The von Neumann inequality for 3$\times$3 matrices in the unit Euclidean ball

Dariusz Piekarz***Partially supported by the NCN grant PRELUDIUM BIS-2: PSP:
K/PBM/000766

ABSTRACT
It is shown that the constant $c_{d,3}$ in von Neumann’s inequality for
$d$-tuples of commutative and row contractive $3\times 3$ matrices, as proved
by Hartz, Richter, and Shalit in [2], is independent of the size of the
$d$-tuple. A numerical estimation of the constant is provided.

##  1\. Introduction

The von Neumann inequality is a classical result in operator theory which
states that, for any bounded and contractive operator $T\in B(H)$ acting on a
Hilbert space $H$ and any polynomial $p\in\mathbb{C}[z]$, the following
inequality holds:

(1.1) $\displaystyle\|p(T)\|\leq\sup_{z\in\mathbb{D}}|p(z)|.$

It is a natural question to ask whether this inequality can be extended to a
$d$-tuple of operators. For $d=2,$ And$\hat{\text{o}}$ answered to this
question positively proving that if $T=(T_{1},T_{2})$ is a pair of commuting,
contractive, bounded operators on a Hilbert space $H$ and
$p\in\mathbb{C}[z,w]$ is a polynomial, then

$\|p(T)\|\leq\sup_{(z,w)\in\mathbb{D}^{2}}|p(z,w)|.$

However, a full generalization of this result is not possible. Davie and Crabb
proved that in a 8-dimensional Hilbert space there exist a triple $T$ of
pairwise commuting, contractive operators, and a homogeneous polynomial $p$ of
degree 3 such that

$\|p(T)\|>1.$

Further details on the proof of And$\hat{\text{o}}$’s inequality and the
Davie-Crabb example can be found in Chapter 1 of [8].

In [5], Knese used And$\hat{\text{o}}$’s inequality and Kosiński’s result on a
solution of the 3-point Pick interpolation problem in the polydisc
$\mathbb{D}^{d}$ to show that the von Neumann inequality holds for $d$-tuples
of $3\times 3$ commuting and contractive matrices in the polydisc
$\mathbb{D}^{d}$.

Recall that a Pick interpolation problem, for a domain
$D\subset\mathbb{C}^{n}$, can be formulated as follows: given distinct points
$z_{1},\ldots,z_{N}$ in $D$ and points $\zeta_{1},\ldots,\zeta_{N}$ in the
unit disc $\mathbb{D}$, decide if there exists a holomorphic function
$F:D\rightarrow\mathbb{D}$ such that $F(z_{j})=\zeta_{j},\ j=1,\ldots,N$.

Additionally, the Pick interpolation is extremal if it is solvable, and there
is no holomorphic function $G:D\rightarrow\mathbb{D}$ such that
$G(z_{j})=\zeta_{j},\ j=1,\ldots,N$ and $G(D)$ is a relatively compact subset
of $\mathbb{D}.$

In the work of Hartz, Richter and Shalit [2] the contractivity property was
replaced by the row contractivity condition, which means that a $d$-tuple of
operators $T$ defined on a Hilbert space satisfies the condition

(1.2) $\displaystyle\sum_{j=1}^{d}T_{j}T^{*}_{j}\leq I,$

where $I$ represents the identity operator, and $T^{*}$ denotes the adjoint
operator of $T$. They proved:

######  Theorem 1.1.

There exists the smallest constant $c_{d,n}\geq 0,$ such that for any
$d-$tuple of commuting, row contractive, $n\times n$ matrices and for any
polynomial $p\in\mathbb{C}[z_{1},...,z_{d}],$ one has

(1.3) $\displaystyle\|p(T)\|\leq c_{d,n}\sup_{z\in\mathbb{B}^{d}}|p(z)|,$

where $d,n\geq 2.$ Additionally, $c_{d,n}>1$ if $n\geq 3,\ d\geq 2$ and
$c_{d,2}=1,$ if $d\geq 2.$

Following Knese’s idea we use the solution to the Pick interpolation problem
in the unit Euclidean ball, developed by Kosiński and Zwonek in [4], to prove
Theorem 1.1 in the particular case:

######  Theorem 1.2.

The constant $c_{d,3}$, described in Theorem 1.1, is independent of $d$, i.e.,

$c_{d,3}=c_{2,3},\ d\geq 2.$

Moreover, one has the following numerical estimation: $1.11767\leq c_{d,3}\leq
3.14626.$

##  2\. Case of a $d$-tuple of $2\times 2$ commuting, row contractive matrices

To present our proof idea for Theorem 1.2, we show a special case of Theorem
1.1, which was proved by Hartz, Richter, and Shalit in [2]:

######  Proposition 2.1.

The von Neumann inequality (1.3) holds for any $d$-tuple of commuting, row
contractive $2\times 2$ matrices, and for any polynomial
$p\in\mathbb{C}[z_{1},\dots,z_{d}],$ where $d\geq 2$ with a constant
$c_{d,2}=1$.

The following lemma is crucial in the proof of Proposition 2.1:

######  Lemma 2.2.

Let

(2.1) $\displaystyle\mathbb{B}^{d}\ni z\mapsto\zeta\in\mathbb{D},$
$\displaystyle\mathbb{B}^{d}\ni w\mapsto\eta\in\mathbb{D},$

be a 2-point extremal Pick interpolation problem in the unit Euclidean ball.
Then, up to composition with automorphisms of the unit Euclidean ball
$\mathbb{B}^{d}$ and the unit disk $\mathbb{D}$, it can be solved by the
projection onto the first coordinate, $\pi(z_{1},\dots,z_{d})=z_{1}$.

