# Backstepping-based Integral Sliding Mode Control with Time Delay Estimation
for Autonomous Underwater Vehicles

Hossein Nejatbakhsh Esfahani, Behdad Aminian, Esten Ingar Grøtli, Sébastien
Gros Authors are with Department of Engineering Cybernetics, Norwegian
University of Science and Technology (NTNU). {hossein.n.esfahani,
behdad.aminian<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

The aim of this paper is to propose a high performance control approach for
trajectory tracking of Autonomous Underwater Vehicles (AUVs). However, the
controller performance can be affected by the unknown perturbations including
model uncertainties and external time-varying disturbances in an undersea
environment. To address this problem, a Backstepping-based Integral Sliding
Mode Control with Time Delay Estimation (BS-ISMC-TDE) is designed. To improve
the performance of a conventional backstepping control algorithm, an Integral
Sliding Mode Control (ISMC) approach is adopted in the backstepping design to
attenuate the steady-state error. Moreover, an adaptive Time Delay Estimation
(TDE) strategy is proposed to provide an estimation of perturbations by
observing the inputs and the states of the AUV one step into the past without
an exact knowledge of the dynamics and the upper bound of uncertainties. From
the simulation results, it is shown that the proposed control approach using
both adaptive and conventional TDE parts outperforms a Backstepping-based
Integral Sliding Mode Control (BS-ISMC).

##  I Introduction

In recent years, underwater vehicles have been extensively developed because
of needs in marine industries and their applications such as nondestructive
test of the marine structures or underwater oil/gas pipeline and ocean
exploration. One of the main reasons that the autonomous marine operation
systems in particular the Autonomous Underwater Vehicles (AUVs) are getting
more attention is potentially dangerous environment and precise tasks which
are difficult and time consuming to accomplish manually [1].

However, the control of the AUVs in a fully autonomous mode is a major
challenging problem since this platform includes a coupled and highly
nonlinear dynamic. There are severe external time-varying disturbances that
affect the tracking performance for such an autonomous platform. Furthermore,
the uncertainties raised from the hydrodynamics forces, the added inertia
matrix, the added Coriolis and centripetal matrix, and the hydrodynamic
damping terms should be considered and tackled.

In [2], an adaptive fuzzy Sliding Mode Control (SMC) approach was proposed
while a neural network compensator was adopted to cope with the uncertainties.
A second-order sliding-mode control algorithm for suppressing the parameter
uncertainties and external disturbances was proposed in [3]. To eliminate the
effects of uncertainties and chattering phenomenon, a hybrid robust control
scheme including the SMC and $H^{\infty}$ was proposed in [4]. A robust model
predictive control of underwater-vehicle manipulator systems was proposed in
[5]. In this robust control scheme, an uncertainty compensator based on fuzzy
logic was designed. In the context of backstepping (BS) algorithm as another
well-known robust control approach for AUVs, some works were conducted in [6]
and [7]. Although the BS is a well-known, simple and useful control approach
for the nonlinear systems, the algorithm is not enough robust against both
matched and mismatched uncertainties. Therefore, people usually combine a BS
with adaptive control approaches or SMC method. Besides, BS requires a full
knowledge of the dynamic and the derivatives of the states, which it is mostly
difficult to have a quite accurate model for an AUV.

To improve the trajectory tracking accuracy of AUV, authors in [8] proposed to
leverage the properties of two robust control schemes as a combined
backstepping-based terminal sliding mode controller. In the SMC-based
backstepping control algorithms, the SMC law is adopted in the first step of
the backstepping design to attenuate the steady-state error and to improve the
robustness in presence of mismatched and state dependant uncertainties.

The advantages of using SMC include fast response and convergence, promising
transient performance and acceptable robustness. The Time Delay Estimation
(TDE) concept is a key part of Time Delay Control (TDC) approach so as to take
advantage of its model-free nature [9, 10, 11]. Indeed, the TDE provides an
estimation of perturbations by observing the inputs and the states of the
dynamical model one step into the past without an exact knowledge of the
dynamics and the upper bound of uncertainties. To address the properties of
the SMC and TDC in a unified controller, researchers proposed to combine these
robust control schemes in [12], [13].

To improve the tracking accuracy for motion control of AUVs under external
time-varying disturbances and uncertainties, we propose to leverage three
robust control approaches including: Backstepping (BS), Integral Sliding Mode
Control (ISMC) and Time Delay Control (TDC) in order to provide a high
performance and sufficiently robust control scheme. We also propose a gain
adaptation strategy for the TDE part to tune a crucial matrix ($\bar{M}$) in
the context of TDC algorithms.

This paper is organized as follows: In Section II, a four degrees of freedom
AUV is modeled and described. Section III, the proposed control approach and a
stability analysis based on Lyapunov theory are presented. The effectiveness
of the proposed robust control scheme for the trajectory tracking scenarios is
investigated and illustrated in Section IV. Finally, conclusions are drawn in
Section V.

##  II AUV Model

The AUV platform is shown in Figure 1. The dynamics can be described in two
reference frames: the Body-fixed frame and the Earth-fixed frame.

Figure 1: AUV model

Note that due to the effect of the buoyancy force, the pitch and roll motions
can be assumed to be intrinsically stable. However, one can assume that there
is higher level controller as autopilot to track the desired roll and pitch
angles. Therefore, the 6-DOF platform can be reduced to the fully actuated
4-DOF model with the following nonlinear equations:

$\displaystyle\dot{\boldsymbol{\mathrm{\eta}}}=J(\boldsymbol{\mathrm{\eta}})\boldsymbol{\mathrm{v}},$
(1a) $\displaystyle
M\dot{\boldsymbol{\mathrm{v}}}+C(\boldsymbol{\mathrm{v}})\boldsymbol{\mathrm{v}}+D(\boldsymbol{\mathrm{v}})\boldsymbol{\mathrm{v}}+\text{g}(\boldsymbol{\mathrm{\eta}})=\boldsymbol{\mathrm{\tau}}+\boldsymbol{\mathrm{\tau}}_{d}$
(1b)

where $\boldsymbol{\mathrm{\eta}}=\left[x,y,z,\psi\right]^{\top}$ is the
position-orientation (yaw angle) vector and
$\boldsymbol{\mathrm{v}}=\left[u,v,w,r\right]^{\top}$ represents the velocity
vector for the surge, sway, heave and yaw motions, respectively.
$M,C(\boldsymbol{\mathrm{v}}),D(\boldsymbol{\mathrm{v}})\in\mathbb{R}^{4\times
4}$ are the inertia matrix (including the effects of added mass), Coriolis-
centripetal matrix and the drag force matrix (hydrodynamic damping),
respectively. The vector of forces and moments induced by the gravity and
buoyancy is labeled by $\text{g}\in\mathbb{R}^{4\times 1}$. The control inputs
and the external time-varying disturbances (winds, waves, and ocean currents)
are labeled as
$\boldsymbol{\mathrm{\tau}},\boldsymbol{\mathrm{\tau}}_{d}\in\mathbb{R}^{4\times
1}$, respectively. The transformation matrix between the reference frames can
be represented in Euler angles as follows:

$\displaystyle
J(\boldsymbol{\mathrm{\eta}})=\begin{bmatrix}cos(\psi)&-sin(\psi)&0&0\\\
sin(\psi)&cos(\psi)&0&0\\\ 0&0&1&0\\\ 0&0&0&1\end{bmatrix}$ (2)

Let $M=diag\left\\{m_{11},m_{22},m_{33},m_{44}\right\\}$ be the inertia
matrix. Then, the diagonal terms are $m_{11}=m-X_{\dot{u}}$,
$m_{22}=m-Y_{\dot{v}}$ , $m_{33}=m-Z_{\dot{w}}$, $m_{44}=I_{z}-N_{\dot{r}}$.
$m$ is the total mass of AUV and $I_{z}$ is the moment of inertia about yaw
motion. The corresponding hydrodynamic coefficients are labeled by
$X_{\star}$, $Y_{\star}$, $Z_{\star}$, $N_{\star}$. Then the
Coriolis–centripetal matrix is expressed as

$\displaystyle
C(\boldsymbol{\mathrm{v}})=\begin{bmatrix}0&0&0&-(m-Y_{\dot{v}})v\\\
0&0&0&(m-X_{\dot{u}})u\\\ 0&0&0&0\\\
(m-Y_{\dot{v}})v&-(m-X_{\dot{u}})u&0&0\end{bmatrix}$ (3)

Let
$D(\boldsymbol{\mathrm{v}})=diag\left\\{d_{11},d_{22},d_{33},d_{44}\right\\}$
be the damping matrix. Then, the diagonal terms are
$d_{11}=-X_{u}-X_{|u|u}|u|$, $d_{22}=-Y_{v}-Y_{|v|v}|v|$ ,
$d_{33}=-Z_{w}-Z_{|w|w}|w|$, $d_{44}=-N_{r}-N_{|r|r}|r|$. The gravity-buoyancy
vector is specified as follows:

$\displaystyle\text{g}=\begin{bmatrix}0\\\ 0\\\ -(G-B)\\\ 0\end{bmatrix}$ (4)

where $G,B$ are the gravity and buoyancy forces, respectively. One can rewrite
(1) as the following equation:

$\displaystyle
M(\boldsymbol{\mathrm{\eta}})\ddot{\boldsymbol{\mathrm{\eta}}}+C(\dot{\boldsymbol{\mathrm{\eta}}},\boldsymbol{\mathrm{\eta}})\dot{\boldsymbol{\mathrm{\eta}}}+D(\dot{\boldsymbol{\mathrm{\eta}}},\boldsymbol{\mathrm{\eta}})\dot{\boldsymbol{\mathrm{\eta}}}+\text{g}(\boldsymbol{\mathrm{\eta}})=\boldsymbol{\mathrm{\tau}}+\boldsymbol{\mathrm{\tau}}_{d}$
(5)

where

$\displaystyle
M(\boldsymbol{\mathrm{\eta}})=J^{-T}MJ^{-1}(\boldsymbol{\mathrm{\eta}}),$ (6a)
$\displaystyle
C(\dot{\boldsymbol{\mathrm{\eta}}},\boldsymbol{\mathrm{\eta}})=J^{-T}\left[C(\boldsymbol{\mathrm{v}})-MJ^{-1}(\boldsymbol{\mathrm{\eta}})\dot{J}(\boldsymbol{\mathrm{\eta}})\right]J^{-1}(\boldsymbol{\mathrm{\eta}}),$
(6b) $\displaystyle
D(\dot{\boldsymbol{\mathrm{\eta}}},\boldsymbol{\mathrm{\eta}})=J^{-T}D(\boldsymbol{\mathrm{v}})J^{-1}(\boldsymbol{\mathrm{\eta}}),$
(6c)
$\displaystyle\bar{\boldsymbol{\mathrm{\tau}}}=J^{-T}\boldsymbol{\mathrm{\tau}},$
(6d)
$\displaystyle\bar{\boldsymbol{\mathrm{\tau}}}_{d}=J^{-T}\boldsymbol{\mathrm{\tau}}_{d}$
(6e)

The above mentioned dynamics include the uncertain parameters practically. Let
$\Delta M,\Delta C,\Delta D,\Delta\text{g}$ be the uncertain parts for each
parameter matrix or vector. Then the entire model of AUV can be rewritten as
follows:

$\displaystyle
M(\boldsymbol{\mathrm{\eta}})\ddot{\boldsymbol{\mathrm{\eta}}}+C(\dot{\boldsymbol{\mathrm{\eta}}},\boldsymbol{\mathrm{\eta}})\dot{\boldsymbol{\mathrm{\eta}}}+D(\dot{\boldsymbol{\mathrm{\eta}}},\boldsymbol{\mathrm{\eta}})\dot{\boldsymbol{\mathrm{\eta}}}+\text{g}(\boldsymbol{\mathrm{\eta}})=\bar{\boldsymbol{\mathrm{\tau}}}+\boldsymbol{\mathrm{p}}$
(7)

where, the perturbation $\boldsymbol{\mathrm{p}}$ involves the external time-
varying disturbances $\boldsymbol{\mathrm{\tau}}_{d}$ and model uncertainties
$\delta$ and then:

$\displaystyle\boldsymbol{\mathrm{p}}=$
$\displaystyle\bar{\boldsymbol{\mathrm{\tau}}}_{d}-J^{-T}\Delta
MJ^{-1}(\boldsymbol{\mathrm{\eta}})\ddot{\boldsymbol{\mathrm{\eta}}}$ (8)
$\displaystyle-J^{-T}\left[\Delta C(\boldsymbol{\mathrm{v}})-\Delta
MJ^{-1}(\boldsymbol{\mathrm{\eta}})\dot{J}(\boldsymbol{\mathrm{\eta}})\right]J^{-1}(\boldsymbol{\mathrm{\eta}})\dot{\boldsymbol{\mathrm{\eta}}}$
$\displaystyle-J^{-T}\Delta
D(\boldsymbol{\mathrm{v}})J^{-1}(\boldsymbol{\mathrm{\eta}})-\Delta\text{g}(\boldsymbol{\mathrm{\eta}})=\bar{\boldsymbol{\mathrm{\tau}}}_{d}-\delta$

Let us define $\boldsymbol{\mathrm{x}}_{1}=\boldsymbol{\mathrm{\eta}}$ and
$\boldsymbol{\mathrm{x}}_{2}=\dot{\boldsymbol{\mathrm{\eta}}}$ as state
variables and then the concatenated state vector is
$\boldsymbol{\mathrm{x}}=\left[\boldsymbol{\mathrm{x}}_{1},\boldsymbol{\mathrm{x}}_{2}\right]^{\top}$.
Then, the system (7) can be described as the following state-space model:

$\displaystyle\left\\{\begin{matrix}\dot{\boldsymbol{\mathrm{x}}}_{1}=\boldsymbol{\mathrm{x}}_{2}\\\
\dot{\boldsymbol{\mathrm{x}}}_{2}=\boldsymbol{\mathrm{f}}(\boldsymbol{\mathrm{x}})+\boldsymbol{\mathrm{u}}+\tilde{\boldsymbol{\mathrm{d}}}\\\
\boldsymbol{\mathrm{y}}=\boldsymbol{\mathrm{x}}_{1}\end{matrix}\right.$ (9)

where

$\displaystyle\boldsymbol{\mathrm{f}}(\boldsymbol{\mathrm{x}})=M^{-1}(\boldsymbol{\mathrm{x}}_{1})\left(-C(\boldsymbol{\mathrm{x}}_{1},\boldsymbol{\mathrm{x}}_{2})\boldsymbol{\mathrm{x}}_{2}-D(\boldsymbol{\mathrm{x}}_{1},\boldsymbol{\mathrm{x}}_{2})\boldsymbol{\mathrm{x}}_{2}-\text{g}\right),$
(10a)
$\displaystyle\boldsymbol{\mathrm{u}}=M^{-1}(\boldsymbol{\mathrm{x}}_{1})\bar{\boldsymbol{\mathrm{\tau}}},$
(10b)
$\displaystyle\tilde{\boldsymbol{\mathrm{d}}}=M^{-1}(\boldsymbol{\mathrm{x}}_{1})\boldsymbol{\mathrm{p}}$
(10c)

##  III Control Approach

In this section, we first propose to leverage the properties of both robust
control approaches Backstepping and ISMC as a Backstepping-based Integral
Sliding Mode (BS-ISMC) and then enhance the performance of BS-ISMC with
considering a TDE part as a compensator to cope with the lumped uncertainties
including the unknown uncertain dynamic in the context of the robust control
design.

###  III-A Backstepping-based Integral Sliding Mode

Let us define the tracking error and its time derivative as follows:

$\displaystyle\boldsymbol{\mathrm{e}}=\boldsymbol{\mathrm{\eta}}-\boldsymbol{\mathrm{\eta}}_{d},$
(11a)
$\displaystyle\dot{\boldsymbol{\mathrm{e}}}=\dot{\boldsymbol{\mathrm{\eta}}}-\dot{\boldsymbol{\mathrm{\eta}}}_{d}$
(11b)

As the first step, the following Lyapunov function is used to ensure the
stability of the system:

$\displaystyle
V_{0}=\frac{1}{2}\boldsymbol{\mathrm{e}}^{\top}\boldsymbol{\mathrm{e}},$ (12a)
$\displaystyle\dot{V}_{0}=\boldsymbol{\mathrm{e}}^{\top}\dot{\boldsymbol{\mathrm{e}}}=\boldsymbol{\mathrm{e}}^{\top}(\boldsymbol{\mathrm{x}}_{2}-\dot{\boldsymbol{\mathrm{\eta}}}_{d})$
(12b)

As an step in the backstepping design, the state variable
$\boldsymbol{\mathrm{x}}_{2}=\dot{\boldsymbol{\mathrm{\eta}}}$ in the state-
space model (9) can be interpreted as a virtual control signal. Then, using
the virtual control
$\boldsymbol{\mathrm{x}}_{2}=\alpha_{0}(\boldsymbol{\mathrm{e}})=-k_{1}\boldsymbol{\mathrm{e}}+\dot{\boldsymbol{\mathrm{\eta}}}_{d}$,
the origin of $\dot{\boldsymbol{\mathrm{e}}}=-k_{1}\boldsymbol{\mathrm{e}}$ is
globally asymptotically stable since
$\dot{V}_{0}=-k_{1}\boldsymbol{\mathrm{e}}^{\top}\boldsymbol{\mathrm{e}}<0,\quad\forall\left(\boldsymbol{\mathrm{e}}\neq
0\right)\in\mathcal{R}$ where $k_{1}$ is a positive diagonal matrix. An
integral action can be added to the virtual control as follows:

$\displaystyle\boldsymbol{\mathrm{x}}_{2}=\alpha(\boldsymbol{\mathrm{e}})=-k_{1}\boldsymbol{\mathrm{e}}-k_{2}\int_{0}^{t}\boldsymbol{\mathrm{e}}(\tau)d\tau+\dot{\boldsymbol{\mathrm{\eta}}}_{d}$
(13)

where $k_{2}$ is also a positive diagonal matrix. Then, the time derivative of
tracking error can be modified as follows:

$\displaystyle\dot{\boldsymbol{\mathrm{e}}}=-k_{1}\boldsymbol{\mathrm{e}}-k_{2}\int_{0}^{t}\boldsymbol{\mathrm{e}}(\tau)d\tau$
(14)

The Lyapunov function addressing the integral action can be expressed as
follows:

$\displaystyle
V_{1}=\frac{1}{2}\boldsymbol{\mathrm{e}}^{\top}\boldsymbol{\mathrm{e}}+\frac{1}{2}k_{2}\left(\int_{0}^{t}\boldsymbol{\mathrm{e}}(\tau)d\tau\right)\left(\int_{0}^{t}\boldsymbol{\mathrm{e}}(\tau)d\tau\right)^{\top}$
(15)

The time derivative of the above function reads as:

$\displaystyle\dot{V}_{1}=\boldsymbol{\mathrm{e}}^{\top}\dot{\boldsymbol{\mathrm{e}}}+k_{2}\left(\int_{0}^{t}\boldsymbol{\mathrm{e}}(\tau)d\tau\right)\boldsymbol{\mathrm{e}}^{\top}$
(16)

Then, substituting (14) to (16), the derivative of Lyapunov function is
delivered as a negative semi-definite
$-k_{1}\boldsymbol{\mathrm{e}}^{\top}\boldsymbol{\mathrm{e}}\leq 0$ and then
the system augmented by an integral action is asymptotically stable in the
first stage of the backstepping design. The sliding mode surface based on the
deviation of the proposed virtual control form its desired signal in the
backstepping control scheme can be defined as follows:

$\displaystyle\boldsymbol{\mathrm{\sigma}}=\dot{\boldsymbol{\mathrm{\eta}}}-\alpha(\boldsymbol{\mathrm{e}})=\dot{\boldsymbol{\mathrm{\eta}}}+k_{1}\boldsymbol{\mathrm{e}}+k_{2}\int_{0}^{t}\boldsymbol{\mathrm{e}}(\tau)d\tau-\dot{\boldsymbol{\mathrm{\eta}}}_{d}$
(17)

Considering the tracking error in (11a) and the above sliding surface, the
time derivative of tracking error can be obtained as follows:

$\displaystyle\dot{\boldsymbol{\mathrm{e}}}=\boldsymbol{\mathrm{\sigma}}-k_{1}\boldsymbol{\mathrm{e}}-k_{2}\int_{0}^{t}\boldsymbol{\mathrm{e}}(\tau)d\tau$
(18)

Let us define a composite Lyapunov function as:

$\displaystyle
V=V_{1}+\frac{1}{2}\boldsymbol{\mathrm{\sigma}}^{\top}\boldsymbol{\mathrm{\sigma}}$
(19)

considering equations (16), (18) and (19):

$\displaystyle\dot{V}$
$\displaystyle=\boldsymbol{\mathrm{e}}^{\top}\dot{\boldsymbol{\mathrm{e}}}+k_{2}\boldsymbol{\mathrm{e}}^{\top}\int_{0}^{t}\boldsymbol{\mathrm{e}}(\tau)d\tau+\boldsymbol{\mathrm{\sigma}}^{\top}\dot{\boldsymbol{\mathrm{\sigma}}}$
$\displaystyle=\boldsymbol{\mathrm{e}}^{\top}\left(\boldsymbol{\mathrm{\sigma}}-k_{1}\boldsymbol{\mathrm{e}}-k_{2}\int_{0}^{t}\boldsymbol{\mathrm{e}}(\tau)d\tau\right)+k_{2}\boldsymbol{\mathrm{e}}^{\top}\int_{0}^{t}\boldsymbol{\mathrm{e}}(\tau)d\tau$
$\displaystyle+\boldsymbol{\mathrm{\sigma}}^{\top}\left(\ddot{\boldsymbol{\mathrm{e}}}+k_{1}\dot{\boldsymbol{\mathrm{e}}}+k_{2}\boldsymbol{\mathrm{e}}\right)$
(20)

Substituting
$\ddot{\boldsymbol{\mathrm{e}}}=\ddot{\boldsymbol{\mathrm{\eta}}}-\ddot{\boldsymbol{\mathrm{\eta}}}_{d}$
and (9) to the above equation:

$\displaystyle\dot{V}$
$\displaystyle=\boldsymbol{\mathrm{e}}^{\top}\left(\boldsymbol{\mathrm{\sigma}}-k_{1}\boldsymbol{\mathrm{e}}-k_{2}\int_{0}^{t}\boldsymbol{\mathrm{e}}(\tau)d\tau\right)+k_{2}\boldsymbol{\mathrm{e}}^{\top}\int_{0}^{t}\boldsymbol{\mathrm{e}}(\tau)d\tau$
$\displaystyle+\boldsymbol{\mathrm{\sigma}}^{\top}\left(\boldsymbol{\mathrm{f}}(\boldsymbol{\mathrm{x}})+\boldsymbol{\mathrm{u}}+\tilde{\boldsymbol{\mathrm{d}}}+k_{1}\dot{\boldsymbol{\mathrm{e}}}+k_{2}\boldsymbol{\mathrm{e}}-\ddot{\boldsymbol{\mathrm{\eta}}}_{d}\right)=-k_{1}\boldsymbol{\mathrm{e}}^{\top}\boldsymbol{\mathrm{e}}$
$\displaystyle+\boldsymbol{\mathrm{\sigma}}^{\top}\left(\boldsymbol{\mathrm{f}}(\boldsymbol{\mathrm{x}})+\boldsymbol{\mathrm{u}}+\tilde{\boldsymbol{\mathrm{d}}}+k_{1}\dot{\boldsymbol{\mathrm{e}}}+(I+k_{2})\boldsymbol{\mathrm{e}}-\ddot{\boldsymbol{\mathrm{\eta}}}_{d}\right)$
(21)

Then, the above Lyapunov function can be negative definite where the following
condition is satisfied:

$\displaystyle\boldsymbol{\mathrm{f}}(\boldsymbol{\mathrm{x}})+\boldsymbol{\mathrm{u}}+\tilde{\boldsymbol{\mathrm{d}}}+k_{1}\dot{\boldsymbol{\mathrm{e}}}+(I+k_{2})\boldsymbol{\mathrm{e}}-\ddot{\boldsymbol{\mathrm{\eta}}}_{d}=-k_{3}\boldsymbol{\mathrm{\sigma}}$
(22)

where $k_{3}$ is a positive diagonal matrix. Therefore, the equivalent control
input can be obtained as follows:

$\displaystyle\boldsymbol{\mathrm{\tau}}_{e}=M(\boldsymbol{\mathrm{\eta}})\left[-\boldsymbol{\mathrm{f}}(\boldsymbol{\mathrm{\eta}})-k_{1}\dot{\boldsymbol{\mathrm{e}}}-(I+k_{2})\boldsymbol{\mathrm{e}}-k_{3}\boldsymbol{\mathrm{\sigma}}+\ddot{\boldsymbol{\mathrm{\eta}}}_{d}\right]$
(23)

Then, the Backstepping-based Integral Sliding Mode Control (BS-ISMC) law is
designed as:

$\displaystyle\boldsymbol{\mathrm{\tau}}=\boldsymbol{\mathrm{\tau}}_{e}-M(\boldsymbol{\mathrm{\eta}})\Gamma\text{sgn}(\boldsymbol{\mathrm{\sigma}})$
(24)

where $\Gamma$ is a positive diagonal matrix and the second part of the above
control law reads as a reaching mode in the context of sliding mode control
algorithms.

###  III-B BS-ISMC with Time Delay Estimation

According to the TDE concept as a part of Time Delay Control (TDC) approach
[12], we first introduce a constant diagonal inertia matrix $\bar{M}$. Then,
the control law (24) can be modified as follows:

$\displaystyle\boldsymbol{\mathrm{\tau}}=$
$\displaystyle\bar{M}\left[-k_{1}\dot{\boldsymbol{\mathrm{e}}}-(I+k_{2})\boldsymbol{\mathrm{e}}-k_{3}\boldsymbol{\mathrm{\sigma}}+\ddot{\boldsymbol{\mathrm{\eta}}}_{d}-\Gamma\text{sgn}(\boldsymbol{\mathrm{\sigma}})\right]$
$\displaystyle+(C_{n}+D_{n}+\text{g}_{n})\dot{\boldsymbol{\mathrm{\eta}}}+\bar{\boldsymbol{\mathrm{\tau}}}_{d}-\delta$
(25)

where the nominal model and perturbation model are described as
$N=(C_{n}+D_{n}+\text{g}_{n})\dot{\boldsymbol{\mathrm{\eta}}}$ and
$\boldsymbol{\mathrm{p}}=\bar{\boldsymbol{\mathrm{\tau}}}_{d}-\delta$,
respectively. Let us define an estimation based on the TDE concept for the
applied perturbations including the model uncertainties and external
disturbances as follows [13]:

$\displaystyle\tilde{\boldsymbol{\mathrm{p}}}=\boldsymbol{\mathrm{p}}_{t-L}=\boldsymbol{\mathrm{\tau}}(t-L)-N(t-L)-\bar{M}\ddot{\boldsymbol{\mathrm{\eta}}}(t-L)$
(26)

where $L$ reads as a small delay time for the TDE part and,

$\displaystyle\ddot{\boldsymbol{\mathrm{\eta}}}(t-L)=\frac{\boldsymbol{\mathrm{\eta}}(t)-2\boldsymbol{\mathrm{\eta}}(t-L)+\boldsymbol{\mathrm{\eta}}(t-2L)}{L^{2}}$
(27)

Substituting (26) to (III-B), the proposed BS-ISMC-TDE control law is designed
as follows:

$\displaystyle\boldsymbol{\mathrm{\tau}}=$
$\displaystyle\boldsymbol{\mathrm{\tau}}(t-L)+\bar{M}\left[-k_{1}\dot{\boldsymbol{\mathrm{e}}}-(I+k_{2})\boldsymbol{\mathrm{e}}-k_{3}\boldsymbol{\mathrm{\sigma}}+\ddot{\boldsymbol{\mathrm{\eta}}}_{d}\right]$
$\displaystyle+\bar{M}\left[-\ddot{\boldsymbol{\mathrm{\eta}}}(t-L)-\Gamma\text{sgn}(\boldsymbol{\mathrm{\sigma}})\right]+N-N(t-L)$
(28)

###  III-C Stability Analysis

A closed-loop system for (9) with the proposed BS-ISMC-TDE control scheme can
be uniformly ultimately bounded (UUB)

###### Lemma 1

The TDE error
$\epsilon_{i}=\boldsymbol{\mathrm{p}}_{i}-\tilde{\boldsymbol{\mathrm{p}}}_{i}$
is bounded
$\epsilon_{i}^{+}\geq\left\|\boldsymbol{\mathrm{p}}_{i}-\tilde{\boldsymbol{\mathrm{p}}}_{i}\right\|_{\infty}$
if the following condition is satisfied:

$\displaystyle\left\|I-M^{-1}(\boldsymbol{\mathrm{\eta}})\bar{M}\right\|<1$
(29)

In practice, $\bar{M}$ is adjusted by increasing its diagonal elements from a
small positive value to a large value before the system approaches the
oscillatory response. Let us obtain the first-time derivative of sliding
surface (17) as follows:

$\displaystyle\dot{\boldsymbol{\mathrm{\sigma}}}=\bar{M}^{-1}\left(\bar{\boldsymbol{\mathrm{\tau}}}+\tilde{\boldsymbol{\mathrm{p}}}+\epsilon-N\right)+k_{1}\dot{\boldsymbol{\mathrm{e}}}+k_{2}\boldsymbol{\mathrm{e}}-\ddot{\boldsymbol{\mathrm{\eta}}}_{d}$
(30)

Substituting (24) to $\bar{\boldsymbol{\mathrm{\tau}}}$ in the above equation,
the closed loop dynamics can be expressed as:

$\displaystyle\dot{\boldsymbol{\mathrm{\sigma}}}=\bar{M}^{-1}\epsilon-\Gamma
sgn(\boldsymbol{\mathrm{\sigma}})-k_{3}\boldsymbol{\mathrm{\sigma}}$ (31)

Then, the first-derivative of the Lyapunov function (19) considering the term
including the sliding surface is expressed as follows:

$\displaystyle\boldsymbol{\mathrm{\sigma}}^{\top}\dot{\boldsymbol{\mathrm{\sigma}}}$
$\displaystyle=-k_{3}\boldsymbol{\mathrm{\sigma}}^{\top}\boldsymbol{\mathrm{\sigma}}+\boldsymbol{\mathrm{\sigma}}^{\top}\left(\bar{M}^{-1}\epsilon-\Gamma
sgn(\boldsymbol{\mathrm{s}})\right)$
$\displaystyle=-k_{3}\boldsymbol{\mathrm{\sigma}}^{\top}\boldsymbol{\mathrm{\sigma}}-\Gamma|\boldsymbol{\mathrm{\sigma}}|+\boldsymbol{\mathrm{\sigma}}^{\top}\bar{M}^{-1}\epsilon$
$\displaystyle\leq-\left(k_{3}+\Gamma\right)|\boldsymbol{\mathrm{\sigma}}|+\boldsymbol{\mathrm{\sigma}}^{\top}\bar{M}^{-1}\epsilon$
(32)

Thus, if ${k_{3}}_{ii}+\Gamma_{ii}>\bar{M}_{ii}^{-1}\epsilon_{i}^{+}$, then
$\dot{V}<0$. However, a proper adjustment of the constant matrix $\bar{M}$ is
a crucial issue to guarantee a stable performance for a trajectory tracking
problem. In this paper, we propose an adaptive mechanism to tune the diagonal
matrix $\bar{M}$. Let us define a Lyapunov function as follows:

$\displaystyle\bar{V}=\frac{1}{2}\boldsymbol{\mathrm{\sigma}}^{\top}\boldsymbol{\mathrm{\sigma}}+\frac{1}{2}\sum_{i=1}^{n}\left(\bar{M}_{ii}\right)^{2}$
(33)

Considering (1), the first derivative of the above Lyapunov function can be
defined by the following inequality:

$\displaystyle\dot{\bar{V}}$
$\displaystyle=\boldsymbol{\mathrm{\sigma}}^{\top}\dot{\boldsymbol{\mathrm{\sigma}}}+\sum_{i=1}^{n}\bar{M}_{ii}\dot{\bar{M}}_{ii}\leq-
k_{3}|\boldsymbol{\mathrm{\sigma}}|+$
$\displaystyle\sum_{i=1}^{n}-\Gamma_{ii}|\boldsymbol{\mathrm{\sigma}}_{i}|+\alpha_{i}\boldsymbol{\mathrm{\sigma}}_{i}\bar{M}_{ii}^{-1}+\bar{M}_{ii}\dot{\bar{M}}_{ii}$
(34)

where $\alpha_{i}$ is a positive scalar value. Hence, the closed loop system
is stable if the following equality as dynamics for $\bar{M}_{ii}$ hold:

$\displaystyle\dot{\bar{M}}_{ii}=\Gamma_{ii}|\boldsymbol{\mathrm{\sigma}}_{i}|-\alpha_{i}\left(\frac{\boldsymbol{\mathrm{\sigma}}_{i}}{\bar{M}_{ii}^{2}}\right)$
(35)

##  IV Computer Simulations

The numerical values of the AUV parameters used in both examples are given by:
$m=54.54kg$, $G=535N$, $B=53.4$, $I_{z}=13.587$, $X_{uu}=2.3e-2$,
$Y_{vv}=5.3e-2$, $Z_{ww}=1.7e-1$, $N_{rr}=2.9e-3$, $X_{\dot{u}}=-7.6e-3$,
$Y_{\dot{v}}=-5.5e-2$, $Z_{\dot{w}}=-2.4e-1$, $N_{\dot{r}}=-3.4e-3$,
$X_{u}=2e-3$, $Y_{v}=-1e-1$, $Z_{w}=-3e-1$, $N_{r}=3e-2$. The parameters of
the proposed controllers are given by: $L=T_{s}=7e-3s$, $\phi=0.2$,
$\Gamma=diag\left\\{1,1,1,1\right\\}$,
$k_{1}=diag\left\\{0.1,0.1,0.1,0.1\right\\}$,
$k_{2}=diag\left\\{0.1,0.1,0.1,0.1\right\\}$,
$k_{3}=diag\left\\{10,10,10,10\right\\}$,
$\bar{M}=diag\left\\{3,3,6,1\right\\}$. To alleviate the chattering induced by
the switching part in (III-B), we propose to use the following saturation
function:

$\displaystyle
sat(\boldsymbol{\mathrm{\sigma}})=\left\\{\begin{matrix}sgn(\boldsymbol{\mathrm{\sigma}})\quad&if\quad|\boldsymbol{\mathrm{\sigma}}|\geq\phi\\\
\boldsymbol{\mathrm{\sigma}}\quad&if\quad\boldsymbol{\mathrm{\sigma}}<\phi\end{matrix}\right.$
(36)

where $\phi$ reads as the boundary layer thickness. In both case studies, we
apply the disturbances shown in Figure 2.

