# Investigation of target in C-ADS and IAEA ADS benchmark 111Supported by
National Natural Science Foundation of China (11045003 & 10975150)

Guojun Hu<EMAIL_ADDRESS>Hongli Wu  Tian Jing
Xiangqi Wang333Corresponding author<EMAIL_ADDRESS>Jingyu Tang
National Synchrotron Radiation Laboratory,USTC  School of Nuclear Science and
Technology, USTC  Institute of High Energy Physics Chinese Academy of Sciences
Department of Modern Physics, USTC

###### Abstract

The spatial and energy distribution of spallation neutrons have an effect on
the performance of Accelerator Driven Subcritical systems. In this work, the
spatial, energy distribution of spallation neutrons and the effect of these
factors on proton efficiency was studied. When the radius of spallation region
increases, backward neutrons were found to have a rather big ratio and have a
positive effect on proton efficiency. By making better use of these neutrons,
we may increase the radius of target to satisfy some other requirements in the
design of subcritical core.

###### keywords:

C-ADS, Spallation target, Backward neutrons

††journal: arXiv.org

##  1 Introduction

China ADS (C-ADS) consists of a high-intensity proton accelerator in CW
(continuous wave) mode with proton energy of 0.6 GeV in phase three and 1.5
GeV in phase four, a subcritical core, and a spallation target of liquid Lead-
Bismuth Eutectic (LBE or Pb-Bi), which is favored in projects for its low
melting point, high boiling point and low neutron absorption cross section
[1].

The target should have such a size that it incepts the main part of the high-
energy cascade and then we get a optimized neutron multiplicity[4]. Neutron
multiplicity is defined to be the number of neutrons produced per beam
particle. Increasing the size of target will affect the spatial and energy
distribution of spallation neutrons and thus affecting the performance of the
total system. So in this work, we discussed the effect of the spatial and
energy distribution of spallation neutrons.

Neutron multiplicity, spatial and energy distribution were simulated with
FLUKA [5, 6] in this work. In these simulations, Intra-nuclear cascade, pre-
equilibrium, evaporation and fission models were activated. Proton efficiency
of the system was partly simulated with RMC(Reactor Monte Carlo code)[7, 8].
IAEA ADS benchmark [9] was an ADS model and was used in these simulations,
shown in Fig. 2.

##  2 Validation of FLUKA

Double-differential distribution of spallation neutrons were measured in the
Laboratoire National Sturne experiment [10]. In order to observe the
availability of FLUKA in simulating spallation reaction, this experiment was
simulated with FLUKA.

Table 1: Parameters of Saturne experiment[10] Beam | Target | Detectors
---|---|---
Projectile | Proton | Material | Lead | $R$ | 800 cm
$E_{\rm{p}}$ | 1.6 GeV | $R_{\rm{s}}$ | 1.5 cm | $dS$ | 1963.5 $\rm{cm}^{2}$
Direction | (0,0,1) | $N$ | $3.30\times 10^{28}\rm{m^{-3}}$ | $d\Omega$ | $3.068\times 10^{-3}\rm{sr}$
$R_{\rm{b}}$ | 1.4 cm | $L_{\rm{s}}$ | 2.0 cm |  |

Fig. 1: Double-differential neutron production cross-section. Data of
different polar angle had been scaled with different ratio marked in the
figure. $1\sigma<5\%$

Parameters of target, beam and detectors in this simulation are listed in
Table 1. Detectors are in a sphere surface with radius of $R=800\ \rm{cm}$.
Detectors are set in polar angle of
$\theta=0^{\circ},10^{\circ},55^{\circ}$,$85^{\circ},130^{\circ},160^{\circ}$.
Double-differential cross-section of neutron production is obtained with
Eq.(1):

$\frac{{\rm d}^{2}\sigma}{{\rm d}\Omega{\rm d}E}=\frac{{\rm
d}N_{\rm{ns}}}{N_{\rm{primary}}\frac{{\rm d}S}{R^{2}}{\rm
d}E}\frac{1}{NL_{\rm{s}}}=\frac{{\rm d}N_{\rm{ns}}}{N_{\rm{primary}}{\rm
d}S{\rm d}E}R^{2}\frac{1}{NL_{\rm{s}}}$ (1)

Where, ${\rm d}S$ , ${\rm d}\Omega={\rm d}S/R^{2}$ , $N$ and $L_{\rm{s}}$ is
area of detector, solid angle, atomic density and thickness of target in Table
1. ${\rm d}N_{\rm{ns}}/({N_{\rm{primary}}{\rm d}S{\rm d}E})$ is scored with
USRBDX card.

Comparison between this simulation and the experiment is shown in Fig. 1. Data
of the experiment was obtained from EXFOR/CRISRS. The result of FLUKA agrees
well with the experiment. There is difference in high energy due to the defect
of physical models and the statistic error of simulation. FlUKA is suitable to
simulate the spallation reaction.

##  3 Description of ADS benchmark and proton efficiency

###  3.1 Description of IAEA ADS benchmark

IAEA ADS benchmark was chosen to simulate the effect of spallation neutrons.
Geometry and parameters of IAEA ADS benchmark is show in Fig. 2 and Table 2.
In regions ”core1” and ”core2”, the ratio between 233-U and 232-Th is 1:9 and
effective multiplication factor is: $k_{\rm{eff}}=0.96455\pm 0.00062$.

Fig. 2: RZ view of IAEA ADS benchmark. The different regions are:(1)Target;
(2)Buffer; (3)Core1; (4)Core2; (5)Core3; (6)Reflector; (7)Beam pipe;
(8),(10)Moderator Table 2: Parameters of ADS benchmark[9] The element of IAEA
ADS benchmark at BOL (20 ∘C) $10^{24}\rm{cm}^{-3}$
---
Nuclides | Core1 | Core2 | Core3 | Reflector | Moderator, Target, Buffer
232-Th |  |  | 7.45 |  |
232-Th,233-U | 6.350 | 7.450 |  |  |
O | 12.70 | 14.90 | 14.90 |  |
Fe | 8.100 | 8.870 | 8.870 | 6.630 |
Cr | 1.120 | 1.060 | 1.060 | 0.800 |
Mn | 0.046 | 0.051 | 0.051 | 0.038 |
W | 0.046 | 0.051 | 0.051 | 0.038 |
Pb | 17.70 | 15.60 | 15.60 | 0.024 | 30.50
The source of IAEA ADS benchmark
Projectile | Proton
Kinetic Energy:$E_{\rm{p}}$ | 1.0 GeV
Beam radius:$R_{\rm{b}}$ | 10.0 cm

Liquid Lead is the material of moderator and target in ADS benchmark.
Spallation region is bounded by the subcritical core and beam pipe. We divided
the spallation region into ”target” and ”buffer” as used in ”Energy
Amplifier”[2]. $R_{\rm{s}}$ and $L_{\rm{s}}$ is the size of spallation region,
see Fig. 2. In simulations of FLUKA, except for region ”target”, all other
regions were filled with vacuum.

###  3.2 Neutron source efficiency and proton efficiency

Neutron flux $\phi_{\rm{s}}$ in the subcritical core is the solution to the
inhomogeneous steady-state neutron transport equation Eq.(2):

$A\phi_{\rm{s}}=F\phi_{\rm{s}}+S_{\rm{n}}$ (2)

Where $F$ is fission operator, $A$ is net neutron loss operator and
$S_{\rm{n}}$ is external neutron source.
The subcritical multiplication factor $k_{\rm{s}}$ [11] is defined as Eq.(3):

$k_{\rm{s}}=\frac{<F\phi_{\rm{s}}>}{<F\phi_{\rm{s}}>+<S_{\rm{n}}>}$ (3)

$k_{\rm{s}}$ describes the number of fission neutrons produced per lost
neutron in the subcritical system.
Neutron source efficiency, usually denoted as $\varphi^{*}$ [11], represents
the efficiency of the external source neutrons and can be expressed as Eq.
(4):

$\varphi^{*}=(\frac{1}{k_{\rm{eff}}-1})\frac{<F\phi_{\rm{s}}>}{S_{\rm{n}}}$
(4)

$<F\phi_{\rm{s}}>$ is total number of neutrons produced by fission.
$<S_{\rm{n}}>$ is total number of external source neutrons. For a given values
of $k_{\rm{eff}}$ and $<S_{\rm{n}}>$, the larger $\varphi^{*}$, the bigger the
fission power. Definition of neutron source efficiency needs the definition of
source neutrons $<S_{\rm{n}}>$. In this work, $<S_{\rm{n}}>$ means neutrons
escaping the spallation region.
Relationship between $k_{\rm{keff}}$ and $k_{\rm{s}}$ [11] is:

$(1-\frac{1}{k_{\rm{keff}}})=\varphi^{*}(1-\frac{1}{k_{\rm{s}}})$ (5)

Proton efficiency[12], denoted as $\psi^{*}$, is defined to be:

$\psi^{*}=\varphi^{*}\frac{<S_{\rm{n}}>}{<S_{\rm{p}}>}=(\frac{1}{k_{\rm{eff}}-1})\frac{<F\phi_{\rm{s}}>}{<S_{\rm{p}}>}$
(6)

$S_{\rm{p}}$ is total number of source protons.

To study proton efficiency of the system, simulations had been performed in
two steps:

  * 1.

protons impact on the target, neutron multiplicity and neutrons escaping the
target are recorded with FLUKA.

  * 2.

the source neutrons recorded in the first step are then processed as an
external neutron source and transported in the system.

In order to reduce the time of simulation, the height and diameter of the
total system was cut to be 360 cm.

##  4 Spatial and energy distribution of spallation neutrons

###  4.1 Description of spatial and energy distribution of spallation neutrons

Spatial and energy distribution of spallation neutrons have an effect on
proton efficiency. 232-Th,238-U, Am can be used as fuels of ADS. An external
neutron hardly ever induce fissions of these fuels when its kinetic energy is
below 1 MeV[12]. The energy spectrum of neutrons in subcritical core is
different from that of spallation neutrons; the former is determined by
geometry and composition of the system, while the later is mostly determined
by the size of spallation region. The spatial distribution of spallation
neutrons determines the distribution of neutron flux in the subcritical core
and thus determining the proton efficiency of the system.

Fig. 3: RZ view of target model used in FLUKA. The different regions are:
(1)Target; (2)Subcritical core; (3)Beam pipe; (4)Moderator; (5)Void. In
simulations of FLUKA, region Moderator and Subcritical core is filled with
vacuum. Spallation neutrons include : Backward(1$\rightarrow$3,4);
Forward(1$\rightarrow$5); Radial(1$\rightarrow$2); Survival(4$\rightarrow$2);
Leak(3,4$\rightarrow$5)

In this work, spatial and energy distribution of spallation neutrons was
simulated with FLUKA. The target model used in FLUKA is shown in Fig. 3.
Neutrons are divided into forward, radial, backward neutrons according to the
surface the neutrons cross when they escape the target. Radial neutrons enter
the subcritical core directly. Forward neutrons hardly ever enter the
subcritical core. Some backward neutrons make it to the subcritical core
through the moderator between the beam pipe and the subcritical core (region
”10” in Fig. 2), called survival neutrons; other backward neutrons enter the
reflector or leak out from the beam pipe, called leak neutrons. A clear
explanation of these neutrons is shown in Fig. 3. Except for region ”target”,
all other regions are filled with vacuum.

As for spatial distribution, what we concern here was the neutron current
surface density $I_{\rm{r}}(Z)$ in the side surface of target. $Z$ is the
distance between a point in the side surface and the top surface of target. We
also cared about the number of backward, forward and radial neutrons. As for
energy distribution, the energy spectrum of radial neutrons were mostly
discussed.

###  4.2 Spatial and energy distribution of radial neutrons in C-ADS

Radial neutrons are the main part of spallation neutrons and their spatial
distribution is closely related to the neutron flux in the subcritical core.
So we studied this factor quantitatively with FlUKA. Remember that we mainly
care about neutron current surface density here.

In this simulation, proton source is a circular uniform beam with radius of
$R_{\rm{b}}=15\ \rm{cm}$ in C-ADS HEBT (High Energy Beam Transport ) [13].
Simulation had been performed with parameters listed in Table 3. To obtain the
neutron current surface density, subroutine mgdraw.f was modified and
activated to record neutrons escaping the ”target” region in Fig. 3. Then we
constructed the spatial and energy distribution of neutrons with the
information recorded. The result of this simulation is shown in Fig. 4.
Average kinetic energy of spallation neutrons in different position was used
in calculating energy distribution: $\overline{E_{\rm{r}}(Z)}$.

Table 3: Parameters of target and beam of C-ADS [13] Target characteristic | Beam
---|---
Material | 50%Pb,50%Bi | Projectile | Proton
$R_{\rm{s}}$ | 25.0 cm | $E_{\rm{p}}$ | 1.5 GeV
$L_{\rm{s}}$ | 100.0 cm | Source | HEBT

As can be seen in Fig. 4, as $Z$ increase: $\overline{E_{\rm{r}}(Z)}$ reaches
a minimum and then saturates; $I_{\rm{r}}(Z)$ reaches a maximum and then
decreases to zero. Based on this result, if the position of maximum
$I_{\rm{r}}(Z)$ locates at the axial center of subcritical core(that is $z=0\
\rm{cm}$), neutron flux $\phi_{\rm{s}}(z)$ in the subcritical core will be
symmetric (with axis at $z=0\ \rm{cm}$). Generally, an symmetric distribution
of $\phi_{\rm{s}}(z)$ is a key factor in designing the system. In this way, we
may optimize the position(relative to the subcritical core) of spallation
target, discussed in Sec. 4.4.1.

Fig. 4: Neutron current surface density $I_{\rm{r}}(Z)$ and average kinetic
energy$\overline{E_{\rm{r}}(Z)}$ as a function of $Z$. $I_{\rm{r}}(Z)$ is
normalized to per source proton.

In this case, $\overline{E_{\rm{r}}(Z)}$ changes slightly,
$(E_{\rm(rmax)}-E_{\rm{rmin}})/E_{\rm(raverage)}\approx 0.141$. Energy
distribution of radial spallation neutrons has a influence on the neutron flux
of the subcritical core, but it is a less important factor than the spatial
distribution of radial neutrons. Detailed effect of the energy distribution
needs further research.

###  4.3 Backward neutrons and radial neutrons as a function of $R_{\rm{s}}$

The size of subcritical core changes with the radius of spallation region. It
was proved that proton efficiency decrease when radius of spallation region
(called target in [12]) increases due to the soft and leakage of neutrons
[12]. Changing the size of the subcritical core will bring to change of
several factors that influence the proton efficiency. In this work, as is
shown in Fig. 5, a ”void” region is inserted between the spallation region and
the subcritical core to study the effect of backward neutrons. Changing the
thickness of this ”void” region, we could simulate different $R_{\rm{s}}$
without changing other parameters, such as the size of the subcritical core.
This ”void” region does not exist in practical design of ADS, but it is useful
to study the effect of backward neutrons.

Fig. 5: RZ view of modified ADS benchmark. The different regions
are:(1)Target; (2)Buffer; (3)Core1; (4)Core2; (5)Core3; (6)Reflector; (7)Beam
pipe; (8)Moderator; (9)Void.

