# Compactness of composition operators on the Bergman spaces of convex domains
and analytic discs

Timothy G. Clos  Timothy G. Clos, Kent State University<EMAIL_ADDRESS>
###### Abstract.

We study the compactness of composition operators on the Bergman spaces of
certain bounded pseudoconvex domains in $\mathbb{C}^{n}$ with non-trivial
analytic discs contained in the boundary. As a consequence we characterize
that compactness of the composition operator with a continuous symbol (up to
the closure) on the Bergman space of the polydisc.

##  1\. Introduction

Let $\Omega\subset\mathbb{C}^{n}$ be a bounded convex domain. Let
$\mathcal{O}(\Omega)$ be the set of all holomorphic functions from $\Omega$
into $\mathbb{C}$. Let $V$ be the Lebesgue volume measure on $\Omega$. For
$p\in[1,\infty)$ we define

$A^{p}(\Omega):=\\{f\in\mathcal{O}(\Omega):\int_{\Omega}|f|^{p}dV<\infty\\}$

to be the $p$-Bergman space. We denote the norm as

$\|f\|_{p,\Omega}:=\left(\int_{\Omega}|f|^{p}dV\right)^{\frac{1}{p}}.$

This paper will consider the $2$-Bergman space, which for brevity is denoted
as the Bergman space. Let $\phi:\Omega\rightarrow\Omega$ be a holomorphic
self-map on $\Omega$. That is, holomorphic in each coordinate function. Then
we define the composition operator with symbol $\phi$ as

$C_{\phi}(f)=f\circ\phi$

for all $f\in A^{p}(\Omega)$. A set $\Delta\subset\mathbb{C}^{n}$ is called an
analytic disc if there exists functions
$f_{j}:\overline{\mathbb{D}}\rightarrow\mathbb{C}$ continuous on
$\overline{\mathbb{D}}$ and holomorphic on $\mathbb{D}$ for
$j\in\\{1,2,...,n\\}$ and $F(\mathbb{D})=\Delta$ where $F=(f_{1},...,f_{n})$.
If $\Delta$ is not a single point, then $\Delta$ is said to be a non-trivial
analytic disc. We define the collection of all non-trivial analytic discs in
$b\Omega$ to be

$\mathcal{L}:=\bigcup\\{F(\mathbb{D})|\,\,F:\mathbb{D}\rightarrow
b\Omega\,,\text{holomorphic and non constant}\\}.$

As we shall see, analytic structure in the boundary of the domain will play a
crucial role in the proof of the main result. The main theorem does not
generalize easily to bounded pseudoconvex domains because of the special
geometry of non-trivial analytic discs in the boundary of the domain. Namely
the following lemma states that on convex domains, non-trivial analytic discs
have a nice form. See also [FS98].

######  Lemma 1 ([ČŞ09]).

Let $\Omega\subset\mathbb{C}^{n}$ be a bounded convex domain. Let
$\Delta\subset b\Omega$ be a non-trivial analytic disc. Then there exists a
complex line $L$ so that the convex hull of $\overline{\Delta}$ is contained
in $L$.

The following definitions are needed for the statement of the main results. We
first consider the notion of an angular derivative, which play an important
role in the theory of composition operators in one and several variables.
Here, $d(.,b\Omega)$ is Euclidean distance to the boundary.

######  Definition 1.

Let $\Omega\subset\mathbb{C}^{n}$ be a bounded domain for $n\geq 1$. Let
$\phi:\Omega\rightarrow\mathbb{C}^{n}$ be a holomorphic self-map. Then for
$p\in b\Omega$, we define the angular derivative at $p$ as

$\lim_{\zeta\in\Omega,\,\zeta\rightarrow
p}\frac{d(\phi(\zeta),b\Omega)}{d(\zeta,b\Omega)},$

provided this limit is finite. If

$\lim_{\zeta\in\Omega,\,\zeta\rightarrow
p}\frac{d(\phi(\zeta),b\Omega)}{d(\zeta,b\Omega)}=\infty,$

we say the angular derivative does not exist at $p$.

Of course, non-existence of the angular derivative at $p$ is equivalent to

$\lim_{\zeta\in\Omega,\,\zeta\rightarrow
p}\frac{d(\zeta,b\Omega)}{d(\phi(\zeta),b\Omega)}=0.$

Please see [Sha93] for more information on angular derivatives and composition
operators in general.

######  Definition 2.

A bounded domain $\Omega\subset\mathbb{C}^{n}$ satisfies the limit point disc
condition if $p\in\overline{\mathcal{L}}$ then there exists a non-trivial
analytic disc $\Delta\subset b\Omega$ so that $p\in\overline{\Delta}$.

It is worth noting there are domains that do not satsify the limit point disc
condition. In fact, any bounded domain $\Omega\subset\mathbb{C}^{n}$
satisfying these properties does not satisfy the limit point disc condition.

  1. (1)

There exists a sequence of non-trivial analytic discs
$\\{\Delta_{j}\\}_{j\in\mathbb{N}}$ so that $\Delta_{j}\subset b\Omega$ for
all $j\in\mathbb{N}$.

  2. (2)

$\overline{\Delta_{J}}\cap\overline{\Delta_{K}}=\emptyset$ for any
$J,K\in\mathbb{N}$.

  3. (3)

There exists $F_{j}:\overline{\mathbb{D}}\rightarrow b\Omega$ holomorphic on
$\mathbb{D}$ and continuous on $\overline{\mathbb{D}}$ so that
$F_{j}(\mathbb{D})=\Delta_{j}$ and
$F_{j}\rightarrow(p_{1},...,p_{n})\in\mathbb{C}^{n}$ as $j\rightarrow\infty$.

A bounded pseudoconvex domain $\Omega\subset\mathbb{C}^{n}$ with
$C^{2}$-smooth boundary has defining function $\rho\in C^{2}(\mathbb{C}^{n})$.
Then for $p\in b\Omega$, the Levi form at $p$ is the complex Hessian

$\left(\frac{\partial^{2}\rho}{\partial
z_{k}\partial\overline{z_{j}}}(p)\right)_{n\times n}$

restricted to the complex tangent space at $p$. See [Str10] for more
information about the Levi form and its important properties. For example, the
Levi form is positive semi-definite on $b\Omega$. Furthermore, if $p$ is in
the closure of a non-trivial analytic disc in $b\Omega$, then $p$ is Levi-flat
(the Levi form has a zero eigenvalue at $p$). Let $\mathcal{F}$ denote the set
of all Levi-flat points in $b\Omega$. The following differential geometric
condition involving the Levi form allows us to control the Levi-flat points.

######  Definition 3.

Let $\Omega\subset\mathbb{C}^{n}$ be a bounded $C^{2}$-smooth convex domain.
We say $\Omega$ has the Levi-flat disc condition if

$\bigcup\\{\overline{F(\mathbb{D})}|\,\,F:\mathbb{D}\rightarrow
b\Omega\,,\text{holomorphic and non constant}\\}=\mathcal{F}.$

######  Lemma 2.

Let $\Omega\subset\mathbb{C}^{n}$ be a bounded $C^{2}$-smooth convex domain.
If $\Omega$ has Levi-flat disc condition, then $\Omega$ satisfies the limit
point disc condition.

###### Proof.

Suppose $\Omega$ has the Levi-flat disc condition. Then for any
$q\in\overline{\mathcal{L}}$, there exists a sequence of points
$\\{q_{j}\\}_{j\in\mathbb{N}}\subset\mathcal{L}$ converging to $q$. Now
$q_{j}$ are all Levi-flat, and the eigenvalues of the Levi form vary
continuously. This implies
$q\in\mathcal{F}=\bigcup\\{\overline{F(\mathbb{D})}|\,\,F:\mathbb{D}\rightarrow
b\Omega\,,\text{holomorphic and non constant}\\}$ by the Levi-flat disc
condition. ∎

Note one can construct a smooth bounded convex Reinhardt domain with non-
trivial analytic discs in the boundary satisfying the Levi-flat disc
condition. Intuitively, one can take a bidisc in $\mathbb{C}^{2}$ and ’smooth
out’ the distinguished boundary into a strongly pseudoconvex part. See [CŞ18]
and [Clo17] for more information on convex Reinhardt domains in
$\mathbb{C}^{2}$.

##  2\. Some Background and Main Results

Compactness of composition operators was studied on the unit disc in
$\mathbb{D}$ in the article [CEGY98]. Here, the authors of [CEGY98] study the
angular derivative of the symbol near the boundary and obtain a compactness
result. They then construct a counterexample to show that the converse of
their theorem does not hold true. The authors of [AGT10] studied the closed
range property of composition operators on the unit disc. Work on essential
norm estimates and compactness of composition operators was studied on the
ball in $\mathbb{C}^{n}$ and on the unit disc in $\mathbb{C}$ by [CM95] and
[HMW11]. On more general bounded strongly pseudoconvex domains in
$\mathbb{C}^{n}$, [ČZ07] studied the essential norm of the composition
operator in terms of the behavior of the norm of the normalized Bergman kernel
composed with the symbol.

In the work [Zhu07], Zhu uses the normalized Bergman kernel as a weakly
convergent sequence. Since these can be explicitly computed on the ball, this
results in a non-trivial characterization of compactness. Zhu in [Zhu07] also
extends these results to bounded strongly pseudoconvex domains. Both these
characterizations involve the angular derivative mentioned earlier. In the
absence of such explicit Bergman kernels (on general bounded convex domains),
we construct weakly convergent sequences using the convexity of the domain in
a significant way, and use estimates on the normalized Bergman kernel near
strongly pseudoconvex points. Here, $K^{\Omega}_{z}$ denotes the Bergman
kernel of $\Omega$, $V$ is the Lebesgue volume measure. We let
$k_{z}^{\Omega}$ be the normalized Bergman kernel of $\Omega$. As notation, we
sometimes write $k_{z}$ as the normalized Bergman kernel of $\Omega$ if there
is no ambiguity about the domain. The following result is well known in the
literature and a proof is given in [Tik15].

######  Proposition 1.

Let $\Omega$ be a smooth bounded pseudoconvex domain. Then the normalized
Bergman kernel $k_{\zeta}^{\Omega}\rightarrow 0$ weakly in $A^{2}(\Omega)$ as
$\zeta\rightarrow b\Omega$.

The following two thereoms are the main results. This first result states that
if $C_{\phi}$ is compact, $\Omega$ is convex, and $\phi\in
C(\overline{\Omega})$ then $\phi(b\Omega)$ stays away from
$\overline{\mathcal{L}}$. We note that this limit point disc condition is an
important geometric assumption, as it allows us to construct a weakly
converging sequence (in the Bergman space) with the required bound from below
near a non-trivial analytic disc in the boundary.

######  Theorem 1.

Let $\Omega\subset\mathbb{C}^{n}$ be a bounded convex domain satisfying the
limit point disc condition. Let $\phi:\Omega\rightarrow\Omega$ be a
holomorphic self map, $\phi\in C(\overline{\Omega})$, and suppose $C_{\phi}$
is compact on $A^{2}(\Omega)$. Then,
$d(\phi(b\Omega),\overline{\mathcal{L}})>0$.

Next we consider symbols that are holomorphic on a neighborhood of
$\overline{\Omega}$ for a strongly convex bounded domain $\Omega$. We let
$J_{\phi}$ be the determinant of the Jacobian matrix of $\phi$.

######  Theorem 2.

Let $\Omega\subset\mathbb{C}^{n}$ be a smooth bounded strongly convex domain.
Let $\phi:\Omega\rightarrow\Omega$ be a holomorphic self-map, where
$\phi:=(\phi_{1},...,\phi_{n})$ and $\phi_{j}$ are holomorphic on a
neighbourhood of $\overline{\Omega}$ for $j=1,...,n$. If $C_{\phi}$ is compact
on $A^{2}(\Omega)$, then $b\Omega\cap b\phi(\overline{\Omega})$ is nowhere
dense in $b\Omega$.

If we assume that the domain is smooth, bounded, convex, and with a certain
boundary condition controlling weakly pseudoconvex points, we can get a
generalization of one implication of [Zhu07]. Recall [Zhu07] showed that
compactness of $C_{\phi}$ implies the angular derivative does not exist at any
boundary point of a bounded strongly pseudoconvex domain (which has no non-
trivial analytic discs in its boundary). Furthermore, [Zhu07] showed the non-
existence of the angular derivative of $\phi$ at any boundary point of a
bounded strongly pseudoconvex domain is a necessary and sufficent condition
for compactness of $C_{\phi}$. The following corollary is consequence of
Theorem 1. Notice we allow the existence of non-trivial analytic discs in the
following corollary.

######  Corollary 1.

Let $\Omega\subset\mathbb{C}^{n}$ be a smooth bounded convex domain satisfying
the Levi-flat disc condition. Let $\phi:\Omega\rightarrow\Omega$ be a
holomorphic self-map, and $\phi\in C(\overline{\Omega})$. Suppose $C_{\phi}$
is compact on $A^{2}(\Omega)$, and $p\in b\Omega$ be a strongly pseudoconvex
point. Then the angular derivative does not exist at $p$. That is,

$\lim_{\zeta\in\Omega\,\,,\zeta\rightarrow
p}\frac{d(\zeta,b\Omega)}{d(\phi(\zeta),b\Omega)}=0.$

###### Proof.

From Theorem 1 and Lemma 2, we may assume $\phi(p)\in b\Omega$ and the
boundary of $\Omega$ near $\phi(p)$ is strongly pseudoconvex. Then one can
show that, for $z$ near $p$,

$\|C^{*}_{\phi}(k_{z})\|^{2}=K(z,z)^{-1}K(\phi(z),\phi(z))\approx\frac{d(z,b\Omega)^{n+1}}{d(\phi(z),b\Omega)^{n+1}}.$

Since $C^{*}_{\phi}$ is compact and $k_{z}\rightarrow 0$ weakly as
$z\rightarrow p$ (refer to Proposition 1), we have

$\lim_{\zeta\in\Omega\,\,,\zeta\rightarrow
p}\frac{d(\zeta,b\Omega)}{d(\phi(\zeta),b\Omega)}=0.$

∎

There are bounded convex domains with the Levi-flat disc condition (namely
bounded strongly convex domains) where the converse of Corollary 1 holds.
However, it remains an open question in general. The following is a corollary
of Theorem 1 and Proposition 2 and concerns the $n$-product of discs.

