# Alternative approach to the critical behavior and microscopic structure of
the power-Maxwell black holes

Ahmad Sheykhi<EMAIL_ADDRESS>Physics Department and Biruni
Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box
55134-441, Maragha, Iran  Mohammad Arab<EMAIL_ADDRESS>Physics
Department and Biruni Observatory, College of Sciences, Shiraz University,
Shiraz 71454, Iran  Department of Physics, Faculty of Science, Bu-Ali Sina
University, Hamedan 65178, Iran  Zeinab Dayyani<EMAIL_ADDRESS>Physics
Department and Biruni Observatory, College of Sciences, Shiraz University,
Shiraz 71454, Iran  Amin Dehyadegari<EMAIL_ADDRESS>Physics Department
and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454,
Iran

###### Abstract

Employing a new approach toward thermodynamic phase space, we investigate the
phase transition, critical behavior and microscopic structure of higher
dimensional black holes in an Anti-de Sitter (AdS) background and in the
presence of Power-Maxwell field. In contrast to the usual extended $P-V$ phase
space where the cosmological constant (pressure) is treated as a thermodynamic
variable, we fix the cosmological constant and treat the charge of the black
hole (or more precisely $Q^{s}$) as a thermodynamic variable. Based on this
new standpoint, we develop the resemblance between higher dimensional
nonlinear black hole and Van der Waals liquid-gas system. We write down the
equation of state as $Q^{s}=Q^{s}(T,\psi)$, where $\psi$ is the conjugate of
$Q^{s}$, and construct a Smarr relation based on this new phase space as
$M=M(S,P,Q^{s})$, while $s=2p/(2p-1)$ and $p$ is the power parameter of the
Power-Maxwell Lagrangian. We obtain the Gibbs free energy of the system and
find a swallowtail behaviour in Gibbs diagrams, which is a characteristic of
first-order phase transition and express the analogy between our system and
van der Waals fluid-gas system. Moreover, we calculate the critical exponents
and show that they are independent of the model parameters and are the same as
those of Van der Waals system which is predicted by the mean field theory.
Finally , we successfully explain the microscopic behavior of the black hole
by using thermodynamic geometry. We observe a gap in the scalar curvature $R$
occurs between small and large black hole. The maximum amount of the gap
increases as the number of dimensions increases. We finally find that
character of the interaction among the internal constituents of the black hole
thermodynamic system is intrinsically a strong repulsive interaction.

##  I introduction

The study of black holes thermodynamics is one of the most important subject
in gravitational physics, which was anticipated by Bekenstein in 1973
Bekenstein (1973). In complete analogy with known non-gravitational
thermodynamic systems, black hole spacetime obeys a version of first law of
thermodynamics.Bekenstein (1973); Hawking (1975). It can be specified by an
entropy $S$ proportional to the horizon area and temperature $T$ proportional
to the surface gravity.

Furthermore, after the advent of AdS/CFT correspondence, phase transition has
gained more attention as a thermodynamical property of black holes in
asymptotically AdS spaces.

A seminal investigation in this relevance was reported in Hawking and Page’s
paper Hawking and Page (1983), where it was demonstrated that in the phase
space of AdS-Schwarzchild black hole, phase transition certainly exists.
Through the AdS/CFT (gage/gravity) duality the Hawking-Page phase transition
can correspond to the confinement/deconfinement phase transition in the dual
quark gluon plasma Witten (1998). In view of this duality, the thermodynamic
phase space of charged AdS black holes exhibits first order SBH/LBH (small
black hole/large black hole) phase transition suggestive of a Van der Waals
liquid/gas phase transition (see e.g Chamblin et al. (1999a, b); Kubiznak and
Mann (2012); Dehghani et al. (2014)).

In most treatments of phase transition in black hole thermodynamics, in an
extended phase space, negative cosmological constant $\Lambda$ is treated as a
thermodynamic variable. In fact, $\Lambda$ is physically thought of as a
pressure and its conjugate variable is considered as a thermodynamic variable
proportional to a volume $V$ Dolan (2011a, b, 2012). In the past few years the
various class of black hole phase transition such as Multiple reentrant phase
transition Frassino et al. (2014), superfluid-like phase transition Hennigar
et al. (2017), zero order phase transition Dehyadegari et al. (2017a) and so
on have been studied in an extended phase space (see e.g.Cvetic et al. (2011);
Urano et al. (2009); Zou et al. (2014); Poshteh et al. (2013); Dehghani et al.
(2016); Dayyani et al. (2017); Majhi and Samanta (2017); Dehyadegari and
Sheykhi (2018)).

The authors of Kastor et al. (2009), by using of the Smarr formula, concluded
that the mass $M$ of AdS black hole should be identified as enthalpy $H$
rather than internal energy of the spacetime . Therefor, regarding the
cosmological constant as a pressure $P$, for a non rotating charged black
hole, the first law of thermodynamics should read,

$\,\mathrm{d}M\equiv\,\mathrm{d}H=T\,\mathrm{d}S+\Phi\,\mathrm{d}Q+V\,\mathrm{d}P,$

where $Q$ and $\Phi$ are charge and electrical potential respectively.

Although, there are some reasons to suppose the cosmological constant as a
variable, but it is more reasonable to hold it as a constant parameter. In
fact, from the physical standpoint it is difficult to consider the
cosmological constant as a thermodynamic variable which can take an arbitrary
value. Also, in general relativity the cosmological constant is understood as
a constant related to the zero point energy of the background spacetime. By
these motivations, the authors of Dehyadegari et al. (2017b) proposed the
cosmological constant as a fixed parameter and consider the charge of the
black hole as a thermodynamic variable. They indicated a phase transition
similar to Van der Waals liquid-gas in the black hole system in this manner.
In this alternative view, the SBH/LBH phase transition of black hole in
$Q^{2}-\psi$ plane exactly correspond to the Van der Walls fluid phase
transition, whereas $\psi$ is thermodynamic variable conjugate to the $Q^{2}$.
Recently, the universality class and critical properties of any AdS black hole
in this alternative approach toward the phase space have been addressed in
Dehyadegari et al. (2019).

Besides, the study of the nonlinear electrodynamic field on the thermodynamic
and phase transition of the black holes has been attracted a lot of attentions
in the literature in the past decade Fernando (2006); Myung et al. (2008a);
Miskovic and Olea (2008); Myung et al. (2008b); Banerjee and Roychowdhury
(2012); Gunasekaran et al. (2012); Hendi et al. (2015a); Sheykhi and Kazemi
(2014); Hendi et al. (2015b); Dayyani et al. (2018). The first one of the
nonlinear electromagnetic field models is Born-Infeld electrodynamics
introduced in 1930’s. Nonlinear behavior in strong electromagnetic field such
as the field in region near a point-like charge, is suggested by Dirac in
$1964$ Dirac (1964). Moreover, nonlinear electromagnetics can be due to the
nonlinear effects of quantum electrodynamics Heisenberg and Euler (1936);
Yajima and Tamaki (2001); Delphenich (2003).

Power-Maxwell invariant field as an important class of the nonlinear
electrodynamics, was introduced in Hassaine and Martinez (2007, 2008); Hendi
and Vahidinia (2013); Hendi et al. (2017); Kord Zangeneh et al. (2015) . It is
worth mentioning that the Lagrangian of the power Maxwell invariant fields are
invariant under the conformal transformation
$g_{\mu\nu}\longrightarrow\Omega^{2}g_{\mu\nu}$, where $g_{\mu\nu}$ is the
metric tensor. In the special case linear electromagnetic can be generated by
reducing of the power Maxwell invariant field. the authors of Hendi and
Vahidinia (2013) have investigated the effect of power Maxwell field on the
$P-V$ criticality of black holes and phase transitions in the extended phase
space.

In the present work, we would like to study the critical behavior of the AdS
black hole in the presence of power-Maxwell field via an alternative
viewpoint. It means we keep the cosmological constant as a fixed parameter and
instead treat the charge of the black hole (or more precisely
$\boldsymbol{Q}_{p}$) as a thermodynamic variable which can vary. The
advantages of this approach is that we do not need to extend the
thermodynamical phase space to see the critical behavior of the system. On the
other side, absorbing or emitting charged particles may cause the change in
the charge of the black hole in reasonable manner which is more logical than
varying the cosmological constant. Phase structure and critical behavior of
AdS balck holes with linear Dehyadegari et al. (2017a), and nonlinear
Dehyadegari and Sheykhi (2018) electrodynamics, Lifshitz dilaton black holes
Dayyani and Sheykhi (2018), where the charge of the system can vary and the
cosmological constant (pressure) is fixed have been investigated. Recently, it
was argued that this method also indeed works for investigating the phase
transition of Gauss-Bonnet black holes GB , which further supports the
viability of this alternative approach.

The outline of this paper is as follows. In the next section we investigate
thermodynamics of the higher dimensional AdS black hole with considering of
the effects of the nonlinear power- Maxwell field and obtain Smarr relation by
the well-known usual thermodynamic quantities. Moreover we introduce a new
phase space and thermodynamic variables $\psi,\boldsymbol{Q}_{p}$ and find the
appropriate Smarr relation in terms of this new variables. In Sec.III, we
write down the equation of state according to the offered variables and
calculate the critical points and critical exponent. Also, the analogy of the
system with Van der Waals liquid gas system is showed in this section. The
Gibbs free energy diagrams are investigated in section IV. The swallowtail
behavior of the Gibbs free energy represents a first-order phase transition in
the system. Finally, in section V, we explore microscopic structure of
thermodynamic system and calculate the Ruppeiner scalar curvature in $4,5$ and
$6$ dimensional spacetime.

##  II Thermodynamic of higher dimensional AdS black hole with a power Maxwell
field

The action of Einstein gravity in $(n+1)$-dimensional spacetime coupled to a
power Maxwell field can be written as Hassaine and Martinez (2008)

$I=-\frac{1}{16\pi}\int{\,\mathrm{d}^{n+1}x\sqrt{-g}}\left[{R}-2\Lambda+(-F_{\mu\nu}F^{\mu\nu})^{p}\right],$
(1)

where $R$ is the Ricci scalar, $p$ is a constant determining the nonlinearity
of the electromagnetic field, $F_{\mu\nu}=\partial{{}_{[\mu}}A_{\nu]}$ is the
electromagnetic field tensor and $A_{\mu}$ is the electromagnetic potential.
Here, $\Lambda$ is the cosmological constant related to the AdS radius as
$l^{2}=-n(n-1)/\left(2\Lambda\right)$.

We apply a spherically symmetric and static metric of $(n+1)$-dimensional
spacetime

$\,\mathrm{d}s^{2}=-f(r)\,\mathrm{d}t^{2}+\frac{\,\mathrm{d}r^{2}}{f(r)}+r^{2}\,\mathrm{d}\Omega_{n-1}^{2},$
(2)

where $\,\mathrm{d}\Omega_{n-1}^{2}$ denotes the metric of spherical
hypersurface with volume $\omega_{n-1}$ and $f(r)$ is given by Hassaine and
Martinez (2008)

$f(r)=1-{\frac{m}{{r}^{n-2}}}+\frac{r^{2}}{l^{2}}+{\frac{{2}^{p}\left(2\,p-1\right)^{2}{q}^{2\,p}}{\left(n-1\right)\left(n-2\,p\right)}{r}^{\left[{(6-2n)p-2}\right]{/}\left[2p-1\right]}}.$
(3)

The quantity $q$ is an integration constant related to the charge $Q$ of the
black hole per unite volume $\omega_{n-1}$ and one can find it by applying the
Gauss law

$Q=\frac{1}{4\pi}\int{r^{n-1}(-F_{\mu\nu}F^{\mu\nu})^{p-1}F_{\mu\nu}n^{\mu}u^{\nu}\,\mathrm{d}r},$
(4)

where $n^{\mu}=f(r)^{-1/2}dt$ and $u^{\nu}=f(r)^{1/2}dr$ are the unit space
like and time like normal to the surface, respectively, and Maxwell invariant
is $F_{\mu\nu}F^{\mu\nu}=-2F_{tr}{}^{2}$. Directly from the generalized
Maxwell equation, nonzero electromagnetic field is
$F_{tr}=q\,\,r^{{(1-n)}/{(2p-1)}}$ Hassaine and Martinez (2008). Thus, Gauss
law relation Eq. (4) reads

$Q=\frac{2^{p-1}q^{2p-1}}{4\pi}.$ (5)

The parameter $m$ is known as the geometrical mass parameter which it can be
expressed in term of the largest horizon radius $r_{+}$ (the largest root of
$f(r_{+})=0$)

$m={r}_{+}^{n-2}+\frac{{r}_{+}^{n}}{l^{2}}+{\frac{{2}^{p}\left(2\,p-1\right)^{2}{q}^{2\,p}}{\left(n-1\right)\left(n-2\,p\right)}{r}_{+}^{\left(n-2p\right)/\left(1-2p\right)}}.$
(6)

Using Brown-York method Brown and York (1993), the total mass of the black
hole per unit volume $\omega_{n-1}$ can be read as follows

$M={\frac{\left(n-1\right)m}{16\pi}}.$ (7)

The electric potential $\Phi$, measured at infinity with respect to the
horizon radius $r_{+}$, is obtained as (details is referred to Gonzalez et al.
(2009))

$\Phi={\frac{p\left(2\,p-1\right){\left(\pi\,Q\right)^{1/\left(2p-1\right)}}}{\left(n-2\,p\right){2}^{\left(p-3\right)/\left(2p-1\right)}}{r}_{+}^{{\left(2p-n\right)/}\left(2p-1\right)}}.$
(8)

In the case $p=1$, the metric function Eq. (3) and the electric potential Eq.
(8) reduce to $n+1$-dimensional Reissner-Nordstrom (RN)-AdS black holes
Kubiznak and Mann (2012).

According to the so-called area law of the entropy, the entropy of the black
hole is a quarter of the event horizon erea. Using this, one can obtain the
entropy of the black hole per unit volume $\omega_{n-1}$ as

$S=\frac{r_{+}^{n-1}}{4}.$ (9)

The Hawking temperature can be calculated as

$T=\frac{f^{\prime}(r_{+})}{4\pi}=\frac{\left(n-2\right)}{4\pi
r_{+}}+\frac{nr_{+}}{4\pi{l}^{2}}-{\frac{{2}^{p-2}\left(2\,p-1\right)q^{2\,p}}{\left(n-1\right)\pi}{r}_{+}^{-\left[2\left(n-2\right)p+1\right]/\left[2p-1\right]}}.$
(10)

One may obtain the generalized Smarr relation for the black hole in the
extended phase space by using the definition of total mass $M$ (7), chrge of
the black hole $Q$ (5) and the entropy $S$ (9). It is a matter of calculation
to show the Smarr formula is

$M\left(S,Q,P\right)=\frac{\left(n-1\right)S^{\left(n-2\right)/\left(n-1\right)}}{2^{2n/\left(n-1\right)}\pi}+\frac{PS^{n/\left(n-1\right)}}{n2^{2n/\left(1-n\right)}}+\frac{\left(2p-1\right)^{2}Q^{2p/\left(2p-1\right)}S^{\left(2p-n\right)/\left[\left(n-1\right)\left(2p-1\right)\right]}}{\left(n-2p\right)\pi^{1/\left(1-2p\right)}2^{\left[n\left(3p-2\right)-7p+4\right]/\left[\left(n-1\right)\left(2p-1\right)\right]}},$
(11)

where $P=-\Lambda/\left(8\pi\right)$.