To study the effect of backward and radial neutrons on proton efficiency with
ADS model, parameters of IAEA ADS benchmark were used, as listed in Table 2;
$L_{\rm{s}}=100\ \rm{cm}$. We use the target model in Fig. 3 to study the
number of backward neutrons and the soft of radial neutrons when $R_{\rm{s}}$
changes. Four different thickness of ”Void” region were chosen to get
$R_{\rm{s}}=12,18,24,30\ \rm{cm}$. The result is listed in Table 4. Remember
that in all simulations with FLUKA, region subcritical core is filled with
vacuum.

Table 4: Characteristic of spallation neutrons, $1\sigma<1.0\%$ Number of
spallation neutrons in different direction
---
$R_{\rm{s}}/\rm{cm}$ | 12 | 18 | 24 | 30
Radial | 20.665 | 21.838 | 21.219 | 20.199
Forward | 0.007 | 0.034 | 0.110 | 0.263
Backward: | 3.729 | 6.368 | 8.702 | 10.767
Survival | 3.530 | 5.697 | 7.270 | 8.281
Leak | 0.199 | 0.671 | 1.432 | 2.486
Ratio of high-energy Radial neutrons
$>100\ \rm{MeV}\%$ | 1.70 | 0.99 | 0.62 | 0.41
$>20\ \rm{MeV}\%$ | 1.26 | 0.86 | 0.58 | 0.39
$>1\ \rm{MeV}\%$ | 59.62 | 48.3 | 38.97 | 31.04

In Table 4, as $R_{\rm{s}}$, we find that: (1) the number of radial neutrons
changes slightly and the spectrum of radial neutrons becomes softer (ratio of
neutrons with kinetic energy higher than 1 MeV decrease from 59.62% to
31.04%); (2) the number of backward neutrons increases greatly (from 3.729 to
10.767); (3) the number of survival neutrons increases considerably (from
3.530 to 8.821). Explanation to these results may be: most neutrons,
especially high energy neutrons are produced near the top surface of target;
as $R_{\rm{s}}$ increases, it is more likely that these neutrons make it to
the top surface of target and then leak out in backward direction. In
conclusion, when the radius of spallation region increases, number of survival
neutrons becomes considerably big, which will contribute a positive effect to
proton efficiency; radial neutrons become softer, which will contribute a
negative effect to proton efficiency. Comparison of these two effects are
discussed below.

In this simulation, material of region ”moderator” in Fig. 3 was vacuum, not
Lead; replacing the vacuum with Lead would bring a slight decrease to the
number of survival neutrons due to the scattering and absorption of neutrons
in the ”moderator”. Another simulation had been performed; for
$R_{\rm{s}}=12,18,24,30\ \rm{cm}$, survival neutrons decrease to 3.409, 5.098,
6.162, 6.838 per source proton. This difference does not severely affect the
fact that a rather big number of backward neutrons will enter the subcritical
core when the radius of spallation region is big.

###  4.4 Effect of spatial and energy distribution of spallation neutrons

####  4.4.1 Optimizing target position with spatial distribution of radial
neutrons

The spatial distribution of radial neutrons may be used to optimize the axial
position of target, as discussed in Sec. 4.2 . In this work, target position
is determined by the top surface of target $Z_{\rm{top}}$. Optimization of
$Z_{\rm{top}}$ had been performed with the following considerations:

  * 1.

$L_{\rm{}s}$ will be larger than the range of proton in Lead, about 100 cm for
1.0-1.5 GeV protons. In this way, $I_{\rm{r}}(Z)$ and neutron multiplicity
will not change when $Z_{\rm{top}}$ changes.

  * 2.

The bottom of target, $Z_{\rm{bottom}}$, remains at 75 cm; $Z_{\rm{top}}$
changes from -60 cm to -15 cm and $L_{\rm{}s}$ varies from 90 cm to 135 cm.

  * 3.

All other parameters do not change, such as $R_{\rm{s}}=32.5\ \rm{cm}$ and
composition of the subcritical core. $k_{\rm{eff}}$ has a little change and
will be considered in the calculation of proton efficiency.

  * 4.

$L_{\rm{s}}$ reaches its saturation, so $<S_{\rm{n}}>/{<S_{\rm{p}}>}$ does not
change when $Z_{\rm{top}}$ changes. $\varphi^{*}$ is equal to $\psi^{*}$.

The position of maximum $I_{\rm{r}}(Z)$ varies with the radius of spallation
region and the energy of protons. For ADS benchmark, $R_{\rm{s}}=32.5\
\rm{cm}$, $L_{\rm{}s}=100\ \rm{cm}$, $E_{\rm{p}}=1.0\ GeV$, the maximum
reaches at $Z\approx 20\ \rm{cm}$. So we expected a maximum proton efficiency
when $Z_{\rm(top)}=-20\ \rm{cm}$. Parameters of this simulation are listed in
Table 3. $\varphi^{*}$ and $\phi_{\rm{s}}(z)$ was obtained as $Z_{\rm{top}}$
changed. The result is shown in Fig. 6 and Fig. 7.

Fig. 6: Neutron source efficiency $\varphi^{*}$, subcritical multiplication
factor $k_{\rm{s}}$ and effective multiplication factor $k_{\rm{eff}}$ as a
function of $Z_{\rm{top}}$
Fig. 7: Neutron flux $\phi_{\rm{s}}(z)$ as a function of $Z_{\rm{top}}$

In Fig. 6, we find that both $k_{\rm{s}}$ and $\varphi^{*}$ reaches a maximum
value at $Z_{\rm{top}}=-25\ \rm{cm}$. This result is a bit smaller than the
position, -20 cm, we expected. This difference may be explained by the fact
that the distribution of $I_{\rm{r}}(Z)$ is not perfectly symmetric, there is
a ”tail” at high $Z$. In Fig. 7, as $Z_{\rm{top}}$ departs from $-25\
\rm{cm}$, the symmetry axis of $\phi_{\rm{s}}(z)$ departs from $z=0\ \rm{cm}$.
This deviation will increase the leakage of neutrons from the subcritical core
to the reflector and thus decreasing the proton efficiency.

Survival neutrons also influence $\varphi^{*}$. The number of survival
neutrons will decrease when $Z_{\rm{top}}$ becomes more negative and this will
bring a decrease to $\varphi^{*}$. The effect of $I_{\rm{r}}(Z)$ and survival
neutrons are coupled. But when $Z_{\rm{top}}$ becomes more positive, the
effect of survival neutrons becomes smaller. In conclusion, both
$\phi_{\rm{s}}(z)$ and $\varphi^{*}$ support the assumption about target
position. The effect of survival neutrons makes the assumption less persuasive
and will be discussed below.

####  4.4.2 Effect of radial and backward neutrons on proton efficiency

Change of $R_{\rm{s}}$ will bring a change to the characteristic of radial and
backward neutrons as discussed in Sec. 4.3. These two factors are coupled.
Based on simulations in Sec. 4.3, proton efficiency $\psi^{*}$ of the modified
ADS benchmark was simulated for different $R_{\rm{s}}$. Parameters of these
simulations are listed in Table 2, the result is listed in Table 5. Changing
the thickness of void region brought to a slight change of $k_{\rm{eff}}$.
This change was considered in the calculation of $\psi^{*}$.

Table 5: Proton efficiency varies with $R_{\rm{s}}$, $1\sigma<1.0\%$ Proton
efficiency $R_{\rm{s}}$
---
$R_{\rm{s}}/\rm{cm}$ | 12 | 18 | 24 | 30
| 0.94828 | 0.94886 | 0.94972 | 0.95148
$k_{\rm{eff}}\pm std$ | 0.00026 | 0.00026 | 0.00025 | 0.00026
$\psi^{*}$ | 26.803 | 27.499 | 27.882 | 28.002

We find that: as $R_{\rm{s}}$ increases, $\psi^{*}$ increases slightly. This
result seems paradox to the conclusion that the bigger of target radius, the
smaller of proton efficiency in [12]. In fact,as is seen in Table 4, it is the
trade-off between the soft of radial neutrons and the increase of survival
neutrons. Source efficiency of backward neutrons in inducing fissions is lower
than these radial neutrons, but the rather big number of survival neutrons
neutralizes the negative effect of radial neutrons. This result does indicate
that the effect of backward neutrons is pronounced. When radius of spallation
region increase, number of neutrons produced per beam particle does increase,
but these extra neutrons are not effectively used. Based on this simulation,
it is difficult to draw the conclusion that the radius of spallation region
should be larger; but if we need to increase radius of spallation region to
satisfy other requirements in designing the system, such as decreasing the
damage of high-energy spallation neutrons to structural material, we may try
to make better use of these backward neutrons and avoid severely affecting the
proton efficiency. At present, we may replace the moderator between the beam
pipe and the subcritical core with fuels and increase the radius of spallation
region; or we may design an annular target (ratio between area of top surface
and side surface is smaller than that of a cylinder target) to reduce the
number of backward neutrons.

##  5 Conclusion

In this work, spallation region and subcritical core in ADS are combined to
study effect of the spatial, energy distribution of spallation neutrons. The
characteristic of radial and backward neutrons were mainly discussed.

As for radial neutrons, neutron current surface density $I_{\rm{r}}(Z)$ was
simulated with FLUKA and used to optimize the position of target. An
assumption about this optimization was made and tested. As for the design of
C-ADS, the considerations made in Sec. 4.4.1 may be used to optimize position
of target. As for backward neutrons, survival neutrons were mainly discussed
and found to have a considerable ratio when radius of spallation region
increases. These survival neutrons have a positive effect on the proton
efficiency. If we make better use of these survival neutrons, we may increase
the radius of spallation region to satisfy other requirements in the design of
subcritical core without affecting proton efficiency severely.

## Acknowledgements

Numerical Simulation received the support and help of Supercomputing Center of
USTC. The authors acknowledge the members in the PDSP group of NSRL for their
help and discussion.

## References

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# Partially isometric Toeplitz operators with operator-valued symbols

Srijan Sarkar  Department of Mathematics, Indian Institute of Science,
Bangalore, 560012, India<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

In this paper, we study partially isometric Toeplitz operators $T_{\Phi}$ on
Hilbert space-valued Hardy spaces $H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$ over
the unit polydisc. We establish the following crucial phenomenon: the range of
partially isometric Toeplitz operators is always a Beurling-type invariant
subspace of $H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$. Using this result, we prove
that partially isometric Toeplitz operators always admit the following
factorization:

$T_{\Phi}=M_{\Gamma}M_{\Psi}^{*},$

where $\Gamma(\bm{z}),\Psi(\bm{z})$ are operator-valued inner functions on
$\mathbb{D}^{n}$, governed by certain conditions that force
$M_{\Gamma}M_{\Psi}^{*}$ to become a Toeplitz operator. Our results are new
even in the case of Hardy spaces over the unit disc, and extend the work of
Brown–Douglas, Deepak–Pradhan–Sarkar on scalar-valued Hardy spaces.

###### Key words and phrases:

Partial isometry, vector-valued Hardy spaces, Toeplitz operators, product of
Toeplitz operators, Beurling type invariant subspaces

###### 2010 Mathematics Subject Classification:

Primary 47B35, 46E40, 47A56, 30H10. Secondary 30J05.

###### Contents

  1. 1 Introduction and main results
  2. 2 Preliminaries
  3. 3 Product of Toeplitz operators
  4. 4 Beurling-type Toeplitz ranges
  5. 5 Partially isometric Toeplitz operators
  6. 6 An alternative proof for Toeplitz operators on $H^{2}(\mathbb{D}^{n})$
  7. 7 Some related questions

##  1\. Introduction and main results

From its inception, Toeplitz operators have played a vital role in the
interplay between operator theory and function theory. This connection has
been instrumental in finding new results across many disciplines like several
complex variables, non-commutative geometry, mathematical physics, and
engineering sciences. In recent times, there has been an active interest in
extending the results of Toeplitz operators on scalar-valued Hardy spaces to
vector-valued spaces. The primary reason has always been to find a deeper
understanding between the operators and their corresponding symbols. Recent
investigations show that analytic Toeplitz operators with operator-valued
symbols have many applications in systems engineering and $H^{\infty}$ control
theory; for instance, see the celebrated monographs [1, 11].

For any Hilbert space $\mathcal{E}$, the $\mathcal{E}$-valued Hardy space on
the unit polydisc $\mathbb{D}^{n}$ in the $n$-dimensional complex plane is
defined by

$H_{\mathcal{E}}^{2}(\mathbb{D}^{n}):=\\{\sum_{\bm{k}\in\mathbb{N}^{n}}a_{\bm{k}}z^{\bm{k}}\in\mathcal{O}(\mathbb{D}^{n},\mathcal{E}):a_{\bm{k}}\in\mathcal{E},\sum_{\bm{k}\in\mathbb{N}^{n}}\|a_{\bm{k}}\|_{\mathcal{E}}^{2}<\infty\\}.$

The space of operator-valued bounded analytic functions on $\mathbb{D}^{n}$ is
denoted by $H_{\mathcal{B}(\mathcal{E})}^{\infty}(\mathbb{D}^{n})$, where
$\mathcal{B}(\mathcal{E})$ denotes the class of all bounded operators on
$\mathcal{E}$. Recall that a Toeplitz operator is defined in the following
manner [10, 22].

######  Definition 1.1.

A bounded linear operator $T$ on $H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$ is said
to be Toeplitz if there exists an operator-valued function $\Phi\in
L_{\mathcal{B}(\mathcal{E})}^{\infty}(\mathbb{T}^{n})$ such that
$T=P_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}L_{\Phi}|_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})},$
where $L_{\Phi}$ is the Laurent operator (multiplication operator) associated
to $\Phi.$ In this case, $T$ is denoted by $T_{\Phi}$, and the function $\Phi$
is called the symbol of the operator $T_{\Phi}.$

It is immediate that when $\Phi(\bm{z})$ is an operator-valued bounded
analytic function on $\mathbb{D}^{n}$, the corresponding Toeplitz operator is
simply the multiplication operator on $H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$.
For this reason, we make a distinction and use the following convention
throughout this article.

$\text{If }\Phi(z)\in
H_{\mathcal{B}(\mathcal{E})}^{\infty}(\mathbb{D}^{n})\text{ then the
corrsponding Toeplitz operator is denoted by }M_{\Phi}.$

It is worth mentioning, that unlike in the case of $H^{2}(\mathbb{D})$, the
theory of Toeplitz operators on both $H^{2}(\mathbb{D}^{n})$ and
$H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$ (where $n>1$) is far from being
complete, as many natural questions remain unanswered. Several challenges
appear whenever we leave the realm of scalar-valued functions and delve into
the world of operator-valued symbols. We refer the reader to the exceptional
monograph [10] by Douglas for results on matrix-valued Toeplitz operators.
Recently, there has been spectacular progress on Halmos’s question of
subnormal Toeplitz operators, for matrix/operator-valued symbols by Curto,
Hwang, and Lee in [5, 6, 7].

In this article, we further explore this theme: the correspondence of Toeplitz
operators with their symbols under operator-theoretic conditions. From the
algebraic characterization of Toeplitz operators developed by Brown and Halmos
in [3], it can be realized that the only possibility for any
$T_{\phi}\in\mathcal{B}(H^{2}(\mathbb{D}))$ to become an isometry is when its
symbol $\phi$ is an inner function (that is, $\phi\in H^{\infty}(\mathbb{D})$,
and $|\phi(e^{it})|=1$ a.e. on $\mathbb{T}$). Brown and Douglas were
interested in the behaviour of Toeplitz operators under the general condition
of it being a partial isometry. We recall that a bounded operator $T$ on a
Hilbert space $\mathcal{H}$ is a partial isometry if $T$ is an isometry on the
orthogonal complement of its kernel. In [2], the authors proved that the only
partially isometric Toeplitz operators on $H^{2}(\mathbb{D})$ are of the form
$T_{\phi}=M_{\theta}$, or else, $T_{\phi}=M_{\theta}^{*}$, where $\theta(z)\in
H^{\infty}(\mathbb{D})$ is an inner function. Recently, this result was
extended by Deepak–Pradhan–Sarkar for Toeplitz operators on
$H^{2}(\mathbb{D}^{n})$, where $n>1$.