######  Corollary 2.

Let $\mathbb{D}^{n}$ be the polydisc in $\mathbb{C}^{n}$. Suppose $\phi\in
C(\overline{\Omega})$ and $\phi:\mathbb{D}^{n}\rightarrow\mathbb{D}^{n}$ is a
holomorphic self map. Then $C_{\phi}$ is compact if and only if
$\phi(\mathbb{D}^{n})$ is relatively compact in $\mathbb{D}^{n}$.

##  3\. Proof of Main Results

First we show that compactness of composition operators is invariant under
biholomorphisms of the domain $\Omega$.

######  Lemma 3.

Let $\Omega_{1},\Omega_{2}\subset\mathbb{C}^{n}$ for $n\geq 2$ be bounded
pseudoconvex domains. Furthermore, assume there exists a biholomorphism
$B:\Omega_{1}\rightarrow\Omega_{2}$ so that $B\in
C^{1}(\overline{\Omega_{1}})$. Suppose
$\phi:=(\phi_{1},\phi_{2},...,\phi_{n}):\Omega_{2}\rightarrow\Omega_{2}$ is
such that the composition operator $C_{\phi}$ is compact on
$A^{2}(\Omega_{2})$. Then, $C_{B^{-1}\circ\phi\circ B}$ is compact on
$A^{2}(\Omega_{1})$.

###### Proof.

Let $g_{j}\in A^{2}(\Omega_{1})$ so that $g_{j}\rightarrow 0$ weakly as
$j\rightarrow\infty$. We will use the fact that $g_{j}\rightarrow 0$ weakly in
$A^{2}(\Omega_{1})$ as $j\rightarrow\infty$ if and only if $\|g_{j}\|$ is a
bounded sequence in $j$ and $g_{j}\rightarrow 0$ uniformly on compact subsets
of $\Omega_{1}$. This fact appears as [ČZ07, lemma 3.5]. Therefore,
$\|g_{j}\|$ is uniformly bounded in $j$ and $g_{j}\rightarrow 0$ uniformly on
compact subsets of $\Omega_{1}$. Then define $h_{j}:=g_{j}\circ B^{-1}\in
A^{2}(\Omega_{2})$. Then using a change of coordinates, one can show
$\|h_{j}\|$ is uniformly bounded in $j$ and $h_{j}\rightarrow 0$ uniformly on
compact subsets of $\Omega_{2}$. Therefore, by [ČZ07, lemma 3.5],
$h_{j}\rightarrow 0$ weakly as $j\rightarrow\infty$. Then we have,

$\displaystyle\|C_{B^{-1}\circ\phi\circ B}(g_{j})\|_{2,\Omega_{1}}^{2}$
$\displaystyle=\|h_{j}\circ\phi\circ B\|_{2,\Omega_{1}}^{2}$
$\displaystyle\leq\sup\\{|J_{B^{-1}}(z)|^{2}:z\in\Omega_{2}\\}\|C_{\phi}(h_{j})\|_{2,\Omega_{2}}^{2}$

This shows that $C_{B^{-1}\circ\phi\circ B}$ is compact on
$A^{2}(\Omega_{1})$.

∎

######  Proposition 2.

Let $\Omega\subset\mathbb{C}^{n}$ be a bounded pseudoconvex domain. Suppose
$\phi$ is a holomorphic self-map on $\Omega$ so that
$\overline{\phi(\Omega)}\subset\Omega$. Then $C_{\phi}$ is compact on
$A^{2}(\Omega)$.

###### Proof.

We let $V_{\phi}:=V\circ\phi^{-1}$ be the pullback measure associated with
$\phi$, where $V$ is the Lebesgue volume measure on $\Omega$. Since
$\phi(\Omega)$ is relatively compact in $\Omega$, the support of $V_{\phi}$ is
relatively compact in $\Omega$. Let $\\{g_{j}\\}_{j\in\mathbb{N}}\subset
A^{2}(\Omega)$ so that $g_{j}\rightarrow 0$ weakly as $j\rightarrow\infty$.
Thus $g_{j}\rightarrow 0$ uniformly on compact subsets of $\Omega$ as
$j\rightarrow\infty$ (see [ČZ07, lemma 3.5]). Since $V_{\phi}$ has compact
support, there exists $M$ relatively compact in $\Omega$ so that

$\|C_{\phi}g_{j}\|^{2}_{L^{2}(\Omega,V)}=\|g_{j}\|^{2}_{L^{2}(\overline{M},V_{\phi})}\leq\|g_{j}\|^{2}_{L^{\infty}(\overline{M})}V_{\phi}(\overline{M})\rightarrow
0$

as $j\rightarrow\infty$.

∎

###### Proof of Theorem 1.

Let us assume $C_{\phi}$ is compact on $A^{2}(\Omega)$ and
$\phi^{-1}(b\Omega)\neq\emptyset$ (otherwise
$d(\phi(b\Omega),\overline{\mathcal{L}})>0$ is vacuously true). Let
$p\in\phi^{-1}(b\Omega)$. The first claim is that $\phi(p)$ cannot be
contained in the closure of a non-trivial analytic disc in $b\Omega$. For the
sake of obtaining a contradiction, suppose
$\phi(p)\in\overline{\mathcal{L}}\subset b\Omega$. By the convexity of
$\Omega$, we may assume $\Omega\subset\\{Re(z_{n})>0\\}$. Since $\Omega$
satisfies the limit point disc condition, we may assume there exists a non-
trivial analytic disc $\Delta\subset b\Omega$ and
$\phi(p)\in\overline{\Delta}$. Then, by Lemma 1, $\Delta\subset L$ for some
complex line $L$. Let $p_{j}\in\Omega$ so that $p_{j}\rightarrow p$ as
$j\rightarrow\infty$. Then we construct $g_{j}$ in a similar way to [CÇŞ18,
Proof of Theorem 2]. That is, by rotating and translating the domain and
appealing to Lemma 3, we may assume $\phi(p)=(0,0,0,...,0)$,
$\Omega\subset\\{Re(z_{n})>0\\}$ and $L\subset\\{Re(z_{n})=0\\}$. We define

$g_{j}(z_{1},...,z_{n}):=\frac{|\phi_{n}(p_{j})|^{k(n)}}{d(p_{j},b\Omega)^{n}\left(z_{n}+\delta_{j}\right)^{k(n)}}.$

The condition that $\overline{\Omega}\subset\\{Re(z_{n})\geq 0\\}$ implies
$g_{j}$ is continuous on $\overline{\Omega}$ for all $j\in\mathbb{N}$. Also,
chose $k(n)\in\mathbb{N}$ is sufficiently large so that $g_{j}\rightarrow 0$
weakly in $A^{2}(\Omega)$ as $j\rightarrow\infty$. One can see this via a
continuity argument since one can chose $k(n)$ and $\delta_{j}>0$ so that
$\\{\|g_{j}\|_{L^{2}(\Omega)}\\}_{j\in\mathbb{N}}$ is bounded and
$g_{j}\rightarrow 0$ uniformly on compact subsets of $\Omega$.

We let

$B_{j}:=\\{z\in\mathbb{C}^{n}:d(z,p_{j})<\frac{1}{2}d(p_{j},b\Omega)\\}\subset\Omega$

and apply the mean value theorem for holomorphic functions. We have,

$\displaystyle\int_{\Omega}|g_{j}\circ\phi(z)|^{2}dV(z)$
$\displaystyle\geq\int_{B_{j}}|g_{j}\circ\phi(z)|^{2}dV(z)$
$\displaystyle=\int_{B_{j}}\left|\frac{|\phi_{n}(p_{j})|^{k(n)}}{d(p_{j},b\Omega)^{n}\left(\phi_{n}(z)+\delta_{j}\right)^{k(n)}}\right|^{2}dV(z)$
$\displaystyle\geq
vol(B_{j})\left|\frac{|\phi_{n}(p_{j})|^{k(n)}}{d(p_{j},b\Omega)^{n}\left(\phi_{n}(p_{j})+\delta_{j}\right)^{k(n)}}\right|^{2}$
$\displaystyle\gtrsim C$

For some $C>0$. This contradicts the compactness of $C_{\phi}$. Therefore,
$\phi(p)$ cannot be contained in the closure of a non-trivial analytic disc. ∎

###### Proof of Theorem 2.

We will show that if $p\in b\Omega\cap b\phi(\overline{\Omega})$, then
$J_{\phi}(p)=0$ and use our assumption that $\Omega$ is strongly convex and
hence has no non-trivial analytic discs in its boundary. Assume that $p\in
b\Omega\cap b\phi(\overline{\Omega})$ and $J_{\phi}(p)\neq 0$. Thus by the
inverse function theorem, $\phi$ has a local inverse $\psi:\phi(U)\rightarrow
U$ for some open set $U$ in $\mathbb{C}^{n}$ containing $p$. Without loss of
generality and appealing to Lemma 3, by rotating and translating the domain,
we may assume that $\phi(p)=(0,0,...,0)$. Let $g_{j}$ be defined as in the
proof of Theorem 1. One can show that for any open set $V$ containing
$(0,...,0)$, there exists $\delta_{V}>0$ so that
$\|g_{j}\|^{2}_{L^{2}(V\cap\Omega)}\geq\delta_{V}$ for all $j\in\mathbb{N}$.
Then we have

$\|C_{\phi}(g_{j})\|^{2}_{L^{2}(U\cap\Omega)}=\||J_{\psi}|^{\frac{1}{2}}g_{j}\|^{2}_{L^{2}(\phi(U\cap\Omega))}=\||J_{\psi}|^{\frac{1}{2}}g_{j}\|^{2}_{L^{2}(V\cap\Omega)}\geq
M_{V}>0$

for some open set $V$, some $M_{V}>0$, and for all $j\in\mathbb{N}$. This is a
contradiction. Therefore, $J_{\phi}(p)=0$. Now $\overline{b\Omega\cap
b\phi(\overline{\Omega})}$ must have empty interior in $b\Omega$ since
$\overline{b\Omega\cap b\phi(\overline{\Omega})}$ contains the zero set of the
Jacobian of $\phi$, a holomorphic function. If $\overline{b\Omega\cap
b\phi(\overline{\Omega})}$ has non-empty interior in $b\Omega$, then one can
show that $b\Omega$ contains a non-trivial analytic disc. This cannot occur
since our domain is strongly pseudoconvex. ∎

This proof shows that the converse of Theorem 1 is false. Let
$\Omega\subset\mathbb{C}^{2}$ be a smoothly bounded convex Reinhardt domain so
that $\mathcal{L}\neq\emptyset$ but with no ’vertical’ non-trivial analytic
discs (see [CŞ18] and [Clo17]). Furthermore, assume $\Omega$ has the Levi-flat
disc condition, $\mathcal{L}\subset\mathbb{C}\times\\{|z_{2}|=1\\}$, and
$(1,0)\in b\Omega$. Define $\phi(z_{1},z_{2}):=(z_{1},\frac{1}{2}z_{2})$.
Since $\Omega$ is convex and Reinhardt, it is a complete Reinhardt domain (see
[CŞ18] and [Clo17]). Therefore, $\phi$ is a holomorphic self-map on $\Omega$
and $\overline{\phi(\Omega)}\cap\overline{\mathcal{L}}=\emptyset$ (so
satisfies the assumptions of the converse of Theorem 1). However, it is clear
the $C_{\phi}$ is not compact, since $J_{\phi}$ is non-vanishing at $(1,0)$
and $\phi(1,0)=(1,0)$. This next example will show the converse of Theorem 2
is not true in general. Let $\Omega=\mathbb{B}^{2}$ be the unit ball in
$\mathbb{C}^{2}$, and let $\phi(z_{1},z_{2}):=(z_{1},\frac{1}{2}z_{2})$ as
before. The claim is that $b\Omega\cap b\phi(\overline{\Omega})$ is nowhere
dense in $b\Omega$ but $C_{\phi}$ is not compact on $A^{2}(\Omega)$, implying
the converse of Theorem 2 is not true. Suppose $|z_{1}|^{2}+|z_{2}|^{2}=1$ and
compute the magnitude of the image of $(z_{1},z_{2})$ under $\phi$. If
$|z_{1}|^{2}+\frac{1}{4}|z_{2}|^{2}=1$, then we get that $z_{2}=0$ and so
$|z_{1}|=1$. This implies $b\Omega\cap b\phi(\overline{\Omega})$ is nowhere
dense in $b\Omega$. It is clear that $C_{\phi}$ is not compact on
$A^{2}(\Omega)$ by [Zhu07] since the angular derivative of $\phi$ exists at
$(1,0)$. The techniques used in the previous proof, along with the maximum
principle for holomorphic functions and Sard’s theorem, can be adapted to show
the following.

######  Proposition 3.

Let $\Omega\subset\mathbb{C}^{n}$ be a smoothly bounded pseudoconvex domain.
Let $\phi:\Omega\rightarrow\Omega$ be a proper holomorphic map where $\phi\in
C^{1}(\overline{\Omega})$. Then $C_{\phi}$ is not compact on $A^{2}(\Omega)$.

###### Proof.

Assume $\phi$ is a proper holomorphic map on $\Omega$, $\phi$ is of class
$C^{1}(\overline{\Omega})$, and $C_{\phi}$ is compact on $A^{2}(\Omega)$. Now
if $\phi$ is proper, then $\phi:\overline{\Omega}\rightarrow\overline{\Omega}$
is surjective and $\phi(b\Omega)=b\Omega$. Therefore, by the proof of Theorem
2, $J_{\phi}=0$ on $b\Omega$. Since $J_{\phi}$ is holomorphic on $\Omega$ and
continuous up to $\overline{\Omega}$, $J_{\phi}=0$ on $\Omega$. This
contradicts Sard’s theorem (see [Sar42]), since the image of the zero set of
the Jacobian has positive volume measure.

