh et al. (2010); Buerger (1945, 1947).
It was argued that iron has a nominal valency of +2.5, due to a “rapid
electron exchange” between Fe2+ and Fe3+ ions in Fe–Fe dimersImbert and
Wintenberger (1967); Greenwood and Whitfield (1968). The Fe2+–Fe3+ ion pairs
are tetrahedrally coordinated by S atoms with two FeS4 tetrahedra sharing
common edges, forming a cluster of paired FeS4 tetrahedra. More structural
details and given below in Section II. We note that 57Fe Mössbauer
measurements also indicate that there is only one Fe site and the data
obtained are explained assuming “rapid electron exchange” between Fe2+
($3d^{6}$) and Fe2+($3d^{5}$), giving an intermediate valency of about 2.5.
Such fast electronic fluctuation would imply metallic conductivity, however,
resistivity measurement on natural single crystal of CuFe2S3 have shown n-type
semiconductor behaviorSleight and Gillson (1973).

Magnetic measurements on CuFe2S3 show that cubanite is canted antiferromagnet
(AFM) at low-temperature phase. Interestingly enough nearest iron neighbors
inside a Fe–Fe dimer are coupled not antiferromagnetically as a conventional
superexchange theory would suggest (see e.g. Ref. [6] or Ref. [15]), but
ferromagnetically. The Fe–Fe dimers are coupled to each other
antiferromagnetically Wintenberger et al. (1974); Townsend et al. (1973). The
local magnetic moment was found to be $\sim$3.2$\mu_{B}$, i.e. somewhat
smaller than what one would expect even for Fe2+. Wintenberger et al. (1974)

Figure 1:  The crystal structure of CuFe2S3. FeS4 (CuS4) tetrahedra are shown
in red (green) colour. Two neighboring FeS4 tetrahedra have a common edge.

Motivated by the above mentioned physical aspects and in view of the current
research activity on CuFe2S3 for several technological applications, we have
investigated the ground state of CuFe2S3 using GGA and GGA+U calculations. We
were able to unveil the ground state properties of CuFe2S3 and show that there
are two types of $d$ orbitals in cubanite. First ones form molecular orbitals
for two neighboring irons and electrons occupying these orbitals belong to
both ions. Second are localized at atomic sites. This not only explains an
intermediate valence 2.5+ of Fe and anomalous results of Mössbauer
measurements, but also elucidates the origin of the unexpected magnetic
structure of this compound.

##  II Crystal structure

The cubanite CuFe2S3 crystallizes within an orthorhombic structure (space
group $Pnma$, a = 6.23 Å, b = 11.11 Å and c = 6.46 Å) with a quasi-hexagonal
stacking of S2- anions where the cations are tetrahedrally coordinated (see
Fig. 1)Fleet (1970). The structure is based upon a hexagonal close-packed
network of S atoms with the cations in ordered, tetrahedral sites; the Cu
atoms and 1/3 of the S atoms occupy the equipoint 4c (mirror planes), and the
Fe atoms and remaining 2/3 of the S atoms are in the general positions, 8d
Fleet (1970). Experimental atomic positions and lattice parameters are taken
from Ref. [11].

In orthorhombic cubanite Cu and Fe atoms are tetrahedrally coordinated by S
atoms with two FeS4 tetrahedra sharing their edges. This gives pairs of Fe
ions with rather short distances between them Szymański (1974); Fleet (1970);
Goh et al. (2010); Buerger (1945, 1947). distance of 2.81 Å, however, is too
long to represent a chemical bond. It is also larger than that in KFeS2 where
tetrahedrally-coordinated iron atoms form chains with the Fe–Fe distance of
2.7 Å Hullige (1968).

##  III Calculation details

Figure 2:  The band structure (a) and the partial densities of states (b) of
CuFe2S3 obtained in non-magnetic GGA calculations. The Fermi level is in zero.
One may note 4 isolated Fe-$3d$ bands between $-1.1$ and $-1.6$ eV (marked in
red lines), which correspond to two bonding molecular states.