One can then define the variables conjugate to $Q$, $S$ and $P$. As mentioned
before, the cosmological constant parameter is treated as pressure $P$. So,
its conjugate variable from the thermodynamic viewpoint should be volume $V$

$V=\frac{\partial M}{\partial P}\biggl{|}_{S,Q}.$ (12)

Likewise, the corresponding conjugate quantity of $S$ and $Q$ are interpreted
as a temperature $T$ and electric potential $\Phi$ respectively

$T=\frac{\partial M}{\partial
S}\biggl{|}_{P,Q},\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\Phi=\frac{\partial
M}{\partial Q}\biggl{|}_{S,P}.$ (13)

It is easy to show that the usual Smarr mass formula can be written as

$M=\frac{n-1}{n-2}\,T\,S+{\frac{\left(n-3\right)p+1}{\left(n-2\right)p}}Q\,\Phi-{\frac{2}{n-2}\,V\,P}.$
(14)

Obviously if we set $n=3$ and $p=1$, Eq.(14) reduces to the well-known Smarr
relation for the $4$-dimensional Einstein-Maxwell black holes Kubiznak and
Mann (2012)

$M=2(T\,S-\,V\,P)+\Phi\ Q.$ (15)

As expected, the obtained thermodynamic quantities satisfy the usual first law
of thermodynamics

$\,\mathrm{d}M=T\,\,\mathrm{d}S+\Phi\,\,\mathrm{d}Q+V\,\,\mathrm{d}P.$ (16)

It is notable that the electric potential $\Phi$ must have a finite value at
infinity. This leads to the following restriction on the parameter $p$

$\frac{1}{2}<p<\frac{n}{2},$ (17)

which obtained it from $\left(2p-n\right)/\left(2p-1\right)<0$ Hassaine and
Martinez (2008).

###  II.1 Alternative phase space

The Van der Waals like critical behavior of various types of AdS black holes
has been studied by considering the cosmological constant as a thermodynamical
pressure in the extended phase spaceDolan (2011a); Kubiznak and Mann (2012).
Also, $Q$-$\Phi$ plane-phase transitions of charged AdS black hole are
investigated by Chamblin et al. (1999a, b). Although the authors claimed that
the phase transition is similar to the Van der Waals fluid system, but the
phase transition behavior exhibits unusual Van der Waals isotherms. In this
approach, a thermodynamic response function $\left(\partial
Q/\partial\Phi\right)_{T}$ does not lead to physically relevant quantity. For
more details, we refer to Ref. Dehyadegari et al. (2017b).

The most recent work that indicates a complete similarity between the charged
AdS black hole and Van der Waals fluid system is important to highlight
Dehyadegari et al. (2017b). In this work, the cosmological constant has been
thought as a fixed parameter and instead, the square of the charge of black
hole $Q^{2}$ has been considered as a thermodynamic independent variable,
where $\Psi=1/(2r_{+})$ is the conjugate of $Q^{2}$ Dehyadegari et al.
(2017b). It allows the definition of new response function $\left(\partial
Q^{2}/\partial\Psi\right)_{T}$ which is clearly characterized stable-unstable
region by its sign Dehyadegari et al. (2017b).

According to this viewpoint, we would like to offer the thermodynamic
variables allowing us to complete the analogy of higher dimensional power
Maxwell-AdS black hole with Van der Waals fluid system. Hence, we consider the
mass of the black hole as a function of $\boldsymbol{Q}_{p}$ where
$\boldsymbol{Q}_{p}\equiv Q^{2p/(2p-1)}$ instead of the standard $Q$.  Our
motivation is that the charge of the black hole appears as $Q_{p}$ in the
expressions of $M$ Eq.(11) and $T$ 10. Another advantage of this choice is to
achieve a physically meaningful response function
$\left(\partial\boldsymbol{Q}_{p}/\partial\Psi\right)_{T}$ . Thus, with the
use of (11) the conjugate of $\boldsymbol{Q}_{p}$ is directly written as

$\displaystyle\Psi$ $\displaystyle=$ $\displaystyle\frac{\partial
M}{\partial\boldsymbol{Q}_{p}}\Big{|}_{S,P}$ $\displaystyle=$
$\displaystyle\frac{(2p-1)^{2}2^{(4-3p)/(2p-1)}}{(n-2p)\pi^{1/(1-2p)}}r_{+}^{\left(n-2p\right)/\left(1-2p\right)}.$

Now we can rewrite the corresponding Smarr mass formula in this new phase
space

$M=\frac{n-1}{n-2}\,T\,S+\,{\frac{2\left(\left(n-3\right)p+1\right)}{\left(2\,p-1\right)\left(n-2\right)}}\boldsymbol{Q}_{p}\,\Psi-{\frac{2}{n-2}\,V\,P}.$
(19)

A simple calculation shows that for $p=1$ and $n=3$, relation (19) is reduced
to the Smarr formula obtained by the authors of Dehyadegari et al. (2017b)

$M=2\,(\,T\,S+\boldsymbol{Q}^{2}\,\Psi-V\,P).$ (20)

Finally, the first law of thermodynamics in this new picture becomes

$\,\mathrm{d}M=T\,\,\mathrm{d}S+\Psi\,\,\mathrm{d}\boldsymbol{Q}_{p}+V\,\,\mathrm{d}P.$
(21)

In what follows, we examine the critical behavior of power Maxwell black hole
in alternative phase space where the AdS radius (or $\Lambda$) is fixed and
the electric charge of black hole can vary.

##  III equation of stats and critical point

In order to determine the critical point, we need to have the equation of
state $\boldsymbol{Q}_{p}\left(T,r_{+}\right)$. Using (5) and (10) one may
write the equation of state as a function of the temperature and horizon
radius

$\boldsymbol{Q}_{p}=\,\frac{\left(n-1\right)2^{5p/\left(1-2p\right)}}{\left(2p-1\right)\pi^{2p/\left(2p-1\right)}}\left(n-2-4\pi
T+\frac{nr_{+}^{2}}{l^{2}}\right)r_{+}^{n-3+\left(n-1\right)/\left(2p-1\right)}.$
(22)

The behavior of the black hole electric charge $\boldsymbol{Q}_{p}$ versus
$\Psi$ are plotted for fixed $l=1$ and different sets of parameter values in
Fig. 1. In Fig. 1, isothermal diagrams show that, for $T=T_{c}$, a second-
order phase transition (critical point) occurs in the point with the following
conditions (inflection point):

$\frac{\partial\boldsymbol{Q}_{p}}{\partial\Psi}\Big{|}_{T_{c}}=0,\quad\frac{\partial^{2}\boldsymbol{Q}_{p}}{\partial\Psi^{2}}\Big{|}_{T_{c}}=0.$
(23)

(a) $n=4$, $p=1$

(b) $n=4$, $p=5/4$

(c) $n=5$, $p=3/2$

Figure 1: The behavior of isothermal $\boldsymbol{Q}_{p}-\Psi$ diagrams of
power-Maxwell black holes for the case $l=1$. The critical points are
indicated by block spot.

Solving the above equations yields the coordinates of the critical point as

$\displaystyle\boldsymbol{Q}_{p_{c}}$ $\displaystyle=$
$\displaystyle{\frac{n\left(2\,p-1\right)\left(n-1\right)2^{5p/\left(1-2p\right)}l^{n-3+\left(n-1\right)/\left(2p-1\right)}}{\left(2p\left(n-2\right)+1\right)\left(p\left(n-3\right)+1\right)\pi^{2p/\left(2p-1\right)}}\left[{\frac{\left(n-2\right)\left(p\left(n-3\right)+1\right)}{np(n-1)}}\right]}^{p(n-1)/(2p-1)},$
$\displaystyle\Psi_{c}$ $\displaystyle=$
$\displaystyle\frac{(2p-1)^{2}2^{\left(3p-4\right)/(2p-1)}}{\left(n-2p\right)\pi^{1/\left(2p-1\right)}}\left[{\frac{\left(n-2\right)\left(p\left(n-3\right)+1\right){l}^{2}}{\left(n-1\right)np}}\right]^{-\left[n-2p\right]/\left[2\left(p-1\right)\right]},$
$\displaystyle T_{c}$ $\displaystyle=$
$\displaystyle{\frac{\sqrt{np\left(n-1\right)\left(n-2\right)\left(p\left(n-3\right)+1\right)}}{\left(2p\left(n-2\right)+1\right)\pi\,l}}.$
(24)

Following the new definition
$\rho_{c}=\Psi_{c}\,\boldsymbol{Q}_{p_{c}}\,T_{c}$ in Dehyadegari et al.
(2017b); Dayyani and Sheykhi (2018), the universal number for black hole at
the critical point is

$\displaystyle\rho_{c}$ $\displaystyle=$
$\displaystyle\Psi_{c}\,\boldsymbol{Q}_{p_{c}}\,T_{c}$ (25) $\displaystyle=$
$\displaystyle{\frac{n(n-1)(n-2)(2p-1)^{3}{l}^{-3+n}}{16(n-2p)(2p(n-2)+1)^{2}\pi^{2}}}\left[\frac{\left(n-2\right)(p(n-3)+1)}{np(n-1)}\right]^{\left(n-1\right)/2},$

according to constrain condition Eq.(17), $\rho_{c}$ is a positive quantity.
Also, it is independent of the AdS radius $l$ only when $n=3$. In the
conformally invariant case $p=\left(n+1\right)/4$, the critical quantities of
the black hole are

$\displaystyle\boldsymbol{Q}_{c}^{c}$ $\displaystyle=$
$\displaystyle{\frac{(n-1)^{(n-1)/2}l^{n-1}}{2^{\left[n+9\right]/\left[2(n-1)\right]}\pi^{(n+1)/(n-1)}}\left[{\frac{\left(n-2\right)}{n(n+1)}}\right]}^{(n+1)/2},\quad\Psi_{c}^{c}=\frac{2^{5/(n-1)-5/2}}{l\pi^{2/(n-1)}}\sqrt{\frac{n(n^{2}+1)}{n-2}}.$
$\displaystyle T_{c}^{c}$ $\displaystyle=$
$\displaystyle{\frac{1}{2\pi\,l}}\sqrt{\frac{(n+1)(n-1)(n-2)}{n}},\quad\rho_{c}^{c}=\frac{\left(n+2/n-3\right)^{(n+1)/2}l^{n-3}}{16\pi^{2}\left(n+1\right)^{(n-1)/2}}.$
(26)

Figure 2: The relation $\boldsymbol{Q}_{p_{c}}$ and $T_{c}$ and $p$ for
different values of dimensions. In (b), as $p$ increases the black hole
follows direction of arrows.

It is remarkable to note that in $3$-dimensional space for a linear Maxwell
$(p=1)$, these critical quantities reduce to those of RN-AdS black holes
Dehyadegari et al. (2017b). To see the effect of $p$ in the range of
$1/2<p<n/2$ on the critical value of black hole, we show the behavior of
$\boldsymbol{Q}_{p_{c}}$ and $T_{c}$ for different dimension $n$ in Fig. 2.
Figure 2 shows, $\boldsymbol{Q}_{p_{c}}$ vanishes near $p=1/2$ in the
different dimension. Also, increasing $p$ makes
$\boldsymbol{Q}_{p_{c}}$higher. According to Fig. 2, the critical temperature
is not almost influenced by the change of $p$.

The existence of oscillating isotherms in Fig. 1 are a consequence of
physically unstable feature which are remedied by the Maxwell equal area
construction Dehyadegari et al. (2017b)

$\oint\Psi dQ^{2}=0.$ (27)

###  III.1 Critical exponent

In the investigation of phase transition phenomena, it is important to study
the scaling behavior of thermodynamic system near the critical point and find
the corresponding universality class. In particular, the behavior of physical
quantities in the vicinity of the critical point can be characterized by the
critical exponents. Hence, we would like to calculate these exponents for the
new approach in this subsection.

In order to calculate the critical exponents, it is convenient to define the
reduced thermodynamic variables

$T_{r}=\frac{T}{T_{c}},~{}~{}~{}~{}~{}~{}~{}~{}\Psi_{r}=\frac{\Psi}{\Psi_{c}},~{}~{}~{}~{}~{}~{}~{}\boldsymbol{Q}_{p_{r}}=\frac{\boldsymbol{Q}_{p}}{\boldsymbol{Q}_{p_{c}}}.$

Also, since the critical exponents should be studied near the critical point,
we rewrite the reduced variables in the form of

$t=T_{r}-1,\qquad\psi=\Psi_{r}-1,\qquad\phi=\boldsymbol{Q}_{p}-1,$ (28)

where $t$, $\psi$ and $\phi$ point out to the deviation from critical point.
Now, we approximate the equation of state (22) around the critical point as

$\phi=At+Bt\psi+Ct\psi^{2}+D\psi^{3}+O(t\psi^{3},\psi^{4}),$ (29)

where $A,B$ and $C$ are constant quantities depend on $n$ and $p$, as follows

$\displaystyle A$ $\displaystyle=$
$\displaystyle-{\frac{4p(n-1)\left(p(n-3)+1\right)}{\left(2p-1\right)^{2}}},\quad
B=\frac{4p(n-1)\left(p(n-3)+1\right)\left(2p(n-2)+1\right)}{\left(2p-1\right)^{2}\left(n-2p\right)},$
$\displaystyle C$ $\displaystyle=$
$\displaystyle-\frac{2p(n-1)(p(n-3)+1)(2p(n-2)+1)(2p(n-3)+n+1)}{\left(2p-1\right)^{2}\left(n-2p\right)^{2}},\quad
D=-\frac{2p(n-1)(p(n-3)+1)(2p(n-2)+1)}{3(n-2p)^{3}}.$ (30)

Due to the fact that during phase transition the charge ($\boldsymbol{Q}_{p}$)
remains constant, we have from Eq.(29)

$Bt\psi_{s}+Ct\psi_{s}^{2}+D\psi_{s}^{3}=Bt\psi_{l}+Ct\psi_{l}^{2}+D\psi_{l}^{3},$
(31)

where $\psi_{s}$, $\psi_{l}$ stand for the small and gas black hole,
respectively. On the other hand, by applying the Maxwell construction Eq.27,
one obtains

$\int_{\psi_{s}}^{\psi_{l}}\psi(Bt+2Ct\psi+3D\psi^{2})\,\mathrm{d}\psi=0.$
(32)

Equation (31) and (32) have a non-trivial solution given by

$\psi_{l,s}=\frac{-Ct\pm\sqrt{3t(C^{2}t-3BD)}}{3D}.$ (33)

So, the corresponding expression for the order parameter near the critical
point becomes

$|\psi_{s}-\psi_{l}|\sim t^{1/2}\Rightarrow\beta=\frac{1}{2}.$

This equations yields to the order parameter near the critical point

$|\psi_{s}-\psi_{l}|\sim t^{1/2}\Rightarrow\beta=\frac{1}{2}.$

The critical exponent $\gamma$ determines the behavior of the parameter
$X_{T}$ as

$\chi_{{}_{T}}=\frac{\partial\Psi}{\partial\boldsymbol{Q}_{p}}\Big{|}_{T}\sim|t|^{-\gamma},$

thus from (29),

$\chi_{{}_{T}}\sim\frac{\Psi_{c}}{B\boldsymbol{Q}_{p_{c}}t}\Longrightarrow\gamma=1.$
(34)

The behavior of charge on the critical isotherm $t=0$ is also explained by
exponent $\delta$. Hence using (29) one can write $\phi=D\psi^{3}$ and so
$\delta=3$.

To find the specific heat behavior, one needs to find the critical exponent
$\alpha$ such that,

$C_{\Psi}=T\frac{\partial S}{\partial T}\Big{|}_{\Psi}\sim|t|^{\alpha}.$

Since the entropy (9) is independent of $t$, $C_{\Psi}=0$ and we can conclude
$\alpha=0$. The values we have found for the set of critical exponents
coincide with those obtained for Van der Waals fluid Kubiznak and Mann (2012).