∎

It would be interesting say whether these techniques generalize to symbols of
less regularity. Also, one can try to relate compactness of composition
operators to angular derivatives (see [Zhu07]) of symbols on more general
bounded convex and bounded pseudoconvex domains.

##  4\. Aknowlegments

I wish to thank Sönmez Şahutoğlu, Akaki Tikaradze, and Trieu Le for useful
conversations and comments on a preliminary version of this manuscript. I also
thank the anonymous referees for their useful suggestions.

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# A New State-of-the-Art Transformers-Based Load Forecaster on the Smart Grid
Domain

André Luiz Farias Novaes, Rui Alexandre de Matos Araújo, José Figueiredo, and
Lucas Aguiar Pavanelli Manuscript received on …. Accepted for publication ….
A.L.F. Novaes, and R.A.M. Araújo are with the Institute for Systems and
Robotics (ISR-UC), and Department of Electrical and Computer Engineering
(DEEC-UC), University of Coimbra, Pólo II, PT-3030-290 Coimbra, Portugal.
(e-mails<EMAIL_ADDRESS>rui@isr.uc.pt). J. Figueredo is with the
Critical Software S.A., Pq. Ind. de Taveiro, Lt 49, PT-3045-504 Coimbra,
Portugal. (e-mail: jose.figueiredo@criticalsoftware.com). L.A. Pavanelli, and
A.L.F. Novaes are with Department of Computer Science, PUC-Rio, Rio de
Janeiro, Brazil (e-mail: lpavanelli@inf.puc-rio.br).

###### Abstract

Meter-level load forecasting is crucial for efficient energy management and
power system planning for Smart Grids (SGs), in tasks associated with
regulation, dispatching, scheduling, and unit commitment of power grids.
Although a variety of algorithms have been proposed and applied on the field,
more accurate and robust models are still required: the overall utility cost
of operations in SGs increases 10 million currency units if the load
forecasting error increases 1%, and the mean absolute percentage error (MAPE)
in forecasting is still much higher than 1%. Transformers have become the new
state-of-the-art in a variety of tasks, including the ones in computer vision,
natural language processing and time series forecasting, surpassing
alternative neural models such as convolutional and recurrent neural networks.
In this letter, we present a new state-of-the-art Transformer-based algorithm
for the meter-level load forecasting task, which has surpassed the former
state-of-the-art, LSTM, and the traditional benchmark, vanilla RNN, in all
experiments by a margin of at least 13% in MAPE.

###### Index Terms:

Transformers, deep learning, long-short term memory, meter-level load
forecasting, short-term load forecasting.

##  I Introduction

Electrical load forecasting plays a crucial role on the Smart Grid (SG)
domain, particularly on efficient energy management and power system planning
[zheng2013smart, avancini2019energy, depuru2011smart]. Consequently, high-
accuracy forecasts are required in multiple time horizons for tasks associated
with regulation, dispatching, scheduling, and unit commitment of power grids
[raza2015review, hafeez2020fast].

The meter-level load forecasting has become an active research topic, with a
variety of algorithms applied on it: [hafeez2020electric, amjady2010short,
ahmad2016accurate, hafeez2020fast, zeng2017switching, sideratos2020novel,
zhang2017short, kong2017short, zhang2016composite, al2013multivariate,
zhang2013short, ghofrani2015hybrid, shi2017deep, kong2017short1,
kaur2019smart, nugaliyadde2019predicting], and others. However, more accurate
and robust models are still required in meter-level load forecasting
[hafeez2020electric], mainly due the following reasons: external factors
influencing forecasts, e.g. climate, calendar events, occupancy patterns,
social conventions; the stochastic and non-linear behavior of consumers; the
sensitive decision making process and operation of the SG. There is a 10
million currency units increase in the overall utility cost if the load
forecasting error increases in 1%.

The Transformer deep learning architecture [vaswani2017attention] has become
the new state-of-the-art forecaster in a variety of tasks
[katharopoulos2020transformers, tay2020efficient, dosovitskiy2020image,
pfeiffer2020adapterhub, zaheer2020big, wolf2020transformers], including on
natural language processing, surpassing alternative neural models such as CNNs
and RNNs. The Transformers’ architecture scales with training data and model
size, facilitates efficient parallel training, and captures long-range
sequence features [wolf2019huggingface]. Unlike sequence-aligned models, the
Transformer does not process data in an ordered sequence manner: it processes
entire sequence of data and uses self-attention mechanisms to learn
dependencies in the sequence. Therefore, Transformer-based models have the
potential to model complex dynamics of time series data that are challenging
for sequence models [wu2020deep].

The main contribution of this letter is to present a new state-of-the-art
Transformer-based algorithm for the meter-level load forecasting task.
Although it has been successfully deployed in many real-world applications and
recently developed for time series forecasting (TSF) [wu2020deep,
li2019enhancing, oh2018learning, lim2019temporal], Transformers has not yet
been applied in the electricity load forecasting community. In this letter, we
fill this gap, specifically applying it on The Smart Meters in London Dataset,
one of the largest public-domain datasets on SGs, the largest one hosted at
Kaggle111https://www.kaggle.com/jeanmidev/smart-meters-in-london.

##  II The Method

Our Transformer-based forecaster follows the one developed in [wu2020deep],
which is in turn based on the original architecture of [vaswani2017attention],
minutely described in Figure 1.

Figure 1: The Transformer Model Architecture [vaswani2017attention].

The Transformer is a encoder-decoder architecture. On a high level, the
encoder maps an input sequence, e.g. a time series sequence, into an abstract
continuous representation that holds all information learned from the input.
The decoder takes that continuous representation and, step-by-step, generates
a single output while also feeding the previous output.

The encoder, the left block of Figure 1, is composed of a stack of $N=6$
identical layers, with two sub-layers each: the first layer is a multi-head
self-attention mechanism; and the second layer is a fully connected feed-
forward network. Residual connections [he2016deep] and normalization
operations [ba2016layer] are applied on each sub-layer.

The decoder, the right block of Figure 1, is similar to the encoder. The main
difference, not the only one, is an extra sub-layer on the stack, Masked
Multi-Head Attention.

The Transformer architecture mainly differentiates itself from other deep
networks created before by self-attention, a mechanism that balances long-
range dependencies modeling and computational efficiency [zhang2019self]. The
full self-attention module calculates response at a position as a weighted sum
of the features at all positions. Unlike the RNN-based methods, Transformers
allow access to any part of the time series history regardless of distance,
potentially grasping recurring patterns with long-term dependencies
[li2019enhancing].

##  III Experiments and Results

The Smart Meters in London Dataset is composed of 5567 London Households
energy consumption readings between November/2011 and February/2014, with a 30
minutes frequency. For this letter, we have selected for forecasting the same
eight houses222The selected houses are: MAC000002, MAC000033, MAC000092,
MAC000156, MAC000246, MAC000450, MAC001074, and MAC003223. as in
[nugaliyadde2019predicting], each of them with its’ own number of samples.

The dataset is split into training and test sets, with a 80%-20% ratio. The
inputs to the prediction model are the measurements of the consumption on the
last $n$ time intervals (TI), from the current TI $k$ to the past, i.e. the
inputs are the consumption measurements on intervals $k-n+1,\ldots,k$; and the
output of the model is the consumption forecast for the next TI, $k+1$, in
kWh.

Experiments were conducted on the Amazon Web Services (AWS) ml.p2.xlarge
instance, 4 vCPUs, 1xK80 GPU, 61 GB of memory, and 12 GB of GPU memory,
totaling approximately 310 hours of GPU-time. The proposed transformer-based
algorithm is compared against the former state-of-the-art in residential load
forecasting [kong2017short1], LSTM, and the vanilla RNN, traditionally used as
a benchmark in the TSF domain. Since all the models rely on random
initialization of network weights, five experiments were performed for each of
the three models, for all eight houses, training using four distinct values
for the number $n$ of TIs, specifically for $n=2,3,6,12$. Therefore, the total
number or runs is 480.

TABLE I: Load Forecasting Summary.

Algorithm |  MAPE Avg. |  Total Train. Time [sec]
---|---|---
Transformer-2TI |  64.87% |  11.53668
LSTM-2TI |  82.62% |  3.04872
RNN-2TI |  78.50% |  2.08464
Transformer-3TI |  62.75% |  10.68004
LSTM-3TI |  75.38% |  3.09136
RNN-3TI |  79.62% |  2.11112
Transformer-6TI |  62.87% |  7.84844
LSTM-6TI |  80.50% |  3.08364
RNN-6TI |  81.25% |  2.10912
Transformer-12TI |  63.50% |  8.42956
LSTM-12TI |  81.88% |  3.09936
RNN-12T |  76.25% |  2.11384

Note: all the time values have been multiplied by $10^{-4}$ in the column
Total Train. Time [sec] .

Overall results are shown in Table I, where MAPE Avg. is the average of the
mean absolute percentage error (MAPE) over all the 40 experiments performed in
the test sets for the configuration (type of model, and number of TIs)
detailed in the corresponding line of Table I, and Total Train. Time [sec] is
the total training time, in seconds, for the same 40 experiments for each
line. The results show that the Transformer-based algorithm outperforms LSTM
and RNN in all experiments by a margin of at least 13% in MAPE.

Forecasting electric load for single residential users is much more changeable
than for aggregated residential loads in a large scale or even industrial
consumers, mainly due the high volatility and uncertainty involved, generated
by distinct daily routines residents’ lifestyle; industrial electricity
consumption patterns, for example, are much more regular than residential ones
[kong2017short]. Results achieved at Table I are comparable in magnitude with
results of leader works in the smart grid forecast domain, as in
[kong2017short].

Figure 2

Figure 2: Forecasting energy for house MAC000002.

shows how each of the methods perform on house MAC000002 in a specific day.
Qualitatively, it is possible to observe that Transformers better responds to
volatile movements in the load series, identified by “Truth”.

##  IV Conclusion

The relevance of a new state-of-the-art Transformer model in the meter-level
load prediction is remarkable. The results indicate that the Transformer
learning model achieves superior performance when compared to LSTM and RNN
models for meter-level load prediction. Created in 2017, Transformers are
allowing new possibilities for TSF in distinct areas, with pure and hybrid
models. Presented to the load forecasting community in 2018 as a new state-of-
the-art [kong2017short1], LSTM networks are still valuable forecasters, with
at least one advantage over Transformers: the smaller amount of training time.
Next research directions will explore ways on developing less costly
Transformers models in terms of training time.

## Acknowledgment

This work was supported by Project CONNECTA-X/2017/33354 co-financed by
PT2020, in the framework of the COMPETE 2020 Programme, and by the European
Union through the European Regional Development Fund (ERDF).


# Isometries and hermitian operators on spaces of vector-valued Lipschitz maps

Shiho Oi  Department of Mathematics, Faculty of Science, Niigata University,
Niigata 950-2181 Japan<EMAIL_ADDRESS>
###### Abstract.

We study hermitian operators and isometries on spaces of vector-valued
Lipschitz maps with the sum norm: $\|\cdot\|_{\infty}+L(\cdot)$. There are two
main theorems in this paper. Firstly, we prove that every hermitian operator
on $\operatorname{Lip}(X,E)$, where $E$ is a complex Banach space, is a
generalized composition operator. Secondly, we give a complete description of
unital surjective complex linear isometries on
$\operatorname{Lip}(X,\mathcal{A})$ where $\mathcal{A}$ is a unital factor
$C^{*}$-algebra. These results improve previous results stated by the author.

###### Key words and phrases:

surjective linear isometry, $C^{*}$-algebra, hermitian operator, Lipschitz
algebra

###### 2020 Mathematics Subject Classification:

46B04, 47B15, 46L52

##  1\. Introduction and Main results

Given a compact metric space $X$ and a complex Banach space
$(E,\|\cdot\|_{E})$, a map $F:X\to E$ is said to be Lipschitz if

$L(F):=\sup_{x\neq y\in
X}\left\\{\frac{\|F(x)-F(y)\|_{E}}{d(x,y)}\right\\}<\infty.$

We denote a space of all E-valued Lipschitz maps on $X$ by
$\operatorname{Lip}(X,E)$. In the case $E=\mathbb{C}$, we simply write
$\operatorname{Lip}(X)$. The Lipschitz space $\operatorname{Lip}(X,E)$ is a
Banach space with the sum norm;

$\|F\|_{L}=\sup_{x\in X}\|F(x)\|_{E}+L(F),\quad F\in\operatorname{Lip}(X,E).$

In particular, $\operatorname{Lip}(X,E)$ endowed with $\|\cdot\|_{L}$ is a
Banach algebra if $E$ is a Banach algebra.