######  Theorem.

[8, Theorem 1.1] Let $\phi$ be a non-zero function in
$L^{\infty}(\mathbb{T}^{n})$. Then $T_{\phi}$ is a partial isometry if and
only if there exist inner functions $\phi_{1},\phi_{2}\in
H^{\infty}(\mathbb{D}^{n})$ such that $\phi_{1}$ and $\phi_{2}$ depends on
different variables and $T_{\phi}=M_{\phi_{1}}^{*}M_{\phi_{2}}$.

In other words, the condition of partial isometry on $T_{\phi}$ forces a
factorization of $\phi=\bar{\phi_{1}}\phi_{2}$ almost everywhere on
$\mathbb{T}^{n}$. An immediate question is finding the corresponding result
for Toeplitz operators with operator-valued symbols. Let us highlight an
immediate obstacle to a straightforward generalization. In scalar-valued
cases, both Brown–Douglas and Deepak–Pradhan–Sarkar’s results show that if we
start with $\phi\in H^{\infty}(\mathbb{D}^{n})$, then the only possibility for
the analytic Toeplitz operator $M_{\phi}$ to be a partial isometry is when
$M_{\phi}$ is an isometry, in other words, $\phi$ must be an inner function.
There are many examples involving block Toeplitz operators which show that
this no longer holds in the setting of operator-valued symbols. First, let us
note that an operator-valued function $\Theta(\bm{z})\in
H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}^{n})$ is said to
be an inner function if $\Theta(z):\mathcal{F}\rightarrow\mathcal{E}$, is an
isometry almost everywhere on $\mathbb{T}^{n}$ [21]. Now, let us consider the
following bounded analytic matrix-valued symbol,

$\Theta(z):=\begin{bmatrix}0&z\\\ 0&0\end{bmatrix}\in
H_{\mathcal{B}(\mathbb{C}^{2})}^{\infty}(\mathbb{D}),$

then for each $\lambda\in\mathbb{T}$,

$\Theta(\lambda)^{*}\Theta(\lambda)=\begin{bmatrix}0&0\\\
\bar{\lambda}&0\end{bmatrix}\begin{bmatrix}0&\lambda\\\
0&0\end{bmatrix}=\begin{bmatrix}0&0\\\ 0&1\end{bmatrix},$

This implies that $\Theta(z)$ is not an inner function. However,

$M_{\Theta}^{*}M_{\Theta}=T_{\Theta^{*}\Theta}=\begin{bmatrix}0&0\\\
0&I_{H^{2}(\mathbb{D})}\end{bmatrix}\in H_{\mathbb{C}^{2}}^{2}(\mathbb{D}),$

shows that $M_{\Theta}$ is indeed a partial isometry. Thus, a non-constant
analytic Toeplitz operator $M_{\Theta}$ can be a partial isometry without
$\Theta(z)$ being an inner function. Motivated by this observation, and
earlier results, we study the following intriguing question in this article:
what are the partially isometric Toeplitz operators on vector-valued Hardy
spaces?

At this moment, let us digress a little to highlight a connection of this
question to the study of invariant subspaces of $M_{z}\oplus M_{z}^{*}$ on
$H_{\mathcal{E}}^{2}(\mathbb{D})\oplus H_{\mathcal{F}}^{2}(\mathbb{D})$ as
observed by Gu and Luo in [15]. This work was again motivated by the
characterization of invariant subspaces of $M_{z}\oplus M_{z}^{*}$ on
$H^{2}(\mathbb{D})\oplus H^{2}(\mathbb{D})$ by Timotin [26]. In their work, Gu
and Luo observed that the invariant subspace of $M_{z}\oplus M_{z}^{*}$ is
related to the range of the following operator.

$V_{\Phi}=\begin{bmatrix}T_{A}&T_{B}\\\
H_{C}&H_{D}\end{bmatrix}:H_{\mathcal{E}}^{2}(\mathbb{D})\oplus
H_{\mathcal{F}}^{2}(\mathbb{D})\rightarrow
H_{\mathcal{E}}^{2}(\mathbb{D})\oplus H_{\mathcal{F}}^{2}(\mathbb{D}),$

where

$\Phi(z)=\begin{bmatrix}A(z)&B(z)\\\
C(z)&D(z)\end{bmatrix}:\mathcal{E}\oplus\mathcal{F}\rightarrow\mathcal{E}\oplus\mathcal{F}\quad(z\in\mathbb{D}),$

and $T_{A},T_{B}$, and $H_{C},H_{D}$ are the corresponding Toeplitz and Hankel
operators, respectively. Furthermore, the authors found that the range of
$V_{\Phi}$ is closed in $H_{\mathcal{E}}^{2}(\mathbb{D})\oplus
H_{\mathcal{F}}^{2}(\mathbb{D})$ if and only if $V_{\Phi}$ is a partial
isometry. This leads to the natural question of when $V_{\Phi}$ becomes a
partial isometry. To this, they made the following statement.

“The above problem seems a difficult one since the partial isometric
characterizations of Toeplitz operator $T_{A}$ (A is not necessarily analytic)
and Hankel operator $H_{D}$ are only known when $A$ and $D$ are scalar
functions in $L^{\infty}$ [2, 23].” In this article, we completely resolve
this issue for Toeplitz operators; see Theorem 1.4 and Theorem 5.2. Thus, we
strongly feel that our main result will have several consequences in
characterizing invariant subspaces of sums of shift operators on vector-valued
Hardy spaces over $\mathbb{D},\mathbb{D}^{n}$, and motivate further
applications.

One of the main difficulties in trying to find an answer to the above question
(in italics) is that we no longer have the commutativity of the symbols, which
is automatic in the scalar cases. Since the proofs in [3, 8] rely crucially on
the commutativity of the symbols, the methods in [3, 8] are not applicable.
Thus, it is essential to find a completely new approach to the above question.
Based on earlier works, we aim to find a factorization of $T_{\Phi}$ into a
product of Toeplitz operators corresponding to inner symbols. For this
purpose, let us first recall that a closed
$(M_{z_{1}},\ldots,M_{z_{n}})$-joint invariant subspace of
$H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$ is said to be Beurling-type if there
exists a Hilbert space $\mathcal{F}$, and an inner function $\Theta(z)\in
H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}^{n})$ such that
$\mathcal{S}=M_{\Theta}H_{\mathcal{F}}^{2}(\mathbb{D}^{n})$. It is well-known,
that given a subspace $\mathcal{S}\subseteq
H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$ if we consider the restriction operators
$R_{i}:=M_{z_{i}}|_{\mathcal{S}}$ for all $i\in\\{1,\ldots,n\\}$, then
$\mathcal{S}$ is Beurling-type invariant subspace if and only if
$[R_{i}^{*},R_{j}]=0$ for all $i\neq j$ ([20, 25]). Thus, if we can show that
the range of Toeplitz operators satisfies the above algebraic conditions
concerning the restriction operators, then we can associate inner functions to
the corresponding symbol. In section 4, we establish this connection by
proving the following result.

######  Theorem 1.2.

If $T_{\Phi}$ is a non-zero partially isometric Toeplitz operator on
$H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$, then $\operatorname{ran}T_{\Phi}$ is a
Beurling-type invariant subspace of $H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$.

Let us note that classically, the kernel of the adjoint of Toeplitz operators
on $H^{2}(\mathbb{D})$ are related with nearly invariant subspaces of
$H^{2}(\mathbb{D})$. This observation was noted by Hayashi in [18], and
further explored by Sarason in [24]. We believe that the above result should
serve as an impetus to study the range of Toeplitz operators as well.

With the above characterization in mind, a curious reader with some attempts
may guess that $T_{\Phi}$ may admit the following factorization.

$T_{\Phi}=M_{\Gamma}M_{\Psi}^{*},$

where $\Gamma,\Psi$ are operator-valued inner functions, but this leads us to
another challenge: under what conditions does $M_{\Gamma}M_{\Psi}^{*}$ become
a Toeplitz operator?

On the scalar-valued Hardy spaces, it is well-known when the product of two
Toeplitz operators is again a Toeplitz operator. Brown and Halmos developed
this result for $H^{2}(\mathbb{D})$ in [3], and Gu proved the corresponding
result for $H^{2}(\mathbb{D}^{n})$ in [13]. However, the answer to this
question for Toeplitz operators with operator-valued symbols is still unclear.
We refer the reader towards some important progress in the case of block
Toeplitz operators by Gu and Zheng in [12, 14]. This is where we make another
significant contribution. Section 3 is dedicated to establishing the following
tractable conditions that make the above product into a Toeplitz operator.

######  Theorem 1.3.

Let $\mathcal{F},\mathcal{E}$ be Hilbert spaces and
$\Gamma(\bm{z}),\Psi(\bm{z})$ be $\mathcal{B}(\mathcal{F},\mathcal{E})$-valued
bounded analytic functions on $\mathbb{D}^{n}$. More precisely, let

$\Gamma(\bm{z}):=\sum_{\bm{l}\in\mathbb{N}^{n}}A_{\bm{l}}\bm{z}^{\bm{l}}\in
H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}^{n});\quad\Psi(\bm{z}):=\sum_{\bm{m}\in\mathbb{N}^{n}}B_{\bm{m}}\bm{z}^{\bm{m}}\in
H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}^{n}),$

where $A_{\bm{k}},B_{\bm{k}}\in\mathcal{B}(\mathcal{F},\mathcal{E})$ for all
$\bm{k}\in\mathbb{N}^{n}$. Then $M_{\Gamma}M_{\Psi}^{*}$ is a Toeplitz
operator on $H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$ if and only if

(1.1)
$A_{\bm{l}+e_{i}}B_{\bm{m}+e_{i}}^{*}=0\quad(\forall\bm{l},\bm{m}\in\mathbb{N}^{n}),$

where $e_{i}=(0,\ldots,0,\underset{\text{i-th position}}{1},0,\ldots,0)$ for
all $i\in\\{1,\ldots,n\\}$.

In section 5, we use Theorem 1.2 and Theorem 1.3, to give a complete
characterization for partially isometric Toeplitz operators on
$H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$.

######  Theorem 1.4.

$T_{\Phi}$ is a non-zero partially isometric Toeplitz operator on
$H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$ if and only if there exist a Hilbert
space $\mathcal{F}$, and inner functions $\Gamma(\bm{z}),\Psi(\bm{z})\in
H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}^{n})$ satisfying
conditions (1.1), such that

$T_{\Phi}=M_{\Gamma}M_{\Psi}^{*}.$

As expected, there are numerous consequences of this result. We end section 5
with the characterization for partially isometric Toeplitz operators which are
analytic (Theorem 5.2) and hyponormal (Theorem 5.3). In section 6, we give a
different proof of the main result in [8].

Let us now describe the plan for the rest of this article. In section 2, we
set notations, and definitions, and establish a few results essential for the
later part. We end this article with section 7, where we have highlighted
several interesting questions worthy of further investigation.

##  2\. Preliminaries

In this section, we set the notations, definitions, and results needed
throughout this article. Let us begin by looking at the equivalent way of
defining $\mathcal{E}$-valued Hardy spaces on $\mathbb{D}^{n}$.

$H_{\mathcal{E}}^{2}(\mathbb{D}^{n}):=\\{f\in\mathcal{O}(\mathbb{D}^{n},\mathcal{E}):\|f\|_{2}^{2}:=\sup_{0<r<1}\int_{\mathbb{T}^{n}}\|f(re^{i\theta_{1}},\ldots,re^{i\theta_{n}})\|_{\mathcal{E}}^{2}d\mu<\infty\\},$

where $\mu$ is the normalized Lebesgue measure on $\mathbb{T}^{n}$. This space
of $\mathcal{E}$-valued analytic functions has a natural collection of shift
operators, namely,

$M_{z_{i}}f:=z_{i}f,\quad(f\in H_{\mathcal{E}}^{2}(\mathbb{D}^{n})),$

for each $i\in\\{1,\ldots,n\\}$. We will denote by
$M_{\bm{z}}=(M_{z_{1}},\ldots,M_{z_{n}})$ as the tuple of shift operators. For
$\bm{k}=(k_{1},\ldots,k_{n})\in\mathbb{N}^{n}$, we set the following
convention,

$M_{\bm{z}}^{\bm{k}}=M_{z_{1}}^{k_{1}}\cdots M_{z_{n}}^{k_{n}},$

and for any
$\bm{k}=(k_{1},\ldots,k_{n}),\bm{l}=(l_{1},\ldots,l_{n})\in\mathbb{N}^{n}$, we
say $\bm{k}\leq\bm{l}$ if $k_{i}\leq l_{i}$ for all $i\in\\{1,\ldots,n\\}$.
For the tuple of shift operators
$L_{e^{i\theta}}=(L_{e^{i\theta_{1}}},\ldots,L_{e^{i\theta_{n}}})$ on
$L^{2}(\mathbb{T}^{n})$, we set

$L_{e^{i\theta}}^{\bm{k}}=L_{e^{i\theta_{1}}}^{k_{1}}\cdots
L_{e^{i\theta_{n}}}^{k_{n}}.$

Brown and Halmos gave an algebraic characterization for Toeplitz operators on
$H^{2}(\mathbb{D})$ [3]. It is well-known that there exists a natural
extension of this result to Toeplitz operators on $H^{2}(\mathbb{D}^{n})$
[19]. The following result shows that such a characterization holds for
$H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$ as well. The result can be proved
verbatim from [19, Theorem 3.1, Theorem 5.2]. For the sake of completion, we
give a sketch of the proof.

######  Theorem 2.1.

A bounded operator $T$ on $H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$ is a Toeplitz
operator if and only if

$M_{z_{i}}^{*}TM_{z_{i}}=T,$

for all $i\in\\{1,\ldots,n\\}$.