##  IV Gibbs free energy

Now, we investigate the black hole thermodynamics by studying the
thermodynamic potential. In particular, the Gibbs free energy as a
thermodynamic potential characterizes the globally stable state at equilibrium
process. Hence, to find the phase transition and classification of its type,
we explore the Gibbs free energy of power Maxwell black holes. In fixed the
AdS radius $l$ ($\Lambda$) regime, the Gibbs free energy is calculated by
Legendre transformation $G=M-TS$ Dehyadegari et al. (2017b). Using (5)-(7) and
(9) the Gibbs free energy per unit volume $\omega_{n-1}$ is obtained as

$G=G(\boldsymbol{Q}_{p},T)=\frac{r_{+}^{n-2}}{16\pi}-\frac{r_{+}^{n-2}}{16\pi
l^{2}}+\frac{(2p-1)\left(2p(n-2)+1\right)\pi^{1/\left(2p-1\right)}\boldsymbol{Q}_{p}}{\left(n-1\right)\left(n-2p\right)2^{4+5p/\left(1-2p\right)}},$
(35)

where $r_{+}=r_{+}(\boldsymbol{Q}_{p},T)$.

(a) $n=4$, $p=1$

(b) $n=4$, $p=5/4$

(c) $n=5$, $p=3/2$

Figure 3: The behavior of $G$ versus $\boldsymbol{Q}_{p}$ for power-Maxwell
black holes corresponding to Fig. 1 with $l=1$. Note curves are shifted for
clarity.

The behavior of the Gibbs free energy $G$ is depicted in Fig. 3. As it is
clear from Fig. 3, for $T<T_{c}$, the Gibbs free energy is single value and
monotonically increasing function of $\boldsymbol{Q}_{p}$. While for
$T>T_{c}$, it becomes multivalued which means that a first-order phase
transition occurs between the small and large black holes. The corresponding
phase diagrams represented as $\boldsymbol{Q}_{p}$ versus $T$ are shown in
Fig. 4. Here, the small and large black holes are distinguished by transition
line (blue line). As one can see from Fig. 4(a), for conformal case
$p=(n+1)/4$, the slope of transition line increases with increasing dimension
$n$. In Figs. 4(b) and 4(c), when we increase $p$, slope of transition line
increases too.

(a) $p=(n+1)/4$

(b) $n=3$

(c) $n=4$

Figure 4: The transition line of the phase transition between small and large
black holes for various values of $n$, $p$ and fixed $l=1$. The critical
points are indicated by block spot.

##  V Thermodynamic geometry and microscopic structure

In this section, we turn to study phase transition structure of power-Maxwell
black holes in AdS space from point of view of thermodynamic state space
geometry. The Ruppeiner geometry has been proposed as new approach to insight
into underlying structure of thermodynamic system from the thermodynamic
fluctuation theory Rupp1 . Indeed, Ricci scalar which is obtained from
Ruppeiner metric, indicates the dominant interaction between possible
molecules by its sign Wei and Liu (2015); comment . In fact, the Ruppeiner
(Ricci) curvature vanishes for the ideal gas, while for attractive (repulsive)
dominant interaction is negative (positive) Ruppeiner (2008); May et al.
(2013); Ruppeiner (2012). Recently, various studies on Ruppeiner geometry have
been carried out in sahay ; 1610.06352 ; Kord Zangeneh et al. (2018).

The components of the Ruppeiner metric in the energy representation are
defined as Rupp1

$g_{\mu\nu}=\frac{1}{T}\frac{\partial^{2}M}{\partial X^{\mu}\partial
X^{\nu}},$ (36)

where $X^{\mu}=(S,\boldsymbol{Q}_{p})$. With Eqs. (10), (11) and (36) at hand,
one can calculate the Ruppeiner scalar curvature

$R=\frac{8p^{\left(n+1\right)/2}\left[\frac{(n-2)(p(n-3)+1)}{n(n-1)}\right]^{(1-n)/2}\left[1+\left(\Psi/\Psi_{c}\right)^{2(2p-1)/(n-2p)}\right]\left(\Psi/\Psi_{c}\right)^{\left(n+1\right)(2p-1)/(n-2p)}}{\left(2p-1\right)\left[1+\frac{\left(n-1\right)}{p(n-3)+1}\left(\Psi/\Psi_{c}\right)^{2(2p-1)/(n-2p)}-\frac{\left(2p-1\right)^{2}\left(\boldsymbol{Q}_{p}/\boldsymbol{Q}_{pc}\right)}{\left(p(n-3)+1\right)\left(2p(n-2)+1\right)}\left(\Psi/\Psi_{c}\right)^{2p(n-1)/(n-2p)}\right]},$
(37)

here $l=1$. As can be seen in Table 1, the positive sign of $R$ is allowed due
to positive temperature, i.e. there is always repulsive interaction. Figure 5
shows Ruppeiner curvature $R/R_{c}$, for conformal case $p=(n+1)/4$, along the
transition line in both the small and large black holes. According to Fig. 5,
the value of Ruppeiner curvature in both small and large black holes is the
same at the critical point. Also, there is a gap in Ruppeiner curvature
between small and large black holes that is increased in higher dimension. The
dependence of the critical Ruppeiner curvature on the allowed range of $p$ is
illustrated in Fig. 5. In Fig. 5 for arbitrary values of dimension, the
critical value of Ruppeiner curvature diverges close to $p=1/2$.

Table 1: The allowed ranges of $R$. | $R$ | $R$
---|---|---
$T$ | Positive | Negative
validity | allowed | not allowed

Figure 5: (a) The behavior of reduced Ruppeiner scalar curvature ($R/R_{c}$)
along the transition line for large and small black holes for
$p=\left(n+1\right)/4$. (b) The dependence of the critical Ruppeiner scalar
curvature $R_{c}$ on the parameter $p$ is depicted for different values of
dimension.

##  VI summary and conclusion

In this paper, we have investigated the critical behavior of higher
dimensional AdS black holes with power-Maxwell nonlinear electrodynamics via
an alternative approach toward the phase space. We have kept the cosmological
constant as a fixed quantity and treated the charge of black hole as
thermodynamic variable. To show the complete analogy between the liquid/gas
phase transition of the Van der Waals fluid and small/large black hole phase
transition, we have investigated the phase space and critical behaviour in
$\boldsymbol{Q}_{p}-\psi$ plane.

We have suggested an algorithmic method to find the charge-independent
thermodynamic variable $\psi$ as a conjugate quantity to
$f(Q)=\boldsymbol{Q}_{p}$ where $s={2p}/{(2p-1)}$. We have also rewritten the
Smarr mass formula in according to the new phase space and emphasized on its
correspondence with standard Smarr relation. Furthermore, we have shown the
behavior of coexistence curve of SBH and LBH in $4,5$ and $6$ dimensional
spacetime. We have calculated the main characteristic properties of the phase
transition such as critical points and critical exponent for all dimensional
cases with power Maxwell field. It was already observed that while the
critical quantities depend on the dimensions of spacetime and nonlinearity
parameter $p$, the critical exponents are independent of the details of the
system and have the same values as those of Van der Waals system. Also, first
order phase transitions are concluded from the swallow tail behaviors of the
Gibbs free energy in the $(n+1)$-dimensional systems. It is interesting to
note that with increasing the dimensionality of the system, the amount of
transition lines gradient ($\partial\boldsymbol{Q}_{p}/\partial T$) is
increasing.

Finally, we have studied the microscopic properties of higher dimensional AdS
black holes by considering the effects of the conformal invariant power
Maxwell field. From the viewpoint of the thermodynamic geometry we have
figured out that the interaction between two micromolecules of black hole is a
strong repulsive interaction. Actually transition from small to large
(n+1)-dimensional black hole is due to this repulsive force. Similar to zero
temperature thermodynamic in Fermi gas, we have introduced a temperature
$T_{0}$. We have seen that at $T>T_{0}$, when $R=0$, large black hole can be
stable and its size is a function of temperature only. Finally, as a result,
the maximum amount of the scalar curvature gap Fig. 5microscopic structur has
been increased by increasing the number of dimensions.

The advantage of the approach presented in this paper is that we do not need
to extend the thermodynamical phase space to see the Van der Wasls behaviour
for black hole systems. The results obtained here further support the argument
given in Dehyadegari et al. (2017b); Dayyani and Sheykhi (2018). Our study
indicates that the approach here is helpful to investigate the critical
behaviour of other gravity theories such as Gauss-Bonnet black hole GB  and
other different electromagnetic fields without needing to consider the
cosmological constant (pressure) as a thermodynamic variable.

Our investigation on thermodynamical properties and microscopic structure of
the asymptotically AdS black holes with power Maxwell field further supports
the viability of this new approach and confirms that this approach is enough
powerful to explore the phase transition of higher dimensional black holes.

###### Acknowledgements.

We thank Shiraz University Research Council. The work of AS has been supported
financially by Research Institute for Astronomy and Astrophysics of Maragha
(RIAAM), Iran.

## References

  * (1)
  * Bekenstein (1973) J. D. Bekenstein, Phys. Rev. D7, 2333 (1973).
  * Hawking (1975) S. W. Hawking, Commun. Math. Phys. 43, 199 (1975).
  * Hawking and Page (1983) S. W. Hawking and D. N. Page, Commun. Math. Phys. 87, 577 (1983).
  * Witten (1998) E. Witten, Adv. Theor. Math. Phys. 2, 505 (1998), eprint hep-th/9803131.
  * Chamblin et al. (1999a) A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, Phys. Rev. D60, 104026 (1999a), eprint hep-th/9904197.
  * Chamblin et al. (1999b) A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, Phys. Rev. D60, 064018 (1999b), eprint hep-th/9902170.
  * Kubiznak and Mann (2012) D. Kubiznak and R. B. Mann, JHEP 07, 033 (2012), eprint 1205.0559.
  * Dehghani et al. (2014) M. H. Dehghani, S. Kamrani, and A. Sheykhi, Phys. Rev. D90, 104020 (2014), eprint 1505.02386.
  * Dolan (2011a) B. P. Dolan, Class. Quant. Grav. 28, 125020 (2011a), eprint 1008.5023.
  * Dolan (2011b) B. P. Dolan, Class. Quant. Grav. 28, 235017 (2011b), eprint 1106.6260.
  * Dolan (2012) B. P. Dolan (INTECH, 2012), pp. 291–316, eprint 1209.1272.
  * Frassino et al. (2014) A. M. Frassino, D. Kubiznak, R. B. Mann, and F. Simovic, JHEP 09, 080 (2014), eprint 1406.7015.
  * Hennigar et al. (2017) R. A. Hennigar, R. B. Mann, and E. Tjoa, Phys. Rev. Lett. 118, 021301 (2017), eprint 1609.02564.
  * Dehyadegari et al. (2017a) A. Dehyadegari, A. Sheykhi, and A. Montakhab, Phys. Rev. D96, 084012 (2017a), eprint 1707.05307.
  * Cvetic et al. (2011) M. Cvetic, G. W. Gibbons, D. Kubiznak, and C. N. Pope, Phys. Rev. D84, 024037 (2011), eprint 1012.2888.
  * Urano et al. (2009) M. Urano, A. Tomimatsu, and H. Saida, Class. Quant. Grav. 26, 105010 (2009), eprint 0903.4230.
  * Zou et al. (2014) D.-C. Zou, Y. Liu, and B. Wang, Phys. Rev. D90, 044063 (2014), eprint 1404.5194.
  * Poshteh et al. (2013) M. B. J. Poshteh, B. Mirza, and Z. Sherkatghanad, Phys. Rev. D88, 024005 (2013), eprint 1306.4516.
  * Dehghani et al. (2016) M. H. Dehghani, A. Sheykhi, and Z. Dayyani, Phys. Rev. D93, 024022 (2016), eprint 1611.08978.
  * Dayyani et al. (2017) Z. Dayyani, A. Sheykhi, and M. H. Dehghani, Phys. Rev. D95, 084004 (2017), eprint 1611.00590.
  * Majhi and Samanta (2017) B. R. Majhi and S. Samanta, Phys. Lett. B773, 203 (2017), eprint 1609.06224.
  * Dehyadegari and Sheykhi (2018) A. Dehyadegari and A. Sheykhi, Phys. Rev. D98, 024011 (2018), eprint 1711.01151.
  * Kastor et al. (2009) D. Kastor, S. Ray, and J. Traschen, Class. Quant. Grav. 26, 195011 (2009), eprint 0904.2765.
  * Dehyadegari et al. (2017b) A. Dehyadegari, A. Sheykhi, and A. Montakhab, Phys. Lett. B768, 235 (2017b), eprint 1607.05333.
  * Dehyadegari et al. (2019) A. Dehyadegari, B. R. Majhi, A. Sheykhi, and A. Montakhab, Phys. Lett. B791, 30 (2019), eprint 1811.12308.
  * Fernando (2006) S. Fernando, Phys. Rev. D74, 104032 (2006), eprint hep-th/0608040.
  * Myung et al. (2008a) Y. S. Myung, Y.-W. Kim, and Y.-J. Park, Phys. Rev. D78, 044020 (2008a), eprint 0804.0301.
  * Miskovic and Olea (2008) O. Miskovic and R. Olea, Phys. Rev. D77, 124048 (2008), eprint 0802.2081.
  * Myung et al. (2008b) Y. S. Myung, Y.-W. Kim, and Y.-J. Park, Phys. Rev. D78, 084002 (2008b), eprint 0805.0187.
  * Banerjee and Roychowdhury (2012) R. Banerjee and D. Roychowdhury, Phys. Rev. D85, 044040 (2012), eprint 1111.0147.
  * Gunasekaran et al. (2012) S. Gunasekaran, R. B. Mann, and D. Kubiznak, JHEP 11, 110 (2012), eprint 1208.6251.
  * Hendi et al. (2015a) S. H. Hendi, S. Panahiyan, and B. Eslam Panah, Int. J. Mod. Phys. D25, 1650010 (2015a), eprint 1410.0352.
  * Sheykhi and Kazemi (2014) A. Sheykhi and A. Kazemi, Phys. Rev. D90, 044028 (2014), eprint 1506.01786.
  * Hendi et al. (2015b) S. H. Hendi, A. Sheykhi, S. Panahiyan, and B. Eslam Panah, Phys. Rev. D92, 064028 (2015b), eprint 1509.08593.
  * Dayyani et al. (2018) Z. Dayyani, A. Sheykhi, M. H. Dehghani, and S. Hajkhalili, Eur. Phys. J. C78, 152 (2018), eprint 1709.06875.
  * Born and Infeld (1934) M. Born and L. Infeld, Proc. Roy. Soc. Lond. A144, 425 (1934).
  * Dirac (1964) P. Dirac, _Lectures on Quantum Mechanics_ , Belfer Graduate School of Science Monographs Series (Belfer Graduate School of Science, Yeshiva University, 1964), URL https://books.google.com/books?id=oQNRAAAAMAAJ.
  * Heisenberg and Euler (1936) W. Heisenberg and H. Euler, Z. Phys. 98, 714 (1936), eprint physics/0605038.
  * Yajima and Tamaki (2001) H. Yajima and T. Tamaki, Phys. Rev. D63, 064007 (2001), eprint gr-qc/0005016.
  * Delphenich (2003) D. H. Delphenich (2003), eprint hep-th/0309108.
  * Hassaine and Martinez (2007) M. Hassaine and C. Martinez, Phys. Rev. D75, 027502 (2007), eprint hep-th/0701058.
  * Hassaine and Martinez (2008) M. Hassaine and C. Martinez, Class. Quant. Grav. 25, 195023 (2008), eprint 0803.2946.
  * Hendi and Vahidinia (2013) S. H. Hendi and M. H. Vahidinia, Phys. Rev. D88, 084045 (2013), eprint 1212.6128.
  * Hendi et al. (2017) S. H. Hendi, B. Eslam Panah, S. Panahiyan, and M. S. Talezadeh, Eur. Phys. J. C77, 133 (2017), eprint 1612.00721.
  * Kord Zangeneh et al. (2015) M. Kord Zangeneh, A. Sheykhi, and M. H. Dehghani, Phys. Rev. D91, 044035 (2015), eprint 1505.01103.
  * Dayyani and Sheykhi (2018) Z. Dayyani and A. Sheykhi, Phys. Rev. D98, 104026 (2018), eprint 1805.00368.
  * (48) H. Yazdikarimi, A. Sheykhi and Z. Dayyani, Phys. Rev. D ??, ?????? (2019).
  * Brown and York (1993) J. D. Brown and J. W. York, Jr., Phys. Rev. D47, 1407 (1993), eprint gr-qc/9209012.
  * Gonzalez et al. (2009) H. A. Gonzalez, M. Hassaine, and C. Martinez, Phys. Rev. D80, 104008 (2009), eprint 0909.1365.
  * (51) G. Ruppeiner, Rev. Mod. Phys. 67, 605 (1995).
  * Wei and Liu (2015) S.-W. Wei and Y.-X. Liu, Phys. Rev. Lett. 115, 111302 (2015), [Erratum: Phys. Rev. Lett.116,no.16,169903(2016)], eprint 1502.00386.
  * (53) M. Kord Zangeneh, A. Dehyadegari and A. Sheykhi, arXiv:1602.03711.
  * Ruppeiner (2008) G. Ruppeiner, Phys. Rev. D78, 024016 (2008), eprint 0802.1326.
  * May et al. (2013) H.-O. May, P. Mausbach, and G. Ruppeiner, Physical Review E 88, 032123 (2013).
  * Ruppeiner (2012) G. Ruppeiner, Physical Review E 86, 021130 (2012).
  * (57) A. Sahay, T. Sarkar, and G. Sengupta, JHEP 1007, 082 (2010).
  * (58) M. Kord Zangeneh, A. Dehyadegari, M. R. Mehdizadeh, B. Wang and A. Sheykhi, Eur. Phys. J. C 77, 423 (2017), 1610.06352.
  * Kord Zangeneh et al. (2018) M. Kord Zangeneh, A. Dehyadegari, A. Sheykhi, and R. B. Mann, Phys. Rev. D97, 084054 (2018), eprint 1709.04432.