Figure 2: Disturbances applied to surge, sway, heave and yaw motions

Case Study 1: In this case study, the trajectories are generated as follows:

$\displaystyle x=4\text{sin}(0.04t)$ (37a) $\displaystyle
y=2.5\text{cos}(0.02t)$ (37b) $\displaystyle
z=2\text{sin}(0.01t)+2\text{cos}(0.02t)$ (37c)
$\displaystyle\psi=0.5\text{cos}(0.01t)-0.5\text{sin}(0.01t)$ (37d)

where initial conditions are set to
$\boldsymbol{\mathrm{x}}_{0}=\left[0,1,2,\frac{\pi}{4}\right]^{\top}$.

Figure 3: Trajectory tracking control of AUV Figure 4: Tracking error signals
Figure 5: Control inputs for the proposed BS-ISMC-TDE Figure 6: Velocity
signals

The differences between the BS-ISMC and a BS-ISMC-TDE in terms of trajectory
tracking performance and the tracking errors are illustrated in Figures 3 and
4.

Case Study 2: In this case, a comparative study on the proposed BS-ISMC-TDE is
accomplished. To investigate the effectiveness of the proposed adaptive TDE
with adjustable $\bar{M}$ shown in Figure 11, we initialize the values of
diagonal matrix with $\bar{M}_{ii}=diag\left([0.03,0.03,0.05,0.02]\right)$ so
that the response is very close to the instability and then let the adaptive
mechanism to tune this matrix. The initial conditions are set to
$\boldsymbol{\mathrm{x}}_{0}=\left[0,1.5,0,\frac{\pi}{4}\right]^{\top}$. The
sampling time $T_{s}$ is equal to the delay time $L=2ms$. The reference
trajectories are generated as follows:

$\displaystyle x=3\text{sin}(0.04t)$ (38a) $\displaystyle
y=1.5\text{cos}(0.04t)$ (38b) $\displaystyle z=0.02t$ (38c)
$\displaystyle\psi=0.04t$ (38d)

As it is observed, the proposed BS-ISMC-TDE with an adaptive gain part can
rapidly bring a poorly-adjusted $\bar{M}$ with an aggressive response of the
controller to a well-adjusted and a smooth and stable behavior shown in
Figures 7 to 10.

Figure 7: Trajectory tracking control of AUV Figure 8: Tracking error signals
Figure 9: Control inputs for the proposed BS-ISMC-TDE Figure 10: Velocity
signals Figure 11: Adjustable matrix $\bar{M}_{ii}$

##  V Conclusion

In this paper, we proposed to leverage three robust control schemes for
improving a trajectory tracking control of autonomous underwater vehicles in
presence of external time-varying disturbances and uncertainties. In the
proposed ,Backstepping-based Integral Sliding Mode Control with Time Delay
Estimation (BS-ISMC-TDE), the backstepping part is responsible for tracking
the desired trajectory while the Integral Sliding Mode Control (ISMC) is
designed for tackling the uncertainties and external time varying
disturbances. To improve the accuracy of the tracking in presence of some
unknown parameters and to compensate the effect of unmodeled dynamics, the
Time Delay Estimation (TDE) is applied to the BS-ISMC. Besides, an adaptive
mechanism for adjusting the TDE part is proposed to address a problem with a
poorly-adjusted $\bar{M}$. As a future work, we will investigate a fully
adaptive BS-ISMC-TDE in which all gains related to the BS and ISMC parts are
autonomously adjusted as well.

## References

  * [1] J. Dowdeswell, J. Evans, R. Mugford, G. Griffiths, S. McPhail, N. Millard, P. Stevenson, M. Brandon, C. Banks, K. Heywood, and et al., “Autonomous underwater vehicles (auvs) and investigations of the ice–ocean interface in antarctic and arctic waters,” _Journal of Glaciology_ , vol. 54, no. 187, p. 661–672, 2008.
  * [2] S. Pezeshki, A. R. Ghiasi, M. A. Badamchizadeh, and K. Sabahi, “Adaptive robust control of autonomous underwater vehicle,” _Autom Electr Syst_ , vol. 27, pp. 250–262, 2016.
  * [3] H. Joe, M. Kim, and S. cheol Yu, “Second-order sliding-mode controller for autonomous underwater vehicle in the presence of unknown disturbances,” _Nonlinear Dyn_ , vol. 78, pp. 183–196, 2014.
  * [4] M. Yarahmadi and Z. T. Sereshki, “A new hybrid robust sliding mode control for an underwater vehicle in dive plane,” _Measurement and Control_ , vol. 52, no. 7-8, pp. 832–843, 2019.
  * [5] H. Nejatbakhsh Esfahani, “Robust model predictive control for autonomous underwater vehicle – manipulator system with fuzzy compensator,” _Polish Maritime Research_ , vol. 26, no. 2, pp. 104–114, 2019.
  * [6] B. N. Rath, B. Subudhi, V. Filaretov, and A. Zuev, “A new backstepping control design method for autonomous underwater vehicle in diving and steering plane,” in _TENCON 2017 - 2017 IEEE Region 10 Conference_ , 2017, pp. 1984–1987.
  * [7] K. D. von Ellenrieder, “Stable backstepping control of marine vehicles with actuator rate limits and saturation,” _IFAC-PapersOnLine_ , vol. 51, no. 29, pp. 262–267, 2018, 11th IFAC Conference on Control Applications in Marine Systems, Robotics, and Vehicles CAMS 2018.
  * [8] C. Yang, F. Yao, and M.-J. Zhang, “Adaptive backstepping terminal sliding mode control method based on recurrent neural networks for autonomous underwater vehicle,” _Chinese Journal of Mechanical Engineering_ , vol. 31, no. 110, pp. 183–196, 2018.
  * [9] S. Roy, J. Lee, and S. Baldi, “A new adaptive-robust design for time delay control under state-dependent stability condition,” _IEEE Transactions on Control Systems Technology_ , vol. 29, no. 1, pp. 420–427, 2021.
  * [10] S. Roy, S. Baldi, P. Li, and V. Narayanan, “Artificial-delay adaptive control for under-actuated euler-lagrange robotics,” _IEEE/ASME Transactions on Mechatronics_ , pp. 1–1, 2021.
  * [11] R. Kumar, C. Kumar, D. Sen, and A. Dasgupta, “Discrete time-delay control of an autonomous underwater vehicle: Theory and experimental results,” _Ocean Engineering_ , vol. 36, no. 1, pp. 74–81, 2009, autonomous Underwater Vehicles.
  * [12] H. Nejatbakhsh Esfahani, V. Azimirad, and M. Danesh, “A time delay controller included terminal sliding mode and fuzzy gain tuning for underwater vehicle-manipulator systems,” _Ocean Engineering_ , vol. 107, pp. 97–107, 2015.
  * [13] H. N. Esfahani, R. Szlapczynski, and H. Ghaemi, “High performance super-twisting sliding mode control for a maritime autonomous surface ship (mass) using adp-based adaptive gains and time delay estimation,” _Ocean Engineering_ , vol. 191, p. 106526, 2019.


# Insights on the Cesàro operator: shift semigroups
and invariant subspaces

Eva A. Gallardo-Gutiérrez  Eva A. Gallardo-Gutiérrez
Departamento de Análisis Matemático y Matemática Aplicada,
Facultad de Matemáticas,
Universidad Complutense de Madrid,
Plaza de Ciencias 3, 28040 Madrid, Spain
and Instituto de Ciencias Matemáticas ICMAT,
Madrid, Spain<EMAIL_ADDRESS>and  Jonathan R. Partington  Jonathan
R. Partington,
School of Mathematics,
University of Leeds,
Leeds LS2 9JT, United Kingdom<EMAIL_ADDRESS>
(Date: June 2022, revised version September 24th, 2022)

###### Abstract.

A closed subspace is invariant under the Cesàro operator $\mathcal{C}$ on the
classical Hardy space $H^{2}(\mathbb{D})$ if and only if its orthogonal
complement is invariant under the $C_{0}$-semigroup of composition operators
induced by the affine maps $\varphi_{t}(z)=e^{-t}z+1-e^{-t}$ for $t\geq 0$ and
$z\in\mathbb{D}$. The corresponding result also holds in the Hardy spaces
$H^{p}(\mathbb{D})$ for $1<p<\infty$. Moreover, in the Hilbert space setting,
by linking the invariant subspaces of $\mathcal{C}$ to the lattice of the
closed invariant subspaces of the standard right-shift semigroup acting on a
particular weighted $L^{2}$-space on the line, we exhibit a large class of
non-trivial closed invariant subspaces and provide a complete characterization
of the finite codimensional ones, establishing, in particular, the limits of
such an approach towards describing the lattice of all invariant subspaces of
$\mathcal{C}$. Finally, we present a functional calculus argument which allows
us to extend a recent result by Mashreghi, Ptak and Ross regarding the square
root of $\mathcal{C}$ and discuss its invariant subspaces.

###### Key words and phrases:

Cesàro operator, composition operator, shift semigroup, invariant subspaces,
functional calculus

###### 2010 Mathematics Subject Classification:

Primary 47A15, 47A55, 47B15

Both authors are partially supported by Plan Nacional I+D grant no.
PID2019-105979GB-I00, Spain. The first author is also supported by the Spanish
Ministry of Science and Innovation, through the “Severo Ochoa Programme for
Centres of Excellence in R&D” (CEX2019-000904-S) and from the Spanish National
Research Council, through the “Ayuda extraordinaria a Centros de Excelencia
Severo Ochoa” (20205CEX001).