###### Proof.

For each $k\in\mathbb{N}$, consider
$\bm{k}_{d}:=(k,\ldots,k)\in\mathbb{N}^{n}$. From the assumption
$M_{z_{i}}^{*}TM_{z_{i}}=T$, we immediately get

$M_{\bm{z}}^{*\bm{k}_{d}}TM_{\bm{z}}^{\bm{k}_{d}}=T.$

This implies that

$\langle Te_{\bm{i}+\bm{k}_{d}}\eta,e_{\bm{i}+\bm{k}_{d}}\zeta\rangle=\langle
TM_{\bm{z}}^{\bm{k}_{d}}e_{\bm{i}}\eta,M_{\bm{z}}^{\bm{k}_{d}}e_{\bm{j}}\zeta\rangle=\langle
Te_{\bm{i}},e_{\bm{j}}\rangle,$

for all $\bm{i},\bm{j}\in\mathbb{N}^{n}$, and $k\in\mathbb{N}$. Now for each
$\bm{l},\bm{m}\in\mathbb{Z}^{n}$ there exists
$\bm{t}=(t_{1},\ldots,t_{n})\in\mathbb{N}^{n}$ such that
$\bm{l}+\bm{k}_{d},\bm{m}+\bm{k}_{d}\in\mathbb{N}^{n}$ for all
$\bm{k}_{d}\geq\bm{t}$. Using this observation, if we set
$A_{k}:=L_{e^{i\theta}}^{*\bm{k}_{d}}TP_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}L_{e^{i\theta}}^{\bm{k}_{d}}$
for $k\in\mathbb{N}\setminus\\{0\\}$, then we get

$\langle
A_{k}e_{\bm{l}},e_{\bm{m}}\rangle_{L_{\mathcal{E}}^{2}(\mathbb{T}^{n})}=\langle
TP_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}e_{\bm{l}+\bm{k}_{d}},e_{\bm{m}+\bm{k}_{d}}\rangle_{L_{\mathcal{E}}^{2}(\mathbb{T}^{n})},$

and therefore, for all $\bm{k}_{d}\geq\bm{t}$

$\langle A_{k}e_{\bm{l}},e_{\bm{m}}\rangle\rightarrow\langle
Te_{\bm{l}+t},e_{\bm{m}+t}\rangle\text{ as }k\rightarrow\infty.$

This lets us define a bounded bilinear form $\textsf{b}(\cdot,\cdot)$ on the
linear span of
$\\{e_{\bm{s}}\zeta:\bm{s}\in\mathbb{Z}^{n},\eta\in\mathcal{E}\\}$, in the
following manner,

$\textsf{b}(e_{\bm{l}}\eta,e_{\bm{m}}\zeta)=\lim_{k\rightarrow\infty}\langle
A_{k}e_{\bm{l}}\eta,e_{\bm{m}}\zeta\rangle,$

for all $\bm{l},\bm{m}\in\mathbb{Z}^{n}$ and $\eta,\zeta\in\mathcal{E}$.
Therefore, there exists an operator
$A_{\infty}\in\mathcal{B}(L_{\mathcal{E}}^{2}(\mathbb{T}^{n}))$ such that

$\langle A_{\infty}f,g\rangle=\lim_{k\rightarrow\infty}\langle
A_{k}f,g\rangle,$

for $f,g\in L_{\mathcal{E}}^{2}(\mathbb{T}^{n})$. Let
$\epsilon_{j}=(0,\ldots,0,\underset{\text{j-th position}}{1},0,\ldots,0)$,
then for all $k$ sufficiently large (depending on $\bm{l},\bm{m}$ and $j$), we
get

$\langle
L_{e^{i\theta}}^{*\bm{k}_{d}}TP_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}L_{e^{i\theta}}^{\bm{k}_{d}}e_{\bm{l}+\epsilon_{j}},e_{\bm{m}+\epsilon_{j}}\rangle_{L_{\mathcal{E}}^{2}(\mathbb{T}^{n})}=\langle
TP_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}e_{\bm{l}+\bm{k}_{d}+\epsilon_{j}},e_{\bm{m}+\bm{k}_{d}+\epsilon_{j}}\rangle_{L_{\mathcal{E}}^{2}(\mathbb{T}^{n})}=\langle
A_{k}e_{\bm{l}},e_{\bm{m}}\rangle_{L_{\mathcal{E}}^{2}(\mathbb{T}^{n})}.$

This immediately gives

$\langle
A_{\infty}e_{\bm{l}+\epsilon_{j}},e_{\bm{m}+\epsilon_{j}}\rangle_{L_{\mathcal{E}}^{2}(\mathbb{T}^{n})}=\lim_{k\rightarrow\infty}\langle
L_{e^{i\theta}}^{*\bm{k}_{d}}TP_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}L_{e^{i\theta}}^{\bm{k}_{d}}e_{\bm{l}+\epsilon_{j}},e_{\bm{m}+\epsilon_{j}}\rangle_{L_{\mathcal{E}}^{2}(\mathbb{T}^{n})}=\langle
A_{\infty}e_{\bm{l}},e_{\bm{m}}\rangle_{L_{\mathcal{E}}^{2}(\mathbb{T}^{n})}.$

This implies that

$A_{\infty}L_{e^{i\theta_{j}}}=L_{e^{i\theta_{j}}}A_{\infty},$

for all $j=1,\ldots,n$. Hence, there exists $\Phi(z)\in
L_{\mathcal{B}(\mathcal{E})}^{\infty}(\mathbb{T}^{n})$ such that

$A_{\infty}=L_{\Phi},$

which further implies that
$T=P_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}L_{\Phi}|_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}$.

Conversely, if we begin with
$T=P_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}L_{\Phi}|_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}$
for some $\Phi(z)\in L_{\mathcal{B}(\mathcal{E})}^{\infty}(\mathbb{T}^{n})$,
then for any $f,g\in H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$ and $j=1,\ldots,n$
we get

$\langle
M_{z_{j}}^{*}TM_{z_{j}}f,g\rangle_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}=\langle\Phi
e^{i\theta_{j}}f,e^{i\theta_{j}}g\rangle_{L_{\mathcal{E}}^{2}(\mathbb{T}^{n})}=\langle\Phi
f,g\rangle_{L_{\mathcal{E}}^{2}(\mathbb{T}^{n})}.$

In other words,

$\langle
M_{z_{j}}^{*}TM_{z_{j}}f,g\rangle_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}=\langle
P_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}L_{\Phi}|_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}f,g\rangle_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})},$

which implies that $M_{z_{j}}^{*}TM_{z_{j}}=T$ for all $j=1,\ldots,n$. This
completes the proof. ∎

This property plays an important role in developing many results on Toeplitz
operators and will be used throughout this article. Let us show an instance
where this is useful.

######  Theorem 2.2.

Let $T_{\Phi}$ be a Toeplitz operator on
$H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$, then $T_{\Phi}$ is an isometry if and
only if $\Phi$ is an inner function in
$H_{\mathcal{B}(\mathcal{E})}^{\infty}(\mathbb{D}^{n})$.

###### Proof.

$T_{\Phi}$ is an isometry gives,

$\displaystyle
I_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}=T_{\Phi}^{*}T_{\Phi}=M_{z_{i}}^{*}T_{\Phi}^{*}M_{z_{i}}M_{z_{i}}^{*}T_{\Phi}M_{z_{i}}$
$\displaystyle=M_{z_{i}}^{*}T_{\Phi}^{*}T_{\Phi}M_{z_{i}}-M_{z_{i}}^{*}T_{\Phi}^{*}P_{\ker
M_{z_{i}}^{*}}T_{\Phi}M_{z_{i}}$
$\displaystyle=I_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}-M_{z_{i}}^{*}T_{\Phi}^{*}P_{\ker
M_{z_{i}}^{*}}T_{\Phi}M_{z_{i}},$

for any $i\in\\{1,\ldots,n\\}$. Thus, we get that
$M_{z_{i}}^{*}T_{\Phi}^{*}P_{\ker M_{z_{i}}^{*}}T_{\Phi}M_{z_{i}}=0$, which
further implies,

$0=P_{\ker
M_{z_{i}}^{*}}T_{\Phi}M_{z_{i}}=(I_{H^{2}(\mathbb{D}^{n})}-M_{z_{i}}M_{z_{i}}^{*})T_{\Phi}M_{z_{i}}=T_{\Phi}M_{z_{i}}-M_{z_{i}}T_{\Phi}.$

This shows that $M_{z_{i}}T_{\Phi}=T_{\Phi}M_{z_{i}}$ for all
$i\in\\{1,\ldots,n\\}$ and hence, $\Phi$ must be in
$H_{\mathcal{B}(\mathcal{E})}^{\infty}(\mathbb{D}^{n})$. Since $T_{\Phi}$ is
an isometry, $\Phi$ must be an inner function in
$H_{\mathcal{B}(\mathcal{E})}^{\infty}(\mathbb{D}^{n})$. The converse
direction is straightforward. This completes the proof. ∎

######  Corollary 2.3.

Let $T_{\Phi}$ be a Toeplitz operator on
$H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$, then $T_{\Phi}$ is a unitary if and
only if $\Phi$ is a constant unitary on $\mathcal{E}$.

###### Proof.

From the assumption, both $T_{\Phi}$ and $T_{\Phi^{*}}=T_{\Phi}^{*}$ are
isometries. Following the above result, both $\Phi$, and $\Phi^{*}$ must be
inner functions in $H_{\mathcal{B}(\mathcal{E})}^{\infty}(\mathbb{D}^{n})$.
Thus, $\Phi$ must be a constant operator on $\mathcal{E}$. Since, $T_{\Phi}$
is unitary, $\Phi$ must be a constant unitary on $\mathcal{E}$. This completes
the proof. ∎

Similar, to the case of $T_{\Phi}$ on $H_{\mathcal{E}}^{2}(\mathbb{D})$, we
can consider the following decomposition on $\mathbb{D}^{n}$,

$T_{FG}=T_{F}T_{G}+H_{F^{*}}^{*}H_{G},$

where
$H_{F}:=JP_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})^{\perp}}L_{\Phi}|_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}:H_{\mathcal{E}}^{2}(\mathbb{D}^{n})\rightarrow
H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$ is the Hankel operator with symbol
$\Phi$, and $J:L_{\mathcal{E}}^{2}(\mathbb{T}^{n})\rightarrow
L_{\mathcal{E}}^{2}(\mathbb{T}^{n})$ is the unitary defined by
$J(f)(\bm{z})=\bar{\bm{z}}I_{\mathcal{E}}f(\bar{\bm{z}})$, where $f\in
L_{\mathcal{E}}^{2}(\mathbb{T}^{n})$. It is well known that Hankel operators
satisfy the following property for each $i\in\\{1,\ldots,n\\}$.

(2.2) $M_{z_{i}}^{*}H_{F}=H_{F}M_{z_{i}}.$

We refer the reader to the monographs [4, 23] for an elaborate discussion of
the above facts. Using the above intertwining property, we can immediately
establish the following well-known fact.

######  Theorem 2.4.

If $T_{F}T_{G}$ is a Toeplitz operator on
$H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$, then $T_{F}T_{G}=T_{FG}$.

###### Proof.

$T_{F}T_{G}=T_{FG}-H_{F^{*}}^{*}H_{G}$. If $T_{F}T_{G}=T_{Y}$, then

$T_{Y}=T_{FG}-H_{F^{*}}^{*}H_{G},$

implies that $T_{Y-FG}=H_{F^{*}}^{*}H_{G}$. But then

$M_{z_{i}}^{*n}H_{F^{*}}^{*}H_{G}M_{z_{i}}^{n}=H_{F^{*}}^{*}H_{G}.$

Using the intertwining identity, we get

$M_{z_{i}}^{*n}H_{F^{*}}^{*}M_{z_{i}}^{*n}H_{G}=H_{F^{*}}^{*}H_{G},$

and therefore, for each $x\in H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$, we get

$\|H_{F^{*}}^{*}H_{G}x\|\leq\|M_{z_{i}}^{*n}H_{G}x\|\underset{n\rightarrow\infty}{\rightarrow}0,$

because $M_{z_{i}}$ is a pure isometry for all $i\in\\{1,\ldots,n\\}$, that
is, $M_{z_{i}}^{*n}\rightarrow 0$ in the strong operator topology. This
implies that $H_{F^{*}}^{*}H_{G}=0$, in other words, $T_{F}T_{G}=T_{FG}$. ∎

Before ending this section, let us briefly recall some concepts on partial
isometries.

######  Definition 2.5.

$T\in\mathcal{B}(\mathcal{H})$ is a partial isometry if $T$ is an isometry on
the orthogonal complement of its kernel inside $\mathcal{H}$.

It is well-known that $T$ is a partial isometry on $\mathcal{H}$ if and only
if $T$ satisfies any one of the following equivalent conditions.
$(i)$ $TT^{*}$ is a projection.
$(ii)$ $TT^{*}T=T$.
From this description, the following useful facts can be deduced.
$(i)$ $T$ is a partial isometry if and only if $T^{*}$ is also a partial
isometry.
$(ii)$ The range of a partially isometric operator must be a closed subspace
of $\mathcal{H}$.
Furthermore, it should be noted that if $T$ is a partial isometry, then
$I_{\mathcal{H}}-TT^{*}=P_{\ker T^{*}}$, that is, the orthogonal projection
onto the closed subspace $\ker T^{*}$. We refer the readers to some
fundamental results on partial isometries by Halmos, McLaughlin, and Wallen in
[16, 17].

##  3\. Product of Toeplitz operators

In this section, we aim to establish necessary and sufficient conditions for
which a certain product of Toeplitz operators corresponding to analytic
symbols is again a Toeplitz operator.

######  Lemma 3.1.

Let $\mathcal{F},\mathcal{E}$ be Hilbert spaces and $\Gamma(\bm{z})\in
H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}^{n})$ and
$\Psi(\bm{z})\in
H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}^{n})$. Then the
following are equivalent
$(i)$ $M_{\Gamma}M_{\Psi}^{*}$ is a Toeplitz operator on
$H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$.
$(ii)$ $M_{z_{i}}^{*}M_{\Gamma}P_{\ker M_{z_{i}^{*}}}M_{\Psi}^{*}M_{z_{i}}=0$,
for all $i\in\\{1,\ldots,n\\}$.

###### Proof.

For proving the equivalence between $(i)$ and $(ii)$, let us note that for any
$i\in\\{1,\ldots,n\\}$

$\displaystyle M_{z_{i}}^{*}M_{\Gamma}M_{\Psi}^{*}M_{z_{i}}$
$\displaystyle=M_{z_{i}}^{*}M_{\Gamma}(M_{z_{i}}M_{z_{i}}^{*})M_{\Psi}^{*}M_{z_{i}}+M_{z_{i}}^{*}M_{\Gamma}(I_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}-M_{z_{i}}M_{z_{i}}^{*})M_{\Psi}^{*}M_{z_{i}}$
$\displaystyle=M_{\Gamma}M_{\Psi}^{*}+M_{z_{i}}^{*}M_{\Gamma}P_{\ker
M_{z_{i}}^{*}}M_{\Psi}^{*}M_{z_{i}}.$

From Theorem 2.1, it follows that $M_{\Gamma}M_{\Psi}^{*}$ is a Toeplitz
operator if and only if for all $i\in\\{1,\ldots,n\\}$,
$M_{z_{i}}^{*}M_{\Gamma}P_{\ker M_{z_{i}}^{*}}M_{\Psi}^{*}M_{z_{i}}=0$, ∎

We can further strengthen the above condition in the following manner.

######  Lemma 3.2.

Let $\mathcal{F},\mathcal{E}$ be Hilbert spaces and $\Gamma(\bm{z})\in
H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}^{n})$ and
$\Psi(\bm{z})\in
H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}^{n})$. Then the
following are equivalent for any $i\in\\{1,\ldots,n\\}$.
$(i)$ $M_{z_{i}}^{*}M_{\Gamma}P_{\ker M_{z_{i}^{*}}}M_{\Psi}^{*}M_{z_{i}}=0$
$(ii)$ $M_{z_{i}}^{*}M_{\Gamma}P_{\mathcal{E}}M_{\Psi}^{*}M_{z_{i}}=0$.