###### Proof of Lemma 2.2.

Let $F$ be a solution of (2.1). It is known that automorphisms of the unit
ball $\mathbb{B}^{d}$ are finite compositions of transformations of the form

$A_{t}(z_{1},...,z_{d})=\left(\sqrt{1-|t|^{2}}\frac{z_{1}}{1+\overline{t}z_{d}},....,\sqrt{1-|t|^{2}}\frac{z_{d-1}}{1+\overline{t}z_{d-1}},\frac{z_{d}+t}{1+\overline{t}z_{d}}\right),\
t\in\mathbb{D}$

as well as unitary mappings. Using automorphisms of the unit ball
$\mathbb{B}^{d}$ and the unit disk $\mathbb{D}$, we can assume that
$w=(0,\dots,0)$, $\eta=0$, and $\zeta>0$. Using additionally a unitary
mapping, we can assume that $z=(x,0,\dots,0)$, where $x>0$. Consider a
function $G:\mathbb{D}\rightarrow\mathbb{D},$ given by
$G(\mu):=F(\mu,0,\dots,0)$. It follows from the Schwarz lemma that
$|G(\mu)|\leq|\mu|.$ Therefore, one has $\zeta\leq x$. Extremality means in
fact that $\zeta=x$. The Schwarz lemma yields that the projection
$\pi:\mathbb{B}^{d}\rightarrow\mathbb{D}$ given by
$\pi(z_{1},\dots,z_{d})=z_{1}$ is the desired solution. ∎

###### Proof of Proposition 2.1.

The set of diagonalizable $d$-tuples of $2\times 2$ matrices is dense in the
set of all $d$-tuples of square $2\times 2$ matrices [1, Paragraph 4, p.21],
so we can assume that the $d$-tuple $T$ of pairwise commuting, row
contractions consists of diagonalizable matrices. In [7, Theorem 2.21.], it
was shown that any family of diagonalizable and commuting matrices is
simultaneously diagonalizable. Hence, we can express $T$ in the form

$T=P\text{diag}(\alpha,\beta)P^{-1},$

where $\text{diag}(\alpha,\beta)$ is defined as the $d$-tuple of diagonal
matrices

$\displaystyle\text{diag}(\alpha,\beta)$
$\displaystyle:=\left(\begin{pmatrix}\alpha_{1}&0\\\
0&\beta_{1}\end{pmatrix},...,\begin{pmatrix}\alpha_{d}&0\\\
0&\beta_{d}\end{pmatrix}\right),$

where $\alpha=(\alpha_{1},...,\alpha_{d})\in\mathbb{B}^{d},\
\beta=(\beta_{1},...,\beta_{d})\in\mathbb{B}^{d}$ and $P$ is a transformation
matrix. Consequently, for any holomorphic function
$f:\mathbb{B}^{d}\rightarrow\mathbb{C}$, we obtain

$f(T)=P\text{diag}(f(\alpha),f(\beta))P^{-1}.$

In particular, this holds for polynomials.

Consider a polynomial $p$ and a $d$-tuple $T$ of commutative, row-contractions
with a joint spectrum $\\{\alpha,\beta\\}$ in $\mathbb{B}^{d}$. We can assume
that $p(\alpha)\neq p(\beta).$ Otherwise $T=p(\alpha)I,$ where $I$ is the
identity matrix. One has in this case
$\|p(T)\|=|p(\alpha)|\leq\sup_{\lambda\in\mathbb{B}^{d}}|p(\lambda)|$ and the
assertion follows. Next, let the interpolation problem

$\displaystyle\alpha\mapsto p(\alpha),$ (2.2) $\displaystyle\beta\mapsto
p(\beta),$

have an analytic solution $f$. Consider the modified problem,

$\displaystyle\alpha\mapsto tp(\alpha),$ (2.3) $\displaystyle\beta\mapsto
tp(\beta),$

where $t\geq 1$. We are interested in finding the greatest $t$ such that there
exists a solution of (2). Note that the set $\tau$ of all such $t$ is bounded
from above as the image of any solution of (2) is a subset of the unit disc
$\mathbb{D}$. Let $(t_{i})_{i}$ be a sequence converging to the supremum
$t_{0}$ of $\tau.$ For any $i$, there exists a holomorphic function
$f_{i}:\mathbb{B}^{d}\rightarrow\mathbb{D}$ such that

(2.4) $\displaystyle f_{i}(\alpha)=t_{i}p(\alpha),\
f_{i}(\beta)=t_{i}p(\beta).$

Using Montel’s theorem, we can pass to a convergent subsequence. Therefore,
there exists a holomorphic function $F:\mathbb{B}^{d}\rightarrow\mathbb{D}$
such that

$F(\alpha)=t_{0}p(\alpha),\ F(\beta)=t_{0}p(\beta).$

Since $t_{0}$ is the greatest possible, the problem is extremal. Due to Lemma
2.2, the function $F$ can be expressed in the form $F=m\circ\pi\circ A,$ where
$m\in\text{Aut}(\mathbb{D}),\ A\in\text{Aut}(\mathbb{B}^{d})$ and $\pi$ is the
natural projection onto the first coordinate. Put $S=A(T)$. Since a
composition of the unit ball automorphism with a $d$-tuple of commutative, row
contractions is a commutative, row contraction, we infer that $S$ is a
$d$-tuple of commutative, row contractions. It follows that

$F(T)=m(\pi(A(T)))=m(S_{1}).$

We have

(2.5)
$\displaystyle\|p(T)\|\leq\|F(T)\|=\|m(S_{1})\|\leq\sup_{z\in\mathbb{D}}|m(z)|=1.$

In the last inequality we applied inequality (1.1) which holds for all
holomorphic functions in the neighbourhood of the unit disc $\mathbb{D}$ due
to uniform approximation by complex polynomials on compact subsets of the unit
disc $\mathbb{D}$. Thus, we have $c_{d,2}=c_{1,2}$. ∎

##  3\. Proof of Theorem 1.2: Constant $c_{d,3}=c_{2,3}$

In [4] Kosiński and Zwonek found a solution to the 3-point Pick interpolation
problem in the unit Euclidean ball.