##  1\. Introduction and preliminaries

Despite the fact that one of the most classical transformations of sequences
is the Cesàro operator $\mathcal{C}$, there are still many questions about it
unsettled. Recall that $\mathcal{C}$ takes a complex sequence
$\textbf{a}=(a_{0},a_{1},a_{2}\dots)$ to that with $n$-th entry:

$(\mathcal{C}\,\textbf{a})_{n}=\frac{1}{n+1}\sum_{k=0}^{n}a_{k},\qquad(n\geq
0).$

Upon identifying sequences with Taylor coefﬁcients of power series,
$\mathcal{C}$ acts _formally_ on $f(z)=\sum_{k=0}^{\infty}a_{k}z^{k}$ as

(1.1)
$\mathcal{C}(f)(z)=\sum_{n=0}^{\infty}\left(\frac{1}{n+1}\sum_{k=0}^{n}a_{k}\right)z^{n}.$

Indeed, if $f$ is a holomorphic function on the unit disc $\mathbb{D}$ so is
$\mathcal{C}(f)$ and moreover, $\mathcal{C}$ is an isomorphism of the Fréchet
space $\mathcal{H}(\mathbb{D})$ of all holomorphic functions on $\mathbb{D}$
endowed with the topology of uniform convergence on compacta.

Nevertheless, this is no longer true when $\mathcal{C}$ is restricted to the
classical Hardy spaces $H^{p}(\mathbb{D})$, $1\leq p<\infty$. A classical
result of Hardy concerning trigonometric series along with M. Riesz’s theorem
yields that $\mathcal{C}$ is bounded on $H^{p}(\mathbb{D})$ for $1<p<\infty$.
Likewise, Siskakis proved that $\mathcal{C}$ is bounded on $H^{1}(\mathbb{D})$
(providing even an alternative proof of the boundedness on $H^{p}(\mathbb{D})$
for $1<p<\infty$; see [34], [35]). However 0 belongs to the spectrum of
$\mathcal{C}$ in $H^{p}(\mathbb{D})$ and hence, $\mathcal{C}$ is not an
isomorphism [34].

Note that (1.1) can be written as

$\mathcal{C}(f)(z)=\left\\{\begin{array}[]{ll}\displaystyle\frac{1}{z}\int_{0}^{z}\frac{f(\xi)}{1-\xi}\,d\xi,&z\in\mathbb{D}\setminus\\{0\\},\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr f(0)&z=0;\end{array}\right.$

for $z\in\mathbb{D}$. There is an extensive literature on the Cesàro operator,
and more general, on integral operators, acting on a large variety of spaces
of analytic functions regarding its boundedness, compactness or spectral
picture (see [1] or [3], for instance).

If we restrict ourselves to the Hilbert space case $H^{2}(\mathbb{D})$, Kriete
and Trutt proved the striking result that the Cesàro operator is _subnormal_ ,
namely, $\mathcal{C}$ on $H^{2}(\mathbb{D})$ has a normal extension. More
precisely, if $I$ denotes the identity operator on $H^{2}(\mathbb{D})$, they
proved that $I-\mathcal{C}$ is unitarily equivalent to the operator of
multiplication by the identity function acting on the closure of analytic
polynomials on the space $L^{2}(\mu,\mathbb{D})$ for a particular measure
$\mu$ (see [24]). An alternative proof of the Kriete and Trutt theorem, based
on the connection between $\mathcal{C}$ and composition operators semigroups,
was later established by Cowen [9].

For $H^{p}(\mathbb{D})$, $1<p<\infty$, Miller, Miller and Smith [29] showed
that $\mathcal{C}$ is _subdecomposable_ , namely, it has a decomposable
extension (the $H^{1}(\mathbb{D})$ case was proved by Persson [32] ten years
later). Decomposable operators were introduced by Foiaş [16] in the sixties as
a generalization of spectral operators in the sense of Dunford, and many
spectral operators in Hilbert spaces as unitary operators, self-adjoint
operators or more generally, normal operators are decomposable (see the
monograph [26] for more on the subject).

Normal operators on Hilbert spaces or more generally, decomposable operators
on Banach spaces have a rich lattice of non-trivial closed invariant subspaces
with a significant description of them. But, very little is known about this
description even for concrete examples of subnormal operators as the Cesàro
operator, and this will be the main motivation of the present manuscript.

In this context, we discuss invariant subspaces of the Cesàro operator
$\mathcal{C}$ on the Hardy space $H^{2}(\mathbb{D})$. Broadly speaking, we
prove a _Beurling–Lax Theorem_ for the Cesàro operator and provide a complete
characterization of the finite codimensional invariant subspaces of
$\mathcal{C}$. The composition semigroup method has turned out to be a
powerful tool to study the Cesàro operator and we will make use of such
technique in Section 2 to link the invariant subspaces of $\mathcal{C}$ to
those of the right-shift semigroup $\\{S_{\tau}\\}_{\tau\geq 0}$ acting on a
particular weighted $L^{2}(\mathbb{R},w(y)dy)$. In particular, we will
establish the limits of our approach towards describing completely the lattice
of the invariant subspaces of $\mathcal{C}$.

In Section 3, we discuss Phillips functional calculus (as in Haase’s book
[22]) which will allow us, in particular, generalize the recent work by
Mashreghi, Ptak and Ross [28] regarding the square roots of $\mathcal{C}$. In
particular, we will discuss their invariant subspaces.

In order to close this introductory section we collect some preliminaries for
the sake of completeness.

###  1.1. Semigroups of composition operators

The study of semigroups of composition operators on various function spaces of
analytic functions has its origins in the work of Berkson and Porta [5], where
they characterize their generators on $H^{p}(\mathbb{D})$, proving, indeed,
that these semigroups are always strongly continuous.

Recall that a one-parameter family $\Phi=\\{\varphi_{t}\\}_{t\geq 0}$ of
analytic self-maps of $\mathbb{D}$ is called a holomorphic flow (or
holomorphic semiflow by some authors) if it is a continuous family that has a
semigroup property with respect to composition, namely

  1. 1)

$\varphi_{0}(z)=z$, for $z\in\mathbb{D}$;

  2. 2)

$\varphi_{t+s}(z)=\varphi_{t}\circ\varphi_{s}(z)$, for $t,s\geq 0$, and
$z\in\mathbb{D}$;

  3. 3)

For any $s\geq 0$ and any $z\in\mathbb{D}$, $\lim_{t\to
s}\varphi_{t}(z)=\varphi_{s}(z)$.

The holomorphic flow $\Phi$ is trivial if $\varphi_{t}(z)=z$ for all $t\geq
0$. Otherwise, we say that $\Phi$ is nontrivial. We refer to the recent
monograph [6] for more on the subject.

Associated to the holomorphic flow $\Phi=\\{\varphi_{t}\\}_{t\geq 0}$ is the
family of composition operators $\\{C_{\varphi_{t}}\\}_{t\geq 0}$, defined on
the space of analytic functions on $\mathbb{D}$ by

$C_{\varphi_{t}}f=f\circ\varphi_{t}.$

Clearly, $\\{C_{\varphi_{t}}\\}_{t\geq 0}$ has the semigroup property:

  1. 1)

$C_{\varphi_{0}}=I$;

  2. 2)

$C_{\varphi_{t}}C_{\varphi_{s}}=C_{\varphi_{t+s}}$ for all $t,s\geq 0$.

Moreover, recall that if an operator semigroup $\\{T_{t}\\}_{t\geq 0}$ acts on
a Banach space $X$, then it is called _strongly continuous_ or _$C_{0}$
-semigroup_, if it satisfies

$\lim_{t\to 0^{+}}T_{t}f=f$

for any $f\in X$. Given a $C_{0}$-semigroup $\\{T_{t}\\}_{t\geq 0}$ on a
Banach space $X$, recall that its generator is the closed and densely defined
linear operator $A$ defined by

$Af=\lim_{t\to 0^{+}}\frac{T_{t}f-f}{t}$

with domain $\mathcal{D}(A)=\\{f\in X:\lim_{t\to
0^{+}}\frac{T_{t}f-f}{t}\enspace\text{exists}\\}$.

##  2\. The lattice of the invariant subspaces of the Cesàro operator

The aim of this section is identifying the lattice of the invariant subspaces
of the Cesàro operator $\mathcal{C}$ acting on the Hardy space
$H^{2}(\mathbb{D})$. In particular, we will characterize the finite
codimensional invariant subspaces of $\mathcal{C}$.

Our first result resembles a _Beurling–Lax Theorem_ for the Cesàro operator.

######  Theorem 2.1.

Let $\Phi=\\{\varphi_{t}\\}_{t\geq 0}$ be the holomorphic flow given by

(2.1) $\varphi_{t}(z)=e^{-t}z+1-e^{-t},\qquad(z\in\mathbb{D}).$

A closed subspace $M$ in $H^{2}(\mathbb{D})$ is invariant under the Cesàro
operator if and only if its orthogonal complement $M^{\perp}$ is invariant
under the semigroup of composition operators induced by $\Phi$, namely,
$\\{C_{\varphi_{t}}\\}_{t\geq 0}$.

Before proceeding with the proof, note that each $\varphi_{t}$ in (2.1) is a
affine map which is a hyperbolic non-automorphism of the unit disc inducing a
bounded composition operator $C_{\varphi_{t}}$ on $H^{2}(\mathbb{D})$ with
norm

(2.2) $\|C_{\varphi_{t}}\|_{2}=e^{\frac{t}{2}},$

(see, for instance, [10, Theorem 9.4]).

Likewise, the generator of the $C_{0}$-semigroup $\\{C_{\varphi_{t}}\\}_{t\geq
0}$ is given by

$Af(z)=(1-z)f^{\prime}(z),\qquad(z\in\mathbb{D}),$

(see the pioneering work by Berkson and Porta [5], for instance).

###### Proof.

First, let us show that the cogenerator of the $C_{0}$-semigroup
$\\{C_{\varphi_{t}}\\}_{t\geq 0}$ given by

$V=(A+I)(A-I)^{-1}$

is a well-defined bounded operator. For such a task, we will prove that
$1\in\rho(A)$, the resolvent of $A$, or equivalently,

$A-I:\mathcal{D}(A)\subset H^{2}(\mathbb{D})\to H^{2}(\mathbb{D})$

is bijective.

Note that $A-I$ is an injective operator in $\mathcal{D}(A)\subset
H^{2}(\mathbb{D})$. Indeed, if $(A-I)f(z)=0$ then $(1-z)f^{\prime}(z)-f(z)=0$,
so $f(z)=C/(1-z)$ for some complex constant $C\in\mathbb{C}$. But for $C\neq
0$ we have that $f\not\in H^{2}(\mathbb{D})$.

We claim that $A-I$ is also surjective. Given $g\in H^{2}(\mathbb{D})$, in
order to find $f\in H^{2}(\mathbb{D})$ such that

$(A-I)f(z)=g(z)\qquad(z\in\mathbb{D}),$

or,

$(1-z)f^{\prime}(z)-f(z)=((1-z)f(z))^{\prime}=g(z)\qquad(z\in\mathbb{D}),$

let

(2.3)
$f(z)=\frac{1}{1-z}\int_{1}^{z}g(u)\,du=\frac{1}{z-1}\int_{z}^{1}g(u)\,du$

for $z\in\mathbb{D}$.