###### Proof.

Let us begin with the direction $(i)\implies(ii)$. Note that for any
$i\in\\{1,\ldots,n\\}$,

$M_{z_{i}}^{*}M_{\Gamma}P_{\ker M_{z_{i}^{*}}}M_{\Psi}^{*}M_{z_{i}}=0,$

implies that

$M_{z_{i}}^{*}M_{\Gamma}\big{(}M_{z_{j}}M_{z_{j}}^{*}+I_{H_{\mathcal{F}}^{2}(\mathbb{D}^{n})}-M_{z_{j}}M_{z_{j}}^{*}\big{)}P_{\ker
M_{z_{i}}^{*}}M_{\Psi}^{*}M_{z_{i}}=0,$

for any $j\in\\{1,\ldots,n\\}\setminus\\{i\\}$. In other words,

$M_{z_{i}}^{*}M_{\Gamma}M_{z_{j}}M_{z_{j}}^{*}P_{\ker
M_{z_{i}}^{*}}M_{\Psi}^{*}M_{z_{i}}+M_{z_{i}}^{*}M_{\Gamma}P_{\ker
M_{z_{j}}^{*}}P_{\ker M_{z_{i}}^{*}}M_{\Psi}^{*}M_{z_{i}}=0.$

Since $j\neq i$, we can use that the commutator $[M_{z_{j}},M_{z_{i}}^{*}]=0$
to get

$M_{z_{j}}M_{z_{i}}^{*}M_{\Gamma}P_{\ker
M_{z_{i}}^{*}}M_{\Psi}^{*}M_{z_{i}}M_{z_{j}}^{*}+M_{z_{i}}^{*}M_{\Gamma}P_{\ker
M_{z_{j}}^{*}}P_{\ker M_{z_{i}}^{*}}M_{\Psi}^{*}M_{z_{i}}=0.$

By our assumption $M_{z_{i}}^{*}M_{\Gamma}P_{\ker
M_{z_{i}}^{*}}M_{\Psi}^{*}M_{z_{i}}=0$, we get

$M_{z_{i}}^{*}M_{\Gamma}P_{\ker M_{z_{j}}^{*}}P_{\ker
M_{z_{i}}^{*}}M_{\Psi}^{*}M_{z_{i}}=0.$

Continuing in the same manner as above, we find that the above condition
implies that for any $k\in\\{1,\ldots,n\\}\setminus\\{i,j\\}$, we get

$M_{z_{i}}^{*}M_{\Gamma}M_{z_{k}}M_{z_{k}}^{*}P_{\ker M_{z_{j}}^{*}}P_{\ker
M_{z_{i}}^{*}}M_{\Psi}^{*}M_{z_{i}}+M_{z_{1}}^{*}M_{\Gamma}P_{\ker
M_{z_{k}}^{*}}P_{\ker M_{z_{j}}^{*}}P_{\ker
M_{z_{i}}^{*}}M_{\Psi}^{*}M_{z_{i}}=0,$

which further implies that

$M_{z_{i}}^{*}M_{\Gamma}P_{\ker M_{z_{k}}^{*}}P_{\ker M_{z_{j}}^{*}}P_{\ker
M_{z_{i}}^{*}}M_{\Psi}^{*}M_{z_{i}}=0.$

Iterating the same procedure for $n-3$ many times with distinct numbers in
$\\{1,\ldots,n\\}\setminus\\{i,j,k\\}$, we can conclude that,

$M_{z_{i}}^{*}M_{\Gamma}P_{\mathcal{E}}M_{\Psi}^{*}M_{z_{i}}=0,$

since $P_{\mathcal{E}}=\underset{i=1}{\overset{n}{\Pi}}P_{\ker
M_{z_{i}}^{*}}$. To prove the opposite direction, let us observe that,

$M_{z_{i}}^{*}M_{\Gamma}P_{\mathcal{E}}M_{\Psi}^{*}M_{z_{i}}=0,$

for some $i\in\\{1,\ldots,n\\}$ will imply that

$M_{z_{i}}^{*}M_{\Gamma}P_{\ker M_{z_{1}}^{*}}P_{\ker
M_{z_{i}}^{*}}\big{(}\underset{k=2;k\neq i}{\overset{n}{\Pi}}P_{\ker
M_{z_{k}}^{*}}\big{)}M_{\Psi}^{*}M_{z_{i}}=0.$

This implies that

$\displaystyle M_{z_{i}}^{*}M_{\Gamma}P_{\ker
M_{z_{i}}^{*}}\big{(}\underset{k=2;k\neq i}{\overset{n}{\Pi}}P_{\ker
M_{z_{k}}^{*}}\big{)}M_{\Psi}^{*}M_{z_{i}}$
$\displaystyle=M_{z_{i}}^{*}M_{\Gamma}M_{z_{1}}M_{z_{1}}^{*}P_{\ker
M_{z_{i}}^{*}}\big{(}\underset{k=2;k\neq i}{\overset{n}{\Pi}}P_{\ker
M_{z_{k}}^{*}}\big{)}M_{\Psi}^{*}M_{z_{i}}$
$\displaystyle=M_{z_{1}}M_{z_{i}}^{*}M_{\Gamma}P_{\ker
M_{z_{i}}^{*}}\big{(}\underset{k=2;k\neq i}{\overset{n}{\Pi}}P_{\ker
M_{z_{k}}^{*}}\big{)}M_{\Psi}^{*}M_{z_{i}}M_{z_{1}}^{*}.$

and therefore, using recursion, for any $m\in\mathbb{N}$ we get

$M_{z_{i}}^{*}M_{\Gamma}P_{\ker M_{z_{i}}^{*}}\Pi_{k=2;k\neq i}^{n}P_{\ker
M_{z_{k}}^{*}}M_{\Psi}^{*}M_{z_{i}}=M_{z_{1}}^{m}M_{z_{i}}^{*}M_{\Gamma}P_{\ker
M_{z_{i}}^{*}}\Pi_{k=2;k\neq i}^{n}P_{\ker
M_{z_{k}}^{*}}M_{\Psi}^{*}M_{z_{i}}M_{z_{1}}^{*m}.$

This will imply that for any $f\in H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$,

$\displaystyle\|M_{z_{i}}^{*}M_{\Gamma}P_{\ker M_{z_{i}}^{*}}\Pi_{k=2;k\neq
i}^{n}P_{\ker M_{z_{k}}^{*}}M_{\Psi}^{*}M_{z_{i}}f\|$
$\displaystyle=\|M_{z_{1}}^{m}M_{z_{i}}^{*}M_{\Gamma}P_{\ker
M_{z_{i}}^{*}}\Pi_{k=2;k\neq i}^{n}P_{\ker
M_{z_{k}}^{*}}M_{\Psi}^{*}M_{z_{i}}M_{z_{1}}^{*m}f\|$
$\displaystyle\leq\|M_{z_{1}}^{*m}f\|\underset{m\rightarrow\infty}{\rightarrow}0.$

Hence, we get

$M_{z_{i}}^{*}M_{\Gamma}P_{\ker M_{z_{i}}^{*}}\Pi_{k=2;k\neq i}^{n}P_{\ker
M_{z_{k}}^{*}}M_{\Psi}^{*}M_{z_{i}}=0.$

We will repeat the same process co-ordinate-wise. For instance, the above will
imply that,

$M_{z_{i}}^{*}M_{\Gamma}P_{\ker M_{z_{2}}^{*}}P_{\ker
M_{z_{i}}^{*}}\Pi_{k=3;k\neq i}^{n}P_{\ker
M_{z_{k}}^{*}}M_{\Psi}^{*}M_{z_{i}}=0,$

which gives,

$M_{z_{i}}^{*}M_{\Gamma}P_{\ker M_{z_{i}}^{*}}\Pi_{k=3;k\neq i}^{n}P_{\ker
M_{z_{k}}^{*}}M_{\Psi}^{*}M_{z_{i}}=M_{z_{2}}M_{z_{i}}^{*}M_{\Gamma}P_{\ker
M_{z_{i}}^{*}}\Pi_{k=3;k\neq i}^{n}P_{\ker
M_{z_{k}}^{*}}M_{\Psi}^{*}M_{z_{i}}M_{z_{2}}^{*}.$

Again as above, for any $m\in\mathbb{N}$ we have

$M_{z_{i}}^{*}M_{\Gamma}P_{\ker M_{z_{i}}^{*}}\Pi_{k=3;k\neq i}^{n}P_{\ker
M_{z_{k}}^{*}}M_{\Psi}^{*}M_{z_{i}}=M_{z_{2}}^{m}M_{z_{i}}^{*}M_{\Gamma}P_{\ker
M_{z_{i}}^{*}}\Pi_{k=3;k\neq i}^{n}P_{\ker
M_{z_{k}}^{*}}M_{\Psi}^{*}M_{z_{i}}M_{z_{2}}^{*m},$

which as in the earlier case, will give

$M_{z_{i}}^{*}M_{\Gamma}P_{\ker M_{z_{i}}^{*}}\Pi_{k=3;k\neq i}^{n}P_{\ker
M_{z_{k}}^{*}}M_{\Psi}^{*}M_{z_{i}}=0.$

Repeating the same process for all co-ordinates in
$\\{3,\ldots,n\\}\setminus\\{i\\}$, will give us

$M_{z_{i}}^{*}M_{\Gamma}P_{\ker M_{z_{i}}^{*}}M_{\Psi}^{*}M_{z_{i}}=0.$

This completes the proof. ∎

We are now ready to establish our main result on the product of Toeplitz
operators.

###### Proof of Theorem 1.3.

From Lemma 3.1 and 3.2, we know that $M_{\Gamma}M_{\Psi}^{*}$ is a Toeplitz
operator on $H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$ if and only if

$T_{\overline{z_{i}}\Gamma}P_{\mathcal{E}}T_{z_{i}\Psi^{*}}=M_{z_{i}}^{*}M_{\Gamma}P_{\mathcal{E}}M_{\Psi}^{*}M_{z_{i}}=0,$

for all $i\in\\{1,\ldots,n\\}$. Now for any $\bm{j},\bm{k}\in\mathbb{N}^{n}$
and $\eta,\zeta\in\mathcal{E}$ we get

$\langle
P_{\mathcal{E}}T_{z_{i}\Psi^{*}}\bm{z}^{\bm{k}}\eta,P_{\mathcal{E}}T_{z_{i}\Gamma^{*}}\bm{z}^{\bm{j}}\zeta\rangle=\langle
P_{\mathcal{E}}T_{z_{i}^{k_{i}+1}\Psi^{*}}\bm{\hat{z}_{i}}^{\bm{\hat{k}_{i}}}\eta,P_{\mathcal{E}}T_{z_{i}^{j_{i}+1}\Gamma^{*}}\bm{\hat{z}_{i}}^{\bm{\hat{j}_{i}}}\zeta\rangle,$

where for any $i\in\\{1,\ldots,n\\}$ and $\bm{m}\in\mathbb{N}^{n}$,

$\bm{\hat{z}_{i}}^{\bm{\hat{m}_{i}}}:=\underset{k=1;k\neq
i}{\overset{n}{\Pi}}z_{k}^{m_{k}}.$

By this notation we have
$\bm{z}^{\bm{m}}=z_{i}^{m_{i}}\bm{\hat{z}_{i}}^{\bm{\hat{m}_{i}}}$. Now note
that from our assumption

$\Gamma(\bm{z}):=\sum_{\bm{l}\in\mathbb{N}^{n}}\bm{z}^{\bm{l}}A_{\bm{l}};\quad\Psi(\bm{z}):=\sum_{\bm{m}\in\mathbb{N}^{n}}\bm{z}^{\bm{m}}B_{\bm{m}},$

where
$A_{\bm{l}}:=A_{(l_{1},\ldots,l_{n})},B_{\bm{m}}:=B_{(m_{1},\ldots,m_{n})}\in\mathcal{B}(\mathcal{F},\mathcal{E})$
for all $\bm{l}=(l_{1},\ldots,l_{n}),$ and $\bm{m}=(m_{1},\ldots,m_{n})$ in
$\mathbb{N}^{n}$. Now,

$\displaystyle
P_{\mathcal{E}}T_{{z_{i}}^{j_{i}+1}\Gamma^{*}}\bm{\hat{z}_{i}}^{\bm{\hat{j}_{i}}}\zeta$
$\displaystyle=P_{\mathcal{E}}P_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}{z_{i}}^{j_{i}+1}\bm{\hat{z}_{i}}^{\bm{\hat{j}_{i}}}\Gamma(\bm{z})^{*}\zeta$
$\displaystyle=P_{\mathcal{E}}\sum_{\bm{l}\in\mathbb{N}^{n}}{z_{i}}^{j_{i}+1}\bm{\hat{z}_{i}}^{\bm{\hat{j}_{i}}}\bar{\bm{z}}^{\bm{l}}A_{\bm{l}}^{*}\zeta$
$\displaystyle=P_{\mathcal{E}}\sum_{\bm{l}\in\mathbb{N}^{n}}{z_{i}}^{j_{i}-l_{i}+1}\bm{\hat{z}_{i}}^{\bm{\hat{j}_{i}}}\bar{\bm{\hat{z}_{i}}}^{\bm{\hat{l}_{i}}}A_{\bm{l}}^{*}\zeta$
$\displaystyle=P_{\mathcal{E}}\sum_{\bm{l}\in\mathbb{N}^{n};l_{i}=j_{i}+1}\bm{\hat{z}_{i}}^{\bm{\hat{j}_{i}-\hat{l}_{i}}}A_{\bm{l}}^{*}\zeta$
$\displaystyle=A_{(j_{1},\ldots,j_{i-1},j_{i}+1,j_{i+1},\ldots,j_{n})}^{*}\zeta.$

Similarly,

$P_{\mathcal{E}}T_{z_{i}\Psi^{*}}\bm{z}^{\bm{k}}\eta=P_{\mathcal{E}}T_{z_{i}^{k_{i}+1}\Psi^{*}}\bm{\hat{z}_{i}}^{\bm{\hat{k}_{i}}}\eta=B_{(k_{1},\ldots,k_{i-1},k_{i}+1,k_{i+1},\ldots,k_{n})}^{*}\eta.$

Hence,

$\langle
P_{\mathcal{E}}T_{z_{i}\Psi^{*}}\bm{z}^{\bm{k}}\eta,P_{\mathcal{E}}T_{z_{i}\Gamma^{*}}\bm{z}^{\bm{j}}\zeta\rangle=\langle
B_{(k_{1},\ldots,k_{i-1},k_{i}+1,k_{i+1},\ldots,k_{n})}^{*}\eta,A_{(j_{1},\ldots,j_{i-1},j_{i}+1,j_{i+1},\ldots,j_{n})}^{*}\zeta\rangle$

for any $\eta,\zeta\in\mathcal{E}$. Therefore, for any $i\in\\{1,\ldots,n\\}$
we get $T_{\overline{z_{i}}\Gamma}P_{\mathcal{E}}T_{z_{i}\Psi^{*}}=0$ if and
only if

$A_{(j_{1},\ldots,j_{i-1},j_{i}+1,j_{i+1},\ldots,j_{n})}B_{(k_{1},\ldots,k_{i-1},k_{i+1},k_{i+1},\ldots,k_{n})}^{*}=0,$

for all $\bm{j},\bm{k}\in\mathbb{N}^{n}$. Using Lemma 3.1 and Lemma 3.2, we
get $M_{\Gamma}M_{\Psi}^{*}$ is a Toeplitz operator if and only if

$A_{\bm{l}+e_{i}}B_{\bm{m}+e_{i}}^{*}=0.$

for all $i\in\\{1,\ldots,n\\}$, and $\bm{l},\bm{m}\in\mathbb{N}^{n}$. This
completes the proof. ∎

######  Remark 3.1.

It is natural to ask what will be the symbol of the Toeplitz operator in the
above case. We will just observe the simple case when $n=1$. From the
conditions in Theorem 1.3, we get that $A_{l+1}B_{m+1}^{*}=0$ for all
$l,m\in\mathbb{N}$. Hence,

$\displaystyle\Gamma(z)\Psi(z)^{*}=\sum_{l\in\mathbb{N}}A_{l}z^{l}\sum_{m\in\mathbb{N}}B_{m}^{*}\bar{z}^{m}$
$\displaystyle=A_{0}\sum_{m\in\mathbb{N}}B_{m}^{*}\bar{z}^{m}+\sum_{l\in\mathbb{N};l\neq
0}A_{l}z^{l}B_{0}^{*}$ $\displaystyle=A_{0}\sum_{m\in\mathbb{N};m\neq
0}B_{m}^{*}\bar{z}^{m}+\sum_{l\in\mathbb{N}}A_{l}z^{l}B_{0}^{*}$
$\displaystyle=A_{0}\big{(}\Psi(z)-\Psi(0)\big{)}^{*}+\Gamma(z)B_{0}^{*}.$

Therefore,

$M_{\Gamma}M_{\Psi}^{*}=T_{\tilde{\Gamma}+\tilde{\Psi}^{*}}=T_{\tilde{\Gamma}}+T_{\tilde{\Psi}^{*}},$

where

$\tilde{\Gamma}(z)=\Gamma(z)B_{0}^{*};\quad\tilde{\Psi}(z)=\big{(}\Psi(z)-\Psi(0)\big{)}A_{0}^{*}.$

##  4\. Beurling-type Toeplitz ranges

In the scalar cases, the main result in [8] shows that a partially isometric
Toeplitz operator $T_{\phi}$ on $H^{2}(\mathbb{D}^{n})$ must be of the
following form.