######  Theorem 3.1 (Kosiński, Zwonek).

If the $3-$point Pick interpolation problem

$\mathbb{B}^{d}\rightarrow\mathbb{D},\ w_{j}\mapsto\lambda_{j},\ j=1,2,3,$

is extremal, then, up to a composition with automorphisms of $\mathbb{B}^{d}$
and $\mathbb{D}$, it is interpolated by a function which belongs to one of the
classes

$\mathcal{F}_{D}=\\{(z_{1},...,z_{n})\mapsto\frac{2z_{1}(1-\tau
z_{1})-\overline{\tau}\omega^{2}z_{2}^{2}}{2(1-\tau
z_{1})-\omega^{2}z_{2}^{2}}:\ |\tau|=1,|\omega|\leq 1\\},$
$\mathcal{F}_{ND}=\\{(z_{1},...,z_{n})\mapsto\frac{z_{1}^{2}+2\sqrt{1-a^{2}}z_{2}}{2-a^{2}}:\
a\in[0,1)\\}.$

We begin with showing that for every integer $d\geq 2$, it is true that
$c_{d,3}=c_{2,3}$. To prove it, we can adopt the same reasoning employed in
the $2\times 2$ matrix case, but instead consider $d$-tuples of diagonalizable
$3\times 3$ matrices (the fact that the set of $d-$tuples of $3\times 3,$
diagonalizable and commuting matrices is dense in set of all $d-$tuples of
commuting square matrices can be found in [6, Lemma 10]). Assuming that $T$
consists of a $d$-tuple of pairwise commuting, diagonalizable, row contractive
matrices, we can represent $T$ as

$T=P\text{diag}(\alpha,\beta,\gamma)P^{-1},$

where $P$ is a transformation matrix and
$\alpha,\beta,\gamma\in\mathbb{B}^{d}.$ As before, let $p$ be a polynomial.
First, consider the case where $p(\alpha),p(\beta),p(\gamma)$ are distinct.
The interpolation problem

$\displaystyle\alpha\mapsto p(\alpha),$ $\displaystyle\beta\mapsto p(\beta),$
$\displaystyle\gamma\mapsto p(\gamma),$

possesses an analytic solution $f$. Next, we consider the modified problem,

$\displaystyle\alpha\mapsto tp(\alpha),$ (3.1) $\displaystyle\beta\mapsto
tp(\beta),$ $\displaystyle\gamma\mapsto tp(\gamma).$

Using the same Montel-type argument as in the proof of Proposition 2.1, there
exists a holomorphic function $F:\mathbb{B}^{d}\rightarrow\mathbb{D}$ that is
an extremal solution of the interpolation problem (3). Theorem 3.1 allows us
to represent $F$ in the form $F=m\circ h\circ A,$ where
$m\in\text{Aut}(\mathbb{D}),\ A\in\text{Aut}(\mathbb{B}^{d})$ and $h$ is a
function belonging either to $\mathcal{F}_{D},$ or $\mathcal{F}_{ND}.$ Put
$S=A(T).$ Then

$F(T)=m(h(A(T)))=m(h(S_{1},S_{2})),$

due to the special forms of functions from classes
$\mathcal{F}_{D},\mathcal{F}_{ND}.$

As in the case of $2\times 2$ matrices, if $p(\alpha)=p(\beta)=p(\gamma),$
then we can proceed as in the proof of Proposition 2.1.

Next, consider the case where only 2 of 3 points
$p(\alpha),p(\beta),p(\gamma)$ are equal, say $p(\alpha)=p(\beta).$ Then, we
can modify the polynomial $p$ replacing $p(\beta)$ by
$p(\beta)-\varepsilon\eta,$ where $\varepsilon>0$ is arbitrarily small number
and $\eta\in\partial\mathbb{D}$ are such that
$p(\beta)-\varepsilon\eta\in\mathbb{D},$ for all such $\varepsilon>0$. Denote
the modified polynomial by $p_{\varepsilon}.$ Points of spectrum for
$p_{\varepsilon}(T)$ are distinct, hence we can argue as above constructing a
holomorphic function $F_{\varepsilon}$ such that
$F_{\varepsilon}(T)=m(h_{\varepsilon}(A(T)))=m(h_{\varepsilon}(S_{1},S_{2})).$
Since this is true for all such $\varepsilon>0$ suitably small, by the
continuity of the norm and the continuity of holomorphic functions we can pass
with $\varepsilon$ to $0$ preserving required properties. To get back to the
von Neumann’s inequality we argue similarly as in the end of the proof of the
Proposition 2.1. Therefore, we infer that $c_{d,3}=c_{2,3}.$