###  1.1. Surjective linear isometries

Let $\mathcal{A}$ be a unital $C^{*}$-algebra. We study unital surjective
linear isometries on $\operatorname{Lip}(X,\mathcal{A})$ with $\|\cdot\|_{L}$.
We explain the motivation for our study. Kadison in [10] obtained the
following characterization of surjective complex linear isometries between
unital $C^{*}$-algebras. Let $\mathcal{A}_{i}$ be unital $C^{*}$-algebras for
$i=1,2$. Let $U:\mathcal{A}_{1}\to\mathcal{A}_{2}$ be a surjective linear
isometry. Then there are a unitary element $u\in\mathcal{A}_{2}$ and a Jordan
$*$-isomorphism $\psi:\mathcal{A}_{1}\to\mathcal{A}_{2}$ such that
$U(a)=u\psi(a)$ for any $a\in\mathcal{A}.$ This has a remarkable and beautiful
consequence that the unital surjective linear isometries between unital
$C^{*}$-algebras are Jordan $*$-isomorphisms. A lot of researchers have been
attracted to consider whether every surjective linear isometry on algebras is
closely related to an isomorphism on the algebras. We deal with surjective
linear isometries on Banach algebras of continuous maps taking values in a
unital $C^{*}$-algebra. For any unital $C^{*}$-algebra $\mathcal{A}$, we
denote by $C(K,\mathcal{A})$ the Banach algebra, with the supremum norm, of
all continuous maps on a compact Hausdorff space $K$ taking values in
$\mathcal{A}$. Let us consider surjective linear isometries between
$C(K,\mathcal{A})$. Since $C(K,\mathcal{A})$ is a unital $C^{*}$-algebra, the
celebrated theorem due to Kadison tells that every unital surjective linear
isometry is a Jordan $*$-isomorphism. In particular, if $\mathcal{A}_{i}$ are
unital factor $C^{*}$-algebras for $i=1,2$, in [8, Corollary 5] they showed
that every surjective linear isometry $U:C(K_{1},\mathcal{A}_{1})\to
C(K_{2},\mathcal{A}_{2})$ is a weighted composition operator of the form

(1) $UF(y)=u\psi_{y}(F(\varphi(y))),$

where $\varphi:K_{2}\to K_{1}$ is a homeomorphism, $\\{\psi_{y}\\}_{y\in
K_{2}}$ is a strongly continuous family of Jordan $*$-isomorphisms from
$\mathcal{A}_{1}$ onto $\mathcal{A}_{2}$, and $u\in C(K_{2},\mathcal{A}_{2})$
is a unitary element. One may wonder whether any surjective linear isometries
from $\operatorname{Lip}(X_{1},\mathcal{A}_{1})$ onto
$\operatorname{Lip}(X_{2},\mathcal{A}_{2})$ is also a weighted composition
operator similar to (1). First we introduce the results by the author in [14].
We showed every hermitian operator on $\operatorname{Lip}(X,E)$ is a
generalized composition operator under the more restrictive condition that $E$
is of finite dimension. Furthermore, we obtained the following theorem by
using the notion of hermitian operators. We denote the Banach algebra of
complex matrices of order $n$ by $M_{n}(\mathbb{C})$.

######  Theorem 1.1 (Theorem 3.3 in [14]).

Let $X_{i}$ be compact metric spaces for $i=1,2$. The map
$U:(\operatorname{Lip}(X_{1},M_{n}(\mathbb{C})),\|\cdot\|_{L})\to(\operatorname{Lip}(X_{2},M_{n}(\mathbb{C})),\|\cdot\|_{L})$
is a linear surjective isometry such that $U(1)=1$ if and only if there exist
a unitary matrix $V\in M_{n}(\mathbb{C})$ and a surjective isometry
$\varphi:X_{2}\to X_{1}$ such that

$(UF)(x)=VF(\varphi(x))V^{-1},\quad
F\in\operatorname{Lip}(X_{1},M_{n}(\mathbb{C})),x\in X_{2}$

or

$(UF)(x)=VF^{t}(\varphi(x))V^{-1},\quad
F\in\operatorname{Lip}(X_{1},M_{n}(\mathbb{C})),x\in X_{2},$

where $F^{t}(y)$ denote transpose of $F(y)$ for $y\in X_{1}$.

Although the arguments of the proof have remained limited to the case that $E$
is finite dimensional, this is the first result on surjective linear
isometries on $\operatorname{Lip}(X,E)$, where $E$ is a non-commutative Banach
algebra. In this framework, it seems natural to ask questions about further
developments. The aim of this paper is to develop our knowledge on hermitian
operators and isometries on $\operatorname{Lip}(X,E)$ and establish a infinite
dimensional version of [14]. More precisely, we prove the next theorem.

######  Theorem 1.2.

Let $X_{i}$ be compact metric spaces and $\mathcal{A}_{i}$ unital factor
$C^{*}$-algebras for $i=1,2$. The map
$U:(\operatorname{Lip}(X_{1},\mathcal{A}_{1}),\|\cdot\|_{L})\to(\operatorname{Lip}(X_{2},\mathcal{A}_{2}),\|\cdot\|_{L})$
is a surjective complex linear isometry such that $U(1)=1$ if and only if
there exist a unital surjective complex linear isometry
$\psi:\mathcal{A}_{1}\to\mathcal{A}_{2}$ and a surjective isometry
$\varphi:X_{2}\to X_{1}$ such that

$UF(y)=\psi(F(\varphi(y))),\quad
F\in\operatorname{Lip}(X_{1},\mathcal{A}_{1}),y\in X_{2}.$

Indeed, Theorem 1.2 also gives an answer to the above question (that is
whether any surjective linear isometries from
$\operatorname{Lip}(X_{1},\mathcal{A}_{1})$ onto
$\operatorname{Lip}(X_{2},\mathcal{A}_{2})$ is also a weighted composition
operator similar to (1)).

In case that $E$ is a finite dimensional Banach space, it follows from [14,
Lemma 2.1] that $\operatorname{Lip}(X)\otimes E=\operatorname{Lip}(X,E)$. If
$E$ is of infinite-dimension, $\operatorname{Lip}(X)\otimes E$ does not
coincide with $\operatorname{Lip}(X,E)$. Moreover it is not known whether
$\operatorname{Lip}(X)\otimes E$ is dense in $\operatorname{Lip}(X,E)$ with
$\|\cdot\|_{L}$ or not. When $E$ is of infinite dimension, the representation
of $\operatorname{Lip}(X,E)$ is more complicated. Thus to describe isometries
and hermitian operators on $\operatorname{Lip}(X,E)$, where $E$ is of infinite
dimension, is much more difficult than the case that $E$ is of finite
dimension. In order to achieve any further progress, we need to make several
improvements and extensions compared to the paper [14].

The paper is organized as follows. In the rest of the introduction we provide
basic background on the study of hermitian operators. In Section 2 we study
hermitian operators on $\operatorname{Lip}(X,E)$. The main theorem of Section
2 is Theorem 2.3. For any a complex Banach space $E$, we prove that every
hermitian operator on $\operatorname{Lip}(X,E)$ is a generalized composition
operator. This is a generalization of the characterization of hermitian
operators on $\operatorname{Lip}(X,E)$ for finite dimensional Banach spaces
$E$ in [14]. In Section 3, we introduce the concept of T-sets due to Myers. By
the notion of T-sets we present properties of the unit ball of the dual space
of $\operatorname{Lip}(X,E)$. Indeed the extreme points of the unit ball of
the dual space of $\operatorname{Lip}(X,E)$ are quite complicated. Thus we
study T-sets instead of extreme points. The main statement in Section 3 is
Proposition 3.6. We show that if a surjective linear isometry between
$\operatorname{Lip}(X,E)$ is a weighted composition operator when restricted
to $\operatorname{Lip}(X)\otimes E$, then it is a weighted composition
operator. Since the representation of $\operatorname{Lip}(X)\otimes E$ is much
easier than that of $\operatorname{Lip}(X,E)$, Proposition 3.6 is successful
in describing the surjective linear isometries on $\operatorname{Lip}(X,E)$.
In Section 4 we present the proof of Theorem 1.2.

###  1.2. Hermitian operators

A bounded operator $T$ on a complex normed space $(V,\|\cdot\|_{V})$ is
hermitian if $[Tv,v]_{V}\in\mathbb{R}$ for any $v\in V$, where
$[\cdot,\cdot]_{V}$ is a semi-inner product on $V$ that is compatible with the
norm $\|\cdot\|_{V}$. The definition does not depend on the choice of semi-
inner products (see [1]). A complete description of hermitian operators on
Banach spaces has been studied for a long period of time. We refer the reader
to [6, 7] for further information about hermitian operators.

Fleming and Jamison in [4] turned their attention to the vector-valued case.
Let $E$ be a complex Banach space. They obtained the first characterization
for hermitian operators between Banach spaces of $E$-valued continuous
functions as follows:

Let $T$ be a hermitian operator on $C(K,E)$, where $K$ is a compact Hausdorff
space. Then for each $t\in K$ there is a hermitian operator $\phi(t)$ on $E$
such that

$TF(t)=\phi(t)(F(t)),\quad t\in K.$

What is the general form of hermitian operators between Banach spaces of
$E$-valued Lipschitz maps? Botelho, Jamison, Jiménez-Vargas and Villegas-
Vallecillos in [2] obtained a characterization for hermitian operators on
$\operatorname{Lip}(X,E)$ with the max norm;
$\|\cdot\|_{M}=\max\\{\|\cdot\|_{\infty},L(\cdot)\\}$ as follows:

Let $X$ be a compact and 2-connected metric space and $E$ a complex Banach
space. Then
$T:(\operatorname{Lip}(X,E),\|\cdot\|_{M}),\to(\operatorname{Lip}(X,E),\|\cdot\|_{M})$
is a hermitian operator if and only if there exists a hermitian operator
$\phi:E\to E$ such that

$TF(x)=\phi(F(x)),\quad F\in\operatorname{Lip}(X,E),\quad x\in X.$

How about the case $\operatorname{Lip}(X,E)$ with $\|\cdot\|_{L}$? One may
think that each feature of the two norms does not make a big difference, but
it is not. The studies of the classes of operators on
$\operatorname{Lip}(X,E)$ depend heavily on the properties of the norm. The
standard approach to the studies of isometries or related operators on Banach
spaces relies on a characterization of the extreme points of the closed unit
ball of the corresponding dual spaces. But the extreme points of the closed
unit ball of the dual space of $(\operatorname{Lip}(X,E),\|\cdot\|_{L})$ are
completely different from those of $(\operatorname{Lip}(X,E),\|\cdot\|_{M})$.
The former is much complicated. For operators on
$(\operatorname{Lip}(X,E),\|\cdot\|_{L})$, it is non trivial to derive a
representation from the action of their adjoints, so we have to work quite
hard to give a representation. Actually, in the case of hermitian operators on
$(\operatorname{Lip}(X,E),\|\cdot\|_{L})$, difficulties to give a
representation remain even if we have a representation of hermitian operators
on $(\operatorname{Lip}(X,E),\|\cdot\|_{M})$. Indeed, Botelho, Jamison,
Jiménez-Vargas and Villegas-Vallecillos proved hermitian operators between
$\operatorname{Lip}(X)$ with $\|\cdot\|_{L}$ in [3] are composition operators.
Recently, the author of this paper generalized to $\operatorname{Lip}(X,E)$,
where $E$ is a finite dimensional complex Banach space in [14]. But it has not
been solved in general. In this paper, we give a complete representation for
any complex Banach space $E$.

###  1.3. Notations and Remarks

Throughout this paper, $X$, $X_{1}$and $X_{2}$ are compact metric spaces, and
$E$, $E_{1}$ and $E_{2}$ are complex Banach spaces. In addition $\mathcal{A}$,
$\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ are unital $C^{*}$-algebras. For a
unital $C^{*}$-algebra $\mathcal{A}$, if its center is trivial, i.e. $A\cap
A^{\prime}=\mathbb{C}1$ we call it a unital factor $C^{*}$-algebra. For Banach
space $E$, we denote the closed unit ball of $E$ by $\mathbb{B}(E)$, and the
closed unit ball of the dual space $E^{*}$ by $\mathbb{B}(E^{*})$. We also
denote the unit sphere of $E$ by $\mathbb{S}(E)$. For any
$f\in\operatorname{Lip}(X)$ and $e\in E$, we define $f\otimes e:X\to E$ by

$(f\otimes e)(x)=f(x)e.$

We have $f\otimes e\in\operatorname{Lip}(X,E)$ such that $\|f\otimes
e\|_{\infty}=\|f\|_{\infty}\|e\|_{E}$ and $L(f\otimes e)=L(f)\|e\|_{E}$. This
implies that $\|f\otimes e\|_{L}=\|f\|_{L}\|e\|_{E}$. We see that $f\otimes e$
is an element of the algebraic tensor product space
$\operatorname{Lip}(X)\otimes E$ with the crossnorm.

Recall that the purpose of this paper is to generalize the theorems in [14].
Although we need new approaches and additional arguments, some arguments
remain valid. Similar arguments may be found in [14], but we adapt these to
our setting and give proofs as accurately as possible.

##  2\. A characterization of hermitian operators on $\operatorname{Lip}(X,E)$

Firstly, we would like to consider hermitian operators on
$(\operatorname{Lip}(X,E),\|\cdot\|_{L})$. We write $\widetilde{X}=\\{(x,y)\in
X^{2};x\neq y\\}$. Let $\beta(\widetilde{X}\times\mathbb{B}(E^{*}))$ be the
Stone-Čech compactification of $\widetilde{X}\times\mathbb{B}(E^{*})$. For any
$F\in\operatorname{Lip}(X,E)$, we denote by
$\widetilde{F}:\beta(\widetilde{X}\times\mathbb{B}(E^{*}))\to\mathbb{C}$ the
unique continuous extension of the bounded continuous function
$((x,y),e^{*})\mapsto e^{*}(\frac{F(x)-F(y)}{d(x,y)})$ on
$\widetilde{X}\times\mathbb{B}(E^{*})$. Since we have
$\|\widetilde{F}\|_{\infty}=L(F)$ for any $F\in\operatorname{Lip}(X,E)$, we
can define a linear isometric embedding
$\Gamma:(\operatorname{Lip}(X,E),\|\cdot\|_{L})\to(C(X\times\beta(\widetilde{X}\times\mathbb{B}(E^{*}))\times\mathbb{B}(E),E),\|\cdot\|_{\infty})$
by $\Gamma(F)(x,\xi,e)=F(x)+\widetilde{F}(\xi)e$. Moreover we define a set
$P_{G}$ by

$P_{G}=\\{t\in
X\times\beta(\widetilde{X}\times\mathbb{B}(E^{*}))\times\mathbb{B}(E);\|\Gamma(G)(t)\|_{E}=\|\Gamma(G)\|_{\infty}=\|G\|_{L}\\}.$

######  Lemma 2.1.

For any $G\in\operatorname{Lip}(X,E)$, we have $P_{G}\neq\emptyset$.