$T_{\phi}=M_{\phi_{1}}^{*}M_{\phi_{2}},$

where $\phi_{1},\phi_{2}$ are inner functions on $\mathbb{D}^{n}$, but
depending on disjoint set of variables. As a consequence of this
factorization, one observes that

$\operatorname{ran}T_{\phi}=M_{\phi_{1}}H^{2}(\mathbb{D}^{n});\quad\operatorname{ran}T_{\phi}^{*}=M_{\phi_{2}}H^{2}(\mathbb{D}^{n}).$

In other words, the range of the partially isometric Toeplitz operator must be
Beurling-type. This section aims to prove that this result is true even for
vector-valued Hardy spaces. Since the commutativity of the symbols is lacking
in vector-valued Hardy spaces, we have to pursue a completely new and
different approach based on the characterization of Beurling-type invariant
subspaces of $H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$ via restriction operators.
We begin by first showing that the range of partially isometric Toeplitz
operators is shift-invariant subspaces of
$H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$.

######  Proposition 4.1.

Let $T_{\Phi}\in\mathcal{B}(H_{\mathcal{B}(\mathcal{E})}^{2}(\mathbb{D}^{n}))$
be a partial isometry, then the range of $T_{\Phi}$ is a
$(M_{z_{1}},\ldots,M_{z_{n}})$-joint invariant closed subspace of
$H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$.

###### Proof.

We already know that $T_{\Phi}$ is a partial isometry if and only if
$T_{\Phi}^{*}$ is a partial isometry. Thus, $T_{\Phi}$ is a partial isometry
implies that both $\operatorname{ran}T_{\Phi}$ and
$\operatorname{ran}T_{\Phi}^{*}$ are closed subspaces of
$H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$. Thus, we only need to show the
$(M_{z_{1}},\ldots,M_{z_{n}})$-joint invariance of
$\operatorname{ran}T_{\Phi}$. To prove this, let us note that for any
$f\in\operatorname{ran}T_{\Phi}$, we have the following inequalities for any
arbitrary but fixed $i\in\\{1,\ldots,n\\}$.

$\|z_{i}f\|\geq\|T_{\Phi}^{*}M_{z_{i}}f\|\geq\|M_{z_{i}}^{*}T_{\Phi}^{*}M_{z_{i}}f\|=\|T_{\Phi}^{*}f\|=\|f\|=\|z_{i}f\|,$

and thus, $\|T_{\Phi}^{*}z_{i}f\|=\|z_{i}f\|$. Since $T_{\Phi}^{*}$ is a
partial isometry we must have $z_{i}f\in\operatorname{ran}T_{\Phi}$. Since $i$
was arbitrarily chosen, this must be true for all $i\in\\{1,\ldots,n\\}$. This
completes the proof. ∎

Before proving our next result, let us highlight some useful facts involving
shift-invariant subspaces of $H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$. Given any
$(M_{z_{1}},\ldots,M_{z_{n}})$-joint invariant closed subspace (say)
$\mathcal{S}\subseteq H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$, we can always
associate the following restriction operators

$R_{i}:=M_{z_{i}}|_{\mathcal{S}},$

for each $i\in\\{1,\ldots,n\\}$. It is well known these operators play a
crucial role in characterizing shift-invariant subspaces of
$H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$. We define the subspace $\mathcal{S}$ to
be doubly commuting if

$[R_{i}^{*},R_{j}]=0$

for all distinct $i,j\in\\{1,\ldots,n\\}$. From [25], we know that the
following equivalence holds:
$(i)$ $\mathcal{S}$ is doubly commuting,
$(ii)$ $\mathcal{S}$ is a Beurling-type subspace of
$H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$, that is, by definition, there exists a
Hilbert space $\mathcal{F}$, and an inner function $\Theta(\bm{z})\in
H_{\mathcal{B}(\mathcal{E})}^{\infty}(\mathbb{D}^{n})$ such that

$\mathcal{S}=M_{\Theta}H_{\mathcal{E}}^{2}(\mathbb{D}^{n}).$

In the following result, we prove that the range of a partially isometric
Toeplitz operator always admits the above description.

###### Proof of Theorem 1.2.

Our approach is to show that if $T_{\Phi}$ is a partial isometry then
$\operatorname{ran}T_{\Phi}^{*}$ is a Beurling-type invariant subspace of
$H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$. This will prove what we want because
then,

$T_{\Phi}\text{ is a partial isometry }\Longleftrightarrow T_{\Phi}^{*}\text{
is a partial isometry }\implies\operatorname{ran}T_{\Phi}\text{ is Beurling-
type}.$

It is evident from the preceding discussion that we need to prove
$\operatorname{ran}T_{\Phi}^{*}$ is doubly commuting. In other words, for any
distinct $i,j\in\\{1,\ldots,n\\}$, the following conditions should be
satisfied.

$R_{i}^{*}R_{j}=R_{j}R_{i}^{*}.$

Since $T_{\Phi}$ is a partial isometry, we can further deduce that,

$R_{i}=M_{z_{i}}|_{\operatorname{ran}T_{\Phi}^{*}}=M_{z_{i}}T_{\Phi}^{*}T_{\Phi}\quad(\forall
i\in\\{1,\ldots,n\\}).$

Now let us establish a few conditions essential for the sequel. From
Proposition 4.1, we know that $\operatorname{ran}T_{\Phi}$ is a
$(M_{z_{1}},\ldots,M_{z_{n}})$-joint invariant subspace, and therefore,

$T_{\Phi}T_{\Phi}^{*}M_{z_{i}}T_{\Phi}=M_{z_{i}}T_{\Phi}.$

Acting on the left by $M_{z_{i}}^{*}$ gives

$M_{z_{i}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{i}}T_{\Phi}=T_{\Phi},$

which further implies that

$\displaystyle T_{\Phi}+M_{z_{i}}^{*}T_{\Phi}P_{\ker
M_{z_{i}}^{*}}T_{\Phi}^{*}M_{z_{i}}T_{\Phi}$
$\displaystyle=T_{\Phi}T_{\Phi}^{*}T_{\Phi}+M_{z_{i}}^{*}T_{\Phi}P_{\ker
M_{z_{i}}^{*}}T_{\Phi}^{*}M_{z_{i}}T_{\Phi}$
$\displaystyle=M_{z_{i}}^{*}T_{\Phi}M_{z_{i}}M_{z_{i}}^{*}T_{\Phi}^{*}M_{z_{i}}T_{\Phi}+M_{z_{i}}^{*}T_{\Phi}P_{\ker
M_{z_{i}}^{*}}T_{\Phi}^{*}M_{z_{i}}T_{\Phi}$
$\displaystyle=M_{z_{i}}^{*}T_{\Phi}\big{(}M_{z_{i}}M_{z_{i}}^{*}+P_{\ker
M_{z_{i}}^{*}}\big{)}T_{\Phi}^{*}M_{z_{i}}T_{\Phi}$
$\displaystyle=M_{z_{i}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{i}}T_{\Phi}$
$\displaystyle=T_{\Phi}.$

Therefore, we get

$M_{z_{i}}^{*}T_{\Phi}P_{\ker M_{z_{i}}^{*}}T_{\Phi}^{*}M_{z_{i}}T_{\Phi}=0,$

which again implies that

$T_{\Phi}^{*}M_{z_{i}}^{*}T_{\Phi}P_{\ker
M_{z_{i}}^{*}}T_{\Phi}^{*}M_{z_{i}}T_{\Phi}=0.$

and hence,

$P_{\ker M_{z_{i}}^{*}}T_{\Phi}^{*}M_{z_{i}}T_{\Phi}=0.$

In other words, since $P_{\ker
M_{z_{i}}^{*}}=I_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}-M_{z_{i}}M_{z_{i}}^{*}$,
we get

(4.3) $M_{z_{i}}T_{\Phi}^{*}T_{\Phi}=T_{\Phi}^{*}M_{z_{i}}T_{\Phi}.$

Since $i$ was arbitrarily chosen, the above identity holds for all
$i\in\\{1,\ldots,n\\}$. Now, if we act on the right of both sides by
$T_{\Phi}^{*}$ and on the left of both sides by $T_{\Phi}$, we get

$T_{\Phi}M_{z_{i}}T_{\Phi}^{*}T_{\Phi}T_{\Phi}^{*}=T_{\Phi}T_{\Phi}^{*}M_{z_{i}}T_{\Phi}T_{\Phi}^{*}.$

Using the $M_{z_{i}}$-invariance of $\operatorname{ran}T_{\Phi}$ we get

(4.4)
$M_{z_{i}}T_{\Phi}T_{\Phi}^{*}=T_{\Phi}M_{z_{i}}T_{\Phi}^{*}\quad(i\in\\{1,\ldots,n\\}).$

and therefore, using identity (4.3), we get

$\displaystyle
R_{i}^{*}R_{j}=T_{\Phi}^{*}T_{\Phi}M_{z_{i}}^{*}M_{z_{j}}T_{\Phi}^{*}T_{\Phi}$
$\displaystyle=T_{\Phi}^{*}M_{z_{i}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{j}}T_{\Phi}=T_{\Phi}^{*}M_{z_{i}}^{*}M_{z_{j}}T_{\Phi}=T_{\Phi}^{*}M_{z_{j}}M_{z_{i}}^{*}T_{\Phi},$

and

$R_{j}R_{i}^{*}=M_{z_{j}}T_{\Phi}^{*}T_{\Phi}M_{z_{i}}^{*}T_{\Phi}^{*}T_{\Phi}=M_{z_{j}}T_{\Phi}^{*}T_{\Phi}M_{z_{i}}^{*}.$

The last equality follows from the fact that $\ker T_{\Phi}^{*}$ is
$(M_{z_{1}}^{*},\ldots,M_{z_{n}}^{*})$-joint invariant. Therefore,
$\operatorname{ran}T_{\Phi}^{*}$ is a doubly commuting shift-invariant
subspace if and only if

(4.5)
$T_{\Phi}^{*}M_{z_{j}}M_{z_{i}}^{*}T_{\Phi}=M_{z_{j}}T_{\Phi}^{*}T_{\Phi}M_{z_{i}}^{*}.$

for all distinct $i,j\in\\{1,\ldots,n\\}$. Using the identity (4.3), we can
again observe that

$\displaystyle T_{\Phi}^{*}M_{z_{j}}P_{\ker
T_{\Phi}^{*}}M_{z_{i}}^{*}T_{\Phi}$
$\displaystyle=T_{\Phi}^{*}M_{z_{j}}M_{z_{i}}^{*}T_{\Phi}-T_{\Phi}^{*}M_{z_{j}}T_{\Phi}T_{\Phi}^{*}M_{z_{i}}^{*}T_{\Phi}$
$\displaystyle=T_{\Phi}^{*}M_{z_{j}}M_{z_{i}}^{*}T_{\Phi}-M_{z_{j}}T_{\Phi}^{*}T_{\Phi}T_{\Phi}^{*}T_{\Phi}M_{z_{i}}^{*}$
$\displaystyle=T_{\Phi}^{*}M_{z_{j}}M_{z_{i}}^{*}T_{\Phi}-M_{z_{j}}T_{\Phi}^{*}T_{\Phi}M_{z_{i}}^{*}.$

Therefore, condition (4.5) is again equivalent to
$T_{\Phi}^{*}M_{z_{j}}P_{\ker T_{\Phi}^{*}}M_{z_{i}}^{*}T_{\Phi}=0$. We will
now show that this identity holds. From the identities (4.3) and (4.4), we can
get for any distinct $i,j\in\\{1,\ldots,n\\}$.

$\displaystyle
M_{z_{i}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{i}}M_{z_{j}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{j}}$
$\displaystyle=M_{z_{i}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{j}}^{*}M_{z_{i}}T_{\Phi}T_{\Phi}^{*}M_{z_{j}}$
$\displaystyle=M_{z_{i}}^{*}T_{\Phi}M_{z_{j}}^{*}T_{\Phi}^{*}T_{\Phi}M_{z_{i}}T_{\Phi}^{*}M_{z_{j}}$
$\displaystyle=M_{z_{i}}^{*}T_{\Phi}M_{z_{j}}^{*}M_{z_{i}}T_{\Phi}^{*}M_{z_{j}}$
$\displaystyle=M_{z_{i}}^{*}T_{\Phi}M_{z_{i}}M_{z_{j}}^{*}T_{\Phi}^{*}M_{z_{j}}$
$\displaystyle=T_{\Phi}T_{\Phi}^{*}.$

Therefore, for any distinct $i,j\in\\{1,\ldots,n\\}$, we get

(4.6)
$M_{z_{i}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{i}}M_{z_{j}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{j}}=T_{\Phi}T_{\Phi}^{*}=M_{z_{j}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{j}}M_{z_{i}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{i}}.$

From this, we can further deduce that

$\displaystyle M_{z_{i}}^{*}P_{\ker T_{\Phi}^{*}}M_{z_{i}}M_{z_{j}}^{*}P_{\ker
T_{\Phi}^{*}}M_{z_{j}}$
$\displaystyle=M_{z_{i}}^{*}M_{z_{i}}M_{z_{j}}^{*}P_{\ker
T_{\Phi}^{*}}M_{z_{j}}-M_{z_{i}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{i}}M_{z_{j}}^{*}P_{\ker
T_{\Phi}^{*}}M_{z_{j}}$ $\displaystyle=M_{z_{j}}^{*}P_{\ker
T_{\Phi}^{*}}M_{z_{j}}-M_{z_{i}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{i}}M_{z_{j}}^{*}P_{\ker
T_{\Phi}^{*}}M_{z_{j}}$ $\displaystyle=M_{z_{j}}^{*}P_{\ker
T_{\Phi}^{*}}M_{z_{j}}-M_{z_{i}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{i}}M_{z_{j}}^{*}M_{z_{j}}$
$\displaystyle+M_{z_{i}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{i}}M_{z_{j}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{j}}$
$\displaystyle=M_{z_{j}}^{*}P_{\ker
T_{\Phi}^{*}}M_{z_{j}}-M_{z_{i}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{i}}+T_{\Phi}T_{\Phi}^{*}.$

Since the right-hand side is self-adjoint, the above implies that for any
distinct $i,j\in\\{1,\ldots,n\\}$,

(4.7) $M_{z_{i}}^{*}P_{\ker T_{\Phi}^{*}}M_{z_{i}}M_{z_{j}}^{*}P_{\ker
T_{\Phi}^{*}}M_{z_{j}}=M_{z_{j}}^{*}P_{\ker
T_{\Phi}^{*}}M_{z_{j}}M_{z_{i}}^{*}P_{\ker T_{\Phi}^{*}}M_{z_{i}}.$

Now let us compute $C_{i}^{*}C_{i}$, where $C_{i}=P_{\ker
T_{\Phi}^{*}}M_{z_{i}}|_{\ker T_{\Phi}^{*}}$ for $i\in\\{1,\ldots,n\\}$.