##  4\. Proof of Theorem 1.2: Estimation of the $c_{2,3}$ constant

According to the previous section, we can assume that $d=2$. Furthermore, we
have a pair $T=(T_{1},T_{2})$ consisting of diagonalizable $3\times 3$
commuting and row contractive matrices. We can assume that the joint spectrum
of $T$ is a set $\\{(a_{1},a_{2}),(0,d_{2}),(0,0)\\}\subset\mathbb{B}^{2}$
with $d_{2}>0$. Indeed, the composition of an automorphism of the unit ball
$\mathbb{B}^{2}$ with a commutative row contraction is itself a commutative
row contraction. Therefore, as shown in the proof of Lemma 2.2, we can compose
$T$ with appropriate automorphisms of the unit ball $\mathbb{B}^{2}.$

Using [7, Theorem 2.23.], we can assume that $T_{i}$’s are upper triangular
matrices, that is,

$T_{1}=\begin{pmatrix}a_{1}&b_{1}&c_{1}\\\ 0&0&e_{1}\\\ 0&0&0\ \end{pmatrix},\
T_{2}=\begin{pmatrix}a_{2}&b_{2}&c_{2}\\\ 0&d_{2}&e_{2}\\\ 0&0&0\
\end{pmatrix}.$

We can represent the matrices $T_{i}$ using a block matrix notation

(4.1) $\displaystyle T_{i}=\begin{pmatrix}A_{i}&\beta_{i}\\\ 0&0\
\end{pmatrix},$

where

$A_{1}=\begin{pmatrix}a_{1}&b_{1}\\\ 0&0\ \end{pmatrix},\
A_{2}=\begin{pmatrix}a_{2}&b_{2}\\\ 0&d_{2}\ \end{pmatrix},\
\beta_{i}=\begin{pmatrix}c_{i}\\\ e_{i}\ \end{pmatrix}\ i=1,2.$

As in the preceding sections, in order to estimate $c_{2,3}$, it is sufficient
to appraise the norm of $(m\circ h\circ A)(T)$, where
$m\in\text{Aut}(\mathbb{D})$, $A\in\text{Aut}(\mathbb{B}^{2})$, and
$h\in\mathcal{F}_{ND}$ or $h\in\mathcal{F}_{D}$.

We split the estimation with respect to the function type.

######  Lemma 4.1.

If $h\in\mathcal{F}_{ND},$ then $\|h(T)\|\leq 2/\sqrt{3}.$

###### Proof.

Consider a function $F_{\alpha}$ from the class $\mathcal{F}_{ND}.$ It is
worth noting that in this case, the inequality given by $(\ref{rc:1})$ can be
restated in an equivalent form as $\|T_{1}z\|^{2}+\|T_{2}z\|^{2}\leq 1,$ where
$z\in\partial\mathbb{B}^{3}.$ This can also be expressed using the block form
as follows:

(4.2) $\displaystyle\|A_{1}x+\beta_{1}y\|^{2}+\|A_{2}x+\beta_{2}y\|^{2}\leq
1,\ z=\begin{pmatrix}x\\\ y\ \end{pmatrix}\in\partial\mathbb{B}^{3},\
x\in\mathbb{B}^{2},\ y\in\mathbb{D}.$

We get

$F_{\alpha}(T_{1},T_{2})=\begin{pmatrix}F_{\alpha}(A_{1},A_{2})&\frac{1}{2-\alpha^{2}}(A_{1}\beta_{1}+2\sqrt{1-\alpha^{2}}\beta_{2})\\\
0&0\ \end{pmatrix}.$

Using the triangle inequality and boundedness of $A_{1}$ we get

(4.3) $\displaystyle\|F_{\alpha}(T_{1},T_{2})\begin{pmatrix}x\\\ y\
\end{pmatrix}\|^{2}$ $\displaystyle\leq$
$\displaystyle\frac{1}{(2-\alpha^{2})^{2}}\left[\|A_{1}x+\beta_{1}y\|+2\sqrt{1-\alpha^{2}}\|A_{2}x+\beta_{2}y\|\right]^{2}.$

We can simplify the estimation of (4.3) by making the substitution
$u=\|A_{1}x+\beta_{1}y\|$ and $v=\|A_{1}x+\beta_{1}y\|$. Then, the estimation
reduces to finding the supremum
$\sup\\{\frac{1}{2-\alpha^{2}}(u+2\sqrt{1-\alpha^{2}}v):\ 0\geq u,v,\
u^{2}+v^{2}\leq 1,\ 0\leq\alpha<1\\}.$ This supremum equals $2/\sqrt{3}$.

∎

Next, we prove

######  Lemma 4.2.

If $h\in\mathcal{F}_{ND},$ and $m\in\text{Aut}(\mathbb{D}),$ then $\|(m\circ
h)(T)\|\leq 3.14626.$

###### Proof.

Let $m_{t}\in\text{Aut}(\mathbb{D}),\ F_{\alpha}\in\mathcal{F}_{ND},$ where
$t\in\mathbb{D},\ \alpha\in[0,1).$ We compute

$(m_{t}\circ
F_{\alpha})(T)=\begin{pmatrix}m_{t}(F_{\alpha}(A))&\frac{1}{2-\alpha^{2}}(\overline{t}m_{t}(F_{\alpha}(A))+I)(A_{1}\beta_{1}+2\sqrt{1-\alpha^{2}}\beta_{2})\\\
0&-t\ \end{pmatrix}.$

Express $z\in\partial\mathbb{B}^{3}$ in the block form $z=\begin{pmatrix}x\\\
y\ \end{pmatrix},$ where $x\in\mathbb{B}^{2},\ y\in\mathbb{D}.$ Then, by the
boundedness property of linear operators