###### Proof.

If $G=0$, we have $(x_{0},\xi,e)\in P_{G}$ for any $(x_{0},\xi,e)\in
X\times\beta(\widetilde{X}\times\mathbb{B}(E^{*}))\times\mathbb{B}(E)$. Thus
let $G\in\operatorname{Lip}(X,E)$ with $G\neq 0$. Since
$\beta(\widetilde{X}\times\mathbb{B}(E^{*}))$ is compact, there exists
$\xi\in\beta(\widetilde{X}\times\mathbb{B}(E^{*}))$ such that
$|\widetilde{G}(\xi)|=\|\widetilde{G}\|_{\infty}=L(G)$. There are $x_{0}\in X$
such that $\|G(x_{0})\|_{E}=\|G\|_{\infty}$ and $\alpha\in\mathbb{C}$ with
$|\alpha|=1$ such that
$\alpha\widetilde{G}(\xi)=\|\widetilde{G}\|_{\infty}=L(G)$. We get

$\begin{split}\|\Gamma(G)(x_{0},\xi,\frac{\alpha}{\|G(x_{0})\|_{E}}G(x_{0}))\|_{E}=\|G(x_{0})+\widetilde{G}(\xi)\frac{\alpha}{\|G(x_{0})\|_{E}}G(x_{0})\|_{E}\\\
=(1+L(G)\frac{1}{\|G\|_{\infty}})\|G\|_{\infty}=\|G\|_{\infty}+L(G)=\|G\|_{L}.\end{split}$

This implies that $(x_{0},\xi,\frac{\alpha}{\|G(x_{0})\|_{E}}G(x_{0}))\in
P_{G}$. ∎

By Lemma 2.1 and the axiom of choice, there exists a choice function

$\Psi:\operatorname{Lip}(X,E)\to
X\times\beta(\widetilde{X}\times\mathbb{B}(E^{*}))\times\mathbb{B}(E)$

such that $\Psi(G)\in P_{G}$ for every $G\in\operatorname{Lip}(X,E)$. Let
$[\cdot,\cdot]_{E}$ on $E$ be a semi-inner product which is compatible with
the norm of $E$. Define a map $[\cdot,\cdot]_{\Psi
L}:\operatorname{Lip}(X,E)\times\operatorname{Lip}(X,E)\to\mathbb{C}$ given by

(2) $[F,G]_{\Psi L}=[\Gamma(F)(\Psi(G)),\Gamma(G)(\Psi(G))]_{E},\quad
F,G\in\operatorname{Lip}(X,E).$

It is easy to check that $[\cdot,\cdot]_{\Psi L}$ is a semi-inner product on
$\operatorname{Lip}(X,E)$ compatible with the norm $\|\cdot\|_{L}$. Now we get
the following lemma. The basic idea of the proof is the same as [14, Lemma
2.3].

######  Lemma 2.2.

Let $T$ be a hermitian operator on $(\operatorname{Lip}(X,E),\|\cdot\|_{L})$.
Then

$T(1\otimes e)\in 1\otimes E$

for any $e\in E$.

###### Proof.

Let $e\in E$. If $e=0$, then $T(1\otimes e)=T(0)=0=1\otimes 0\in 1\otimes E$.
Thus we assume that $0\neq e\in\mathbb{B}(E)$. Fix $x^{\prime}\in X$,
$(x,y)\in\widetilde{X}$ and $e^{*}\in\mathbb{B}(E^{*})$. Let
$\theta\in[0,2\pi)$, we obtain

(3) $\hbox to0.0pt{$\displaystyle\Gamma(1\otimes
e)(x^{\prime},((x,y),e^{i\theta}e^{*}),e)$\hss}\\\ =(1\otimes
e)(x^{\prime})+e^{i\theta}e^{*}\left(\frac{(1\otimes e)(x)-(1\otimes
e)(y)}{d(x,y)}\right)e=e+0e=e.$

This implies that

$\|\Gamma(1\otimes e)(x^{\prime},((x,y),e^{i\theta}e^{*}),e)\|_{E}=\|1\otimes
e\|_{L}.$

Thus we get $(x^{\prime},((x,y),e^{i\theta}e^{*}),e)\in P_{1\otimes e}$.
Choose a choice function $\Psi_{\theta}:\operatorname{Lip}(X,E)\to
X\times\beta(\widetilde{X}\times\mathbb{B}(E^{*}))\times\mathbb{B}(E)$ such
that

$\Psi_{\theta}(1\otimes e)=(x^{\prime},((x,y),e^{i\theta}e^{*}),e)$

and define a semi-inner product $[\cdot,\cdot]_{\Psi_{\theta}L}$ on
$\operatorname{Lip}(X,E)$ in the manner as in (2). Since $T$ is a hermitian
operator, we have $[T(1\otimes e),1\otimes e]_{\Psi_{\theta}L}\in\mathbb{R}$.
By (3), it follows that

(4) $\begin{split}\mathbb{R}&\ni[T(1\otimes e),1\otimes e]_{\Psi_{\theta}L}\\\
&=[\Gamma(T(1\otimes e))(\Psi_{\theta}(1\otimes e)),\Gamma(1\otimes
e)(\Psi_{\theta}(1\otimes e))]_{E}\\\ &=[T(1\otimes
e)(x^{\prime})+e^{i\theta}e^{*}\left(\frac{T(1\otimes e)(x)-T(1\otimes
e)(y)}{d(x,y)}\right)e,e]_{E}\\\ &=[T(1\otimes
e)(x^{\prime}),e]_{E}+e^{i\theta}e^{*}\left(\frac{T(1\otimes e)(x)-T(1\otimes
e)(y)}{d(x,y)}\right)\|e\|^{2}_{E}.\end{split}$

As $e\neq 0$, we see that $\|e\|^{2}_{E}>0$. Since $\theta\in[0,2\pi)$ is
arbitrary, it must be

(5) $e^{*}\left(\frac{T(1\otimes e)(x)-T(1\otimes e)(y)}{d(x,y)}\right)=0$

for any $e^{*}\in\mathbb{B}(E^{*})$. This implies

$\frac{T(1\otimes e)(x)-T(1\otimes e)(y)}{d(x,y)}=0$

for any $(x,y)\in\widetilde{X}$. Thus we deduce $L(T(1\otimes e))=0$.
Therefore, there exists $e_{0}\in E$ such that $T(1\otimes e)=1\otimes e_{0}$.
∎

Applying Lemma 2.2 we define a map $\phi:E\to E$ by

(6) $T(1\otimes e)=1\otimes\phi(e)$

for each $e\in E$. By (4) and (5) we have $\mathbb{R}\ni[T(1\otimes
e),1\otimes e]_{\Psi_{\theta}L}=[T(1\otimes e)(x^{\prime}),e]_{E}$. This
implies that $[\phi(e),e]_{E}\in\mathbb{R}$ for any $e\in E$. Since $T$ is a
bounded linear operator, we get $\phi$ is a hermitian operator on $E$.

We give a complete description of hermitian operators on
$\operatorname{Lip}(X,E)$ with $\|\cdot\|_{L}$, where $E$ is any complex
Banach space (without assuming that $E$ is of a finite dimension).

######  Theorem 2.3.

Let $X$ be a compact metric space and $E$ a complex Banach space. Then
$T:(\operatorname{Lip}(X,E),\|\cdot\|_{L}),\to(\operatorname{Lip}(X,E),\|\cdot\|_{L})$
is a hermitian operator if and only if there exists a hermitian operator
$\phi:E\to E$ such that

(7) $TF(x)=\phi(F(x)),\quad F\in\operatorname{Lip}(X,E),\quad x\in X.$

###### Proof of Theorem 2.3.

Suppose that $T$ is of the form described as (7) in the statement of Theorem
2.3. To prove that $T$ is a hermitian operator, we apply the fact that $T$ is
a hermitian if and only if $e^{itT}$ is a surjective isometry for every
$t\in\mathbb{R}$, see [6, Theorem 5.2.6]. Let $t\in\mathbb{R}$. By the
definition of $T$, we have

$e^{itT}F(x)=e^{it\phi}(F(x))$

for any $F\in\operatorname{Lip}(X,E)$ and $x\in X$. Since $\phi$ is a
hermitian on $E$, $e^{it\phi}$ is a surjective isometry. This implies that
$\|e^{itT}F\|_{\infty}=\|F\|_{\infty}$ and $L(e^{itT}F)=L(F).$ Thus we deduce
$\|e^{itT}F\|_{L}=\|F\|_{L}$ for any $F\in\operatorname{Lip}(X,E)$. Since
$e^{itT}$ is a surjective isometry for every $t\in\mathbb{R}$, we conclude $T$
is a hermitian operator.

We prove the converse. Suppose that
$T:\operatorname{Lip}(X,E)\to\operatorname{Lip}(X,E)$ is a hermitian operator.
Let $\phi$ be the operator defined by (6). A similar argument with above
yields an operator from $\operatorname{Lip}(X,E)$ into itself given by
$F\mapsto\phi\circ F$ is a hermitian operator. Hence we define a hermitian
operator $T_{0}:\operatorname{Lip}(X,E)\to\operatorname{Lip}(X,E)$ by

$(T_{0}F)(x)=(TF)(x)-\phi(F(x))$

for all $F\in\operatorname{Lip}(X,E)$ and $x\in X$. We shall prove that
$T_{0}=0$ on $\operatorname{Lip}(X,E)$ in two steps.
Step1. For any $f\in\operatorname{Lip}(X)$ and $e\in E$ we have
$T_{0}(f\otimes e)=0$.
Note that the same idea with [14, Theorem 2.2] is valid even if we replace a
finite dimensional Banach space $E$ with a Banach space $E$.

By [6, p. 10] there is a semi-inner product $[\cdot,\cdot]_{E}$ on $E$
compatible with the norm such that $[e_{1},\lambda
e_{2}]_{E}=\bar{\lambda}[e_{1},e_{2}]_{E}$ for any $e_{i}\in E$ and
$\lambda\in\mathbb{C}$. Let $e\in\mathbb{S}(E)$. We define a map
$S_{e}:\operatorname{Lip}(X)\to\operatorname{Lip}(X)$ by

$S_{e}(f)(x)=[T_{0}(f\otimes e)(x),e]_{E},\quad
f\in\operatorname{Lip}(X),\,\,x\in X.$

By simple calculations we have $S_{e}$ is a bounded linear operator with
$\|S_{e}\|\leq\|T_{0}\|$. Moreover we shall prove $S_{e}$ is a hermitian
operator. Let $t\in\mathbb{R}$. By the definition of $S_{e}$ we get
$(I+itS_{e})(1)(x)=1$ for any $x\in X$. This implies that

(8) $1\leq\|I+itS_{e}\|.$

On the other hand, let $f\in\operatorname{Lip}(X)$. We obtain for any $x,y\in
X$,

$|(I+itS_{e})(f)(x)|\leq\|(I+itT_{0})(f\otimes e)\|_{\infty}$

and

$|(I+itS_{e})(f)(x)-(I+itS_{e})(f)(y)|\leq L((I+itT_{0})(f\otimes e))d(x,y).$

Therefore, we get

$\begin{split}\|(I+itS_{e})(f)\|_{L}&\leq\|(I+itT_{0})(f\otimes
e)\|_{\infty}+L((I+itT_{0})(f\otimes e))\\\ &\leq\|I+itT_{0}\|\|f\otimes
e\|_{L}=\|I+itT_{0}\|\|f\|_{L}\end{split}$

for any $f\in\operatorname{Lip}(X)$. We conclude that

(9) $\|I+itS_{e}\|\leq\|I+itT_{0}\|.$

Since $T_{0}$ is a hermitian operator on $\operatorname{Lip}(X,E)$, we have
$\|I+itT_{0}\|=1+o(t)$ by [6, Theorem 5.2.6]. By (8) and (9), we see that

$1\leq\|I+itS_{e}\|\leq\|I+itT_{0}\|=1+o(t).$

This implies that $S_{e}:\operatorname{Lip}(X)\to\operatorname{Lip}(X)$ is a
hermitian operator. By [3, Theorem 3.1.] we have $S_{e}$ is a real multiple of
the identity. Since $S_{e}(1)(x)=[T_{0}(1\otimes e)(x),e]=0$, we deduce
$S_{e}(f)(x)=0f(x)=0$ for any $f\in\operatorname{Lip}(X)$ and $x\in X$. This
implies that $[T_{0}(f\otimes e)(x),e]_{E}=0$ for all
$f\in\operatorname{Lip}(X)$ and $x\in X$. As $e\in\mathbb{S}(E)$ is arbitrary,
we obtain

(10) $[T_{0}(f\otimes e)(x),e]_{E}=0,\quad e\in E,\quad
f\in\operatorname{Lip}(X),\quad x\in X.$

Let $f\in\operatorname{Lip}(X)$ and $x\in X$. Then we define a map
$S_{fx}:E\to E$ by $S_{fx}(e)=T_{0}(f\otimes e)(x)$ for any $e\in E$. Since
$T_{0}$ is a bounded linear operator, $S_{fx}$ is also a bounded linear
operator with $\|S_{fx}\|\leq\|T_{0}\|\|f\|_{L}$. By (10) we have
$[S_{fx}(e),e]_{E}=[T_{0}(f\otimes e)(x),e]_{E}=0$ for all $e\in E$. Applying
[11, Theorem 5], we have $T_{0}(f\otimes e)(x)=S_{fx}(e)=0$ for any $e\in E$.
As $f\in\operatorname{Lip}(X)$ and $x\in X$ are arbitrary, we conclude Step 1.
Step 2. For any $F\in\operatorname{Lip}(X,E)$, we have $T_{0}(F)=0$.
If $F\in\operatorname{Lip}(X)\otimes E$, Step1 yields that $T_{0}(F)=0$ by the
linearity of $T_{0}$. Thus it suffices to show $T_{0}(F)=0$ holds for any
$F\in\operatorname{Lip}(X,E)\setminus\operatorname{Lip}(X)\otimes E$. Let
$F\in\operatorname{Lip}(X,E)\setminus\operatorname{Lip}(X)\otimes E$ with
$F(x_{0})=0$. For any $e\in\mathbb{S}(E)$, put