$C_{i}^{*}C_{i}=P_{\ker T_{\Phi}^{*}}M_{z_{i}}^{*}P_{\ker
T_{\Phi}^{*}}M_{z_{i}}P_{\ker T_{\Phi}^{*}}=M_{z_{i}}^{*}P_{\ker
T_{\Phi}^{*}}M_{z_{i}}P_{\ker T_{\Phi}^{*}}=M_{z_{i}}^{*}P_{\ker
T_{\Phi}^{*}}M_{z_{i}}.$

The second and last equalities follow by using the fact that
$\operatorname{ran}T_{\Phi}$ is $(M_{z_{1}},\ldots,M_{z_{n}})$-joint
invariant. Now, let us observe that

$\displaystyle I_{\ker T_{\Phi}^{*}}-C_{i}^{*}C_{i}=I_{\ker
T_{\Phi}^{*}}-M_{z_{i}}^{*}P_{\ker T_{\Phi}^{*}}M_{z_{i}}$
$\displaystyle=I_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}-T_{\Phi}T_{\Phi}^{*}-M_{z_{i}}^{*}P_{\ker
T_{\Phi}^{*}}M_{z_{i}}$
$\displaystyle=I_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}-T_{\Phi}T_{\Phi}^{*}-I_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}+M_{z_{i}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{i}}$
$\displaystyle=M_{z_{i}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{i}}-T_{\Phi}T_{\Phi}^{*}.$

By compressing the above identity with respect to $P_{\ker T_{\Phi}^{*}}$, we
get,

$I_{\ker T_{\Phi}^{*}}-C_{i}^{*}C_{i}=P_{\ker
T_{\Phi}^{*}}M_{z_{i}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{i}}P_{\ker T_{\Phi}^{*}},$

Therefore, for each $i\in\\{1,\ldots,n\\}$, there exists isometries,

$Y_{i}:\overline{(I_{\ker
T_{\Phi}^{*}}-C_{i}^{*}C_{i})^{\frac{1}{2}}\mathcal{H}}\rightarrow\overline{T_{\Phi}^{*}M_{z_{i}}P_{\ker
T_{\Phi}^{*}}\mathcal{H}}$

such that

$Y_{i}(I_{\ker
T_{\Phi}^{*}}-C_{i}^{*}C_{i})^{\frac{1}{2}}=T_{\Phi}^{*}M_{z_{i}}P_{\ker
T_{\Phi}^{*}}.$

Now using conditions (4.3), (4.4), and (4.6), we deduce that

$\displaystyle(I_{\ker T_{\Phi}^{*}}-C_{i}^{*}C_{i})(I_{\ker
T_{\Phi}^{*}}-C_{j}^{*}C_{j})$
$\displaystyle=\big{(}M_{z_{i}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{i}}-T_{\Phi}T_{\Phi}^{*}\big{)}\big{(}M_{z_{j}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{j}}-T_{\Phi}T_{\Phi}^{*}\big{)}$
$\displaystyle=M_{z_{i}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{i}}M_{z_{j}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{j}}-M_{z_{i}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{i}}T_{\Phi}T_{\Phi}^{*}$
$\displaystyle-
T_{\Phi}T_{\Phi}^{*}M_{z_{j}}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{j}}+T_{\Phi}T_{\Phi}^{*}$
$\displaystyle=T_{\Phi}T_{\Phi}^{*}-M_{z_{i}}^{*}T_{\Phi}M_{z_{i}}T_{\Phi}^{*}T_{\Phi}T_{\Phi}^{*}$
$\displaystyle-
T_{\Phi}M_{z_{j}}^{*}T_{\Phi}^{*}T_{\Phi}T_{\Phi}^{*}M_{z_{j}}+T_{\Phi}T_{\Phi}^{*}$
$\displaystyle=T_{\Phi}T_{\Phi}^{*}-T_{\Phi}T_{\Phi}^{*}-T_{\Phi}T_{\Phi}^{*}+T_{\Phi}T_{\Phi}^{*}$
$\displaystyle=0.$

Since, both $(I_{\ker T_{\Phi}^{*}}-C_{i}^{*}C_{i})$ and $(I_{\ker
T_{\Phi}^{*}}-C_{j}^{*}C_{j})$ are non-negative operators, therefore, we have

$(I_{\ker T_{\Phi}^{*}}-C_{i}^{*}C_{i})^{\frac{1}{2}}(I_{\ker
T_{\Phi}^{*}}-C_{j}^{*}C_{j})^{\frac{1}{2}}=0.$

This implies that

$T_{\Phi}^{*}M_{z_{j}}P_{\ker T_{\Phi}^{*}}M_{z_{i}}^{*}T_{\Phi}=Y_{j}(I_{\ker
T_{\Phi}^{*}}-C_{j}^{*}C_{j})^{\frac{1}{2}}(I_{\ker
T_{\Phi}^{*}}-C_{i}^{*}C_{i})^{\frac{1}{2}}Y_{i}^{*}=0.$

This completes the proof. ∎

##  5\. Partially isometric Toeplitz operators

This section will prove that partially isometric Toeplitz operators always
admit a factorization into Toeplitz operators corresponding to inner symbols.

###### Proof of Theorem 1.4.

From Theorem 1.2, we know that $T_{\Phi}$ is a partial isometry implies that
both $\operatorname{ran}T_{\Phi}$ and $\operatorname{ran}T_{\phi}^{*}$ are
Beurling-type invariant subspace of $H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$.
Thus, there exist Hilbert spaces $\mathcal{F},\mathcal{G}$ and inner functions
$\Gamma(\bm{z})\in
H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}^{n})$,
$\Psi(\bm{z})\in
H_{\mathcal{B}(\mathcal{G},\mathcal{E})}^{\infty}(\mathbb{D}^{n})$ such that

$\operatorname{ran}T_{\Phi}=M_{\Gamma}H_{\mathcal{F}}^{2}(\mathbb{D}^{n}),$

and

$\operatorname{ran}T_{\Phi}^{*}=M_{\Psi}H_{\mathcal{G}}^{2}(\mathbb{D}^{n}).$

From the above identities, it follows that,

$T_{\Phi}T_{\Phi}^{*}=M_{\Gamma}M_{\Gamma}^{*};\quad
T_{\Phi}^{*}T_{\Phi}=M_{\Psi}M_{\Psi}^{*}.$

Since $T_{\Phi}$ is partial isometry, it further implies that

$M_{\Psi}M_{\Psi}^{*}=T_{\Phi}^{*}T_{\Phi}=T_{\Phi}^{*}T_{\Phi}T_{\Phi}^{*}T_{\Phi}=T_{\Phi}^{*}M_{\Gamma}M_{\Gamma}^{*}T_{\Phi},$

and therefore, $\|M_{\Psi}^{*}h\|=\|M_{\Gamma}^{*}T_{\Phi}h\|$ for any $h\in
H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$. Further note that since $M_{\Psi}$ is an
isometry, we have
$\operatorname{ran}M_{\Psi}^{*}=H_{\mathcal{G}}^{2}(\mathbb{D}^{n})$. Thus, we
can define an isometry $X:H_{\mathcal{G}}^{2}(\mathbb{D}^{n})\rightarrow
H_{\mathcal{F}}^{2}(\mathbb{D}^{n})$ by

$XM_{\Psi}^{*}h=M_{\Gamma}^{*}T_{\Phi}h,$

in other words, $XM_{\Psi}^{*}=M_{\Gamma}^{*}T_{\Phi}$ on
$H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$. Using this map, we can observe that

$T_{\Phi}=M_{\Gamma}M_{\Gamma}^{*}T_{\Phi}=M_{\Gamma}XM_{\Psi}^{*}.$

This further implies that

$X=M_{\Gamma}^{*}T_{\Phi}M_{\Psi},$

and hence,

$M_{z_{i}}^{*}XM_{z_{i}}=M_{z_{i}}^{*}M_{\Gamma}^{*}T_{\Phi}M_{\Psi}M_{z_{i}}=M_{\Gamma}^{*}M_{z_{i}}^{*}T_{\Phi}M_{z_{i}}M_{\Psi}=M_{\Gamma}^{*}T_{\Phi}M_{\Psi}=X,$

for all $i\in\\{1,\ldots,n\\}$. Moreover,

$XX^{*}=M_{\Gamma}^{*}T_{\Phi}M_{\Psi}M_{\Psi}^{*}T_{\Phi}^{*}M_{\Gamma}=M_{\Gamma}^{*}T_{\Phi}T_{\Phi}^{*}T_{\Phi}T_{\Phi}^{*}M_{\Gamma}=M_{\Gamma}^{*}T_{\Phi}T_{\Phi}^{*}M_{\Gamma}=M_{\Gamma}^{*}M_{\Gamma}M_{\Gamma}^{*}M_{\Gamma}=I_{H_{\mathcal{F}}^{2}(\mathbb{D}^{n})},$

shows that $X$ is a unitary between $H_{\mathcal{G}}^{2}(\mathbb{D}^{n})$ and
$H_{\mathcal{F}}^{2}(\mathbb{D}^{n})$. Again, for any $i\in\\{1,\ldots,n\\}$,
we get

$I_{H_{\mathcal{F}}^{2}(\mathbb{D}^{n})}=XX^{*}=M_{z_{i}}^{*}XM_{z_{i}}M_{z_{i}}^{*}X^{*}M_{z_{i}}=I_{H_{\mathcal{F}}^{2}(\mathbb{D}^{n})}-M_{z_{i}}^{*}XP_{\ker
M_{z_{i}}^{*}}X^{*}M_{z_{i}}.$

This implies that $M_{z_{i}}^{*}XP_{\ker M_{z_{i}}^{*}}X^{*}M_{z_{i}}=0$, and
hence
$(I_{H_{\mathcal{F}}^{2}(\mathbb{D}^{n})}-M_{z_{i}}M_{z_{i}}^{*})X^{*}M_{z_{i}}=0$
and therefore,

(5.8) $X^{*}M_{z_{i}}=M_{z_{i}}X^{*},$

for all $i\in\\{1,\ldots,n\\}$. Similarly, for any $i\in\\{1,\ldots,n\\}$ we
have

$I_{H_{\mathcal{G}}^{2}(\mathbb{D}^{n})}=X^{*}X=M_{z_{i}}^{*}X^{*}M_{z_{i}}M_{z_{i}}^{*}XM_{z_{i}}=I_{H_{\mathcal{G}}^{2}(\mathbb{D}^{n})}-M_{z_{i}}^{*}X^{*}P_{\ker
M_{z_{i}}^{*}}XM_{z_{i}}.$

This implies that $M_{z_{i}}^{*}X^{*}P_{\ker M_{z_{i}}^{*}}XM_{z_{i}}=0$ and
thus, we get

(5.9) $XM_{z_{i}}=M_{z_{i}}X,$

for all $i\in\\{1,\ldots,n\\}$. From conditions (5.8) and (5.9), it implies
that $X$ must be a constant unitary from $H_{\mathcal{G}}^{2}(\mathbb{D})$ to
$H_{\mathcal{F}}^{2}(\mathbb{D})$. Therefore, we can re-write the map $X$ as
$I_{H^{2}(\mathbb{D}^{n})}\otimes X$, where
$X:\mathcal{G}\rightarrow\mathcal{F}$ is a unitary. Based on these
observations, we can now write

$T_{\Phi}=M_{\Gamma}(I_{H_{\mathcal{E}}^{2}(\mathbb{D}^{n})}\otimes
X)M_{\Psi}^{*}=M_{\Gamma}M_{\tilde{\Psi}}^{*},$

where $\tilde{\Psi}(\bm{z}):=\Psi(\bm{z})X^{*}\in
H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}^{n})$ is an inner
function. Since $T_{\Phi}=M_{\Gamma}M_{\tilde{\Psi}}^{*}$ is a Toeplitz
operator, from Theorem 1.3, we must have

$A_{\bm{l}+e_{i}}XB_{\bm{m}+e_{i}}^{*}=0\quad(\bm{m},\bm{n}\in\mathbb{N}^{n}),$

where $e_{i}=(0,\ldots,0,\underset{\text{ ith position}}{1},0,\ldots,0)$ for
all $i\in\\{1,\ldots,n\\}$ and

$\Gamma(\bm{z}):=\sum_{\bm{l}\in\mathbb{N}^{n}}A_{\bm{l}}z^{\bm{l}}\in
H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}^{n});\quad\Psi(\bm{z}):=\sum_{\bm{m}\in\mathbb{N}^{n}}B_{\bm{m}}z^{\bm{m}}\in
H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}^{n}).$

Conversely, if the Toeplitz operator admits a factorization like
$T_{\Phi}=M_{\Gamma}M_{\Psi}^{*}$. Then

$T_{\Phi}T_{\Phi}^{*}=M_{\Gamma}M_{\Psi}^{*}M_{\Psi}M_{\Gamma}^{*}=M_{\Gamma}M_{\Gamma}^{*},$

shows that $T_{\Phi}$ is a partial isometry. This completes the proof. ∎

The above result can be used to see direct correspondence between partially
isometric Toeplitz operators and their symbols.

######  Corollary 5.1.

If $T_{\Phi}$ is a partially isometric Toeplitz operator on
$H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$ then $\Phi(\bm{z})$ is a partial
isometry almost everywhere on $\mathbb{T}^{n}$.