(4.4) $\displaystyle\|m_{t}(F_{\alpha}(T_{1},T_{2}))\begin{pmatrix}x\\\ y\
\end{pmatrix}\|^{2}$
$\displaystyle\leq(\|m_{t}(F_{\alpha}(A_{1},A_{2}))\|\|x\|$
$\displaystyle+\frac{|y|}{2-\alpha^{2}}\|\overline{t}m_{t}(F_{\alpha}(A_{1},A_{2}))+I\|\|A_{1}\beta_{1}$
$\displaystyle+2\sqrt{1-\alpha^{2}}\beta_{2}\|)^{2}+|ty|^{2}.$

According to Proposition 2.1, we have that

$\|m_{t}(F_{\alpha}(A_{1},A_{2}))\|\leq 1.$

By applying the triangle inequality, we can estimate the right-hand side of
inequality (4.4) by

(4.5)
$\displaystyle(\|x\|+\frac{2|y|}{2-\alpha^{2}}\|A_{1}\beta_{1}+2\sqrt{1-\alpha^{2}}\beta_{2}\|)^{2}+|y|^{2}.$

We may utilize numerical computations to obtain an upper bound for the
expression (4.5). Because of the maximum modulus principle, we may also assume
that $\|x\|^{2}+|y|^{2}=1$. Hence, the right-hand side of inequality (4.4) may
be estimated from above by a function that is dependent on the coefficients of
matrices $T_{1}$ and $T_{2}$, as well as the parameters $y$ and $\alpha$. All
variables and parameters satisfy the constraints given by the equations
$|a_{1}|^{2}+|b_{1}|^{2}+|c_{1}|^{2}+|c_{2}|^{2}\leq 1,\
|e_{1}|^{2}+|e_{2}|^{2}\leq 1,\ |y|\leq 1,\ 0\leq\alpha<1.$ By imposing these
constraints on our function, we obtain

$\|m_{t}(F_{\alpha}(T_{1},T_{2}))\|\leq 3.14626.$

∎

Finally, we consider the composition of a unit disc automorphism with a
function from the class $\mathcal{F}_{D}$. We prove

######  Lemma 4.3.

If $h\in\mathcal{F}_{D}$ and $m\in\text{Aut}(\mathbb{D}),$ then $\|(m\circ
h)(T)\|\leq 3.14626.$

###### Proof.

Consider a function $F_{\tau,\omega}\in\mathcal{F}_{D}$. Note that it can be
expressed as

$F_{\tau,\omega}(z,w)=\overline{\tau}F_{1,1}(\tau z,\omega w),$

where $|\tau|\leq 1,\ |\omega|=1.$ The function $F_{1,1}$ can be obtained as a
limit of functions from the class $\mathcal{F}_{ND}$, composed with
appropriate automorphisms of the unit disc $\mathbb{D}$ and the unit ball
$\mathbb{B}^{2}$. In particular, we have

$F_{1,1}(z,w)=\lim_{\mathbb{R}\cap\mathbb{D}\ni t\rightarrow 1^{-}}(m_{t}\circ
F_{0}\circ A_{t}\circ u)(z,w),$

where

  * •

$m_{t}(z)=-\frac{z-t}{1-tz}\in\text{Aut}(\mathbb{D}),\ t\in(0,1),$

  * •

$A_{t}(z,w)=\large(\sqrt{1-t^{2}}\frac{z}{1+tw},\frac{w+t}{1+tw}\large)\in\text{Aut}(\mathbb{B}^{2}),\
t\in(0,1),$

  * •

$u(z,w)=(z,-w)$ is an unitary map,

  * •

$F_{0}\in\mathcal{F}_{ND}.$

Consider a function $F_{\tau,\omega}\in\mathcal{F}_{D}$, where
$\tau,\omega\in\partial\mathbb{D}$. Let
$m_{s}(z)=\frac{z-s}{1-\overline{s}z}\in\text{Aut}(\mathbb{D}),$ for some
$s\in\mathbb{D},$ and $A\in\text{Aut}(\mathbb{B}^{2}).$ Then, the mapping
$U_{\tau,\omega}:\mathbb{B}^{2}\ni(z,w)\mapsto(\tau z,\omega
w)\in\mathbb{B}^{2}$ is unitary.

Recall that the composition of a commutative, row contraction with a unitary
map or a unit Euclidean ball automorphism is itself a commutative, row
contraction. Therefore, if $T$ is a commutative, row contraction, then so is
$U_{\tau,\omega}(T)$. In estimating $\|(m_{s}\circ F_{\tau,\omega}\circ
A)(T)\|$ we are considering all pairs $T$ of commuting, row contractions.
Hence, we can replace any composition of a unitary map or a unit ball
automorphism with $T$ by $T$ itself. Thus, we have

$\|(m_{s}\circ F_{\tau,\omega}\circ
A)(T)\|=\|(m_{s}\circ\overline{\tau}F_{1,1}\circ U_{\tau,\omega}\circ
A)(T)\|=\|(m_{s}\circ\overline{\tau}F_{1,1})(T)\|,$

where $s\in\mathbb{D},$ and $\omega,\tau\in\partial\mathbb{D}.$ Hence, using
the continuity of a norm, we can estimate the norm of
$(m_{s}\circ\overline{\tau}F_{1,1})(T)$ as follows.