$G_{e}=(\|F\|_{\infty}-|F|)\otimes e+F,$

where $|F|(x):=\|F(x)\|$ and $|F|\in\operatorname{Lip}(X)$. Then we have

$G_{e}(x_{0})=\|F\|_{\infty}e$

and

$\|G_{e}(x)\|=\|(\|F\|_{\infty}-\|F(x)\|)\otimes e+F(x)\|\\\
\leq\|F\|_{\infty}-\|F(x)\|+\|F(x)\|=\|F\|_{\infty}$

for any $x\in X$. Thus we obtain
$\|G_{e}(x_{0})\|=\|F\|_{\infty}=\|G_{e}\|_{\infty}$. As
$\beta(\widetilde{X}\times\mathbb{B}(E^{*}))$ is compact, there are
$\xi\in\beta(\widetilde{X}\times\mathbb{B}(E^{*}))$ and $\alpha\in\mathbb{C}$
with $|\alpha|=1$ such that $\alpha\widetilde{G_{e}}(\xi)=L(G_{e})$. This
implies that $(x_{0},\xi,\alpha e)\in P_{G_{e}}$. We choose a choice function
$\Psi_{e}:\operatorname{Lip}(X,E)\to
X\times\beta(\widetilde{X}\times\mathbb{B}(E^{*}))\times\mathbb{B}(E)$ such
that $\Psi_{e}(G_{e})=(x_{0},\xi,\alpha e)$ and define a semi-inner product
$[\cdot,\cdot]_{\Psi_{e}L}$ in the manner as in (2). Since
$T_{0}:\operatorname{Lip}(X,E)\to\operatorname{Lip}(X,E)$ is a hermitian
operator, we get

$\mathbb{R}\ni[T_{0}(G_{e}),G_{e}]_{\Psi_{e}L}=[T_{0}(F),G_{e}]_{\Psi_{e}L}\\\
=[T_{0}(F)(x_{0})+\alpha\widetilde{T_{0}(F)}(\xi)e,\|F\|_{\infty}e+L(G_{e})e]_{E}\\\
=(e^{*}(T_{0}(F)(x_{0}))+\alpha\widetilde{T_{0}(F)}(\xi))\|G_{e}\|_{L},$

where $e^{*}\in\mathbb{B}(E^{*})$ with $e^{*}(e)=1$ for any
$e\in\mathbb{S}(E)$. We have

(11) $e^{*}(T_{0}(F)(x_{0}))+\alpha\widetilde{T_{0}(F)}(\xi)\in\mathbb{R}.$

On the other hand, there exists $y_{0}\in X$ such that
$\|F(y_{0})\|=\|F\|_{\infty}\neq 0$ and there is $f_{y_{0}}\in\mathbb{S}(E)$
such that $F(y_{0})=\|F\|_{\infty}f_{y_{0}}$. We get

$G_{e}(y_{0})=F(y_{0})=\|F\|_{\infty}f_{y_{0}}.$

This implies that
$\|G_{e}(y_{0})\|=\|F(y_{0})\|=\|F\|_{\infty}=\|G_{e}\|_{\infty}$. We have

$\|\Gamma G_{e}(y_{0},\xi,\alpha
f_{y_{0}})\|_{E}=\|G_{e}(y_{0})+\alpha\widetilde{G_{e}}(\xi)f_{y_{0}}\|_{E}\\\
=\|\|F\|_{\infty}f_{y_{0}}+L(G_{e})f_{y_{0}}\|_{E}=\|F\|_{\infty}+L(G_{e})=\|G_{e}\|_{L}.$

Thus we get $(y_{0},\xi,\alpha f_{y_{0}})\in P_{G_{e}}$. In the same manner,
there is a choice function $\Psi_{f_{y_{0}}}:\operatorname{Lip}(X,E)\to
X\times\beta(\widetilde{X}\times\mathbb{B}(E^{*}))\times\mathbb{B}(E)$ such
that $\Psi_{f_{y_{0}}}(G_{e})=(y_{0},\xi,\alpha f_{y_{0}})$ and we can define
a semi-inner product $[\cdot,\cdot]_{\Psi_{f_{y_{0}}}L}$ on
$\operatorname{Lip}(X,E)$. It follows that

$\mathbb{R}\ni[T_{0}(G_{e}),G_{e}]_{\Psi_{f_{y_{0}}}L}=[T_{0}(F),G_{e}]_{\Psi_{f_{y_{0}}}L}\\\
=[T_{0}(F)(y_{0})+\alpha\widetilde{T_{0}(F)}(\xi)f_{y_{0}},\|G_{e}\|_{L}f_{y_{0}}]_{E}\\\
=({f_{y_{0}}}^{*}(T_{0}(F)(y_{0}))+\alpha\widetilde{T_{0}(F)}(\xi))\|G_{e}\|_{L},$

where ${f_{y_{0}}}^{*}\in\mathbb{B}(E^{*})$ with
${f_{y_{0}}}^{*}(f_{y_{0}})=1$. We obtain

(12)
${f_{y_{0}}}^{*}(T_{0}(F)(y_{0}))+\alpha\widetilde{T_{0}(F)}(\xi)\in\mathbb{R}.$

By (11) and (12) we get
$e^{*}(T_{0}(F)(x_{0}))-{f_{y_{0}}}^{*}(T_{0}(F)(y_{0}))\in\mathbb{R}$. Since
$e\in\mathbb{S}(E)$ is arbitrary, it follows that $T_{0}F(x_{0})=0$. Let
$F\in\operatorname{Lip}(X,E)\setminus\operatorname{Lip}(X)\otimes E$ and $x\in
X$. We define $F_{x}=F-1\otimes F(x)$. Since $F_{x}(x)=0$, we get

$0=T_{0}(F_{x})(x)=T_{0}F(x)-T_{0}(1\otimes F(x))(x)=T_{0}(F)(x).$

Thus we have $T_{0}(F)=0$ for any $F\in\operatorname{Lip}(X,E)$ and conclude
Step 2.

Therefore we obtain $TF(x)=\phi(F(x))$ for any $F\in\operatorname{Lip}(X,E)$.
This completes the proof. ∎

##  3\. An extension of isometries on $\operatorname{Lip}(X)\otimes E$

We define the notation of T-sets which is introduced by Myers in [13].

######  Definition 3.1.

Let $(A,\|\cdot\|_{A})$ be a semi-normed space. For a subset $\mathbb{U}$ of
$A$, we call $\mathbb{U}$ a T-set of $A$ with respect to $\|\cdot\|_{A}$ if
$\mathbb{U}$ satisfies the property that for any finite collection
$a_{1},\cdots,a_{n}\in\mathbb{U}$,
$\|\Sigma^{n}_{i=1}a_{i}\|_{A}=\Sigma^{n}_{i=1}\|a_{i}\|_{A}$ and such that
$\mathbb{U}$ is a maximal with respect to the property. If no confusion is
possible, we will refer to T-set of $A$ with respect to $\|\cdot\|_{A}$ as
T-set of $A$.

######  Lemma 3.2.

Let $(A,\|\cdot\|_{A})$ be a Banach space and $\mathbb{U}$ a T-set of $A$ with
respect to $\|\cdot\|_{A}$. If $a\in\mathbb{U}$ then $\lambda a\in\mathbb{U}$
for any $\lambda\geq 0$.

###### Proof.

We conclude this Lemma by the Hahn$-$Banach theorem immediately. ∎

######  Lemma 3.3.

Let $N_{i}$ be normed spaces for $i=1,2$. Suppose that $U:N_{1}\to N_{2}$ is a
surjective isometry with $U(0)=0$. Then $U$ maps T-set of $N_{1}$ to T-set of
$N_{2}$.

###### Proof.

It follows from the Mazur$-$Ulam theorem that every surjective isometry $U$
between two normed spaces with $U(0)=0$ is a real linear isometry. By the
maximality of T-set and surjectivity of $U$, we conclude that $U$ preserves a
T-set. ∎

Let $(E,\|\cdot\|_{E})$ be a Banach space. Let $x\in X$, $\mathbb{U}$ be a
T-set of $E$ with respect to $\|\cdot\|_{E}$ and $\mathbb{T}$ be a T-set of
$\operatorname{Lip}(X,E)$ with respect to $L(\cdot)$. We write

$S(x,\mathbb{U},\mathbb{T})=\\{F\in\operatorname{Lip}(X,E)|F(x)\in\mathbb{U},\|F(x)\|_{E}=\|F\|_{\infty},F\in\mathbb{T}\\}.$

######  Lemma 3.4.

Let $x\in X$, $\mathbb{U}$ be a T-set of $E$ and $\mathbb{T}$ be a T-set of
$\operatorname{Lip}(X,E)$ with respect to $L(\cdot)$. Then for any finite
collection $F_{1},\cdots,F_{n}\in S(x,\mathbb{U},\mathbb{T})$, we have
$\|\Sigma^{n}_{i=1}F_{i}\|_{L}=\Sigma^{n}_{i=1}\|F_{i}\|_{L}$.

###### Proof.

For any $F_{1},\cdots,F_{n}\in S(x,\mathbb{U},\mathbb{T})$, we have
$F_{i}(x)\in\mathbb{U}$ and $\|F_{i}(x)\|_{E}=\|F_{i}\|_{\infty}$ for any
$i=1,\cdots,n$. We get

$\|\Sigma^{n}_{i=1}F_{i}\|_{\infty}\leq\Sigma^{n}_{i=1}\|F_{i}\|_{\infty}=\Sigma^{n}_{i=1}\|F_{i}(x)\|=\|\Sigma^{n}_{i=1}F_{i}(x)\|\leq\|\Sigma^{n}_{i=1}F_{i}\|_{\infty}.$

This implies that
$\|\Sigma^{n}_{i=1}F_{i}\|_{\infty}=\Sigma^{n}_{i=1}\|F_{i}\|_{\infty}$. Since
$F_{i}\in\mathbb{T}$ for any $i=1,\cdots,n$, we also get
$L(\Sigma^{n}_{i=1}F_{i})=\Sigma^{n}_{i=1}L(F_{i})$. This implies that
$\|\Sigma^{n}_{i=1}F_{i}\|_{L}=\Sigma^{n}_{i=1}\|F_{i}\|_{L}$. ∎

######  Proposition 3.5.

Let $\mathcal{S}$ be a T-set of $\operatorname{Lip}(X,E)$ with respect to
$\|\cdot\|_{L}$. Then there is $x\in X$ and there are $\mathbb{U}$ and
$\mathbb{T}$, where $\mathbb{U}$ is a T-set of $E$ and $\mathbb{T}$ is a T-set
of $\operatorname{Lip}(X,E)$ with respect to $L(\cdot)$, such that
$\mathcal{S}=S(x,\mathbb{U},\mathbb{T})$.

###### Proof.

For any $F\in\mathcal{S}$ we write $P(F):=\\{x\in
X|\|F(x)\|_{E}=\|F\|_{\infty}\\}$. We shall show that
$\bigcap_{F\in\mathcal{S}}P(F)\neq\emptyset$. For any finite collection
$F_{1},\cdots,F_{n}\in\mathcal{S}$, since
$\|\Sigma^{n}_{i=1}F_{i}\|_{L}=\Sigma^{n}_{i=1}\|F_{i}\|_{L}$ we have
$\|\Sigma^{n}_{i=1}F_{i}\|_{\infty}=\Sigma^{n}_{i=1}\|F_{i}\|_{\infty}$. Since
$\Sigma^{n}_{i=1}F_{i}\in\operatorname{Lip}(X,E)$, there is $x\in X$ such that
$\|(\Sigma^{n}_{i=1}F_{i})(x)\|=\|\Sigma^{n}_{i=1}F_{i}\|_{\infty}$. Thus we
get

$\Sigma^{n}_{i=1}\|F_{i}\|_{\infty}=\|\Sigma^{n}_{i=1}F_{i}\|_{\infty}=\|(\Sigma^{n}_{i=1}F_{i})(x)\|\leq\Sigma^{n}_{i=1}\|F_{i}(x)\|.$

This implies that $\|F_{i}\|_{\infty}=\|F_{i}(x)\|$ for any $i=1,\cdots,n$ and
$x\in\bigcap^{n}_{i=1}P(F_{i})$. Since $X$ is compact and $P(F)$ is a closed
set for each $F\in\mathcal{S}$, we have
$\bigcap_{F\in\mathcal{S}}P(F)\neq\emptyset$ by the finite intersection
property.

Let $x\in\bigcap_{F\in\mathcal{S}}P(F)$. We consider the set
$R_{x}(\mathcal{S}):=\\{F(x)\in E|F\in\mathcal{S}\\}$. Choose any finite
collection $F_{1}(x),\cdots,F_{n}(x)\in R_{x}(\mathcal{S})$. Since
$\Sigma^{n}_{i=1}F_{i}\in\mathcal{S}$, we have $x\in
P(\Sigma^{n}_{i=1}F_{i})$. This implies

$\Sigma^{n}_{i=1}\|F_{i}\|_{\infty}=\|\Sigma^{n}_{i=1}F_{i}\|_{\infty}=\|\Sigma^{n}_{i=1}F_{i}(x)\|\leq\Sigma^{n}_{i=1}\|F_{i}(x)\|=\Sigma^{n}_{i=1}\|F_{i}\|_{\infty}.$

Thus we have $\|\Sigma^{n}_{i=1}F_{i}(x)\|=\Sigma^{n}_{i=1}\|F_{i}(x)\|$,
which means that there is a T-set $\mathbb{U}$ of $E$ such that
$R_{x}(\mathcal{S})\subset\mathbb{U}$. Therefore for any $F\in\mathcal{S}$ we
have $F(x)\in\mathbb{U}$ and $\|F(x)\|=\|F\|_{\infty}$.