$S\in\mathcal{S}_{d}({\mathcal{F}},{\mathcal{Y}})$ is McCT-inner (referring to
the authors of the seminal paper [39]) if the operator
$M_{S}:\,{\mathcal{H}}_{\mathcal{F}}(k_{d})\to{\mathcal{H}}_{\mathcal{Y}}(k_{d})$
is a partial isometry.

If we specify Theorem 4.3 to the case where ${\mathcal{U}}=\\{0\\}$ (as we did
in Section 3.5 in the non-commutative setting), then $\mathfrak{A}$ collapses
to the analytic function $\Phi(\lambda)$ given by the formula (3.62). The
assumption $P:={\mathcal{O}}_{E,{\bf T}}^{*}{\mathcal{O}}_{E,{\bf T}}\succ 0$
still implies that ${\bf T}$ is strongly stable, i.e., that

$\lim_{N\to\infty}\sum_{{\bf n}\in\mathbb{Z}_{+}^{d}:|{\bf n}|=N}\frac{{\bf
n}!}{|{\bf n}|!}\|{\bf T}^{{\bf
n}}x\|^{2}_{{\mathcal{H}}_{{}_{\mathcal{Y}}}(k_{d})}=0\quad\mbox{for
all}\;\;x\in{\mathcal{X}},$

which this time guarantees only $\Phi$ be McCT-inner; see [16, Section 5]. The
rest is the same as in Theorem 3.18.

######  Theorem 4.4.

Given an output stable, exactly observable pair $(E,{\bf
T})\in{\mathcal{L}}({\mathcal{X}},{\mathcal{Y}})\times{\mathcal{L}}({\mathcal{X}})^{d}$
(i.e., $P:={\mathcal{O}}_{E,{\bf T}}^{*}{\mathcal{O}}_{E,{\bf T}}\succ 0$)
with $\mathbf{T}$ a commutative tuple, there exists an auxiliary Hilbert space
${\mathcal{F}}$ and a McCT-inner function
$\Phi(\lambda)\in{\mathcal{S}}_{d}({\mathcal{F}},{\mathcal{Y}})$ such that

$\frac{I_{{\mathcal{Y}}}-\Phi(\lambda)\Phi(\eta)^{*}}{1-\langle\lambda,\eta\rangle}=E(I-Z(\lambda)T)^{-1}P^{-1}(I-T^{*}Z(\eta)^{*})^{-1}E^{*}.$

Furthermore, for a given vector ${\bf x}\in{\mathcal{X}}$, a function
$f\in{\mathcal{H}}_{{\mathcal{Y}}}(k_{d})$ satisfies the interpolation
condition

$(E^{*}f)^{\wedge L}(\mathbf{T}^{*}):={\mathcal{O}}_{E,\mathbf{T}}^{*}f={\bf
x}$

if and only if it is of the form

$f(\lambda)=E(I-Z(\lambda)T)^{-1}P^{-1}{\bf x}+\Phi(\lambda)h(\lambda),\quad
h\in{\mathcal{H}}_{{\mathcal{F}}}(k_{d}).$

The homogeneous version of the latter theorem contains the
${\mathcal{H}}_{\mathcal{Y}}(k_{d})$-version of the Beurling-Lax theorem [39].
Still writing ${\bf R}_{z}$ for the $d$-tuple of coordinate-variable
multipliers $R_{\lambda_{j}}$ in ${\mathcal{H}}_{\mathcal{Y}}(k_{d})$, we note
that their adjoints are now given by formulas

$R_{\lambda_{j}}^{*}:\;\sum_{{\bf n}\in\mathbb{Z}_{+}^{d}}f_{{\bf
n}}\lambda^{{\bf n}}\mapsto\sum_{{\bf n}\in\mathbb{Z}_{+}^{d}}f_{{\bf
n}+e_{j}}\lambda^{{\bf n}}\quad\mbox{for}\quad j=1,\ldots,d$

where $e_{j}$ is the element in $\mathbb{Z}_{+}^{d}$ having the $j$-th partial
index equal to one and all other partial indices equal to zero. If we let
${\bf ev}_{0}:\,f(\lambda)\to f(0)$ denote the zero-evaluation operator on
${\mathcal{H}}_{\mathcal{Y}}(k_{d})$, then it is easily verified that ${\bf
ev}_{0}{\bf R}_{\lambda}^{{\bf n}}\colon\sum_{{\bf
n}^{\prime}\in{\mathbb{Z}}^{d}_{+}}f_{{\bf n}^{\prime}}\lambda^{{\bf
n}^{\prime}}\to f_{{\bf n}}$ for all ${\bf n}\in\mathbb{Z}_{d}^{+}$ and
subsequently, that the observability operator ${\mathcal{O}}_{{\bf
ev}_{0},{\bf R}_{\lambda}}$ equals the identity operator on
${\mathcal{H}}_{\mathcal{Y}}(k_{d})$. Indeed, for any
$f\in{\mathcal{H}}_{\mathcal{Y}}(k_{d})$, we have

${\mathcal{O}}_{{\bf ev}_{0},{\bf R}_{\lambda}^{*}}f=\sum_{{\bf
n}\in\mathbb{Z}_{+}^{d}}({\bf ev}_{0}{\bf R}_{\lambda}^{*{{\bf
n}}}f)\lambda^{{\bf n}}=\sum_{{\bf n}\in\mathbb{Z}_{+}^{d}}f_{{\bf
n}}\lambda^{{\bf n}}=f.$

If ${\mathcal{M}}$ is a closed subspace of
${\mathcal{H}}_{\mathcal{Y}}(k_{d})$ which is ${\bf R}_{\lambda}$-invariant,
then the restricted output pair $(E,{\bf
T})\in{\mathcal{L}}({\mathcal{X}},{\mathcal{Y}})\times{\mathcal{L}}({\mathcal{X}})^{d}$
given by

${\mathcal{X}}={\mathcal{M}}^{\perp},\quad E={\bf
ev}_{0}|_{\mathcal{X}},\quad{\bf T}={\bf R}_{\lambda}^{*}|_{\mathcal{X}}$

is exactly observable and
${\mathcal{M}}^{\perp}=\operatorname{Ran}{\mathcal{O}}_{E,{\bf T}}$.
Therefore, ${\mathcal{M}}=\operatorname{Ker}{\mathcal{O}}_{E,{\bf T}}^{*}$
coincides with the solution set of the homogeneous Operator Argument
interpolation Problem ${\bf OAP}_{{\mathcal{H}}_{\mathcal{Y}}(k_{d})}$ with
interpolation conditions

$(E^{*}f)^{\wedge L}(\mathbf{T}^{*}):={\mathcal{O}}_{E,\mathbf{T}}^{*}f=0.$

Then it follows by Theorem 4.4 that there exist a Hilbert space
${\mathcal{F}}$ and a McCT-inner multiplier
$\Phi\in{\mathcal{S}}_{d}({\mathcal{F}},{\mathcal{Y}})$ such that
${\mathcal{M}}=\Phi\cdot{\mathcal{H}}_{\mathcal{F}}(k_{d})$.

##  5\. An operator theoretical view

In this section we present a general, purely operator theoretical perspective
on the problems considered above, relying on what we will call a generalized
de Branges-Rovnyak space. Our approach in this section is to view the problem
in the context of a Douglas Factorization Problem with respect to the lifted
norms from the generalized de Branges-Rovnyak spaces. For that purpose we also
derive some results on the classical Douglas factorization problem.

###  5.1. Generalized de Branges-Rovnyak spaces

With a given contraction operator
$T\in{\mathcal{L}}({\mathcal{U}},{\mathcal{Y}})$ we may associate the operator
range of the positive semidefinite operator $I_{{\mathcal{Y}}}-TT^{*}\succeq
0$:

${\mathcal{H}}(T):=\operatorname{Ran}(I-TT^{*})^{\frac{1}{2}}\subset{\mathcal{Y}}$

with the lifted norm

(5.1)
$\|(I-TT^{*})^{\frac{1}{2}}y\|_{{\mathcal{H}}(T)}=\|(I-\pi)y\|_{{\mathcal{Y}}}$

where $\pi$ is the orthogonal projection onto
$\operatorname{Ker}(I-TT^{*})^{\frac{1}{2}}$. It follows from (5.1) that
$\|y\|_{{\mathcal{H}}(T)}\geq\|y\|_{{\mathcal{Y}}}$ for every
$y\in{\mathcal{H}}(T)$ and thus ${\mathcal{H}}(T)$ is contractively included
in ${\mathcal{Y}}$. Upon letting $y=(I-TT^{*})^{\frac{1}{2}}y^{\prime}$ in the
last formula we get

(5.2)
$\|(I-TT^{*})y^{\prime}\|_{{\mathcal{H}}(T)}=\langle(I-TT^{*})y^{\prime},\,y^{\prime}\rangle_{{\mathcal{Y}}}.$

The original characterization of ${\mathcal{H}}(T)$ as the space of all
vectors in $y\in{\mathcal{Y}}$ such that

$\kappa(y):=\sup_{u\in{\mathcal{U}}}\big{\\{}\|y+Tu\|^{2}_{{\mathcal{Y}}}-\|u\|^{2}_{{\mathcal{U}}}\big{\\}}$

is finite and the identity $\|y\|^{2}_{{\mathcal{H}}(T)}=\kappa(y)$ is due to
de Branges and Rovnyak [27]. Let us note that de Branges and Rovnyak worked
with the special case where ${\mathcal{U}}$ and ${\mathcal{Y}}$ are replaced
by Hardy spaces $H^{2}_{\mathcal{U}}$ and $H^{2}_{\mathcal{Y}}$ and the
operator $T\colon H^{2}_{\mathcal{U}}\to H^{2}_{\mathcal{Y}}$ is the operator
$T=M_{S}\colon f(\lambda)\mapsto S(\lambda)f(\lambda)$ of multiplication by a
Schur-class function $S\in{\mathcal{S}}({\mathcal{U}},{\mathcal{Y}})$. The
spaces ${\mathcal{H}}(T)$ (operator- rather than function-theoretic) were
proposed by Sarason [43] and will here be referred to as generalized de
Branges-Rovnyak spaces.

###  5.2. The solutions to Douglas Factorization Problem

Consider $A\in{\mathcal{L}}({\mathcal{Y}},{\mathcal{X}})$ and
$B\in{\mathcal{L}}({\mathcal{U}},{\mathcal{X}})$. The following well-known
result due to Douglas [30] describes when there exists an operator
$Y\in{\mathcal{L}}({\mathcal{U}},{\mathcal{Y}})$ that satisfies

(5.3) $AY=B\quad\text{\em and}\quad\|Y\|\leq 1.$

######  Lemma 5.1.

There exists a $Y\in{\mathcal{L}}({\mathcal{U}},{\mathcal{Y}})$ satisfying
(5.3) if and only if $AA^{*}\succeq BB^{*}$. In this case, there exists a
unique $Y$ satisfying (5.3) and the additional constraints
$\operatorname{Ran}Y\subset\operatorname{Ran}A^{*}$ and
$\operatorname{Ker}Y=\operatorname{Ker}B$.

The next proposition contains several characterizations of the operators
$Y\in{\mathcal{L}}({\mathcal{U}},{\mathcal{Y}})$ that satisfy (5.3).

######  Proposition 5.2.

Given $A\in{\mathcal{L}}({\mathcal{U}},{\mathcal{X}})$ and
$B\in{\mathcal{L}}({\mathcal{Y}},{\mathcal{X}})$, let us assume that

(5.4) $Q:=AA^{*}-BB^{*}\succeq 0.$

Then there exist unique contractions

$X_{1}\in{\mathcal{L}}({\mathcal{U}},\overline{\operatorname{Ran}}A)\quad\mbox{and}\quad
X_{2}\in{\mathcal{L}}({\mathcal{Y}},\overline{\operatorname{Ran}}A)$

so that

(5.5) $(AA^{*})^{\frac{1}{2}}X_{1}=B,\ \ (AA^{*})^{\frac{1}{2}}X_{2}=A,\ \
\operatorname{Ker}X_{1}=\operatorname{Ker}B,\ \
\operatorname{Ker}X_{2}=\operatorname{Ker}A,$

with $X_{2}$ being a coisometry. Given
$Y\in{\mathcal{L}}({\mathcal{U}},{\mathcal{Y}})$, define

$F_{{}_{Y}}:=A^{*}-YB^{*}\in{\mathcal{L}}({\mathcal{X}},{\mathcal{Y}}).$

Then the following statements are equivalent:

  1. (1)

$Y$ satisfies conditions (5.3).

  2. (2)

The operator

$\mathbb{P}=\begin{bmatrix}Q&F_{{}_{Y}}^{*}\\\
F_{{}_{Y}}&I_{{\mathcal{Y}}}-YY^{*}\end{bmatrix}:\;\begin{bmatrix}{\mathcal{X}}\\\
{\mathcal{Y}}\end{bmatrix}\to\begin{bmatrix}{\mathcal{X}}\\\
{\mathcal{Y}}\end{bmatrix}$

is positive semidefinite, or, which is the same, $Y$ is a contraction and
$F_{{}_{Y}}$ maps ${\mathcal{X}}$ into ${\mathcal{H}}(Y)$ with the property

$\|F_{{}_{Y}}x\|_{{\mathcal{H}}(Y)}\leq\|Q^{\frac{1}{2}}x\|_{\mathcal{X}}\quad\mbox{for
every}\quad x\in{\mathcal{X}}.$

  3. (3)

$Y$ is a contraction and $F_{{}_{Y}}$ maps ${\mathcal{X}}$ into
${\mathcal{H}}(Y)$ with the property

(5.6)
$\|F_{{}_{Y}}x\|_{{\mathcal{H}}(Y)}=\|Q^{\frac{1}{2}}x\|_{\mathcal{X}}\quad\mbox{for
every}\quad x\in{\mathcal{X}}.$

  4. (4)

$Y$ is of the form

(5.7)
$Y=X_{2}^{*}X_{1}+(I-X_{2}^{*}X_{2})^{\frac{1}{2}}K(I-X_{1}^{*}X_{1})^{\frac{1}{2}}$

where $X_{1}$ and $X_{2}$ are defined as in (5.5) and where the parameter $K$
is an arbitrary contraction from
$\overline{\operatorname{Ran}}(I-X_{1}^{*}X_{1})$ into
$\overline{\operatorname{Ran}}(I-X_{2}^{*}X_{2})$.