(4.6) $\displaystyle\|(m_{s}\circ\overline{\tau}F_{1,1})(T)\|$
$\displaystyle=\lim_{\mathbb{R}\cap\mathbb{D}\ni t\rightarrow
1^{-}}\|(m_{s}\circ\overline{\tau}m_{t}\circ F_{0}\circ A_{t}\circ u)(T)\|$
$\displaystyle=\lim_{\mathbb{R}\cap\mathbb{D}\ni t\rightarrow
1^{-}}\|(m_{\tilde{s}}(F_{0}(T)))\|\leq 3.14626,$

where $\tilde{s}\in\mathbb{D}$ is such that
$m_{\tilde{s}}=m_{s}\circ\overline{\tau}m_{t}.$ Thus we have the upper bound
on the constant $c_{2,3}$ for $\tau,\omega\in\partial\mathbb{D}$.

Consider the mapping
$\mathbb{D}\ni\tau\mapsto(m_{s}\circ\overline{\tau}F_{1,1})(T)$. Using the
estimation (4.6) and the maximum modulus principle, we obtain

$\|(m_{s}\circ\overline{\tau}F_{1,1})(T)\|\leq 3.14626,$

for $\tau\in\overline{\mathbb{D}}$ and $\omega\in\partial\mathbb{D}.$ The
proof of the Lemma 4.3 is completed. ∎

Moreover, by computational simulations, one can obtain a lower bound estimate.
The sample considered for estimation of the lower bound had a size $\sim
5\cdot 10^{6}$ per each of classes $\mathcal{F}_{ND}$ and $\mathcal{F}_{D}.$

######  Example 4.4.

Let $\alpha=0.707107$ and

$D_{1}=\textup{diag}(0.289+0.31i,0.02-0.08i,-0.22-0.12i),$ $\
D_{2}=\textup{diag}(0.008+0.18i,-0.03-0.08i,0.08+0.13i),$

be diagonal matrices and let

$P=\begin{pmatrix}0.1-0.15i&0.15+0.58i&0.48+0.4i\\\
-0.01-0.67i&0.11+0.53i&0.17+0.64i\\\ 0.18+0.69i&-0.07-0.57i&-0.26-0.3i\
\end{pmatrix}.$

be a transition matrix. Consider the tuple $T=(PD_{1}P^{-1},PD_{2}P^{-1}).$
Then, $T$ consists of commutative, row contractions and
$\|F_{\alpha}(T)\|=1.11767,$ where $F_{\alpha}\in\mathcal{F}_{ND}.$

It is worth to notice that the best numerical hits were found for functions
from the class $\mathcal{F}_{ND}.$ For the other class the highest norm we
found was $\sim 1.06.$ This let us believe that the optimal estimate of
$c_{2,3}$ is the estimate we found in the Lemma 4.1 i.e. $2/\sqrt{3}\approx
1.1547.$ It would be of interest to compare the estimations presented above
with those obtained by Hartz, Richter, and Shalit in their work [2].

## Acknowledgments

I would like to express my sincere gratitude to the anonymous reviewer for
their valuable comments and suggestions, which significantly contributed to
improving the quality of this article.

## References

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# AGN’s Deadness Over Cosmic Time: $UVJ$ Diagrams of X-Ray-Selected AGN

Rachel Cionitti  Department of Physics & Astronomy, University of Kansas
1450 Jayhawk Blvd
Lawrence KS 66045, USA  Brandon Coleman  Department of Physics & Astronomy,
University of Kansas
1450 Jayhawk Blvd
Lawrence KS 66045, USA  Allison Kirkpatrick  Department of Physics &
Astronomy, University of Kansas
1450 Jayhawk Blvd
Lawrence KS 66045, USA  Greg Troiani  Department of Physics & Astronomy,
University of Kansas
1450 Jayhawk Blvd
Lawrence KS 66045, USA

###### Abstract

Active Galactic Nuclei (AGN) are intensely accreting supermassive black holes
at the centers of massive galaxies. Though these objects occupy little spatial
extent of the galaxy itself, they are thought to have far reaching affects,
impacting the galaxy’s star formation, and possibly it’s lifespan until it
becomes ’red and dead’. Typical galaxies demonstrate that, over cosmic time,
they tend to separate into a bimodal distribution of ’red and dead’ or blue
and star forming. We examine whether active galaxies evolve over cosmic time
in a similar way, and whether this can reveal anything about the complexities
of the relationship between an AGN and the host galaxy. We use the Stripe82X
survey to identify 3940 X-ray AGN spanning $z=0-2.5$, and we measure the rest-
frame $UVJ$ colors of each galaxy. We classify AGN as star-forming or
quiescent based on their location in a $UVJ$ color diagram. We find that there
is not a clear bimodal distribution between AGN in star forming and quiescent
galaxies. Furthermore, the most luminous X-ray sources tend to lie in the star
forming region, which may indicate a correlation between central engine
activity and increased rates of star formation.

##  1 Introduction

It is generally thought that active supermassive black holes at the center of
galaxies impact the rest of the galaxy’s evolution and star formation rates
through feedback, which is the mechanism by which the active center heats,
cools or pressurizes the galaxy’s gas content. However, the exact details of
feedback are the subject of much debate (for a review, see Kormendy & Ho,
2013). Quasars are the brightest subclass of active galactic nuclei (AGN), so
the effect of the feedback may be more pronounced, making them ideal objects
for studying the relationship between nuclear activity and star formation
rate. This AGN feedback on star formation can be positive or negative,
depending on the ways in which the AGN is affecting the surrounding gas.
Positive feedback occurs when activity around the central black hole causes
outward shocks of radiation pressure, which compress gas and trigger star
formation (e.g. Silk, 2013). Negative feedback is the opposite–activity in the
nucleus causes winds that blow out  the excess gas and dust needed to generate
new stars (Shin et al., 2019); AGN radiation can also heat up gas in the
interstellar medium, potentially suppressing the process of star formation
(McKinney et al., 2021). Past theoretical work has predicted that the star
forming phase occurs early on in the quasar’s life (Hopkins & Hernquist,
2006). However, observational studies have shown that there is no clear
correlation with star formation rate (SFR) and declining X-ray activity, where
X-ray activity indicates the galactic nucleus is active (Coleman et al., 2022;
Stanley et al., 2015).