Since $L(\Sigma^{n}_{i=1}F_{i})=\Sigma^{n}_{i=1}L(F_{i})$ for any finite
collection $F_{1},\cdots,F_{n}\in\mathcal{S}$, there exists a T-set
$\mathbb{T}$ of $\operatorname{Lip}(X,E)$ with respect to $L(\cdot)$ such that
$\mathcal{S}\subset\mathbb{T}$. This implies that $\mathcal{S}\subset
S(x,\mathbb{U},\mathbb{T})$. By Lemma 3.4 and maximality of $\mathcal{S}$, we
conclude that $\mathcal{S}=S(x,\mathbb{U},\mathbb{T})$. ∎

######  Proposition 3.6.

Let $X_{i}$ be a compact metric space and $E_{i}$ be a Banach space for
$i=1,2$. Let
$U:\operatorname{Lip}(X_{1},E_{1})\to\operatorname{Lip}(X_{2},E_{2})$ be a
surjective complex linear isometry. Suppose that there is a surjective complex
linear isometry $\psi:E_{1}\to E_{2}$ and there is a surjective isometry
$\varphi:X_{2}\to X_{1}$ such that $U(f\otimes e)(y)=\psi(f(\varphi(y))e)$ for
any $f\in\operatorname{Lip}(X_{1})$ and $e\in E_{1}$. Then

$UF(y)=\psi(F(\varphi(y)))$

for any $F\in\operatorname{Lip}(X_{1},E_{1})$.

In the rest of this section, we assume that a surjective complex linear
isometry $U:\operatorname{Lip}(X_{1},E_{1})\to\operatorname{Lip}(X_{2},E_{2})$
satisfies the assumption of Proposition 3.6. To prove Proposition 3.6 we first
show the following lemma.

######  Lemma 3.7.

Let $x_{0}\in X_{1}$ and $F\in\operatorname{Lip}(X_{1},E_{1})$ with
$\|F\|_{\infty}=1$ and $F(x_{0})=0$. Then $UF(y_{0})=0$, where
$y_{0}=\varphi^{-1}(x_{0})$.

###### Proof.

Suppose that $UF(y_{0})\neq 0$. Put $a=\frac{UF(y_{0})}{\|UF(y_{0})\|}$. The
map from $\mathbb{S}(E_{2})$ to $\mathbb{R}$ defined by
$e\mapsto\|UF(y_{0})+(\|UF\|_{\infty}+1)e\|$ is continuous. Since
$\|UF(y_{0})\|\neq 0$, we have

$\|UF(y_{0})+(\|UF\|_{\infty}+1)a\|=\|\frac{UF(y_{0})}{\|UF(y_{0})\|}(\|UF(y_{0}))\|+\|UF\|_{\infty}+1)\|\\\
=\|UF(y_{0}))\|+\|UF\|_{\infty}+1>\|UF\|_{\infty}+1.$

There exists $\delta>0$ such that if $e\in\mathbb{S}(E_{2})$ with
$\|a-e\|<\delta$ then $\|UF(y_{0})+(\|UF\|_{\infty}+1)e\|>\|UF\|_{\infty}+1$.
We choose $\theta\in(0,2\pi)$ such that $|e^{i\theta}-1|<\delta$. We write
$e_{\theta}:=\psi^{-1}(e^{i\theta}a)$. This implies that

(13) $\|UF(y_{0})+(\|UF\|_{\infty}+1)\psi(e_{\theta})\|>\|UF\|_{\infty}+1.$

For any $n\in\mathbb{N}$, we define $g_{n}\in\operatorname{Lip}(X_{1})$ by

$g_{n}(x)=(\|UF\|_{\infty}+1)\max\\{1-nL(F)d(x,x_{0}),0\\},\quad x\in X_{1}.$

There is $\mathcal{S}_{n}$ which is a T-set of
$\operatorname{Lip}(X_{1},E_{1})$ with respect to $\|\cdot\|_{L}$ such that
$F+g_{n}\otimes e_{\theta}\in\mathcal{S}_{n}$. Since we have

$(F+g_{n}\otimes e_{\theta})(x_{0})=(\|UF\|_{\infty}+1)e_{\theta}.$

When $x\neq x_{0}$ and $1-nL(F)d(x,x_{0})\geq 0$, we have

$\|(F+g_{n}\otimes e_{\theta})(x)\|=\|F(x)-F(x_{0})+(g_{n}\otimes
e_{\theta})(x)\|\\\ \leq
L(F)d(x,x_{0})+(\|UF\|_{\infty}+1)(1-nL(F)d(x,x_{0}))\\\
=(1-n(\|UF\|_{\infty}+1))L(F)d(x,x_{0})+\|UF\|_{\infty}+1<\|UF\|_{\infty}+1.$

When $1-nL(F)d(x,x_{0})\leq 0$, we have

$\|(F+g_{n}\otimes e_{\theta})(x)\|=\|F(x)\|\leq 1<\|UF\|_{\infty}+1.$

Thus we obtain $P(F+g_{n}\otimes e_{\theta})=\\{x_{0}\\}$. By Proposition 3.5
there are T-set $\mathbb{U}_{n}\subset E_{1}$ and T-set
$\mathbb{T}_{n}\subset\operatorname{Lip}(X_{1},E_{1})$ such that
$F+g_{n}\otimes
e_{\theta}\in\mathcal{S}_{n}=S(x_{0},\mathbb{U}_{n},\mathbb{T}_{n})$. In
particular we have

$(\|UF\|_{\infty}+1)e_{\theta}=(F+g_{n}\otimes
e_{\theta})(x_{0})\in\mathbb{U}_{n}.$

By Lemma 3.2, $e_{\theta}\in\mathbb{U}_{n}$. Since $U$ is a surjective
isometry, Lemma 3.3 shows that there are $y_{n}\in X_{2}$, T-set
$\mathbb{V}_{n}\subset E_{2}$ and T-set
$\mathbb{T^{\prime}}_{n}\subset\operatorname{Lip}(X_{2},E_{2})$ with respect
to $L(\cdot)$ such that
$U(S(x_{0},\mathbb{U}_{n},\mathbb{T}_{n}))=S(y_{n},\mathbb{V}_{n},\mathbb{T^{\prime}}_{n})$.
Since $e_{\theta}\in\mathbb{U}_{n}$, we have $1\otimes e_{\theta}\in
S(x_{0},\mathbb{U}_{n},\mathbb{T}_{n})$. By the assumption, we have
$U(1\otimes e_{\theta})=1\otimes\psi(e_{\theta})\in
S(y_{n},\mathbb{V}_{n},\mathbb{T^{\prime}}_{n})$. It implies that
$\psi(e_{\theta})\in\mathbb{V}_{n}$ for any $n\in\mathbb{N}$. For any $y\in
X_{2}$

$U(F+g_{n}\otimes e_{\theta})(y)=UF(y)+\psi(g_{n}(\varphi(y))e_{\theta})\\\
=UF(y)+(\|UF\|_{\infty}+1)\max\\{1-nL(F)d(\varphi(y),x_{0}),0\\}\psi(e_{\theta}).$

We shall show that the sequence $\\{y_{n}\\}$ converges $y_{0}$ as
$n\to\infty$. Suppose that there exists $n\in\mathbb{N}$ such that
$1-nL(F)d(\varphi(y_{n}),x_{0})<0$. Then we have

$\|UF\|_{\infty}\geq\|UF(y_{n})\|=\|(UF+U(g_{n}\otimes
e_{\theta}))(y_{n})\|=\|UF+U(g_{n}\otimes e_{\theta})\|_{\infty}$

and

$\|UF+U(g_{n}\otimes
e_{\theta})\|_{\infty}\geq\|UF(y_{0})+(\|UF\|_{\infty}+1)\psi(e_{\theta})\|>\|UF\|_{\infty}+1.$

by (13). This is a contradiction. Thus for every $n\in\mathbb{N}$ we have

$1-nL(F)d(\varphi(y_{n}),x_{0})\geq 0.$

Thus we get
$\frac{1}{nL(F)}>d(\varphi(y_{n}),x_{0})=d(\varphi(y_{n}),\varphi(y_{0}))=d(y_{n},y_{0})$.
This implies that $y_{n}\to y_{0}$ as $n\to\infty$. Since
$UF\in\operatorname{Lip}(X_{2},E_{2})$ we get $UF(y_{n})\to UF(y_{0})$.

Because we obtain $0\leq 1-nL(F)d(\varphi(y_{n}),x_{0})\leq 1$, the sequence
$\\{1-nL(F)d(\varphi(y_{n}),x_{0})\\}$ has a convergent subsequence. Without
loss of generality we can assume that the sequence converges to
$\beta\in[0,1]$ as $n\to\infty$. We write

$c_{n}:=U(F+g_{n}\otimes
e_{\theta})(y_{n})=UF(y_{n})+(\|UF\|_{\infty}+1)(1-nL(F)d(\varphi(y_{n}),x_{0}))\psi(e_{\theta})$

and

(14) $c_{0}:=UF(y_{0})+(\|UF\|_{\infty}+1)\beta\psi(e_{\theta}).$

We obtain that

(15) $\|c_{n}-c_{0}\|_{E_{2}}\to 0\ \quad\text{if}\quad n\to\infty.$

As $U(F+g_{n}\otimes e_{\theta})\in
S(y_{n},\mathbb{V}_{n},\mathbb{T^{\prime}}_{n})$, we get
$c_{n}\in\mathbb{V}_{n}$. Since $\psi(e_{\theta})\in\mathbb{V}_{n}$, we have
$\|c_{n}+\psi(e_{\theta})\|=\|c_{n}\|+\|\psi(e_{\theta})\|$. By (15) we get
$\|c_{0}+\psi(e_{\theta})\|=\|c_{0}\|+\|\psi(e_{\theta})\|$. As
$\psi(e_{\theta})=e^{i\theta}a$ we obtain

$\|e^{-i\theta}c_{0}+a\|=\|e^{-i\theta}c_{0}\|+\|a\|.$

Thus there is $\tau\in E_{2}^{*}$ such that $\|\tau\|=1$,
$\tau(e^{-i\theta}c_{0})=\|c_{0}\|$ and $\tau(a)=\|a\|=1$. By (14) and
$a=\frac{UF(y_{0})}{\|UF(y_{0})\|}$, we have

$e^{i\theta}\|c_{0}\|=\tau(c_{0})=\tau(UF(y_{0}))+\tau((\|UF\|_{\infty}+1)\beta
e^{i\theta}a)\\\ =\|UF(y_{0})\|+e^{i\theta}(\|UF\|_{\infty}+1)\beta.$

We obtain that
$\|UF(y_{0})\|=e^{i\theta}(\|c_{0}\|-(\|UF\|_{\infty}+1)\beta)$. As
$\theta\in(0,2\pi)$ and $\|c_{0}\|-(\|UF\|_{\infty}+1)\beta\in\mathbb{R}$ , we
conclude $UF(y_{0})=0$. ∎

###### The proof of Proposition 3.6.

By the assumption, it suffices to show that $UF(y)=\psi(F(\varphi(y)))$ holds
for any $F\in\operatorname{Lip}(X_{1},E_{1})$ which $F$ is not a constant map.
For any $x\in X_{1}$, we define $G:=F-1\otimes F(x)$. Then, we have $G(x)=0$.
As $G\neq 0$, without loss of generality, we assume that $\|G\|_{\infty}=1$.
By Lemma 3.7, we obtain $(UG)(\varphi^{-1}(x))=0$. This implies that
$(UF)(\varphi^{-1}(x))=U(1\otimes F(x))=\psi(1(x)F(x))=\psi(F(x))$. ∎

##  4\. Proof of Theorem 1.2

Let $\mathcal{A}_{i}$ be unital $C^{*}$-algebras for $i=1,2$. In this section
we consider unital surjective complex isometries with respect to the norm
$\|\cdot\|_{L}$ from $\operatorname{Lip}(X_{1},\mathcal{A}_{1})$ onto
$\operatorname{Lip}(X_{2},\mathcal{A}_{2})$. Although we apply the similar
arguments as [14], we show a proof without omitting it because this is a
generalization for [14, Theorem 3.3]. We say that a bounded operator $D$ on a
unital $C^{*}$-algebra $\mathcal{A}$ is a $*$-derivation if

(16) $\begin{split}&D(ab)=D(a)b+aD(b),\\\ &D(a^{*})=D(a)^{*}\\\ \end{split}$

for every pair $a,b\in\mathcal{A}$. By the definition, it is easy to see that
$D(1)=1$ for any $*$-derivation on $\mathcal{A}$. For each $a\in\mathcal{A}$,
a left multiplication operator $M_{a}:\mathcal{A}\to\mathcal{A}$ is defined by
$M_{a}b=ab$ for every $b\in\mathcal{A}$. We denote the set of all hermitian
elements of $\mathcal{A}$ by $H(\mathcal{A})$.

The following is the characterization of hermitian operators on a unital
$C^{*}$-algebra.

######  Theorem 4.1 (Sinclair [15]).

Let $\mathcal{A}$ be a unital $C^{*}$-algebra. A bounded operator $T$ on
$\mathcal{A}$ is a hermitian operator if and only if there exist $h\in
H(\mathcal{A})$ and a $*$-derivation $D$ on $\mathcal{A}$ such that
$T=M_{h}+iD$.

We introduce the notation to characterize hermitian operators on
$\operatorname{Lip}(X,\mathcal{A})$.

######  Definition 4.2.

For any $h\in H(\mathcal{A})$, we define a multiplication operator
$M_{1\otimes
h}:\operatorname{Lip}(X,\mathcal{A})\to\operatorname{Lip}(X,\mathcal{A})$ by

$M_{1\otimes h}(F)=(1\otimes h)F,\quad F\in\operatorname{Lip}(X,\mathcal{A}).$

For any $*$-derivation $D:\mathcal{A}\to\mathcal{A}$, we define a map
$\widehat{D}:\operatorname{Lip}(X,\mathcal{A})\to\operatorname{Lip}(X,\mathcal{A})$
by

$\widehat{D}(F)(x)=D(F(x)),\quad F\in\operatorname{Lip}(X,\mathcal{A}),\quad
x\in X.$

Combining Theorem 4.1 and Theorem 2.3 we obtain the following.

######  Proposition 4.3.

Suppose that
$T:\operatorname{Lip}(X,\mathcal{A})\to\operatorname{Lip}(X,\mathcal{A})$ is a
map. Then $T$ is a hermitian operator if and only if there exists $h\in
H(\mathcal{A})$ and a $*$-derivation $D$ on $\mathcal{A}$ such that

(17) $T=M_{1\otimes h}+i\widehat{D}.$

The following Proposition is a well-known fact.