The ab initio band structure calculations of CuFe2S3 were carried out within
the framework of density functional theory (DFT) Hohenberg and Kohn (1964)
implemented in VASP package Kresse and Furthmuller (1996). We used the
generalized gradient approximation (GGA) Perdew et al. (1996) and projector
augmented wave (PAW) method Blöchl (1994). The exchange-correlation functional
in Perdew-Burker-Ernzerhof (PBE) form was utilized Perdew et al. (1997). The
cut-off energy was chosen to be 600 eV and the mesh of $6\times 3\times 6$ was
used for integration over the Brillouin zone according to Monkhorst-Pack
scheme Monkhorst and Pack (1976). A non-interacting GGA Hamiltonian for the
estimation of hopping integrals inside the Fe $3d$ states was generated using
the Wannier projection procedure Korotin et al. (2008) in Quantum Espresso
code Gianozzi et al. (2009) on the same k-point grid. The correlation effects
were taken into account via GGA+$U$ approach as introduced in Ref. [27]. We
chose the on-site Coulomb repulsion parameter to be $U$=$8$ eV and $U$=$6$ eV
for Cu and Fe respectively, while the Hund’s rule coupling parameter ($J_{H}$)
was taken as $J_{H}$=$0.95$ eV for both $3d$ metal transition ions Ushakov et
al. (2017); Pesant and Cote (2011). The occupation numbers of Fe-$3d$ states
were obtained by integration within atomic sphere with radius $1.302$ Å. The
crystal structure in GGA+$U$ calculations was relaxed unless the interatomic
forces were larger than 0.005 eV/Å.

##  IV Calculation results

We start with the simplest non-magnetic calculations, which results are
presented in Fig. 2. One might see that the bands extending from -2.7 to -1.5
eV correspond to Cu-$3d$ states, while Fe-$3d$ bands are between -1.5 to 2 eV.
S-$3p$ bands are below -3 eV (not shown in Fig. 2a). There are 4 formula units
in the unit cell used in the calculation. All twenty (twice degenerate due to
spin) Cu-$d$ bands are below the Fermi energy and therefore we see that Cu
must be 1+ ($3d^{10}$) in cubanite.

Figure 3:  The partial charge density represented 4 separated Fe-$3d$ bands on
[$-1.1$, $-1.6$] eV interval in a band structure on Fig. 2(a).

The second important fact, which can be extracted from this type of
calculations is that we see four lowest isolated Fe-$3d$ bands in energy range
from -1.1 to -1.6 eV (Fig. 2a). Thus, there is one such band per each formula
unit. A partial charge density corresponding to these bands shows that these
are $e_{g}$ states ($x^{2}-y^{2}$ orbitals in the local coordinate system,
where the axis points to the centers of tetrahedron edges), see Fig. 3. One
can also see that these are exactly two $d$ orbitals directed to each other in
the common edge geometry of two FeS4 tetrahedra.

This explains Mössbauer data, which do not show charge disproportionation
between two irons, but observe a “rapid electron exchange”Imbert and
Wintenberger (1967); Greenwood and Whitfield (1968), which is obviously a
consequence of molecular orbitals formation. Moreover, one can see that there
are two types of $d$ electrons in CuFe2S3 those forming molecular orbitals
($x^{2}-y^{2}$) and others, which are expected to be localized at atomic
sites. Thus, cubanite could be an example of materials demonstrating orbital-
selective behavior Streltsov and Khomskii (2014, 2016).