Galaxy SFR can be traced through the use of the $UVJ$ diagram (Williams et
al., 2009). The ’colors’ here are the $V-J$, or the difference of brightness
between the $V$ and $J$ bands, and the $U-V$, which is identical but for the
$U$ and $J$ bands. This color-color diagram combines rest-frame $V-J$ with
$U-V$ to separate star-forming galaxies from quiescent galaxies, which have
quenched their star formation. These diagrams have been shown to be effective
to $z\sim 2$ (Williams et al., 2009). As the stellar populations initially age
in a galaxy, the $U$-band emission will weaken first. This causes a galaxy to
move upward in $UVJ$ colorspace. Increasing the age of the population will
then begin to affect the $V-J$ color, causing galaxies to move to the right in
colorspace. Dust can also affect the $V-J$ color, causing galaxies to appear
redder and further to the right.

In this work, we explore how AGN strength correlates with star formation
through the use of the $UVJ$ diagram and X-ray luminosities of our sample. In
Section 2, we describe our survey data from Stripe82X, and discuss how the
rest-frame $UVJ$ photometry is calculated. In Section 3, we show how X-ray
luminosity correlates with location in $UVJ$ space from $z=0-2.5$. We present
our Conclusions in Section 4. In this analysis, we assume a standard cosmology
of H0 = 70 Mpc/km/s, $\Lambda$ = 0.7, and $\Omega_{m}$ = 0.3.

##  2 Data & Analysis

###  2.1 Sample Selection

The data in this work comes from the Accretion History of AGN
survey111https://project.ifa.hawaii.edu/aha/. This survey contains X-ray
observations of $>$6000 galaxies from the Stripe 82X survey (LaMassa et al.,
2016). The X-ray data covers 31.3 deg2 of the Sloan Digital Sky Survey (SDSS)
Stripe 82 Legacy Field, with observations coming from  the X-ray Multi-Mirror
Mission (XMM-Newton) and the Chandra X-ray Observatory. The X-ray survey data
has been crossmatched with many other multi-wavelength observations for a
final sample of 6181 sources (Ananna et al., 2017). Of the 6181 sources
detected at X-ray wavelengths, 6044 have corresponding multi-wavelength
associations. For this work, we make use of $ugriz$ photometry from SDSS and
$JHK$ photometry from the United Kingdom Infrared Telescope (UKIRT) Infrared
Deep Sky Survey (UKIDSS).

For Stripe82X, spectroscopic redshifts are available from SDSS for 53% of our
sources. For the remainder of the galaxies, photometric redshifts were
measured in Ananna et al. (2017) by fitting a suite of templates to the
optical photometry. These templates are empirical and include starburst
galaxies, Type 1 and Type 2 AGN, quasars, Seyferts, spirals and ellipticals.
There are 206 galaxies which are not well-fit with any of the templates and
which do not have a measured spectroscopic redshift. We remove these from the
sample.

We further select all galaxies with observations in at least two bands from
the $ugriz$ and $JHK$ photometry for a sample of 5514. We limit our analysis
to $z<2.5$, as this is the redshift range for which the $UVJ$ diagram has been
calibrated (Williams et al., 2009). There are 5180 of these sources with
$0<z<2.5$, where 60% are spectroscopic redshifts. These galaxies span a hard
X-ray luminosity range of $L_{2-10\,{\rm keV}}=2.7*10^{34}-7.7*10^{45}$erg/s.
3029 have $L_{2-10\,{\rm keV}}\geq 10^{42}$erg/s, which is the typical
threshold for declaring a source an AGN. For the remainder of this work,
$L_{X}$ refers to the hard X-ray luminosity, $L_{2-10\,{\rm keV}}$.

###  2.2 Measuring Rest Frame UVJ Photometry

We fit the visible and infrared (ugriz, JHK) photometry for each source with
the suite of empirical templates from (Ananna et al., 2017) after shifting all
photometry to the rest-frame and converting magnitudes to fluxes.

We use the following equation to fit the templates:

$f=N*L_{\nu,template}*\lambda^{\alpha}$ (1)

$f$ represents the SED template after it has been fitted to best match the
observed fluxes. $N$ and $\alpha$ are free parameters. $N$ must be positive,
but $\alpha$ is allowed to be positive, zero, or negative. This form can allow
for a more precise fitting than simply a scaling factor alone. We are
exclusively concerned with achieving the best fit possible to the data, to
allow for the most accurate calculation of the rest-frame $UVJ$ photometry. As
such, we are not concerned with a physical interpretation of $\alpha$, nor do
we draw any inference from the best-fit template (which can be star-forming,
elliptical, Seyfert, etc.). We select the best-fitting model by calculating an
$R^{2}$ parameter for each fit. The $R^{2}$ statistic measures the ratio of
the variance in the residuals (in this case, measured photometry vs. model-
predicted photometry) to the variance in the data (photometry). The model with
$R^{2}$ closest to 1 is then used to calculate the $UVJ$ photometry.