######  Proposition 4.4.

Let $\mathcal{B}_{j}$ be Banach algebras for $j=1,2$. Suppose that $U$ is a
surjective complex linear isometry from $\mathcal{B}_{1}$ onto
$\mathcal{B}_{2}$ and $T$ is a hermitian operator on $\mathcal{B}_{1}$. Then
the map $UTU^{-1}$ is a hermitian operator on $\mathcal{B}_{2}$.

In the rest of this section we consider a surjective complex linear isometry
$U:(\operatorname{Lip}(X_{1},\mathcal{A}_{1}),\|\cdot\|_{L})\to(\operatorname{Lip}(X_{2},\mathcal{A}_{2}),\|\cdot\|_{L})$
with $U(1)=1$.

######  Lemma 4.5.

For any $h\in H(\mathcal{A}_{1})$, there exists $h^{\prime}\in
H(\mathcal{A}_{2})$ such that

$U(1\otimes h)=1\otimes h^{\prime}.$

###### Proof.

Let $h\in H(\mathcal{A}_{1})$. By Proposition 4.3 and Proposition 4.4,
$UM_{1\otimes h}U^{-1}$ is a hermitian operator on
$\operatorname{Lip}(X_{2},\mathcal{A}_{2})$. Thus there exists $h^{\prime}\in
H(\mathcal{A}_{2})$ and a $*$-derivation $D$ on $\mathcal{A}_{2}$ such that
$UM_{1\otimes h}U^{-1}=M_{1\otimes h^{\prime}}+i\widehat{D}$. For any $y\in
X_{2}$, we have

$(UM_{1\otimes h}U^{-1})(1)(y)=(UM_{1\otimes h}1)(y)=U(1\otimes h)(y)$

and

$h^{\prime}(1(y))+iD(1(y))=h^{\prime}+i0=h^{\prime}.$

This implies that $U(1\otimes h)=1\otimes h^{\prime}$. ∎

By Lemma 4.5, we define a map $\psi_{0}:H(\mathcal{A}_{1})\to
H(\mathcal{A}_{2})$ by

$U(1\otimes h)=1\otimes\psi_{0}(h).$

######  Lemma 4.6.

The map $\psi_{0}$ is a real linear isometry from $H(\mathcal{A}_{1})$ onto
$H(\mathcal{A}_{2})$ such that $\psi_{0}(1)=1$.

###### Proof.

For any $h_{2}\in H(\mathcal{A}_{2})$, we have that $U^{-1}M_{1\otimes
h_{2}}U$ is a hermitian operator on
$\operatorname{Lip}(X_{1},\mathcal{A}_{1})$. By Proposition 4.3 there are
$h_{1}\in H(\mathcal{A}_{1})$ and a $*$-derivation $D_{1}$ on
$\mathcal{A}_{1}$ such that

$U^{-1}M_{1\otimes h_{2}}U=M_{1\otimes h_{1}}+i\widehat{D_{1}}.$

Since we have $M_{1\otimes h_{1}}=U^{-1}M_{1\otimes h_{2}}U-i\widehat{D_{1}}$,
we get

$UM_{1\otimes h_{1}}U^{-1}(1)=U(U^{-1}M_{1\otimes
h_{2}}U-i\widehat{D_{1}})U^{-1}(1)\\\ =M_{1\otimes
h_{2}}(1)-U(i\widehat{D_{1}}(1))=1\otimes h_{2}-iU(0)=1\otimes h_{2}.$

We obtain $U(1\otimes h_{1})=1\otimes h_{2}$ and $\psi_{0}(h_{1})=h_{2}$. It
follows that $\psi_{0}$ is surjective. For any $h\in H(\mathcal{A}_{1})$, we
get $\|\psi_{0}(h)\|=\|1\otimes\psi_{0}(h)\|_{L}=\|U(1\otimes
h)\|_{L}=\|1\otimes h\|_{L}=\|h\|$. Thus, we have $\psi_{0}$ is an isometry.
Since $U$ is a linear map, it is easy to see that $\psi_{0}$ is real linear.
Moreover $U(1)=1$, we get $\psi_{0}(1)=1$. ∎

For any $a\in\mathcal{A}_{1}$, there are $h_{1},h_{2}\in H(\mathcal{A}_{1})$
such that $a=h_{1}+ih_{2}$. Thus, we define a map
$\psi:\mathcal{A}_{1}\to\mathcal{A}_{2}$ by

$\psi(a)=\psi(h_{1}+ih_{2}):=\psi_{0}(h_{1})+i\psi_{0}(h_{2}).$

By a simple calculation, we have

(18) $U(1\otimes a)=1\otimes\psi(a)$

for any $a\in\mathcal{A}_{1}$.

######  Lemma 4.7.

The map $\psi$ is a surjective complex linear isometry from $\mathcal{A}_{1}$
onto $\mathcal{A}_{2}$ such that $\psi(1)=1$.

###### Proof.

By (18), we have $\psi$ is a complex linear isometry with $\psi(1)=1$.
Therefore it suffices to show $\psi$ is surjective. For any
$a\in\mathcal{A}_{2}$, there exists $h_{1}$, $h_{2}\in H(\mathcal{A}_{2})$
such that $a=h_{1}+ih_{2}$. Since Lemma 4.6 shows that
$\psi_{0}:H(\mathcal{A}_{1})\to H(\mathcal{A}_{2})$ is surjective, there are
$h^{{}^{\prime}}_{1},h^{{}^{\prime}}_{2}\in H(\mathcal{A}_{1})$ such that
$\psi_{0}(h^{{}^{\prime}}_{1})=h_{1}$ and
$\psi_{0}(h^{{}^{\prime}}_{2})=h_{2}$. Then we get
$a^{\prime}=h^{{}^{\prime}}_{1}+ih^{{}^{\prime}}_{2}\in\mathcal{A}_{1}$. This
implies that

$\psi(a^{\prime})=\psi_{0}(h^{{}^{\prime}}_{1})+i\psi_{0}(h^{{}^{\prime}}_{2})=h_{1}+ih_{2}=a.$

This completes the proof. ∎

######  Lemma 4.8.

Suppose that $\mathcal{A}_{i}$ is a unital factor $C^{*}$-algebra for $i=1,2$.
Then there exists a surjective isometry $\varphi:X_{2}\to X_{1}$ such that

$U(f\otimes 1)(y)=f(\varphi(y))\otimes 1$

for all $f\in\operatorname{Lip}(X_{1})$ and $y\in X_{2}$.

###### Proof.

For any $b\in\mathcal{A}_{2}$ with $b^{*}=-b$, we define a $*$-derivation $D$
on $\mathcal{A}_{2}$ by

$D(a)=ba-ab,\quad a\in\mathcal{A}_{1}.$

Note that Proposition 4.3 shows that the map
$i\widehat{D}:\operatorname{Lip}(X_{2},\mathcal{A}_{2})\to\operatorname{Lip}(X_{2},\mathcal{A}_{2})$
defined by

$(i\widehat{D})(F)(y)=iD(F(y))\quad
F\in\operatorname{Lip}(X_{2},\mathcal{A}_{2}),\quad y\in X_{2},$

is a hermitian operator on $\operatorname{Lip}(X_{2},\mathcal{A}_{2})$. Since
the map $U$ is an isometry, $U^{-1}i\widehat{D}U$ is a hermitian operator on
$\operatorname{Lip}(X_{1},\mathcal{A}_{1})$. By Proposition 4.3 there exists
$h\in H(\mathcal{A}_{1})$ and $*$-derivation $D^{\prime}$ on $\mathcal{A}_{1}$
such that

$U^{-1}i\widehat{D}U=M_{1\otimes h}+i\widehat{D^{\prime}}.$

As $U(1)=1$, we get

$(U^{-1}i\widehat{D}U)(1)=i(U^{-1}\widehat{D}U)(1)=iU^{-1}\widehat{D}(1)=iU^{-1}(0)=0.$

This implies that

$0=(U^{-1}i\widehat{D}U)(1)=(M_{1\otimes h}+i\widehat{D^{\prime}})(1)\\\
=1\otimes h+i\widehat{D^{\prime}}(1)=1\otimes h+i0=1\otimes h.$

Thus we have $U^{-1}i\widehat{D}U=i\widehat{D^{\prime}}$. This implies that
for any $f\in\operatorname{Lip}(X_{1})$, we have
$(U^{-1}i\widehat{D}U)(f\otimes 1)(x)=i\widehat{D^{\prime}}(f\otimes 1)(x)=0$.
In addition by the definition of $D$, we get

(19) $\begin{split}(U^{-1}i\widehat{D}U)(f\otimes
1)&=U^{-1}(i\widehat{D}U(f\otimes 1))\\\ &=iU^{-1}(1\otimes bU(f\otimes
1)-U(f\otimes 1)1\otimes b).\end{split}$

Therefore we have

$U^{-1}(1\otimes bU(f\otimes 1)-U(f\otimes 1)1\otimes b)=0.$

Since $U$ is surjective, we have

(20) $1\otimes bU(f\otimes 1)=U(f\otimes 1)1\otimes b.$

Note we choose $b\in\mathcal{A}_{2}$ with $b^{*}=-b$ arbitrary. For each
$a\in\mathcal{A}_{2}$ there exist unique elements
$b_{1},b_{2}\in\mathcal{A}_{2}$ such that $b_{i}^{*}=-b_{i}$ for $i=1,2$ and
$a=-ib_{1}+b_{2}$. By applying (20), we have

$aU(f\otimes 1)(y)=U(f\otimes 1)(y)a$

for any $a\in\mathcal{A}_{2}$ and $y\in X_{2}$. We get $U(f\otimes
1)(y)\in\mathcal{A}\cap\mathcal{A}^{\prime}=\mathbb{C}1$. Thus there is
$g(y)\in\mathbb{C}$ such that $U(f\otimes 1)(y)=g(y)1$. Since $U(f\otimes
1)\in\operatorname{Lip}(X_{2},\mathcal{A}_{2})$, we get
$g\in\operatorname{Lip}(X_{2})$ and

$U(f\otimes 1)=g\otimes 1.$

Thus we can define a map
$P_{U}:\operatorname{Lip}(X_{1})\to\operatorname{Lip}(X_{2})$ by

$U(f\otimes 1)=P_{U}(f)\otimes 1,\quad f\in\operatorname{Lip}(X_{1}).$

It is easy to see that $P_{U}$ is a surjective complex linear isometry.
Applying [9, Corollary 15], there is a surjective isometry $\varphi:X_{2}\to
X_{1}$ such that

$U(f\otimes 1)(y)=P_{U}(f)(y)\otimes 1=f(\varphi(y))\otimes 1,\quad
f\in\operatorname{Lip}(X_{1}),\,\,y\in X_{2}.$

∎

###### Proof of Theorem 1.2.

A simple calculation shows that the map $U$ from
$\operatorname{Lip}(X_{1},\mathcal{A}_{1})$ onto
$\operatorname{Lip}(X_{2},\mathcal{A}_{2})$, which has the form of the theorem
is a unital surjective linear isometry. We show the converse. For any $h\in
H(\mathcal{A}_{1})$, there exists $\psi_{0}(h)\in H(\mathcal{A}_{2})$ and
$*$-derivation $D$ on $\mathcal{A}_{2}$ such that

$UM_{1\otimes h}U^{-1}=M_{1\otimes\psi_{0}(h)}+i\widehat{D}.$

Let $f\in\operatorname{Lip}(X_{1})$. By Lemma 4.8, we have

$\begin{split}U(f\otimes h)(y)&=U(M_{1\otimes h}(f\otimes 1))(y)=UM_{1\otimes
h}U^{-1}U(f\otimes 1)(y)\\\
&=(M_{1\otimes\psi_{0}(h)}+i\widehat{D})(U(f\otimes 1))(y)\\\
&=\psi_{0}(h)(U(f\otimes 1)(y))=f(\varphi(y))\psi_{0}(h)\end{split}$

for any $y\in X_{2}$. For any $a\in\mathcal{A}_{1}$, there exist
$h_{1},h_{2}\in H(\mathcal{A}_{1})$ such that $a=h_{1}+ih_{2}$ and we get

$\begin{split}U(f\otimes a)(x)&=U(f\otimes(h_{1}+ih_{2}))(x)=U(f\otimes
h_{1})(x)+iU(f\otimes h_{2})(x)\\\
&=f(\varphi(x))\psi_{0}(h_{1})+if(\varphi(x))\psi_{0}(h_{2})\\\
&=f(\varphi(x))\psi(a)=\psi((f\otimes
a)(\varphi(x)))=\psi(f(\varphi(y))a)\end{split}$

for any $f\in\operatorname{Lip}(X_{1})$ and $a\in\mathcal{A}_{1}$. Applying
Proposition 3.6 we obtain

$UF(y)=\psi(F(\varphi(y))),\quad
F\in\operatorname{Lip}(X_{1},\mathcal{A}_{1}),y\in X_{2}.$

∎

##  5\. Concluding comments and remarks

Let us look at further problems related to Theorem 1.2. It is natural to
investigate the following questions; What is the general form of unital
surjective linear isometries between $\operatorname{Lip}(X,\mathcal{A})$,
where $\mathcal{A}$ is a unital $C^{*}$-algebra? What is a complete
description of surjective linear isometries on
$\operatorname{Lip}(X,\mathcal{A})$ without the assumption that isometries
preserve the identity? In fact, less is known about surjective linear
isometries on Banach spaces of all vector-valued Lipschitz maps with
$\|\cdot\|_{L}$. The author suspects the reason relies on a lack of a complete
characterization of the extreme points of
$\mathbb{B}((\operatorname{Lip}(X,E))^{*})$. Thus we believe Theorem 2.3 is
one of crucial tools in investigating our questions. This might be an
interesting direction for further research. These are left as research
problems in the future.

### Acknowledgments

This work was supported by JSPS KAKENHI Grant Numbers JP21K13804.

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