There are important implications of orbital-selective physics in CuFe2S3. E.g.
in the case of odd number of electrons on $d$ shell the mechanism like double
exchange stabilizes ferromagnetic exchange interaction for nearest neighbor Fe
ions (those forming dimers Fe2S6 with a common edge of FeS4 tetrahedra).
Neglecting conventional superexchange one can readily see that the
ferromagnetic configuration shown in Fig. 4a has the lowest total energy
$E_{a}=-20J_{H}-t$, where $t$ is a hopping parameter for electrons forming
molecular orbitals and $J_{H}$ is the intra-atomic exchange. Antiferromagnetic
order is impossible in this situation since one might have only one electron
with given spin per site. This state is always lower in energy than the
configuration with site-localized electrons of Fig. 4b (only antiferro version
is shown; the ferro one has the same energy if one neglects super-exchange),
$E_{b}=-20J_{H}$. There also will be the configurations with two electrons
occupying the molecular orbital, one of them is shown in Fig. 4c, it is energy
is $E_{c}=-16.5J_{H}-2t$. Thus, we see that ferromagnetic “intra-dimer” order
wins if $3.5J_{H}>t$, which is exactly the case of CuFe2S3, since $J_{H}\sim
1$ eV and $t=0.4$ eV as follows from our DFT calculations (the value of $t$
was taken from Wannier function projection).

Figure 4:  The sketch illustrating stabilization of the ferromagnetic order,
case (a), due to the formation of molecular orbitals ($x^{2}-y^{2}$) in Fe–Fe
pair. (b) and (c) demonstrate two other possible solutions with all electrons
localized at sites (case b) and two electrons occupying the molecular orbital
(case c). It is assumed that bonding-antibonding splitting is large and one
can neglect superexchange between localized electrons.

This idea is very similar to a conventional double exchange: itinerant
electrons with the $x^{2}-y^{2}$ symmetry move from one Fe site to another and
thereby substantially lowers the total energy (Fig. 4a). This is only possible
if both Fe sites have the same spin projection. If there would be no such
fluctuations, i.e. the system will be in configurations shown in Fig. 4b or
Fig. 4c, there will be no such energy gain.

Table 1:  The total energies (Etot) and energy gap (Egap) for the insulating states as obtained from GGA and GGA+$U$ ($U$(Cu)=8 eV, $U$(Fe)=6 eV) calculations for the different magnetic structures of CuFe2S3 (per formula unit). Notation are the same as in Ref. [17], corresponding magnetic configurations are presented in Fig. 5. Magnetic | GGA | GGA+$U$
---|---|---
config. | Etot, meV | Egap, eV | Etot, meV | Egap, eV
A | 0 | 0.3 | 0 | 0.8
C | 191.6 | 0.05 | 98.1 | 0.51
AF dimers | 262.6 | no | 179.6 | no
B | 621.2 | no | 335.4 | 0.35

This discussion is in accord with direct magnetic GGA calculations, which
results are presented in Table 1. The lowest in total energy is the
configuration with Fe spins ordered ferromagnetically inside Fe–Fe dimers and
maximally antiferromagnetic between them (configuration A). This result agrees
with analisys of magnetic data performed in Ref. [17]. It also has to be
mentioned a conventional superexchange mechanism between localized electrons
tends to stabilize AFM order and is indeed operative for inter-dimer exchange
interaction, while intra-dimer ferromagnetism is due to the formation of
molecular orbitals, i.e. orbital-selective behavior.

Figure 5:  The schematic representation of the possible magnetic structures of
CuFe2S3 discussed in Ref.[ 17] with FM order in Fe–Fe dimers and one more with
AFM order in Fe–Fe dimers considered in this paper.

Moreover we see that our GGA calculations are able to reproduce the correct
magnetic ground state, and CuFe2S3 becomes insulating already in GGA
approximation. The energy gap in configuration A is about 0.3 eV. Thus the
ferromagnetic order of Fe spin moments in Fe-Fe dimers retains an additional
itinerant electron for two irons ions within a dimer cluster

In order to take into account correlation effects in CuFe2S3, we carried out
GGA+$U$ calculations. The results are shown in Fig. 6 and displayed in Table
1. One can see that the band gap in GGA+$U$ increases up to about 0.8 eV in
configuration A. The total energy calculations demonstrate that the state with
FM dimer remains the lowest, see Table 1. The decrease of the energy
difference between various configurations is related to the fact that now not
Stoner-like exchange, but a much larger Hubbard $U$ correction defines the
superexchange processes (in the denominator for the superexchange
interaction).