We convolve $UVJ$ transmission curves with each best fit model to calculate
synthetic, rest-frame photometry for each galaxy. We use the Bessel
transmission curves which have central wavelengths of 3543 Å, 5510 Å, and
12000 Å for the $U,V$, and $J$ filters, respectively. We visually inspected
the best fit template, galaxy photometry, and new synthetic photometry to
ensure the results were consistent with the real galaxy observations.

##  3 Results & Discussion

Figure 1: The $UVJ$ diagrams for all objects in the sample at each redshift
bin. The quiescent region falls in the upper left and is indicated by the
solid black line. The black and white is a 2D histogram representing the
density of sources at each region. The colored points represent the exact
locations of the most luminous X-ray sources ($\log L_{X}>44.5$). The color
bar demonstrates intensity in log(erg/s) in the hard X-ray band. In each
panel, we list the number of sources and the percentage of quiescent galaxies.
We also list the percentage of galaxies in the star forming region that are
X-ray luminous and the percentage in the quiescent region that are X-ray
luminous.

Figure 1 shows the $UVJ$ plots of our sources using the synthetic rest-frame
photometry. The sources are split by redshift intervals of $\Delta z=0.5$. The
quiescent region is set by Williams et al. (2009). The black and white
portions on the plot represent a 2D histogram of the density of sources in
each position in color space, where the darker the bin is, the more populated
that region is. The lower redshift bins contain $>$1000 sources each, while
the highest redshift bin has only $\sim$300 galaxies. Each panel in Figure 1
indicates the quiescent fraction of X-ray sources at each epoch. In general,
the quiescent fraction increases with decreasing redshift, similar to typical
galaxies (Williams et al. (2009), Madau & Dickinson (2014)). At $z<0.5$, 35%
of X-ray sources are quiescent, compared with 6% at $z=1.5-2.0$. This number
increases to 12% at $z>2$, but the location of the quiescent region grows more
uncertain at higher redshift. This is due to two reasons. First, star-forming
galaxies at $z>2$ are dustier as gas fractions increase (e.g., Whitaker et
al., 2017), which could contaminate the right edge of the $UVJ$ quiescent
region. Second, the calculation of rest-frame photometry becomes more
uncertain as it relies heavily on extrapolation (the $K$-band corresponds to
$\lambda_{\rm rest}=7000$ Å at $z=2.0$, which is a lower wavelength than the
rest-frame $J$ photometry that we are estimating.

Our sources span a range of X-ray luminosities and therefore could include
non-AGN, which could bias the results. To test this, we consider the location
of only the brightest sources with $L_{X}>10^{44.5}$ erg/s. The highly
luminous X-ray sources are indicated by the shaded dots. There is no clear
trend of X-ray luminous sources preferentially residing in the quiescent
region, as might be expected if quasar activity comes after star formation. In
general, the highly X-ray luminous sources tend to be slightly biased toward
residing in the star forming region, indicative of positive feedback.

Notably, neither the full X-ray population nor the X-ray luminous sources
demonstrate as clear of a bimodal distribution as seen in non-AGN. This could
be due to a significant portion of AGN residing in the “green valley”, the
transition region in colorspace between star forming and quiescent galaxies
(e.g., Mahoro et al., 2017). The position of AGN in $UVJ$ space may indicate
that AGN do not preferentially reside in any type of galaxy; AGN can be found
in star forming, quiescent, or transitioning galaxies. This non-preference
could be due to the timescales of AGN variability making a correlation between
AGN luminosity and host galaxy properties difficult to idenfity. It could also
be due to the complicated relationship between AGN and host, with feedback
both quenching and triggering star formation.

This apparent star forming bias for the highly X-ray luminous sources (and
also for the full population) indicates that there may be a correlation
between central engine activity, which is signalled by extreme X-ray
luminosity, and star formation in the host galaxy. This is interesting because
it is suggesting a sort of positive feedback, where central engine activity
may cause phenomena such as shocks that can compress the surrounding gas and
increase star formation. This result is in line with what is seen in Kocevski
et al. (2017) where the bulk of X-ray-selected AGN at $z\sim 2$ in the EGS and
UDS fields lie in galaxies that are still star forming, but are more compact
than normal star forming galaxies. However, this skew of the highly luminous
sources toward the star forming region of the diagram is not obvious or
especially strong.

Finally, we note that it is possible the AGN can contaminate the optical
portion of a galaxy’s spectral energy distribution, so the $UVJ$ is tracing
the accretion disk, not the galaxy. However, X-ray emission will mimic
elliptical galaxies most strongly, so we anticipate that the quiescent
fractions will be even lower if AGN emission is a major source of
contamination.

##  4 Conclusion

This work set out to assess whether X-ray selected AGN were preferentially
star forming using an empirical $UVJ$ color-color diagram, which indicates
whether a source is star forming or quiescent. We calculated rest-frame $UVJ$
colors for 3940 X-ray AGN from $z=0-2.5$ and classified galaxies as quiescent
based on their location in $UVJ$ colorspace. The results indicate that active
galaxies trend toward quiescence with decreasing redshift, similar to normal
star forming galaxies. However, we also find that there does not seem to be a
clear correlation between central engine activity as probed by X-ray
luminosity and the location of the sources on the $UVJ$ diagram. There is a
near ubiquitous trend where the highly X-ray luminous sources tend to be more
star forming than quiescent, but this is not by a significant amount. However,
the possibility of a correlation can not be ruled out either, and can be honed
with further studies. This work shows that X-ray AGN seem to reside in star
forming galaxies, quiescent galaxies, and those transitioning to a quiescent
phase.

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