Moreover, there is a unique operator
$Y\in{\mathcal{L}}({\mathcal{U}},{\mathcal{Y}})$ satisfying (5.3) if and only
if $X_{1}$ is isometric on ${\mathcal{U}}$ or $X_{2}$ is isometric on
${\mathcal{Y}}$. Moreover, for $Y$ as in (5.7) we have

(5.8)
$\|Yh\|^{2}=\|X_{2}^{*}X_{1}h\|^{2}+\|(I-X_{2}^{*}X_{2})^{\frac{1}{2}}K(I-X_{1}^{*}X_{1})^{\frac{1}{2}}h\|^{2},\quad
h\in{\mathcal{U}}$

so that $X_{2}^{*}X_{1}$ is the minimal norm solution to the problem (5.3).

###### Proof.

The existence and uniqueness of the contractions $X_{1}$ and $X_{2}$
satisfying (5.5) is a direct consequence of Lemma 5.1. That $X_{2}$ is a
coisometry can be seen as a consequence of the identity

$(AA^{*})^{\frac{1}{2}}(I_{\overline{\operatorname{Ran}}A}-X_{2}X_{2}^{*})(AA^{*})^{\frac{1}{2}}=0$

which in turn is a consequence of the second equation in (5.5).

The equivalence of (1), (2) and (4) was established in [18, Lemma 2.2] via
Schur-complement arguments. The central observation there was that
$Y\in{\mathcal{L}}({\mathcal{U}},{\mathcal{Y}})$ satisfies (5.3) if and only
if it satisfies the condition

(5.9) $\begin{bmatrix}I_{{\mathcal{U}}}&B^{*}&Y^{*}\\\ B&AA^{*}&A\\\
Y&A^{*}&I_{{\mathcal{Y}}}\end{bmatrix}\succeq 0.$

The implication (3) $\Rightarrow$ (2) is trivial. We verify
(1)$\Rightarrow$(3) as follows. If $Y$ satisfies conditions (5.3), then for
every $x\in{\mathcal{X}}$ we have

$F_{{}_{Y}}x=(A^{*}-YB^{*})x=(A^{*}-YY^{*}A^{*})x=(I-YY^{*})A^{*}x.$

We now apply the formula (5.2) to $y^{\prime}=A^{*}x$ and then make subsequent
use of (5.3) and (5.4) to get

$\displaystyle\|F_{{}_{Y}}x\|_{{\mathcal{H}}(Y)}$
$\displaystyle=\langle(I-YY^{*})A^{*}x,\,A^{*}x\rangle_{{\mathcal{Y}}}$
$\displaystyle=\langle A(I-YY^{*})A^{*}x,\,x\rangle_{{\mathcal{X}}}$
$\displaystyle=\langle(AA^{*}-BB^{*})x,\,x\rangle_{{\mathcal{X}}}=\langle
Qx,\,x\rangle_{{\mathcal{X}}}=\|Q^{\frac{1}{2}}x\|_{{\mathcal{X}}},$

which is equivalent to (5.6).

Finally, to see (5.8) simply note that $(I-X_{2}^{*}X_{2})^{\frac{1}{2}}$ is
the orthogonal projection onto
${\mathcal{Y}}\ominus\operatorname{Ker}A={\mathcal{Y}}\ominus\operatorname{Ker}X_{1}$
since $X_{2}$ is a coisometry. ∎

######  Remark 5.3.

The equivalence of conditions (5.3) and (5.9) means that the Douglas
factorization problem can be reformulated as a matrix completion problem:
given a partially defined matrix

$\begin{bmatrix}I_{{\mathcal{H}}_{1}}&B^{*}&?^{*}\\\ B&AA^{*}&A\\\
?&A^{*}&I_{{\mathcal{H}}_{2}}\end{bmatrix},$

find an operator $Y$ such that plugging in $Y$ for ? leads to a positive
semidefinite operator matrix. This at various times has been an active area of
research in its own right (see e.g. [37]).

###  5.3. Douglas Factorization Problem in generalized de Branges-Rovnyak
spaces

We next consider a problem similar to (5.3) but now $Y$ is contractive with
respect to the norms induced by two generalized de Branges-Rovnyak spaces:
given operators $A\in{\mathcal{L}}({\mathcal{Y}},{\mathcal{X}})$,
$B\in{\mathcal{L}}({\mathcal{U}},{\mathcal{X}})$ and contractions
$T_{1}\in{\mathcal{L}}(\widetilde{{\mathcal{U}}},{\mathcal{U}})$ and
$T_{2}\in{\mathcal{L}}(\widetilde{{\mathcal{Y}}},{\mathcal{Y}})$, find an
operator $Y:\;{\mathcal{H}}(T_{1})\to{\mathcal{H}}(T_{2})$ such that

(5.10) $AY=B|_{{\mathcal{H}}(T_{1})}\quad\mbox{and}\quad\|Y\|\leq 1.$

If $A$ and $B$ are a priori considered as operators in
${\mathcal{L}}({\mathcal{H}}(T_{2}),{\mathcal{X}})$ and
${\mathcal{L}}({\mathcal{H}}(T_{1}),{\mathcal{X}})$ respectively (or,
equivalently, $T_{1}=0$ and $T_{2}=0$), then this is the classical Douglas
Factorization Problem from the previous subsection. The following lemma shows
that the case where $A\in{\mathcal{L}}({\mathcal{Y}},{\mathcal{X}})$,
$B\in{\mathcal{L}}({\mathcal{U}},{\mathcal{X}})$ also can be translated to the
classical case. Given a contraction
$T\in{\mathcal{L}}(\widetilde{{\mathcal{H}}},{\mathcal{H}})$ and an operator
$C\in{\mathcal{L}}({\mathcal{H}},{\mathcal{X}})$, the range of the operator

(5.11) $F_{{{}_{C}}}=(I-TT^{*})C^{*}:\;{\mathcal{X}}\to{\mathcal{H}}$

is contained in ${\mathcal{H}}(T)\subset{\mathcal{H}}$, and therefore,
$F_{{{}_{C}}}$ can also be viewed as an operator in
${\mathcal{L}}({\mathcal{X}},{\mathcal{H}}(T))$. Following Remark 3.2, we
write $F_{{{}_{C}}}^{*}$ for the adjoint of $F_{{{}_{C}}}$ in
${\mathcal{L}}({\mathcal{H}},{\mathcal{X}})$ and $F_{{{}_{C}}}^{[*]}$ for the
adjoint of $F_{{{}_{C}}}$ in ${\mathcal{L}}({\mathcal{H}}(T),{\mathcal{X}})$.

######  Lemma 5.4.

Given $C\in{\mathcal{L}}({\mathcal{H}},{\mathcal{X}})$ and a contraction
$T\in{\mathcal{L}}(\widetilde{{\mathcal{H}}},{\mathcal{H}})$, define
$F_{{{}_{C}}}$ as in (5.11). Then

(5.12) $F_{{{}_{C}}}^{[*]}g=Cg\;\;\mbox{for
all}\;\;g\in{\mathcal{H}}(T)\quad\mbox{and}\quad
F_{{{}_{C}}}^{[*]}F_{{{}_{C}}}=C(I-TT^{*})C^{*}.$

###### Proof.

For any $g\in{\mathcal{H}}(T)$ and $x\in{\mathcal{X}}$, we have

$\displaystyle\langle x,\,F_{{}_{C}}^{[*]}g\rangle_{{\mathcal{X}}}$
$\displaystyle=\langle
F_{{}_{C}}x,\,g\rangle_{{\mathcal{H}}(T)}=\langle(I-TT^{*})C^{*}x,\,g\rangle_{{\mathcal{H}}(T)}$
$\displaystyle=\langle C^{*}x,\,g\rangle_{{\mathcal{H}}}=\langle
x,\,Cg\rangle_{{\mathcal{X}}},$

and the first equality in (5.12) follows. Letting
$g:=F_{{{}_{C}}}\widetilde{x}=(I-TT^{*})C^{*}\widetilde{x}$ in this equality
we get

$F_{{{}_{C}}}^{[*]}F_{{{}_{C}}}\widetilde{x}=C(I-TT^{*})C^{*}\widetilde{x}\quad\mbox{for
all}\quad\widetilde{x}\in{\mathcal{X}}$

which justifies the second equality in (5.12). ∎

Given operators $A\in{\mathcal{L}}({\mathcal{Y}},{\mathcal{X}})$,
$B\in{\mathcal{L}}({\mathcal{U}},{\mathcal{X}})$ and contractions
$T_{1}\in{\mathcal{L}}(\widetilde{{\mathcal{U}}},{\mathcal{U}})$ and
$T_{2}\in{\mathcal{L}}(\widetilde{{\mathcal{Y}}},{\mathcal{Y}})$, define the
operators

(5.13) $F_{{{}_{A}}}:=(I-T_{2}T_{2}^{*})A^{*}\quad\mbox{and}\quad
F_{{{}_{B}}}:=(I-T_{2}T_{2}^{*})B^{*}$

which can also be viewed as operators in
${\mathcal{L}}({\mathcal{X}},{\mathcal{H}}(T_{2}))$ and
${\mathcal{L}}({\mathcal{X}},{\mathcal{H}}(T_{1}))$, respectively. Using Lemma
5.4 it follows that (5.10) can be rewritten as

$F_{{{}_{A}}}^{[*]}Y=F_{{{}_{B}}}^{[*]}\quad\mbox{and}\quad\|Y\|\leq 1.$

In this form, Lemmas 5.1 and 5.2 are applicable with $A$ and $B$ replaced by
$F_{{{}_{A}}}^{[*]}$ and $F_{{{}_{B}}}^{[*]}$, respectively, leading to the
following result.

######  Proposition 5.5.

The problem (5.10) has a solution if and only if

$Q:=A(I-T_{2}T_{2}^{*})A^{*}-B(I-T_{1}T_{1}^{*})B^{*}\succeq 0.$

If this is the case, then the following are equivalent:

  1. (1)

$Y\in{\mathcal{L}}({\mathcal{H}}(T_{1}),{\mathcal{H}}(T_{2}))$ satisfies
(5.10).

  2. (2)

The operator

$\mathbb{P}=\begin{bmatrix}Q&F^{[*]}_{{{}_{A}}}-F^{[*]}_{{{}_{B}}}Y^{*}\\\
F_{{{}_{A}}}-YF_{{{}_{B}}}&I_{{\mathcal{H}}(T_{2})}-YY^{*}\end{bmatrix}:\;\begin{bmatrix}{\mathcal{X}}\\\
{\mathcal{H}}(T_{2})\end{bmatrix}\to\begin{bmatrix}{\mathcal{X}}\\\
{\mathcal{H}}(T_{2})\end{bmatrix}$

is positive semidefinite or, which is the same, $Y$ is a contraction and
$F_{{}_{Y}}:=F_{{{}_{A}}}-YF_{{{}_{B}}}$ maps ${\mathcal{X}}$ into
${\mathcal{H}}(Y)$ with the property

$\|F_{{}_{Y}}x\|_{{\mathcal{H}}(Y)}\leq\|Q^{\frac{1}{2}}x\|_{\mathcal{X}}\quad\mbox{for
every}\quad x\in{\mathcal{X}}.$

  3. (3)

$Y$ is a contraction and $F_{{}_{Y}}$ maps ${\mathcal{X}}$ into
${\mathcal{H}}(Y)$ with the property

$\|F_{{}_{Y}}x\|_{{\mathcal{H}}(Y)}=\|Q^{\frac{1}{2}}x\|_{\mathcal{X}}\quad\mbox{for
every}\quad x\in{\mathcal{X}}.$

  4. (4)

$Y$ is of the form

(5.14)
$Y=X_{2}^{*}X_{1}+(I-X_{2}^{*}X_{2})^{\frac{1}{2}}K(I-X_{1}^{*}X_{1})^{\frac{1}{2}}$

where $X_{1}$ the contraction in
${\mathcal{L}}({\mathcal{H}}(T_{1}),(\operatorname{Ker}(I-T_{2}T_{2}^{*})A^{*})^{\perp})$
and $X_{2}$ the contraction in
${\mathcal{L}}({\mathcal{H}}(T_{2}),(\operatorname{Ker}(I-T_{2}T_{2}^{*})A^{*})^{\perp})$
that are uniquely determined by

$\displaystyle(A(I-T_{2}T_{2}^{*})A^{*})^{\frac{1}{2}}X_{1}g_{1}=Bg_{1},$
$\displaystyle\ \mbox{ for }\ g_{1}\in{\mathcal{H}}(T_{1}),$
$\displaystyle(A(I-T_{2}T_{2}^{*})A^{*})^{\frac{1}{2}}X_{1}g_{2}=Ag_{2},$
$\displaystyle\ \mbox{ for }\ g_{2}\in{\mathcal{H}}(T_{2}),$

with $X_{2}$ being a coisometry, and where the parameter $K$ is an arbitrary
contraction from $\overline{\operatorname{Ran}}(I-X_{1}^{*}X_{1})$ into
$\overline{\operatorname{Ran}}(I-X_{2}^{*}X_{2})$.

Moreover, there is a unique operator
$Y\in{\mathcal{L}}({\mathcal{H}}(T_{1}),{\mathcal{H}}(T_{2}))$ satisfying
(5.10) if and only if $X_{1}$ is isometric on ${\mathcal{H}}(T_{1})$ or
$X_{2}$ is isometric on ${\mathcal{H}}(T_{2})$. Furthermore, for $Y$ as in
(5.14) we have

$\|Yh\|^{2}=\|X_{2}^{*}X_{1}h\|^{2}+\|(I-X_{2}^{*}X_{2})^{\frac{1}{2}}K(I-X_{1}^{*}X_{1})^{\frac{1}{2}}h\|^{2},\quad
h\in{\mathcal{H}}(T_{1})$

so that $X_{2}^{*}X_{1}$ is the minimal norm solution to the problem (5.10) .

Of particular interest for this paper is the special case of the de Branges-
Rovnyak Douglas Factorization Problem (5.10) with
${\mathcal{H}}_{1}=\widetilde{{\mathcal{H}}}_{1}={\mathbb{C}}$ and $T_{1}$ the
zero operator. Then ${\mathcal{H}}(T_{1})={\mathbb{C}}$ and we identify
${\mathcal{L}}({\mathcal{H}}(T_{1}),{\mathcal{H}}(T_{2}))$ with
${\mathcal{H}}(T_{2})$ and ${\mathcal{L}}({\mathcal{H}}_{1},{\mathcal{X}})$
with ${\mathcal{X}}$. Then we arrive at the following problem: for
$A\in{\mathcal{L}}({\mathcal{H}}_{2},{\mathcal{X}})$, ${\bf
x}\in{\mathcal{X}}$ and a contraction
$T\in{\mathcal{L}}(\widetilde{{\mathcal{H}}}_{2}),{\mathcal{H}}_{2}$, find a
vector $g\in{\mathcal{H}}_{2}$ so that

(5.15) $g\in{\mathcal{H}}(T_{2}),\quad Ag={\bf
x},\quad\|g\|_{{\mathcal{H}}(T_{2})}\leq 1.$

Hence we consider the problem (5.10) with $B={\bf x}$ and $Y=g$. We split the
result of Proposition 5.5 for the case at hand into two results.