###### Proof.

From Theorem 1.4 and Theorem 2.4, we get

$T_{\Phi}=M_{\Gamma}M_{\Psi}^{*}=T_{\Gamma\Psi^{*}}.$

This implies that $\Phi(\bm{z})=\Gamma(\bm{z})\Psi(\bm{z})^{*}$ a.e. on
$\mathbb{T}^{n}$. Since $\Gamma(\bm{z}),\Psi(\bm{z})$ are inner functions it
implies that $\Gamma(\bm{z)}\Gamma(\bm{z)}^{*}$ is projection-valued and
$\Psi(\bm{z})^{*}\Psi(\bm{z})=I_{\mathcal{F}}$ a.e. on $\mathbb{T}^{n}$ and
therefore,

$\Phi(\bm{z})\Phi(\bm{z})^{*}=\Gamma(\bm{z})\Psi^{*}(\bm{z})\Psi(\bm{z})\Gamma^{*}(\bm{z})=\Gamma(\bm{z})\Gamma^{*}(\bm{z})\quad(\text{a.e.
on }\mathbb{T}^{n}).$

This shows that $\Phi(\bm{z})$ is a partial isometry almost everywhere on
$\mathbb{T}^{n}$. This completes the proof. ∎

######  Remark 5.1.

The converse direction is not true and we present an example here. Let
$\theta(z)=\frac{z}{2}$, and let us consider the symbol $\Phi\in
L_{\mathcal{B}(\mathbb{C}^{2})}^{\infty}(\mathbb{T})$ defined by

$\Phi(e^{it}):=\begin{bmatrix}\theta(e^{it})&(1-|\theta(e^{it})|^{2})^{\frac{1}{2}}\\\
0&0\end{bmatrix}=\begin{bmatrix}\theta(e^{it})&\frac{\sqrt{3}}{2}\\\
0&0\end{bmatrix}\in\mathcal{B}(\mathbb{C}^{2}).$

It is clear that

$\Phi(e^{it})^{*}\Phi(e^{it})=\begin{bmatrix}1&0\\\ 0&0\end{bmatrix}.$

Thus, $\Phi(e^{it})$ is a partial isometry on $\mathbb{T}$. However, the
corresponding Toeplitz operator

$T_{\Phi}=\begin{bmatrix}T_{\theta}&T_{\frac{\sqrt{3}}{2}}\\\
0&0\end{bmatrix}\in H_{\mathbb{C}^{2}}^{2}(\mathbb{D}),$

is not partially isometric. One can easily that if we want
$T_{\Phi}T_{\Phi}^{*}T_{\Phi}=T_{\Phi}$, then a necessary condition is
$T_{\theta}T_{\theta}^{*}=T_{\frac{1}{4}}$. This is not possible as
$\theta(z)$ is analytic.

We will now completely characterize partially isometric Toeplitz operators
with analytic symbols.

######  Theorem 5.2.

Let $\Phi(\bm{z})\in H_{\mathcal{B}(\mathcal{E})}^{\infty}(\mathbb{D}^{n})$,
then $M_{\Phi}$ is a partial isometry on $H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$
if and only if there exists a Hilbert space $\mathcal{F}$, an inner function
$\Gamma(\bm{z})\in H_{\mathcal{B}(\mathcal{E})}^{\infty}(\mathbb{D}^{n})$, and
an isometry $V:\mathcal{F}\rightarrow\mathcal{E}$ such that

$\Phi(\bm{z})=\Gamma(\bm{z})V^{*}\quad(\bm{z}\in\mathbb{D}^{n}).$

###### Proof.

If $M_{\Phi}$ is a partial isometry, then by Theorem 1.4, we will get the
following factorization

$M_{\Phi}=M_{\Gamma}M_{\Psi}^{*},$

for some Hilbert space $\mathcal{F}$, and inner functions
$\Gamma(\bm{z}),\Psi(\bm{z})\in
H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}^{n})$. Since
$\Phi(\bm{z})$ is a bounded analytic function, we know

$M_{z_{i}}M_{\Phi}=M_{\Phi}M_{z_{i}},$

which gives,

$M_{z_{i}}M_{\Gamma}M_{\Psi}^{*}=M_{\Gamma}M_{\Psi}^{*}M_{z_{i}},$

and therefore,

$M_{\Gamma}M_{z_{i}}M_{\Psi}^{*}=M_{\Gamma}M_{\Psi}^{*}M_{z_{i}}.$

Using the fact that $M_{\Gamma}$ is an isometry we get

$M_{z_{i}}M_{\Psi}^{*}=M_{\Psi}^{*}M_{z_{i}}.$

This implies that $M_{\Psi}$ must be a constant isometry from
$H_{\mathcal{F}}^{2}(\mathbb{D})$ to $H_{\mathcal{E}}^{2}(\mathbb{D})$. Thus,
$M_{\Psi}=I_{H^{2}(\mathbb{D}^{n})}\otimes V$ for some isometry
$V:\mathcal{F}\rightarrow\mathcal{E}$, and using this we get

$M_{\Phi}=M_{\Gamma}(I_{H^{2}(\mathbb{D}^{n})}\otimes V^{*})=M_{\Gamma
V^{*}}.$

Conversely, if the bounded analytic function admits the factorization
$\Phi(\bm{z})=\Gamma(\bm{z})V^{*}$ for some inner function $\Gamma(\bm{z})\in
H_{\mathcal{B}(\mathcal{E})}^{\infty}(\mathbb{D}^{n})$, and some isometry
$V:\mathcal{F}\rightarrow\mathcal{E}$, then

$T_{\Phi}T_{\Phi}^{*}=M_{\Gamma}(I_{H^{2}(\mathbb{D}^{n})}\otimes
V^{*})(I_{H^{2}(\mathbb{D}^{n})}\otimes
V)M_{\Gamma}^{*}=M_{\Gamma}M_{\Gamma}^{*},$

shows that $T_{\Phi}$ is a partial isometry. This completes the proof. ∎

We will now characterize partially isometric Toeplitz operators which are
hyponormal. Let us recall that a bounded operator $T$ on a Hilbert space
$\mathcal{H}$ is said to be hyponormal if $T^{*}T\geq TT^{*}$. The following
result is an extension of [8, Corollary 5.1], and the initial part of the
proof follows from their method.

######  Theorem 5.3.

Let $T_{\Phi}$ be a partially isometric Toeplitz operator on
$H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$. Then the following statements are
equivalent.
$(i)$ $T_{\Phi}$ is hyponormal,
$(ii)$ there exists Hilbert spaces $\mathcal{F},\mathcal{G}$ and inner
functions $\Psi(\bm{z})\in
H_{\mathcal{B}(\mathcal{F},\mathcal{E})}^{\infty}(\mathbb{D}^{n})$ and
$\Theta(\bm{z})\in
H_{\mathcal{B}(\mathcal{F},\mathcal{G})}^{\infty}(\mathbb{D}^{n})$ such that

$T_{\Phi}=M_{\Psi}M_{\Theta}M_{\Psi}^{*}.$

###### Proof.

$T_{\Phi}$ is a partial isometry will imply that
$T_{\Phi}=M_{\Gamma}M_{\Psi}^{*}$. Using this identity we get,

$T_{\Phi}^{*}T_{\Phi}-T_{\Phi}T_{\Phi}^{*}=M_{\Psi}M_{\Psi}^{*}-M_{\Gamma}M_{\Gamma}^{*}.$

Now, hyponormality of $T_{\Phi}$ implies that

$M_{\Gamma}M_{\Gamma}^{*}\leq M_{\Psi}M_{\Psi}^{*}.$

By Douglas’s lemma [9], there must exist a contraction
$Z:H_{\mathcal{F}}^{2}(\mathbb{D}^{n})\rightarrow
H_{\mathcal{G}}^{2}(\mathbb{D}^{n})$ such that

$M_{\Gamma}=M_{\Psi}Z.$

This implies that for any $i\in\\{1,\ldots,n\\}$

$M_{\Psi}M_{z_{i}}Z=M_{z_{i}}M_{\Psi}Z=M_{z_{i}}M_{\Gamma}=M_{\Gamma}M_{z_{i}}=M_{\Psi}ZM_{z_{i}},$

which further implies that $ZM_{z_{i}}=M_{z_{i}}Z$. Thus, there exists
$\Theta(\bm{z})\in
H_{\mathcal{B}(\mathcal{F},\mathcal{G})}^{\infty}(\mathbb{D}^{n})$ such that

$Z=M_{\Theta},$

Furthermore, $\Theta(\bm{z})$ is an inner function because so are
$\Gamma(\bm{z})$ and $\Psi(\bm{z})$. Thus, we get

$T_{\Phi}=M_{\Gamma}M_{\Psi}^{*}=M_{\Psi}M_{\Theta}M_{\Psi}^{*}.$

Conversely, if $T_{\Phi}=M_{\Psi}M_{\Theta}M_{\Psi}^{*}$, then

$T_{\Phi}T_{\Phi}^{*}=M_{\Psi}M_{\Theta}M_{\Psi}^{*}M_{\Psi}M_{\Theta}^{*}M_{\Psi}^{*}=M_{\Psi}M_{\Theta}M_{\Theta}^{*}M_{\Psi}^{*}\leq
M_{\Psi}M_{\Psi}^{*}=T_{\Phi}^{*}T_{\Phi},$

implies that $T_{\Phi}$ is hyponormal. This completes the proof. ∎

##  6\. An alternative proof for Toeplitz operators on $H^{2}(\mathbb{D}^{n})$

This section will adapt the preceding results’ ideas to characterize partially
isometric Toeplitz operators on scalar-valued Hardy space on unit polydisc.
This is an alternative approach to recently obtained results by
Deepak–Pradhan–Sarkar in [8].

######  Theorem 6.1.

Let $\zeta,\psi$ be bounded analytic functions $\mathbb{D}^{n}$.Then the
following are equivalent
$(i)$ $M_{\zeta}M_{\psi}^{*}$ is a Toeplitz operator,
$(ii)$ $M_{\zeta}$ and $M_{\psi}^{*}$ depend on disjoint set of variables.
$(iii)$ $[M_{\zeta},M_{\psi}^{*}]=0$.

###### Proof.

Let us denote the bounded analytic functions $\zeta,\psi$ on $\mathbb{D}^{n}$
in the following form

$\zeta(\bm{z}):=\sum_{\bm{l}\in\mathbb{N}^{n}}a_{\bm{l}}z^{\bm{l}};\quad\psi(\bm{z}):=\sum_{\bm{m}\in\mathbb{N}^{n}}b_{\bm{m}}z^{\bm{m}}\quad(z\in\mathbb{D}^{n}).$

From Theorem 1.3, we know that $M_{\zeta}M_{\psi}^{*}$ is a Toeplitz operator
if and only if,

$a_{\bm{l}+e_{i}}\bar{b}_{\bm{m}+e_{i}}=0.$

for all $\bm{l},\bm{m}\in\mathbb{N}^{n}$ and $i\in\\{1,\ldots,n\\}$. From this
identity it follows that for any given $i\in\\{1,\ldots,n\\}$, either
$a_{\bm{l}+e_{i}}=0$ for all $\bm{l}\in\mathbb{N}^{n}$, or
$b_{\bm{m}+e_{i}}=0$ for all $\bm{m}\in\mathbb{N}^{n}$ or both. If
$a_{\bm{l}+e_{i}}=0$, for all $\bm{l}\in\mathbb{N}^{n}$, then it implies that
$\zeta(\bm{z})$ does not depend on the $z_{i}$-variable. Similarly, if
$b_{\bm{m}+e_{i}}=0$, for all $\bm{m}\in\mathbb{N}^{n}$, then it will imply
that $\psi(\bm{z})$ does not depend on the $z_{i}$-variable. Thus, from this
discussion it follows that if $M_{\zeta}M_{\psi}^{*}$ is a Toeplitz operator
then for each $i\in\\{1,\ldots,n\\}$, any one of the following cases can hold:
$(i)$ $\zeta(\bm{z})$ depend on the variable $z_{i}$,but $\psi(\bm{z})$ does
not depend on the variable $z_{i}$,
$(ii)$ $\psi(\bm{z})$ depend on the variable $z_{i}$,but $\zeta(\bm{z})$ does
not depend on the variable $z_{i}$,
$(iii)$ both $\zeta(\bm{z})$ and $\psi(\bm{z})$ does not depend on the
variable $z_{i}$.
This completes the $(i)\implies(ii)$ proof. The other directions
$(ii)\implies(iii)$, and $(iii)\implies(i)$ can be easily verified. ∎

We are now ready to give a new proof for the main result in [8, Theorem 1.1].

######  Theorem 6.2.

A Toeplitz operator $T_{\phi}$ on $H^{2}(\mathbb{D}^{n})$ is a partial
isometry if and only if there exist inner functions $\zeta,\psi\in
H^{\infty}(\mathbb{D}^{n})$ depending on the disjoint set of variables such
that

$T_{\phi}=M_{\psi}^{*}M_{\zeta}.$

###### Proof.

We proceed as in the proof of Theorem 1.4. Since $T_{\phi}$ is a partial
isometry, we know from Theorem 1.2, that both $\operatorname{ran}T_{\phi}$ and
$\operatorname{ran}T_{\phi}^{*}$ are Beurling type invariant subspaces of
$H^{2}(\mathbb{D}^{n})$. This means that there must exist inner functions
$\gamma,\psi\in H^{\infty}(\mathbb{D}^{n})$ such that

$\operatorname{ran}T_{\phi}=M_{\gamma}H^{2}(\mathbb{D}^{n});\quad\operatorname{ran}T_{\phi}^{*}=M_{\psi}H^{2}(\mathbb{D}^{n}),$

If we follow the proof of Theorem 1.4, we can construct an unitary
$X:\mathbb{C}\rightarrow\mathbb{C}$ such that

$T_{\phi}=M_{\gamma}XM_{\psi}^{*}.$

In other words, $T_{\phi}=M_{\zeta}M_{\psi}^{*}$, where
$\zeta(\bm{z}):=\lambda\gamma(\bm{z})$ for some uni-modular constant
$\lambda\in\mathbb{C}$. From the above Theorem 6.1, $M_{\zeta}M_{\psi}^{*}$ is
a Toeplitz operator if and only if $\zeta$ and $\psi$ depends on the disjoint
set of variables. Thus, we can write,

$T_{\phi}=M_{\psi}^{*}M_{\zeta}.$

Conversely, if the Toeplitz operator admits the factorization
$T_{\phi}=M_{\psi}^{*}M_{\zeta}$ for some inner function $\psi,\zeta$
depending on disjoint set of variables, then

$T_{\phi}T_{\phi}^{*}=M_{\psi}^{*}M_{\zeta}M_{\zeta}^{*}M_{\psi}=M_{\psi}^{*}M_{\psi}M_{\zeta}M_{\zeta}^{*}=M_{\zeta}M_{\zeta}^{*},$

shows that $T_{\phi}$ is a partial isometry. This completes the proof. ∎

##  7\. Some related questions

Based on the results of this article, we can ask several interesting questions
for Toeplitz operators on vector-valued Hardy spaces worthy of further
investigation.

$(\textbf{A})\text{ Characterize Toeplitz operators with shift-invariant range
spaces}.$

A significant contribution of this article is to show that if we start with a
somewhat nice Toeplitz operator, then the range is Beurling-type. This gives
rise to a fascinating problem.

$(\textbf{B})\text{ Characterize Toeplitz operators with Beurling-type range
spaces.}$

In Corollary 5.1, we have seen that a partially isometric Toeplitz operator
has a partially isometric symbol a.e. on $\mathbb{T}^{n}$, but what can we say
about the converse?

$\hskip 25.60747pt(\textbf{C})\text{ Characterize symbols }\Phi\in
L_{\mathcal{B}(\mathcal{E})}^{\infty}(\mathbb{T}^{n})\text{ such that
}T_{\Phi}\text{ is a partial isometry.}$

Unlike in the scalar cases, it is still not clear when partially isometric
Toeplitz operators on $H_{\mathcal{E}}^{2}(\mathbb{D}^{n})$ become power
partial isometries, so we end with the following question.

$\hskip 17.07164pt(\textbf{D})~{}\text{ Characterize Toeplitz operators which
are power partial isometries. }$

## Acknowledgement

The Department of Science and Technology supports the author via the INSPIRE
Faculty fellowship DST/INSPIRE/04/2019/000769.

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