Note that the adjoint of the Cesàro operator $\mathcal{C}^{*}$ has the
following matrix with respect to the canonical orthonormal basis of
$H^{2}(\mathbb{D})$:

$\begin{pmatrix}1&\frac{1}{2}&\frac{1}{3}&\frac{1}{4}&\ldots\\\
0&\frac{1}{2}&\frac{1}{3}&\frac{1}{4}&\ldots\\\
0&0&\frac{1}{3}&\frac{1}{4}&\ldots\\\ 0&0&0&\frac{1}{4}&\ldots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{pmatrix}$

Writing

$Tg(z)=\frac{1}{z-1}\int_{z}^{1}g(u)\,du,$

for $g\in H^{2}(\mathbb{D})$, observe that

$Tz^{n}=\frac{1}{z-1}\left(\frac{1-z^{n+1}}{n+1}\right)=-\frac{1+z+\ldots+z^{n}}{n+1}.$

Accordingly, $T$ is a well-defined operator in $H^{2}(\mathbb{D})$ (as it is
-$\mathcal{C}^{*}$). This in particular implies that the function $f$ in (2.3)
belongs to $H^{2}(\mathbb{D})$ and hence $A-I$ is surjective.

Accordingly, the cogenerator $V$ of the $C_{0}$-semigroup
$\\{C_{\varphi_{t}}\\}_{t\geq 0}$ is a well-defined bounded operator on
$H^{2}(\mathbb{D})$.

Now, having in mind the norm estimate (2.2), we observe that the
$C_{0}$-semigroup $\\{e^{-t}C_{\varphi_{2t}}\\}_{t\geq 0}$ is contractive on
$H^{2}(\mathbb{D})$ and its generator is $2A-I$. Since

$V-I=2(A-I)^{-1}={4}((2A-I)-I)^{-1}=-2\mathcal{C}^{*}$

the invariant subspaces of the cogenerator are simply the common invariant
subspaces of the semigroup (see [17, Chap. 10, Theorem 10.9]) and the
statement of the theorem follows. ∎

First, let us remark that a similar argument in the context of
$C_{0}$-semigroups of analytic 2-isometries was also used in [19]. Likewise,
recalling that the Hardy space $H^{p}(\mathbb{D})$, $1\leq p<\infty$, consists
of holomorphic functions $f$ on $\mathbb{D}$ for which the norm

$\|f\|_{p}=\left(\sup_{0\leq
r<1}\int_{0}^{2\pi}|f(re^{i\theta})|^{p}\,\frac{d\theta}{2\pi}\right)^{1/p}$

is finite, we note that the previous proof also works in
$H^{p}(\mathbb{D})$-spaces ($1<p<\infty$) with the natural identification of
the dual space $H^{p}(\mathbb{D})^{*}\cong H^{p^{\prime}}(\mathbb{D})$ where
$p^{\prime}$ is the conjugate exponent: $\frac{1}{p}+\frac{1}{p^{\prime}}=1$.
In this case, the bounded composition operator $C_{\varphi_{t}}$ on
$H^{p}(\mathbb{D})$ has norm

(2.4) $\|C_{\varphi_{t}}\|_{p}=e^{\frac{t}{p}}$

for $1\leq p<\infty$ (see [30, Exercise 3.12.5, p. 56-57] and [23, Part Two,
Ch. 1, Section 5 1. B p. 165-166], for instance). Accordingly, the
$C_{0}$-semigroup $\\{e^{-(p^{\prime}-1)t}C_{\varphi_{p^{\prime}t}}\\}_{t\geq
0}$ is also contractive on $H^{p}(\mathbb{D})$ with generator
$p^{\prime}A-(p^{\prime}-1)I$ and cogenerator
$I+2(p^{\prime}A-(p^{\prime}-1)I-I)^{-1}=I+2(p^{\prime}(A-I))^{-1}$.
Therefore, a closed subspace $M$ in $H^{p}(\mathbb{D})$ for $1<p<\infty$ is
invariant under the Cesàro operator if and only if its annihilator $M^{\perp}$
in $H^{p^{\prime}}(\mathbb{D})$ is invariant under the $C_{0}$-semigroup
$\\{C_{\varphi_{t}}\\}_{t\geq 0}$.

In this regard, it is worth noting that $(1-z)^{-1}\not\in H^{p}(\mathbb{D})$
for any $1\leq p\leq\infty$ since $(1-e^{i\theta})^{-1}\not\in
L^{p}(\mathbb{T})$.

######  Remark 2.2.

By Dunford and Schwartz [15, Thm. 11, p. 622], the resolvent can be expressed
in terms of the Laplace transform of the semigroup; that is,

$(A-I)^{-1}f(z)=\int_{0}^{\infty}e^{-t}C_{\varphi_{t}}f(z)\,dt,\qquad(z\in\mathbb{D}).$

So

$\mathcal{C}^{*}f(z)=-\int_{0}^{\infty}e^{-t}f(e^{-t}z+1-e^{-t})\,dt,,\qquad(z\in\mathbb{D}).$

Indeed, a consequence of the previous formula is the following:

######  Corollary 2.3.

A closed subspace $M$ in $H^{2}(\mathbb{D})$ is invariant under the Cesàro
operator if and only if it is invariant under the semigroup
$\\{C_{\varphi_{t}}^{*}\\}_{t\geq 0}$.

Alternatively, one may use that the adjoint of the generator is the generator
of the adjoint semigroup in the context of Hilbert spaces (see [17, Chap. 10],
for instance).

Finally, note that the adjoint $C_{\varphi_{t}}^{*}$ in $H^{2}(\mathbb{D})$ in
Corollary 2.3 may be explicitly computed as a weighted composition operator
(see [10, Theorem 9.2]). Indeed, expressing $\varphi_{t}(z)=e^{-t}z+1-e^{-t}$
in its normal form, namely $\varphi_{t}(z)=(a_{t}z+b_{t})/(0z+a_{t}^{-1})$
where

$a_{t}=e^{-t/2},\qquad b_{t}=\frac{1-e^{-t}}{e^{-t/2}}$

we deduce that $C_{\varphi_{t}}^{*}=T_{g_{t}}C_{\sigma_{t}}T_{h_{t}}^{*}$
where

$g_{t}(z)=\frac{1}{-b_{t}z+a_{t}^{-1}}=\frac{e^{-t/2}}{1-(1-e^{-t})z},\qquad(z\in\mathbb{D}),$
$\sigma_{t}(z)=\frac{a_{t}z}{-b_{t}z+a_{t}^{-1}}=\frac{e^{-t}z}{1-(1-e^{-t})z},\qquad(z\in\mathbb{D}),$

and

$h_{t}(z)=a_{t}^{-1}=e^{t/2};$

and $T_{g_{t}}$ and $T_{h_{t}}$ denotes the analytic Toeplitz operators acting
on $H^{2}(\mathbb{D})$ induced by the symbols $g_{t}$ and $h_{t}$
respectively.

Accordingly,

$C_{\varphi_{t}}^{*}f(z)=\frac{1}{1-(1-e^{-t})z}f\left(\frac{e^{-t}z}{1-(1-e^{-t})z}\right),\qquad(z\in\mathbb{D})$

for $f\in H^{2}(\mathbb{D})$ and every $t\geq 0$.

###  2.1. Shift semigroups

In order to provide a characterization of the finite codimensional invariant
subspaces of $\mathcal{C}$, we will make use of a semigroup of operators
acting on the Hardy space of the right half-plane $\mathbb{C}_{+}$. Recall
that the Hardy space $H^{2}(\mathbb{C}_{+})$ consists of the functions $F$
analytic on $\mathbb{C}_{+}$ with finite norm

$\|F\|_{H^{2}(\mathbb{C}_{+})}=\left\\{\sup_{0<x<\infty}\int_{-\infty}^{\infty}|F(x+iy)|^{2}\,dy\right\\}^{1/2}.$

The classical Paley–Wiener Theorem (see [33], for instance) states that
$H^{2}(\mathbb{C}_{+})$ is isomorphic under the Laplace transform to
$L^{2}(\mathbb{R}_{+})$, the space of measurable functions square-integrable
over $(0,\infty)$. More precisely, to each function $F\in
H^{2}(\mathbb{C}_{+})$ there corresponds a function $f\in
L^{2}(\mathbb{R}_{+})$ such that

$F(s)=({\mathcal{L}}f)(s):=\int_{0}^{\infty}f(x)e^{-sx}\,dx\,,\qquad(s\in\mathbb{C}_{+}),$

and

$\|F\|^{2}_{H^{2}(\mathbb{C}_{+})}=2\pi\int_{0}^{\infty}|f(x)|^{2}\,dx\,.$

A first observation already stated in [8, Lemma 4.2] is that each
$\varphi_{t}(z)=e^{-t}z+1-e^{-t}$ for $z\in\mathbb{D}$ and $t>0$ induces a
composition operator in $H^{2}(\mathbb{D})$ which is similar under an
isomorphism $\mathcal{U}$ (indeed unitarily equivalent up to a constant) to
$e^{t}\,C_{\phi_{t}}$ in $H^{2}(\mathbb{C}_{+})$, where

$\phi_{t}(s)=e^{t}s+(e^{t}-1),\qquad(s\in\mathbb{C}_{+}).$

Namely,

(2.5)
$\mathcal{U}C_{\varphi_{t}}\mathcal{U}^{-1}=e^{t}C_{\phi_{t}},\qquad(t\geq
0).$

Since we are interested in studying invariant subspaces, either for the entire
semigroup or individual elements, we may disregard factors of the form
$e^{\lambda t}$ for a fixed $\lambda\in\mathbb{R}$.

By means of the inverse Laplace transform we are led to consider the semigroup
on $L^{2}(\mathbb{R}_{+})$

(2.6) $V_{t}g(x)=e^{-t}e^{-(1-e^{-t})x}g(e^{-t}x),\qquad(x>0,\ t\geq 0).$

Now, proceeding as in [18], we may find a further equivalence with an operator
on $L^{2}(\mathbb{R})$ using the unitary mapping $T:L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R}_{+})$ defined by

$Th(x)=x^{-1/2}h(\log x),\qquad(x>0)$

and

$T^{-1}g(y)=e^{y/2}g(e^{y}),\qquad(y\in\mathbb{R}).$

Accordingly,

(2.7)
$T^{-1}V_{t}Th(y)=e^{-t/2}e^{-(1-e^{-t})e^{y}}h(y-t),\qquad(y\in\mathbb{R}),$

for $t\geq 0$.

Denoting by $\\{S_{t}:\,t\geq 0\\}$ the right-shift semigroup on
$L^{2}(\mathbb{R})$:

$S_{t}f(y)=f(y-t),\qquad(y\in\mathbb{R}),$

and recalling that if $w$ denotes a positive measurable function in
$\mathbb{R}$ the space $L^{2}(\mathbb{R},w(y)dy)$ consists of measurable
functions in $\mathbb{R}$ square-integrable respect to the measure $w(y)dy$, a
key observation is the following

######  Proposition 2.4.

The semigroup $\\{\sigma_{t}:\,t\geq 0\\}$ in $L^{2}(\mathbb{R})$ given by

(2.8) $\sigma_{t}h(y)=e^{-(1-e^{-t})e^{y}}h(y-t),\qquad(y\in\mathbb{R}),$

for $h\in L^{2}(\mathbb{R})$ is unitarily equivalent to the right-shift
semigroup $\\{S_{t}:\,t\geq 0\\}$ acting on the weighted Lebesgue space
$L^{2}(\mathbb{R},e^{-2(e^{y}-1)}\,dy)$.