######  Lemma 5.6.

Let $A\in{\mathcal{L}}({\mathcal{H}}_{2},{\mathcal{X}})$, ${\bf
x}\in{\mathcal{X}}$ and
$T_{2}\in{\mathcal{L}}(\widetilde{{\mathcal{H}}}_{2},{\mathcal{H}}_{2})$ with
$T_{2}$ a contraction. Define $F_{{}_{A}}$ as in (5.13). Then there exists a
vector $g\in{\mathcal{H}}_{2}$ satisfying (5.15) if and only if
$P:=F_{{}_{A}}^{[*]}F_{{}_{A}}=A(I-T_{2}T_{2}^{*})A^{*}\succeq{\bf x}{\bf
x}^{*}$. Furthermore, a vector $g\in{\mathcal{H}}_{2}$ satisfies (5.15) if and
only if

$\mathbb{P}=\begin{bmatrix}P-{\bf x}{\bf x}^{*}&F_{{}_{A}}^{[*]}-{\bf
x}g^{[*]}\\\ F_{{}_{A}}-g{\bf
x}^{*}&I_{{\mathcal{H}}(T)}-gg^{[*]}\end{bmatrix}\succeq 0$

or equivalently, if and only if

$\begin{bmatrix}1&{\bf x}^{*}&g^{[*]}\\\ {\bf x}&P&F_{{}_{A}}^{[*]}\\\
g&F_{{}_{A}}&I_{{\mathcal{H}}(T)}\end{bmatrix}\succeq 0.$

In the second result we provide the parametrization of the solutions. For this
purpose, note that in case $F_{{}_{A}}^{[*]}F_{{}_{A}}\succeq{\bf x}{\bf
x}^{*}$, then by Lemma 5.1 there exist unique $\widetilde{\bf
x}\in(\operatorname{Ker}F_{{}_{A}})^{\perp}$ and
$\widetilde{F}_{{}_{A}}\in{\mathcal{L}}((\operatorname{Ker}F_{{}_{A}})^{\perp},{\mathcal{H}}(T_{2}))$
so that

(5.16) ${\bf x}=(F_{{}_{A}}^{[*]}F_{{}_{A}})^{\frac{1}{2}}\widetilde{\bf
x}\quad\mbox{and}\quad
F_{{}_{A}}=\widetilde{F}_{{}_{A}}(F_{{}_{A}}^{[*]}F_{{}_{A}})^{\frac{1}{2}},$

with $\widetilde{F}_{{}_{A}}$ being an isometry. The space

(5.17)
${\mathcal{N}}:=\overline{\operatorname{Ran}}\big{(}I_{{\mathcal{H}}(T_{2})}-\widetilde{F}_{{}_{A}}\widetilde{F}_{{}_{A}}^{[*]}\big{)}^{\frac{1}{2}}={\mathcal{H}}(\widetilde{F}_{{}_{A}})$

is isometrically included in ${\mathcal{H}}(T_{2})$, and its orthogonal
complement ${\mathcal{N}}^{\perp}$ in ${\mathcal{H}}(T_{2})$ can be
characterized as

(5.18)
${\mathcal{N}}^{\perp}=\\{F_{{}_{A}}x:\,x\in{\mathcal{X}}\\}\quad\mbox{with
norm}\quad\|F_{{}_{A}}x\|_{{\mathcal{H}}(T)}=\|(F_{{}_{A}}^{(*)}F_{{}_{A}})^{\frac{1}{2}}x\|_{{}_{\mathcal{X}}}.$

######  Theorem 5.7.

Let $A\in{\mathcal{L}}({\mathcal{H}}_{2},{\mathcal{X}})$, ${\bf
x}\in{\mathcal{X}}$ and
$T_{2}\in{\mathcal{L}}(\widetilde{{\mathcal{H}}}_{2},{\mathcal{H}}_{2})$ with
$T_{2}$ a contraction. Define $F_{{}_{A}}$ as in (5.13) and assume that
$F_{{}_{A}}^{(*)}F_{{}_{A}}\succeq{\bf x}{\bf x}^{*}$. Define $\widetilde{\bf
x}$ and $\widetilde{F}_{{}_{A}}$ by (5.16) and ${\mathcal{N}}$ as in (5.17).
Then all solutions $g$ to the problem (5.15) are given by the formula

(5.19) $g=F_{{}_{A}}\widetilde{\bf x}+h$

where $h$ is a free parameter from the subspace
${\mathcal{N}}={\mathcal{H}}(\widetilde{F}_{{}_{A}})\subset{\mathcal{H}}(T_{2})$
subject to

(5.20) $\|h\|_{{\mathcal{H}}(T_{2})}\leq\sqrt{1-\|\widetilde{\bf
x}\|^{2}_{{\mathcal{X}}}}.$

Furthermore, the problem has a unique solution if and only if
$\|\widetilde{\bf x}\|=1$ or
$\widetilde{F}_{{}_{A}}\widetilde{F}_{{}_{A}}^{[*]}=I_{{\mathcal{H}}(T)}$.
Furthermore, we have

$\|g\|_{{\mathcal{H}}(T_{2})}^{2}=\|F_{{}_{A}}\widetilde{\bf
x}\|_{{\mathcal{H}}(T_{2})}^{2}+\|h\|_{{\mathcal{H}}(T_{2})}^{2}=\|\widetilde{\bf
x}\|_{{\mathcal{X}}}^{2}+\|h\|_{{\mathcal{H}}(T_{2})}^{2}.$

###### Proof.

In the present framework, the operators $X_{1}$ and $X_{2}$ in Proposition 5.5
amount to $\widetilde{\bf x}$ and $\widetilde{F}_{{}_{T}}^{[*]}$ respectively,
and therefore, the parametrization formula (5.7) takes the form

(5.21) $g=F_{{}_{A}}\widetilde{\bf
x}+\big{(}I_{{\mathcal{H}}(T_{2})}-\widetilde{F}_{{}_{A}}\widetilde{F}_{{}_{A}}^{[*]}\big{)}^{\frac{1}{2}}K\sqrt{1-\widetilde{\bf
x}^{*}\widetilde{\bf x}}$

where now $K$ is an arbitrary vector in ${\mathcal{N}}$ with
$\|K\|_{{\mathcal{N}}}=\|K\|_{{\mathcal{H}}(T_{2})}\leq 1$. Therefore, the
second term on the right side of (5.21) represents a generic vector
$h\in{\mathcal{N}}$ subject the inequality (5.20). The uniqueness statement
follows immediately from (5.21). ∎

######  Remark 5.8.

We remark that the second term $h$ on the right hand side of (5.19) represents
in fact the general solution of the homogeneous interpolation problem (with
interpolation condition $F_{{}_{A}}^{[*]}g=0$). If $h$ runs through the whole
space ${\mathcal{H}}(T_{2})$, then formula (5.19) produces all
$g\in{\mathcal{H}}(T_{2})$ such that $F_{{}_{A}}^{[*]}g={\bf x}$. This
unconstrained interpolation problem has a solution if and only if ${\bf
x}\in\operatorname{Ran}F_{{}_{A}}^{(*)}F_{{}_{A}}^{\frac{1}{2}}$ and has a
unique solution if and only if
$\widetilde{F}_{{}_{A}}\widetilde{F}_{{}_{A}}^{[*]}=I_{{\mathcal{H}}(T_{2})}$.
If $\widetilde{F}_{{}_{T}}\widetilde{F}_{{}_{T}}^{[*]}\neq
I_{{\mathcal{H}}(T)}$, then the unconstrained problem has infinitely many
solutions, and $F_{{}_{A}}\widetilde{\bf x}$ has the minimal possible
${\mathcal{H}}(T_{2})$-norm, which is $\|\widetilde{\bf x}\|$, by (5.18).
Thus, if $\|\widetilde{\bf x}\|=1$, then uniqueness occurs since the minimal
norm solution already has unit norm.

##  Appendix A A Kreĭn space lemma

In this section we prove a general Kreĭn space lemma. See the end of the
introduction for the basic definitions of Kreĭn spaces and some classes of
Kreĭn space operators. Before we state the result, some further preliminaries
are required.

Given two Kreĭn spaces $({\mathcal{X}}^{\prime},J^{\prime})$ and
$({\mathcal{X}},J)$ and given an operator
$T\in{\mathcal{L}}({\mathcal{X}}^{\prime},{\mathcal{X}})$, we define the
$(J^{\prime},J)$-adjoint of $T$ to be the operator
$T^{[*]}\in{\mathcal{L}}({\mathcal{X}},{\mathcal{X}}^{\prime})$ so that

$[Tx,y]_{J}=[x,T^{[*]}y]_{J^{\prime}}\text{ for all
}x\in{\mathcal{X}}^{\prime},\,y\in{\mathcal{X}}.$

The elementary computation

$\displaystyle[x,T^{[*]}y]_{J^{\prime}}$ $\displaystyle=[Tx,y]_{J}=\langle
JTx,y\rangle_{\mathcal{X}}=\langle
x,T^{*}Jy\rangle_{{\mathcal{X}}^{\prime}}=\langle
J^{\prime}x,J^{\prime-1}T^{*}Jy\rangle_{{\mathcal{X}}^{\prime}}$
$\displaystyle=[x,(J^{\prime-1}T^{*}J)y]_{J^{\prime}}$

shows that

$T^{[*]}=J^{\prime-1}T^{*}J.$

In case $({\mathcal{X}}^{\prime},J^{\prime})=({\mathcal{X}},J)$ we use the
term $J$-adjoint rather than $(J,J)$-adjoint for $T^{[*]}$. If it happens that
furthermore $T=T^{[*]}$, we say that $T$ is $J$-self-adjoint. Note that
elsewhere in this paper the notation $T^{[*]}$ is also used for the adjoint
with respect to a de Branges-Rovnyak space (cf., Remark 3.2). We use the same
notation here for a Kreĭn space adjoint since this notation is standard and is
not used outside the current appendix.

Using the Kreĭn space adjoint notation we note that $T$ is a
$(J,J^{\prime})$-isometry if $T^{[*]}T=I_{{\mathcal{X}}}$. We then say that
$T$ is a $(J,J^{\prime})$-coisometry if $TT^{[*]}=I_{{\mathcal{X}}^{\prime}}$.
The latter corresponds to the identity $TJ^{\prime-1}T^{*}=J^{-1}$, which
corresponds to the second entry in (1.6) with the inequality replaced by an
equality in case $J$ and $J^{{}^{\prime}}$ are signature operators. As before,
$T$ is $(J,J^{\prime})$-unitary if $T$ is both $(J,J^{\prime})$-isometric and
$(J,J^{\prime})$-coisometric, i.e., when $T$ is invertible with
$T^{-1}=T^{[*]}$.

A well-known property for the Hilbert-space case is that a surjective isometry
is unitary. This property extends to the Kreĭn space setting, as we record in
the following remark.

######  Remark A.1.

An onto $(J^{\prime},J)$-isometry is in fact $(J^{\prime},J)$-unitary. Indeed,
suppose that $V\colon{\mathcal{X}}^{\prime}\to{\mathcal{X}}$ is an onto
$(J^{\prime},J)$-isometry. To show that $V$ is $(J^{\prime},J)$-coisometric we
must show that $VJ^{\prime-1}V^{*}=J^{-1}$. Given $x\in{\mathcal{X}}$, we can
find $y\in{\mathcal{X}}^{\prime}$ so that $x=JVy$. We then compute

$VJ^{\prime-1}V^{*}x=VJ^{\prime-1}V^{*}(JVy)=VJ^{\prime-1}(V^{*}JV)y=VJ^{\prime-1}J^{\prime}y=Vy=J^{-1}x$

implying that $V$ is also ($J,J^{\prime})$-coisometric as claimed.

Finally let us say that two invertible self-adjoint operators $J$ on
${\mathcal{X}}$ and $J^{\prime}$ on ${\mathcal{X}}^{\prime}$ are congruent if
there exists a linear bijection
$R\in{\mathcal{L}}({\mathcal{X}}^{\prime},{\mathcal{X}})$ so that
$R^{*}JR=J^{\prime}$. By taking inverses in this equation we see that if $R$
implements a congruence from $J^{\prime}$ to $J$, then $R^{*-1}$ implements a
congruence from $J^{\prime-1}$ to $J^{-1}$.

The following lemma shows how a Kreĭn-space isometry $V$ has a row extension
$\begin{bmatrix}V&W\end{bmatrix}$ to a Kreĭn space unitary operator. It is a
variation on [9, Theorem 2.3 (3)] in a more general setting. Let us also
mention that there is an extensive discussion in Section 2.3 of [31] of row
extensions of a Kreĭn-space bicontraction operator $T$ to a Kreĭn space
bicontraction operator on a larger space $\begin{bmatrix}T&S\end{bmatrix}$.
For simplicity, we give a self-contained direct proof of the special case of
interest here.

######  Lemma A.2.

Suppose that $({\mathcal{X}}^{\prime}_{1},J_{1}^{\prime})$ and
$({\mathcal{X}},J)$ are Kreĭn spaces. Let
$V\in{\mathcal{L}}({\mathcal{X}}^{\prime}_{1},{\mathcal{X}})$ be a
$(J^{\prime}_{1},J)$-isometry. Then there exist a Kreĭn spaces
$({\mathcal{X}}^{\prime}_{2},J^{\prime}_{2})$ and an operator
$W\in{\mathcal{L}}({\mathcal{X}}^{\prime}_{2},{\mathcal{X}})$ so that $J$ is
congruent with $J^{\prime}:=\left[\begin{smallmatrix}J^{\prime}_{1}&0\\\
0&J_{2}^{\prime}\end{smallmatrix}\right]$ and
$\begin{bmatrix}V&W\end{bmatrix}$ is $\big{(}J,J^{\prime}\big{)}$-unitary.
Moreover, given a Kreĭn space $({\mathcal{X}}^{\prime}_{2},J^{\prime}_{2})$,
an operator $W\in{\mathcal{L}}({\mathcal{X}}^{\prime}_{2},{\mathcal{X}})$
completes $V$ to a $(J,J^{\prime})$-unitary operator if and only if $W$ is an
injective solution to the following indefinite-metric Cholesky factorization
problem:

(A.1) $WJ_{2}^{\prime-1}W^{*}=J^{-1}-VJ_{1}^{\prime-1}V^{*}.$

###### Proof.

Since $V$ is a $(J^{\prime}_{1},J)$-isometry, we have
$V^{*}JV=J_{1}^{\prime}$. For $x,y\in{\mathcal{X}}_{1}^{\prime}$ we then have
that

$[Vx,Vy]_{J}=\langle JVx,Vy\rangle_{\mathcal{X}}=\langle
V^{*}JVx,y\rangle_{{\mathcal{X}}_{1}^{\prime}}=\langle
J_{1}^{\prime}x,y\rangle_{{\mathcal{X}}_{1}^{\prime}}=[x,y]_{J_{1}^{\prime}}.$

Hence the restriction of the indefinite inner product of $({\mathcal{X}},J)$
to ${\mathcal{M}}:=\operatorname{Ran}V$ defines a Kreĭn space on
${\mathcal{M}}$, that is, ${\mathcal{M}}$ is a regular (or Kreĭn) subspace of
${\mathcal{X}}$. Another formulation of regularity of the subspace
${\mathcal{M}}$ as a subspace of ${\mathcal{K}}$ (see e.g. [31, Theorem
1.1.1]) is that ${\mathcal{M}}$ be the range of a $J$-orthogonal projection
operator, i.e., a bounded $J$-self-adjoint idempotent operator on
${\mathcal{X}}$. Explicitly for the case here with
${\mathcal{M}}=\operatorname{Ran}V$, one can check that
${\mathcal{M}}=\operatorname{Ran}P$ where the $J$-orthogonal projection
operator $P$ is given by

$P=VJ_{1}^{\prime-1}V^{*}J,$

i.e., one can check that $P=J^{-1}P^{*}J=P^{[*]}$, $P^{2}=P$ and $PV=V$
(implying when combined with the definition of $P$ that
$\operatorname{Ran}P=\operatorname{Ran}V$).