The detailed analysis of the occupation matrices shows that the charge
disproportionation does not occur even if all symmetry information is removed
in the calculations. This suggests that molecular orbitals are not destroyed
completely by the Hubbard correction (which tends to localize all electrons on
atomic orbitals), but a more accurate methods such as e.g. cluster DFT+DMFT
calculationsBiroli and Kotliar (2002) should be used to study this effect.
Moreover, lattice optimization also does not break the charge homogeneous
state. Interestingly enough, we were able to obtain charge ordered solution in
GGA+$U$ approach when Fe ions are coupled AFM in a Fe2S6 dimers, but its total
energy is higher on $180$ meV.

In order to justify the validity of chosen $U$ parameters we repeat the
calculations using smaller $U$(Fe)=4.5 eV, as e.g. in Ref.Pchelkina and
Streltsov (2013), and $U$(Cu)=8 eV ($J_{H}$ on both ions is 0.95 eV). The
results remain the same: the magnetic configuration A corresponds to the
ground state, which is insulating with a band gap is of $\sim$1 eV. We obtain
for other configurations have total energies of: 725 (B), 125 (C), and 400
(AFM dimers) meV. The ground state also does not change if one ignores Hubbard
$U$ on Cu (it also retains insulating behavior).

Figure 6:  The band structure (a) and partial densities of states (b) of
CuFe2S3 obtained in GGA+U approximation ($U$(Cu) = $8$ eV, $U$(Fe) = $6$ eV,
$J_{H}$(Cu, Fe) = $0.95$ eV). The bands with bonding-antibonding splitting of
molecular orbitals are marked by red color, their character was deduced from
the corresponding charge-density plots.

##  V Conclusions

We used ab initio calculations to study the physical properties of CuFe2S3. In
transition metal compounds containing structural clusters of metals there can
be realized a special state, when some of the electrons form singlet ($S=0$)
pairs, while others are effectively decoupled and may give e.g. a long-range
magnetic order or stay paramagnetic Streltsov and Khomskii (2014, 2016). Our
GGA calculations show that there are indeed two types of $3d$-electrons in
cubanite: those forming molecular orbitals ($x^{2}-y^{2}$) and others, which
are expected to be localized at atomic sites. This fact determines CuFe2S3 as
a system with orbital-selective behavior. This in turn strongly affects
magnetic properties of cubanite making exchange interaction between nearest
neighbors strong and ferromagnetic (by “double-exchange-like” mechanism),
while magnetic ordering between other irons is governed by the superexchange
path.

The orbital-selective behavior also explains puzzling Mössbauer data, where no
Fe separation onto Fe2+ and Fe3+, but a “rapid electron exchange” has been
observed. We feel that our findings could be applied to explain the metallic
high pressure phase of CuFe2S3, where Fe ions are surrounded by sulphur
octahedra, forming dimers with a common faceRozenberg et al. (1997). In such a
geometry a1g orbitals may take part in a strong molecular bonding leaving
other electrons ($e_{g}^{\pi}$) site localized as it occurs in e.g.
Ba5AlIr2O11 Terzic et al. (2015); Streltsov et al. (2017) or hexagonal
structure such as Ba3Me$TM$O9 (where $TM$ is a transition metal and Me is
metal like Li, Na, La etc.)Nguyen and Cava (2021). Of course this has to be
checked by corresponding calculations on the high pressure phase of CuFe2S3.

##  VI Acknowledgments

We acknowledge support of the Russian Science Foundation via project
20-62-46047 and also thank Prof. A. Fujimori for useful discussions on
physical properties of this material.

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