###### Proof.

Let us denote the weight $w(y)=e^{-2(e^{y}-1)}$ for $y\in\mathbb{R}$ and
consider the unitary mapping $W:L^{2}(\mathbb{R})\to
L^{2}(\mathbb{R},w(y)\,dy)$ given by

$Wh(y)=h(y)/\sqrt{w(y)},\qquad(y\in\mathbb{R}),$

for $h\in L^{2}(\mathbb{R})$. A computation shows that for any function $f\in
L^{2}(\mathbb{R},w(y)\,dy)$ and $t>0$

$\displaystyle W\sigma_{t}W^{-1}f(y)$
$\displaystyle=W\sigma_{t}\,f(y)e^{-(e^{y}-1)}$
$\displaystyle=We^{-(1-e^{-t})e^{y}}f(y-t)e^{-(e^{y-t}-1)}$
$\displaystyle=e^{e^{y}-1}e^{-(1-e^{-t})e^{y}}f(y-t)e^{-(e^{y-t}-1)}$
$\displaystyle=f(y-t),$

for $y\in\mathbb{R}$. This yields the statement of the proposition. ∎

Figure 1 shows a plot of the weight function described in Proposition 2.4.

Figure 1. The weight $e^{-2(e^{y}-1)}$ as a function of $y$.

As a by-product of equations (2.5), (2.6), (2.7), Proposition 2.4 and Theorem
2.1, if we denote by $\mathfrak{F}$ the unitary isomorphism
$\mathfrak{F}=WT^{-1}\mathcal{L}^{-1}\mathcal{U}$ from $H^{2}(\mathbb{D})$
onto $L^{2}(\mathbb{R},e^{-2(e^{y}-1)}\,dy)$, the following result regarding
the lattice of invariant subspaces of the Cesàro operator holds:

######  Theorem 2.5.

A closed subspace $M$ in $H^{2}(\mathbb{D})$ is invariant under the Cesàro
operator if and only if $\mathfrak{F}M^{\perp}$ in
$L^{2}(\mathbb{R},e^{-2(e^{y}-1)}\,dy)$ is invariant under the right-shift
semigroup $\\{S_{t}:\,t\geq 0\\}$.

Accordingly, characterizing the lattice of invariant subspaces of the Cesàro
operator in the Hardy space reduces to characterize the lattice of the right-
shift semigroup in $L^{2}(\mathbb{R},e^{-2(e^{y}-1)}\,dy)$.

Though the lattice of the invariant subspaces of the right-shift semigroup
acting on weighted Lebesgue spaces is only characterized for a very restricted
subclass of weights (for instance the Beurling–Lax Theorem provides a
characterization in $L^{2}(\mathbb{R}_{+})$, where the weight is the
characteristic function $\chi_{(0,+\infty)}$ of $(0,+\infty)$), the question
if such a lattice contains non-standard invariant subspaces has been
extensively studied (see [11, 12, 13, 14], [20, 21], [27] or [31], for
instance).

Recall that given $a\in\mathbb{R}\cup\\{-\infty\\}\cup\\{\infty\\}$, the
“standard invariant subspaces” of $\\{S_{t}:\,t\geq 0\\}$ are given by

$L^{2}((a,\infty),w(y)\,dy)=\\{f\in L^{2}(\mathbb{R},w(y)\,dy):f(y)=0\mbox{
for a.~{}e. }y\leq a\\}.$

In [11, Equation (8)], Domar proved that if the weight satisfies

$\displaystyle{\underline{\lim}}_{y\to-\infty}\frac{\log w(y)}{y}>-\infty$

then the lattice of invariant subspaces of $\\{S_{t}:\,t\geq 0\\}$ in
$L^{2}(\mathbb{R},w(y)\,dy)$ contains non-standard invariant subspaces.

_A word about notation_ : Domar denotes by $L^{2}(\mathbb{R},w(y)\,dy)$ the
space of measurable functions $f$ in $\mathbb{R}$ such that $f\,w\in
L^{2}(\mathbb{R})$. Note that this does not affect the previous equation since
it is enough to consider the positive function $w^{1/2}$.

In our case, $w(y)=e^{-2(e^{y}-1)}$ for $y\in\mathbb{R}$ and consequently,
$\\{S_{t}:\,t\geq 0\\}$ has non-standard invariant subspaces in
$L^{2}(\mathbb{R},e^{-2(e^{y}-1)}\,dy)$. Indeed, it is possible to exhibit
many non-standard invariant subspaces in this case. In order to show them
recall that, by means of the unitary equivalence
${\mathcal{L}}:L^{2}(\mathbb{R}_{+},\sqrt{2\pi}\,dt)\to
H^{2}(\mathbb{C}_{+})$, the Beurling–Lax Theorem asserts that a closed
subspace $\mathcal{M}$ of $L^{2}(\mathbb{R}_{+})$ is invariant under every
truncated right-shift to $L^{2}(\mathbb{R}_{+})$

$S_{\mathbb{R}_{+},\,\tau}f(t)=\left\\{\begin{array}[]{ll}0&\mbox{ if }0\leq
t\leq\tau,\\\ f(t-\tau)&\mbox{ if }t>\tau;\end{array}\right.$

$\tau\geq 0$, if and only if there exists an inner function $\Theta\in
H^{\infty}(\mathbb{C}_{+})$ such that ${\mathcal{L}}\mathcal{M}=\Theta
H^{2}(\mathbb{C}_{+})$ (see [30, Cor. 6.5.5(2), p.149], for instance). Here,
recall that an inner function $\Theta$ is an analytic function in
$\mathbb{C}_{+}$ with $|\Theta(z)|\leq 1$ for $z\in\mathbb{C}_{+}$, such that
the non-tangential limits exist and are of modulus 1 almost everywhere on the
imaginary axis.

######  Example 2.6.

Let $T\in\mathbb{R}$ fixed and write

$L^{2}(\mathbb{R},w(y)\,dy)=L^{2}((-\infty,T),w(y)\,dy)\oplus
L^{2}((T,\infty),w(y)\,dy),$

the orthogonal direct sum of closed subspaces. Note that

  1. (1)

$\exp(-2(e^{T}-1))\leq w(y)\leq e^{2}$ on $(-\infty,T)$, so
$L^{2}((-\infty,T),w(y)\,dy)$ is naturally isomorphic to $L^{2}(-\infty,T)$.

  2. (2)

If $M$ is a closed subspace of $L^{2}((-\infty,T),w(y)\,dy)$ invariant under
all truncated right-shifts on $L^{2}((-\infty,T),w(y)\,dy)$, i.e.,

$S_{(-\infty,T),\,\tau}f(y)=\left\\{\begin{array}[]{ll}0&\mbox{ if
}y-\tau>T,\\\ f(y-\tau)&\mbox{ if }y-\tau\leq T,\end{array}\right.$

for $\tau\geq 0$, then $M\oplus L^{2}((T,\infty),w(y)\,dy)$ is a closed
subspace of $L^{2}(\mathbb{R},w(y)\,dy)$ invariant under all right shifts.

Now, the Beurling–Lax theorem provides a large class of nonstandard invariant
subspaces $M$: take the “twisted” Laplace transform

(2.9) $\tilde{\mathcal{L}}f(s)=\int_{-T}^{\infty}e^{-su}f(-u)\,du$

which gives an isomorphism from $L^{2}((-\infty,T),w(y)\,dy)$ onto
$e^{sT}H^{2}(\mathbb{C}_{+})$. Then any subspace of the form
$\widetilde{\mathcal{L}}^{-1}e^{sT}K_{\Theta}$ is invariant under all
truncated right shifts $S_{(-\infty,T),\,\tau}$ where $\Theta\in
H^{\infty}(\mathbb{C}_{+})$ is inner and
$K_{\Theta}=H^{2}(\mathbb{C}_{+})\ominus\Theta H^{2}(\mathbb{C}_{+})$ is the
associated model space (these calculations are easiest to follow when $T=0$,
and the general case is a shifted version.)

As an explicit example, let $K_{\Theta}$ be spanned by the reproducing kernel
$s\mapsto 1/(s+\overline{\lambda})$ for $\lambda\in\mathbb{C}_{+}$, so that
$M$ is the one-dimensional space of $L^{2}((-\infty,T),w(y)\,dy)$ spanned by
$e^{\overline{\lambda}t}$; then $M\oplus L^{2}((T,\infty),w(y)\,dy)$ is a non-
standard invariant subspace for all right shifts.

While a theorem of Aleman and Koreblum [2] asserts that the analytic Volterra
operator is unicellular in $H^{p}$-spaces, as consequence of the previous
considerations we have a new deduction of the following known result:

######  Corollary 2.7.

The Cesàro operator $\mathcal{C}$ is not a unicellular operator in
$H^{2}(\mathbb{D})$.

In this regard, the feature that the Cesàro operator $\mathcal{C}$ is not a
unicellular operator on $H^{2}(\mathbb{D})$ can be also deduced from [7], as
the referee kindly pointed out to us. Indeed, it follows from the result that
the point spectrum of $I-\mathcal{C}^{*}$ in $H^{2}(\mathbb{D})$ is
$\mathbb{D}$ along with the fact that any operator on a Hilbert space which
has at least two eigenvalues cannot be unicellular. Likewise, in [25,
Corollary 6], the authors constructed two nonzero invariant subspaces of
$\mathcal{C}$ whose intersection is zero space.

On the other hand, it is worth pointing out that the classifying the invariant
subspaces turns out to be completely different if one considers other
semigroups studied in the context of Cesàro-like operators, as in the
following remark:

######  Remark 2.8.

In [4], the authors considered the composition operator group on
$H^{2}(\mathbb{C}_{+})$ corresponding to the flow on $\mathbb{C}_{+}$ given by

$\phi_{t}(s)=e^{-t}s,\qquad(s\in\mathbb{C}_{+})$

$t\in\mathbb{R}$, in a broader context of studying Cesàro-like operators.

Proceeding similarly as before, the transformed semigroup on $L^{2}(0,\infty)$
is given by

$\tilde{V}_{t}g(x)=e^{t}g(e^{t}x),\qquad(x>0,\ t\in\mathbb{R}\mbox{ and }g\in
L^{2}(0,\infty)),$

which transferred to $L^{2}(\mathbb{R})$ is

$T^{-1}\tilde{V}_{t}Th(y)=e^{t/2}h(y+t),\qquad(y\in\mathbb{R},\
t\in\mathbb{R}),$

for $h\in L^{2}(\mathbb{R})$.

The subspaces $M$ invariant under the group
$(\tau_{t})_{t\in\mathbb{R}}=(T^{-1}\tilde{V}_{t}T)_{t\in\mathbb{R}}$ were
essentially classified by Lax – the factors $e^{t/2}$ are irrelevant – and can
be found, with a slightly different notation in [30, Cor 6.5.4, p. 149]. There
are two types:

  1. (1)

$1$-invariant subspaces, i.e., $\tau_{t}M\subset M$ for all $t<0$ but not for
all $t\in\mathbb{R}$. These have the form $M=\mathcal{F}qH^{2}(\Pi^{+})$,
where $q$ is measurable with $|q|=1$ almost everywhere and here $\Pi^{+}$
denotes the upper half-plane;

  2. (2)

$2$-invariant subspaces, i.e., $\tau_{t}M\subset M$ for all $t\in\mathbb{R}$.
These have the form $M=\mathcal{F}\chi_{E}L^{2}(\mathbb{R})$ for some
measurable subset $E\subset\mathbb{R}$.