Another characterization of regularity of ${\mathcal{M}}$ (see e.g. Theorem
1.3 in [32]) is that ${\mathcal{X}}$ is obtained as the direct sum of
${\mathcal{M}}$ and the Kreĭn space orthogonal complement
${\mathcal{M}}^{\perp_{J}}:=\\{x\in{\mathcal{X}}\colon[x,y]_{J}=0\text{ for
all }y\in{\mathcal{M}}\\}$ of ${\mathcal{M}}$ in ${\mathcal{X}}$. Explicitly
the associated $J$-orthogonal projection operator $Q$ such that
$\operatorname{Ran}Q={\mathcal{M}}^{\perp_{J}}$ is given as the $J$-orthogonal
projection operator $Q$ $J$-complementary to $P$, namely

(A.2) $Q=I_{\mathcal{X}}-P=I_{\mathcal{X}}-VJ_{1}^{\prime-1}V^{*}J.$

Indeed one can check directly that

$Q=Q^{[*]},\quad Q^{2}=Q,\quad QV=0,\quad P+Q=I_{\mathcal{X}}.$

We can get another formula for the $J$-orthogonal projection operator with
range equal to ${\mathcal{M}}^{\perp_{J}}$ as follows. Set
$\widetilde{\mathcal{X}}^{\prime}_{2}={\mathcal{M}}^{\perp_{J}}$ and let
$\widetilde{W}\colon\widetilde{\mathcal{X}}_{2}\to{\mathcal{X}}$ be the
inclusion map of $\widetilde{\mathcal{X}}^{\prime}_{2}$ into ${\mathcal{X}}$.
Hence by construction $\widetilde{W}$ is injective. Define
$\widetilde{J}^{\prime}_{2}$ by

$\widetilde{J}^{\prime}_{2}=\widetilde{W}^{*}J\widetilde{W}.$

Since $\operatorname{Ran}\widetilde{W}={\mathcal{M}}^{\perp_{J}}$ is a regular
subspace of ${\mathcal{X}}$, it follows that $\widetilde{J}^{\prime}_{2}$ is
an invertible self-adjoint operator on $\widetilde{\mathcal{X}}^{\prime}_{2}$.
By an argument parallel to that done above for the operator $V$ having
$\operatorname{Ran}V={\mathcal{M}}$, we see that $Q$ (the $J$-orthogonal
projection operator with range equal to ${\mathcal{M}}^{\perp_{J}}$) is given
by

(A.3) $Q=\widetilde{W}\widetilde{J}_{2}^{\prime-1}\widetilde{W}^{*}J.$

Combining (A.2) with (A.3) gives us
$\widetilde{W}\widetilde{J}_{2}^{\prime-1}\widetilde{W}^{*}J=I_{\mathcal{X}}-VJ_{1}^{\prime-1}V^{*}J$,
or in more Hermitian form,

$\widetilde{W}\widetilde{J}_{2}^{\prime-1}\widetilde{W}^{*}=J^{-1}-VJ_{1}^{\prime-1}V^{*},$

i.e., $\widetilde{W}$ arises as an injective solution of an indefinite-metric
Cholesky factorization problem of the type (A.1).

Since $\operatorname{Ran}V={\mathcal{M}}$ is $J$-orthogonal to
$\operatorname{Ran}\widetilde{W}={\mathcal{M}}^{\perp_{J}}$, we certainly have

$0=\widetilde{W}^{*}JV,\quad 0=V^{*}J\widetilde{W}.$

Let us next compute

$\begin{bmatrix}V^{*}\\\
\widetilde{W}^{*}\end{bmatrix}J\begin{bmatrix}V&\widetilde{W}\end{bmatrix}=\begin{bmatrix}V^{*}JV&V^{*}J\widetilde{W}\\\
\widetilde{W}^{*}JV&\widetilde{W}^{*}J\widetilde{W}\end{bmatrix}=\begin{bmatrix}J_{1}^{\prime}&0\\\
0&\widetilde{J}_{2}^{\prime}\end{bmatrix},$

i.e., $\begin{bmatrix}V&\widetilde{W}\end{bmatrix}$ is a
$\big{(}\left[\begin{smallmatrix}J_{1}^{\prime}&0\\\
0&\widetilde{J}_{2}^{\prime}\end{smallmatrix}\right],J\big{)}$-isometry. But
we also know that
$\operatorname{Ran}\begin{bmatrix}V&\widetilde{W}\end{bmatrix}={\mathcal{M}}+{\mathcal{M}}^{\perp_{J}}={\mathcal{X}}$,
so by Remark A.1 it follows that in fact
$\begin{bmatrix}V&\widetilde{W}\end{bmatrix}$ is a
$\big{(}\left[\begin{smallmatrix}J_{1}^{\prime}&0\\\
0&\widetilde{J}_{2}^{\prime}\end{smallmatrix}\right],J\big{)}$-unitary
operator, and we conclude that $\widetilde{W}$ implements the desired
completion of $V$ to a Kreĭn space unitary operator.

Now suppose that $W$ is any injective solution of the $J$-Cholesky
factorization problem (A.1). As observed in the first part of the proof, the
operator to be factored on the right-hand side of (A.1) multiplied by $J$ on
the right, namely,
$(J^{-1}-VJ_{1}^{\prime-1}V^{*})J=I_{\mathcal{X}}-VJ_{1}^{\prime-1}V^{*}J$, is
equal to the $J$-orthogonal projection operator with range equal to
$(\operatorname{Ran}V)^{\perp_{J}}$, and hence has a fairly substantial
kernel, namely

$\operatorname{Ker}\,(I_{\mathcal{X}}-VJ_{1}^{\prime-1}V^{*}J)=\operatorname{Ran}V.$

Hence the Hermitian operator $J^{-1}-VJ_{1}^{\prime-1}V^{*}$ has closed range
and its restriction to
$(\operatorname{Ker}(J^{-1}-VJ_{1}^{\prime-1}V^{*}))^{\perp}$ is bounded
below. Using spectral theory we can then find a factorization

(A.4) $J^{-1}-VJ_{1}^{\prime-1}V^{*}=WJ_{2}^{\prime-1}W^{*}$

with $J_{2}^{\prime}$ a bounded, invertible signature operator on
${\mathcal{X}}^{\prime}_{2}=(\operatorname{Ker}(J^{-1}-VJ_{1}^{\prime-1}V^{*}))^{\perp}$,
i.e., the pair $(W,J^{\prime}_{2})$ is a solution of the $J$-Cholesky
factorization problem (A.1) with $W$ injective. Note that any other choice of
$J_{2}$ in such a factorization is then uniquely determined up to a congruence
with any particular such $J_{2}^{\prime}$, with a corresponding adjustment of
$\widetilde{W}$ to get $W$:

$J^{\prime}_{2}=X\widetilde{J}^{\prime}_{2}X^{*},\quad
W=\widetilde{W}X^{-1}\text{ for some invertible
}X\colon\widetilde{\mathcal{X}}^{\prime}_{2}\to{\mathcal{X}}^{\prime}_{2}.$

The equality (A.4) then can be interpreted as saying that
$\operatorname{Ran}W={\mathcal{M}}^{\perp_{J}}$ (where
${\mathcal{M}}=\operatorname{Ran}V\subset{\mathcal{X}}$). We may then proceed
as in the first part of the proof to see that
$\begin{bmatrix}V&W\end{bmatrix}$ is
$\big{(}\left[\begin{smallmatrix}J_{1}^{\prime}&0\\\
0&J_{2}^{\prime}\end{smallmatrix}\right],J\big{)}$-unitary as expected. ∎

The following corollary adds some additional information which will be needed
to complete the proof of Theorem 3.14 in Section 3.4. We shall make use of the
following notation: for an invertible self-adjoint operator $J$ on some
separable Hilbert space ${\mathcal{X}}$,

$\operatorname{Inertia}(J)=(\kappa_{+}(J),\kappa_{-}(J))$

means that $\kappa_{+}(J)$ and $\kappa_{-}(J)$ denote the respective
dimensions (with $\infty$ allowed) of the positive (respectively negative)
spectral subspaces of $J$. Note that two Kreĭn spaces $({\mathcal{X}},J)$ and
$({\mathcal{X}}^{\prime},J^{\prime})$ are Kreĭn-space isomorphic if and only
if $\operatorname{Inertia}(J)=\operatorname{Inertia}(J^{\prime})$.

######  Corollary A.3.

Suppose that $({\mathcal{X}}_{1},J_{1})$, $({\mathcal{X}}_{2},J_{2})$,
$({\mathcal{X}}^{\prime}_{1},J^{\prime}_{1})$,
$({\mathcal{X}}^{\prime}_{2},J^{\prime}_{2})$ are Kreĭn spaces such that

  * (i)

$\big{(}{\mathcal{X}}_{1}\oplus{\mathcal{X}}_{2},\left[\begin{smallmatrix}J_{1}&0\\\
0&J_{2}\end{smallmatrix}\right]\big{)}$ is Kreĭn-space isomorphic to
$\big{(}{\mathcal{X}}^{\prime}_{1}\oplus{\mathcal{X}}^{\prime}_{2},\left[\begin{smallmatrix}J^{\prime}_{1}&0\\\
0&J^{\prime}_{2}\end{smallmatrix}\right]\big{)}$, and

  * (ii)

$({\mathcal{X}}_{1},J_{1})$ is Kreĭn-space isomorphic to
$({\mathcal{X}}^{\prime}_{1},J^{\prime}_{1})$.

Then:

  1. (1)

If $\dim{\mathcal{X}}_{j}<\infty$ and $\dim{\mathcal{X}}^{\prime}_{j}<\infty$
for $j=1,2$, then $({\mathcal{X}}_{2},J_{2})$ is Kreĭn-space isomorphic to
$({\mathcal{X}}^{\prime}_{2},J^{\prime}_{2})$, i.e.,

$\kappa_{+}(J_{2})=\kappa_{+}(J^{\prime}_{2})\quad\text{and}\quad\kappa_{-}(J_{2})=\kappa_{-}(J^{\prime}_{2}).$

  2. (2)

If $\kappa_{-}(J_{1})=\kappa_{-}(J_{1})=0$, i.e., $(X_{1},J_{1})$ and
$(X^{\prime}_{1},J^{\prime}_{1})$ are Hilbert spaces of the same dimension,
then we can at least still conclude that

$\displaystyle\kappa_{+}(J_{1})+\kappa_{+}(J_{2})$
$\displaystyle=\kappa_{+}(J^{\prime}_{1})+\kappa_{+}(J^{\prime}_{2}),$
$\displaystyle\kappa_{-}(J_{2})$ $\displaystyle=\kappa_{-}(J^{\prime}_{2}).$

###### Proof.

In case (1), all the indices $\kappa_{\pm}(J_{j})$and
$\kappa_{\pm}(J^{\prime}_{j})$ ($j=1,2$) are finite. Note that hypothesis (i)
is the statement that

$\displaystyle\kappa_{+}(J_{1})+\kappa_{+}(J_{2})=\kappa_{+}(J^{\prime}_{1})+\kappa_{+}(J^{\prime}_{2}),$
(A.5)
$\displaystyle\kappa_{-}(J_{1})+\kappa_{-}(J_{2})=\kappa_{-}(J^{\prime}_{1})+\kappa_{-}(J^{\prime}_{2}),$

while the hypothesis in (ii) is that

(A.6)
$\kappa_{+}(J_{1})=\kappa_{+}(J^{\prime}_{1}),\quad\kappa_{-}(J_{1})=\kappa_{-}(J^{\prime}_{1}).$

Subtraction of the first of (A.6) from the first of (A.5) and of the second of
(A.6) from the second of (A.5) and using that all these integers are finite
leads us to the two identities

$\kappa_{+}(J_{2})=\kappa_{+}(J^{\prime}_{2}),\quad\kappa_{-}(J_{2})=\kappa_{-}(J^{\prime}_{2})$

which in turn is that statement that $({\mathcal{X}}_{2},J_{2})$ is Kreĭn-
space isomorphic to $({\mathcal{X}}^{\prime}_{2},J^{\prime}_{2})$.

Take now the extra hypothesis to be as in case (2). In this case the quantity
$\kappa_{+}(J_{1})+\kappa_{+}(J_{2})$ can be identified as the dimension of a
maximal positive subspace in
$\big{(}{\mathcal{X}}_{1}\oplus{\mathcal{X}}_{2},\left[\begin{smallmatrix}J_{1}&0\\\
0&J_{2}\end{smallmatrix}\right]\big{)}$ while the quantity
$\kappa_{+}(J^{\prime}_{1})+\kappa_{+}(J^{\prime}_{2})$ can be identified as
the dimension of a maximal positive subspace in
$\big{(}{\mathcal{X}}^{\prime}_{1}\oplus{\mathcal{X}}^{\prime}_{2},\left[\begin{smallmatrix}J^{\prime}_{1}&0\\\
0&J^{\prime}_{2}\end{smallmatrix}\right]\big{)}$. The assumption that
$\big{(}{\mathcal{X}}_{1}\oplus{\mathcal{X}}_{2},\left[\begin{smallmatrix}J_{1}&0\\\
0&J_{2}\end{smallmatrix}\right]\big{)}$ is Kreĭn-space isomorphic to
$\big{(}{\mathcal{X}}^{\prime}_{1}\oplus{\mathcal{X}}^{\prime}_{2},\left[\begin{smallmatrix}J^{\prime}_{1}&0\\\
0&J^{\prime}_{2}\end{smallmatrix}\right]\big{)}$ then implies that these two
quantities must be the same (finite or infinite). Similarly, one can identify
$\kappa_{-}(J_{2})$ as the dimension of a maximal negative subspace in
$\big{(}{\mathcal{X}}_{1}\oplus{\mathcal{X}}_{2},\left[\begin{smallmatrix}J_{1}&0\\\
0&J_{2}\end{smallmatrix}\right]\big{)}$ (since ${\mathcal{X}}_{1}$ is a
Hilbert space by assumption) and similarly $\kappa_{-}(J^{\prime}_{2})$ is the
dimension of a maximal negative subspace in
$\big{(}{\mathcal{X}}^{\prime}_{1}\oplus{\mathcal{X}}^{\prime}_{2},\left[\begin{smallmatrix}J^{\prime}_{1}&0\\\
0&J^{\prime}_{2}\end{smallmatrix}\right]\big{)}$. Again, the assumed Kreĭn-
space isomorphism between
$\big{(}{\mathcal{X}}_{1}\oplus{\mathcal{X}}_{2},\left[\begin{smallmatrix}J_{1}&0\\\
0&J_{2}\end{smallmatrix}\right]\big{)}$ and
$\big{(}{\mathcal{X}}^{\prime}_{1}\oplus{\mathcal{X}}^{\prime}_{2},\left[\begin{smallmatrix}J^{\prime}_{1}&0\\\
0&J^{\prime}_{2}\end{smallmatrix}\right]\big{)}$ implies that we must have the
equality $\kappa_{-}(J_{2})=\kappa_{-}(J^{\prime}_{2})$ (finite or infinite).
∎

##  Appendix B Schur multipliers and their adjoints

As is well known, any Schur-class function
$S\in\mathcal{S}({\mathcal{U}},{\mathcal{Y}})$ gives rise to a Schur-class
function
$S^{\sharp}(\lambda):=S(\overline{\lambda})^{*}\in\mathcal{S}({\mathcal{Y}},{\mathcal{U}})$.
As a consequence of (3.45) and (3.49), it follows that a similar result holds
in multivariable settings.

######  Theorem B.1.

If
$S(z)={\displaystyle\sum_{\alpha\in{\mathbb{F}}^{+}_{d}}S_{\alpha}z^{\alpha}}$
belongs to ${\mathcal{S}}_{{\rm nc},d}({\mathcal{U}},{\mathcal{Y}})$, then
$S^{\sharp}(z):={\displaystyle\sum_{\alpha\in{\mathbb{F}}^{+}_{d}}S^{*}_{\alpha}z^{\alpha^{\top}}}$
belongs to ${\mathcal{S}}_{{\rm nc},d}({\mathcal{Y}},{\mathcal{U}})$.

###### Proof.

Since $S\in\mathcal{S}({\mathcal{U}},{\mathcal{Y}})$, i.e., the formal kernel
(3.28) is positive, it follows (see e.g., [15, Theorem 3.1]) that there exists
a Hilbert space ${\mathcal{X}}$ and a unitary connection operator
${\mathbf{U}}$ of the form

(B.1) ${\mathbf{U}}=\begin{bmatrix}A&B\\\
C&D\end{bmatrix}=\begin{bmatrix}A_{1}&B_{1}\\\ \vdots&\vdots\\\ A_{d}&B_{d}\\\
C&D\end{bmatrix}\colon\begin{bmatrix}{\mathcal{X}}\\\
{\mathcal{U}}\end{bmatrix}\to\begin{bmatrix}{\mathcal{X}}\\\ \vdots\\\
{\mathcal{X}}\\\ {\mathcal{Y}}\end{bmatrix}$

so that $S(z)$ can be realized as a formal power series in the form

$\displaystyle S(z)$
$\displaystyle=D+\sum_{j=1}^{d}\sum_{\alpha\in{\mathbb{F}}^{+}_{d}}C{\bf
A}^{\alpha}B_{j}z^{\alpha}\cdot z_{j}=D+C(I-Z(z)A)^{-1}Z(z)B,$

Then

$\displaystyle S^{\sharp}(z)$
$\displaystyle=D^{*}+\sum_{j=1}^{d}\sum_{\alpha\in{\mathbb{F}}^{+}_{d}}B_{j}^{*}{\bf
A}^{*\alpha^{\top}}C^{*}z_{j}z^{\alpha^{\top}}$
$\displaystyle=D^{*}+\sum_{j=1}^{d}\sum_{\alpha\in{\mathbb{F}}^{+}_{d}}B_{j}^{*}z_{j}{\bf
A}^{*\alpha}C^{*}z^{\alpha}$
$\displaystyle=D^{*}+B^{*}Z(z)^{\top}(I-A^{*}Z(z)^{\top})^{-1}C^{*}=D^{*}+B^{*}(I-Z(z)^{\top}A^{*})^{-1}Z(z)^{\top}C^{*},$

and since ${\bf U}$ is unitary, we have

(B.2) $\displaystyle K_{S^{\sharp}}(z,\zeta):=k_{\rm
Sz}(z,\zeta)I_{{\mathcal{U}}}-S^{\sharp}(z)(k_{\rm
Sz}(z,\zeta)I_{{\mathcal{U}}})S^{\sharp}(\zeta)^{*}$
$\displaystyle=B^{*}(I-Z(z)^{\top}A^{*})^{-1}\big{(}k_{\rm
Sz}(z,\zeta)I_{{\mathcal{X}}^{d}}-Z(z)^{\top}k_{\rm
Sz}(z,\zeta)Z(\overline{\zeta})\big{)}(I-AZ(\overline{\zeta}))^{-1}B.$

As we have seen in the proof of Theorem 3.14, the kernel

$k_{\rm Sz}(z,\zeta)I_{{\mathcal{X}}^{d}}-Z(z)^{\top}k_{\rm
Sz}(z,\zeta)Z(\overline{\zeta})$

is positive; therefore the kernel $K_{S^{\sharp}}(z,\zeta)$ is also positive
and hence $S^{\sharp}\in{\mathcal{S}}_{{\rm
nc},d}({\mathcal{Y}},{\mathcal{U}})$. ∎

In [7, p. 110], it was erroneously suggested that for an
$S\in\mathcal{S}_{d}({\mathcal{U}},{\mathcal{Y}})$, the function $S^{\sharp}$
does not have to belong to $\mathcal{S}_{d}({\mathcal{Y}},{\mathcal{U}})$. It
was stated that for the Schur multiplier
$S(\lambda)=\begin{bmatrix}\lambda_{1}&\lambda_{2}\end{bmatrix}$, the
associated function
$S^{\sharp}(\lambda)=\left[\begin{smallmatrix}\lambda_{1}\\\
\lambda_{2}\end{smallmatrix}\right]$ is not a Schur multiplier, since the
kernel

(B.3) $K^{S^{\sharp}}(\lambda,w)=\frac{\left[\begin{array}[]{cc}1&0\\\
0&1\end{array}\right]-\left[\begin{array}[]{c}\lambda_{1}\\\
\lambda_{2}\end{array}\right]\begin{bmatrix}\overline{w}_{1}&\overline{w}_{2}\end{bmatrix}}{1-\lambda_{1}\overline{w}_{1}-\lambda_{2}\overline{w}_{2}}$

is not positive on ${\mathbb{B}}^{d}$, as the matrix

$\left[\begin{array}[]{cc}K^{S^{\sharp}}(w^{(1)},w^{(1)})&K^{S^{\sharp}}(w^{(1)},w^{(2)})\\\
K^{S^{\sharp}}(w^{(2)},w^{(1)})&K^{S^{\sharp}}(w^{(2)},w^{(2)})\end{array}\right],\quad
w^{(1)}=\left(\frac{1}{2},0\right),\;\;w^{(1)}=\left(0,\frac{1}{2}\right)$

is not positive semidefinite. More careful later inspection revealed that the
latter matrix equals

$\left[\begin{array}[]{rccr}1&0&1&-\frac{1}{4}\\\ 0&\frac{4}{3}&0&1\\\
1&0&\frac{4}{3}&0\\\ -\frac{1}{4}&1&0&1\end{array}\right]$

and is positive definite. Moreover, the kernel (B.3) is actually positive on
$\mathbb{B}^{2}$ by the following abelianized version of Theorem B.1.

######  Theorem B.2.

If $S(\lambda)$ belongs to ${\mathcal{S}}_{d}({\mathcal{U}},{\mathcal{Y}})$,
then $S^{\sharp}(\lambda):=S(\overline{\lambda})^{*}$ belongs to
${\mathcal{S}}_{d}({\mathcal{Y}},{\mathcal{U}})$.

##### Acknowledgments

This work is based on research supported in part by the National Research
Foundation of South Africa (NRF) and the DST-NRF Centre of Excellence in
Mathematical and Statistical Sciences (CoE-MaSS). Any opinion, finding and
conclusion or recommendation expressed in this material is that of the authors
and the NRF and CoE-MaSS do not accept any liability in this regard.

## References

  * [1] J. Agler and J. McCarthy, Complete Nevanlinna-Pick kernels, J. Func. Anal. 175 (2000) no. 1, 111-124.
  * [2] D. Alpay and V. Bolotnikov, On tangential interpolation in reproducing kernel Hilbert modules and applications, in: Topics in interpolation theory, pp. 37–68, Oper. Theory Adv. Appl. 95, Birkhäuser, Basel, 1997.
  * [3] D. Alpay, V. Bolotnikov and H.T. Kaptanoğlu, The Schur algorithm and reproducing kernel Hilbert spaces in the ball, Linear Algebra Appl. 343 (2002), 163–186.
  * [4] A. Arias and G. Popescu, Noncommutative interpolation and Poisson transforms, Israel J. Math. 115 (2000), 205–234.
  * [5] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950), 337–404.
  * [6] D.Z. Arov and H. Dym, $J$-contractive matrix valued functions and related topics, Encyclopedia of Mathematics and its Applications 116, Cambridge University Press, Cambridge, 2008.
  * [7] J.A. Ball and V. Bolotnikov, _On a bitangential interpolation problem for contractive-valued functions on the unit ball_ , Linear Algebra Appl. 353 (2002), 107–147.
  * [8] J.A. Ball and V. Bolotnikov, _A bitangential interpolation problem on the closed unit ball for multipliers of the Arveson space_ , Integr. Equ. Oper. Theory 46 (2003), 125–164.
  * [9] J.A. Ball and V. Bolotnikov, Interpolation problems for Schur multipliers on the Drury-Arveson space: from Nevanlinna-Pick to Abstract Interpolation Problem, Integr. Equ. Oper. Theory 62 (2008), 301–349.
  * [10] J.A. Ball and V. Bolotnikov, Interpolation in Sub-Bergman Spaces, in: Advances in Structured Operator Theory and Related Areas, pp. 17–39, Oper. Theory Adv. Appl. 237, Birkhäuser, Basel, 2013.
  * [11] J.A. Ball and V. Bolotnikov, The bitangential matrix Nevanlinna-Pick interpolation problem revisited, in: Indefinite inner product spaces, Schur analysis, and differential equations, pp. 107–161, Oper. Theory Adv. Appl. 263, Birkhäuser, Cham, 2018.
  * [12] J.A. Ball and V. Bolotnikov, Interpolation by contractive multipliers between Fock spaces, in: Complex function theory, operator theory, Schur analysis and systems theory, 79–138, Oper. Theory Adv. Appl. 280, Birkhäuser, Cham, 2020.
  * [13] J.A. Ball and V. Bolotnikov, Noncommutative function-theoretic operator theory and applications, Cambridge Tracts in Mathematics 225, Cambridge University Press, Cambridge, 2022.
  * [14] J.A. Ball, V. Bolotnikov, and Q. Fang, Multivariable backward-shift-invariant subspaces and observability operators, Multidimens. Syst. Signal Process. 18 (2007), 191–248.
  * [15] J.A. Ball, V. Bolotnikov, and Q. Fang, Schur-class multipliers on the Fock space: de Branges-Rovnyak reproducing kernel spaces and transfer-function realizations, Operator theory, structured matrices, and dilations, 85–114, Theta Ser. Adv. Math., 7, Theta, Bucharest, 2007.
  * [16] J.A. Ball, V. Bolotnikov, and Q. Fang, Schur-class multipliers on the Arveson space: De Branges-Rovnyak reproducing kernel spaces and commutative transfer-function realizations, J. Math. Anal. Appl. 341 (2008), 519–539.
  * [17] J.A. Ball, V. Bolotnikov and S. ter Horst, Interpolation in de Branges-Rovnyak spaces, Proc. Amer. Math. Soc. 139 (2011), 609-618.
  * [18] J.A. Ball, V. Bolotnikov and S. ter Horst, Abstract interpolation in vector-valued de Branges-Rovnyak spaces, Integr. Equ. Oper. Theory 70 (2011), 227-263.
  * [19] J.A. Ball, I. Gohberg, and L. Rodman, _Interpolation of rational matrix functions_ , Oper. Theory Adv. Appl. 45, Birkhäuser Verlag, Basel, 1990.
  * [20] J.A. Ball and M.W. Raney, Discrete-time dichotomous well-posed linear systems and generalized Schur-Nevanlinna-Pick interpolation, Complex Anal. Oper. Theory 1 (2007), 1–54.
  * [21] J.A. Ball, T.T. Trent, and V. Vinnikov, Interpolation and commutant lifting for multipliers on reproducing kernel Hilbert spaces, in: Operator theory and analysis (Amsterdam, 1997), pp. 89–138, Oper. Theory Adv. Appl. 122, Birkhäuser, Basel, 2001.
  * [22] J.A. Ball and V. Vinnikov, Zero-pole interpolation for meromorphic matrix functions on an algebraic curve and transfer functions of 2D systems, Acta Applicandae Mathematicae 45 (1996), 239-316.
  * [23] J.A. Ball and V. Vinnikov, Zero-pole interpolation for matrix meromorphic functions on a compact Riemann surface and a matrix Fay trisecant identity, Amer. J. Math. 121 no. 4 (1999), 841-888.
  * [24] J.A. Ball and V. Vinnikov, _Formal reproducing kernel Hilbert spaces: the commutative and noncommutative settings_ , in: Reproducing Kernel Spaces and Applications, pp. 77–134, Oper. Theory Adv. Appl. 143, Birkhäuser, Basel, 2003.
  * [25] F. Beatrous and J. Burbea, _Positive-definiteness and its applications to interpolation problems for holomorphic functions_ , Trans. Amer. Math. Soc. 284 (1984), 247–270.
  * [26] J. Bognár, Indefinite inner product spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete 78, Springer-Verlag, New York-Heidelberg, 1974.
  * [27] L. de Branges and J. Rovnyak, _Square Summable Power Series_ , Holt, Rinehart and Winston, New-York, 1966.
  * [28] K. Davidson and D. Pitts, Nevanlinna-Pick interpolation for non-commutative analytic Toeplitz algebras, Integr. Equ. Oper. Theory 31 (1998), 321–337.
  * [29] B. Davis and J.E. McCarthy, Multipliers of de Branges spaces, Michigan Math. J. 38 (1991), 225–240.
  * [30] R.G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413–415.
  * [31] M. A. Dritschel and J. Rovnyak, Extension theorems for contraction operators on Kreĭn spaces, in: Extension and interpolation of linear operators and matrix functions, pp. 221–305, Oper. Theory Adv. Appl. 47, Birkhäuser, Basel, 1990.
  * [32] M. A. Dritschel and J. Rovnyak, Operators on indefinite inner product spaces, in: Lectures on operator theory and its applications (Waterloo, ON, 1994), pp. 141–232, Fields Inst. Monogr. 3, Amer. Math. Soc., Providence, RI, 1996.
  * [33] C. Foias and A. E. Frazho, The Commutant Lifting Approach to Interpolation Problems, Oper. Theory Adv. Appl. 44, Birkhäuser Verlag, Basel, 1990.
  * [34] C. Foias, A.E. Frazho, I. Gohberg, and M.A. Kaashoek, Metric constrained interpolation, commutant lifting and systems, Oper. Theory Adv. Appl. 100, Birkhäuser Verlag, Basel, 1998.
  * [35] A.E. Frazho, S. ter Horst, and M.A. Kaashoek, All solutions to an operator Nevanlinna-Pick interpolation problem, in: Operator theory in different settings and related applications, pp. 139–220, Oper. Theory Adv. Appl. 262, Birkhäuser, Cham, 2018.
  * [36] H. Gross, Quadratic forms in infinite-dimensional vector spaces, Progress in Mathematics 1, Birkhäuser, Boston, Mass., 1979.
  * [37] C.R. Johnson, Matrix completion problems: a survey, in: Matrix Theory and Applications (Phoenix, AZ, 1989), pp. 171–198, Proc. Sympos. Appl. Math. 40, AMS Short Course Lecture Notes, Amer. Math. Soc., Providence, RI, 1990.
  * [38] B. Lotto and D. Sarason, Multipliers in de Branges-Rovnyak spaces, Indiana Uni. Math. J. 42 (1993), 907–920.
  * [39] S. McCullough and T.T. Trent, Invariant subspaces and Nevanlinna-Pick kernels, J. Funct. Anal. 178 (2000), 226–249.
  * [40] B. Sz.-Nagy, C. Foias, H. Bercovici, L. Kérchy, Harmonic Analysis of Operators on Hilbert Space, revised and enlarged edition, Universitext, Springer, New York (2010).
  * [41] G. Popescu, Isometric dilations for infinite sequences of noncommuting operators, Trans. Amer. Math. Soc. 316 (1989), 523–536.
  * [42] M. Rosenblum and J. Rovnyak, _Hardy Classes and Operator Theory_ , Oxford University Press, 1985.
  * [43] D. Sarason, Sub-Hardy Hilbert Spaces in the Unit Disk, Wiley, New York, 1994.