Here $\mathcal{F}$ denotes the Fourier transform but, alternatively, one can
use the bilateral Laplace transform and express the subspaces in terms of
$L^{2}(i\mathbb{R})$ and the space $H^{2}(\mathbb{C}_{+})$ of the right half-
plane.

Likewise, in this case the invariant subspaces of the form
$\tilde{\mathcal{L}}^{-1}K_{\Theta}\oplus L^{2}(\mathbb{R}_{+})$ are
1-invariant subspaces as described above, corresponding in
$L^{2}(i\mathbb{R})$ to $\overline{\Theta}K_{\Theta}\oplus
H^{2}(\mathbb{C}_{+})=\overline{\Theta}H^{2}(\mathbb{C}_{+})$.

Finally, as an application of Theorem 2.5, we present a characterization of
the finite codimensional invariant subspaces of the Cesàro operator
$\mathcal{C}$ in $H^{2}(\mathbb{D})$. Of particular relevance will be a
theorem of Domar [12] which states that the lattice of the invariant subspaces
of $\\{S_{\tau}:\tau\geq 0\\}$ consists of just the standard invariant
subspaces in $L^{2}(\mathbb{R}_{+},w(x)\,dx)$ whenever:

  1. (1)

$w$ is a positive continuous function in $\mathbb{R}_{+}$ such that $\log w$
is concave in $[c,\infty)$ for some $c>0$.

  2. (2)

$\displaystyle\lim_{x\to\infty}\frac{-\log w(x)}{x}=\infty.$

  3. (3)

$\displaystyle\lim_{x\to\infty}\frac{\log|\log w(x)|-\log x}{\sqrt{\log
x}}=\infty.$

######  Theorem 2.9.

A finite codimensional closed subspace $M$ in $H^{2}(\mathbb{D})$ is invariant
under the Cesàro operator if and only if $\mathfrak{F}M^{\perp}$ in
$L^{2}(\mathbb{R},e^{-2(e^{y}-1)}\,dy)$ is spanned by a finite subset of
functions given by

$\bigcup_{\lambda\in\Lambda}\\{y^{k}e^{\lambda
y}:\;k=0,1,2,\ldots,n_{\lambda}\\}$

where $\Lambda\subset\mathbb{C}_{+}$ is a finite set and $n_{\lambda}\geq 0$
for each $\lambda\in\Lambda$.

Before proceeding with the proof, let us introduce the notation

$f_{\lambda,k}(y)=y^{k}e^{\lambda y}$

for $y\in\mathbb{R},\lambda\in\mathbb{C}_{+}$ and $k=0,1,2,\dots$ Note that

$(S_{t}f_{\lambda,k})(y)=f_{\lambda,k}(y-t)=\sum_{j=0}^{k}\binom{k}{j}(-t)^{k-j}e^{-\lambda
t}y^{j}e^{\lambda y},$

so

(2.10) $S_{t}f_{\lambda,k}=\sum_{j=0}^{k}\binom{k}{j}(-t)^{k-j}e^{-\lambda
t}f_{\lambda,j}$

for any $\lambda\in\mathbb{C}_{+}$ and $k=0,1,2,\dots$

###### Proof.

Suppose first that $M$ is a finite codimensional closed subspace in
$H^{2}(\mathbb{D})$ invariant under $\mathcal{C}$. Theorem 2.5 yields that
$N=\mathfrak{F}M^{\perp}$ is a finite dimensional subspace of
$L^{2}(\mathbb{R},e^{-2(e^{y}-1)}\,dy)$ invariant under all right shifts.
Thus, $P_{-}N$, the projection onto
$L^{2}((-\infty,0),e^{-2(e^{y}-1)}\,dy)\cong L^{2}(-\infty,0)$, is a finite
dimensional subspace invariant under all the truncated right shifts.

Thus, by the Beurling–Lax Theorem, $P_{-}N$ corresponds to a model space and
in particular is spanned by a finite set of functions of the form

$y^{k}e^{\lambda y},\mbox{ for }y\in(-\infty,0)$

where $k=0,1,2,\ldots,n_{\lambda}$ for
$\lambda\in\Lambda\subset\mathbb{C}_{+}$. We now show that $N$ is spanned by
what we shall call the “natural extension” to $\mathbb{R}$ of such functions
as elements of $L^{2}(\mathbb{R},e^{-2(e^{y}-1)}\,dy)$, namely,
$y^{k}e^{\lambda y}\mbox{ for }y\in\mathbb{R}$.

Observe that since N is ﬁnite dimensional, by the Domar theorem
aforementioned,

$N\cap L^{2}((0,\infty),e^{-2(e^{y}-1)}\,dy)=\\{0\\}.$

Therefore, there exist $h_{\lambda,k}\in N$ such that
$P_{-}f_{\lambda,k}=h_{\lambda,k}$ and $N$ is spanned by $h_{\lambda,k}$. Let
$N_{1}$ be spanned by $f_{\lambda,k}$ with the same $k$ and $\lambda$. By
(2.10), $N_{1}$ is invariant under all right shifts $S_{t}$. Then
$\operatorname*{span\;}\\{N,N_{1}\\}$ is a finite dimensional invariant
subspace of all right shifts $S_{t}$. Upon applying Domar’s theorem again,

$\operatorname*{span\;}\\{N,N_{1}\\}\cap
L^{2}((0,\infty),e^{-2(e^{y}-1)}\,dy)=\\{0\\}.$

Since

$h_{\lambda,k}-f_{\lambda,k}\in\operatorname*{span\;}\\{N,N_{1}\\}\cap
L^{2}((0,\infty),e^{-2(e^{y}-1)}\,dy),$

we conclude that $h_{\lambda,k}=f_{\lambda,k}$. Thus, $N$ is spanned by
$f_{\lambda,k}$.

For the converse, assume $N=\mathfrak{F}M^{\perp}$ in
$L^{2}(\mathbb{R},e^{-2(e^{y}-1)}\,dy)$ is a finite dimensional subspace
spanned by a finite subset of the form
$\bigcup_{\lambda\in\Lambda}\\{y^{k}e^{\lambda
y}:\;k=0,1,2,\ldots,n_{\lambda}\\}$ where
$\lambda\in\Lambda\subset\mathbb{C}_{+}$ is finite. Equation (2.10) yields
that $N$ is invariant under all right shifts $S_{t}$. Now, Theorem 2.5 and the
fact that $\mathfrak{F}$ is an isomorphism yield that $M$ is a finite
codimensional closed subspace invariant under $\mathcal{C}$, which completes
the proof. ∎

######  Remark 2.10.

Note that in the case that $N=\mathfrak{F}M^{\perp}$ is infinite-dimensional,
the arguments above giving the structure of $N$ in terms of the structure of
$P_{-}N$ fail because we can no longer assume that $P_{-}N$ is closed.
However, it is of interest to note that the closure $\overline{P_{-}N}$ in
$L^{2}(-\infty,0)$ has the same property of invariance under all truncated
right shifts, and corresponds to a model space.

For instance, if $B$ is a Blaschke product in $\mathbb{C}_{+}$ with the set of
zeros $\Lambda$ and multiplicities $n_{\lambda}+1$ for $\lambda\in\Lambda$
with $n_{\lambda}\in\\{0,1,2,\dots\\}$ and we set

$N_{B}=\overline{\operatorname*{span\;}\\{f_{\lambda,k}:\,\lambda\in\Lambda,k=0,1,\dots,n_{\lambda}\\}}^{L^{2}(\mathbb{R},e^{-2(e^{y}-1)}\,dy)},$

clearly $N_{B}$ is invariant under all right shifts. Nevertheless,

$\overline{\tilde{\mathcal{L}}P_{-}N_{B}}=\overline{\operatorname*{span\;}\\{\widetilde{\mathcal{L}}P_{-}f_{\lambda,k}:\lambda\in\Lambda,k=0,1,\dots,n_{\lambda}\\}}^{H^{2}(\mathbb{C}_{+})}=K_{B},$

where $\tilde{\mathcal{L}}$ is defined in (2.9) with $T=0$. Consequently,
$N_{B}\neq L^{2}(\mathbb{R},e^{-2(e^{y}-1)}\,dy)$, and if $B_{1}\neq B_{2}$
are two Blaschke products, then $N_{B_{1}}\not=N_{B_{2}}$.

###  2.2. A final remark regarding the lattice of the invariant subspaces of
$\mathcal{C}$

As we have just noted, the approach addressed in the previous theorem fails if
$P_{-}N$ is not closed. Indeed, the following example shows that $P_{-}N$ need
not be closed even if $N$ is a closed shift-invariant subspace of
$L^{2}(\mathbb{R},e^{-2(e^{y}-1)}\,dy)$ showing, somehow, the limits of such
an approach.

Let $\lambda>0$ and denote by $e_{\lambda}$ the function
$e_{\lambda}:y\in\mathbb{R}\mapsto e^{\lambda y}$. Since
$1<e^{-2(e^{y}-1)}<e^{2}$ for $y<0$, we have

(2.11)
$\|e_{\lambda}\|^{2}_{L^{2}((-\infty,0),e^{-2(e^{y}-1)}\,dy)}\approx\|e_{\lambda}\|^{2}_{L^{2}(-\infty,0)}\approx
1/\lambda.$

On the other hand,

(2.12)
$\displaystyle\|e_{\lambda}\|^{2}_{L^{2}(\mathbb{R},e^{-2(e^{y}-1)}\,dy)}$
$\displaystyle\geq$ $\displaystyle\int_{1}^{2}e^{2\lambda
y}e^{-2(e^{y}-1)}\,dy\geq e^{2\lambda}e^{-2(e^{2}-1)}.$

Now take $N$ to be the closed linear span in
$L^{2}(\mathbb{R},e^{-2(e^{y}-1)}\,dy)$ of $\\{e_{n^{2}}:n\in\mathbb{N}\\}$,
that is,

$N=\overline{\operatorname*{span\;}\\{e_{n^{2}}:n\in\mathbb{N}\\}}^{L^{2}(\mathbb{R},e^{-2(e^{y}-1)}\,dy)}.$

Now the twisted Laplace transform given in (2.9), with $T=0$, provides an
isomorphism from $L^{2}((-\infty,0),e^{-2(e^{y}-1)}\,dy)$ onto
$H^{2}(\mathbb{C}_{+})$ transforming $e_{\lambda}$ to $1/(s+\lambda)$. By an
argument similar to that used in proving the classical Müntz–Szász theorem it
follows that $\overline{P_{-}N}\subsetneq
L^{2}((-\infty,0),e^{-2(e^{y}-1)}\,dy)$ since there are functions orthogonal
to each $1/(s+n^{2})$, for example, $1/(s+2)$ times a Blaschke product with
zeros at $\\{n^{2}:n\in\mathbb{N}\\}$.

Now $(\mathop{\rm ker}\nolimits P_{-})\cap N$ is a closed shift-invariant
subspace of $L^{2}((0,\infty),e^{-2(e^{y}-1)}\,dy)$ and hence, by Domar’s
theorem, a standard subspace. It must be $\\{0\\}$ (again this follows from a
Műntz–Szász argument) so the restriction $P_{-}:N\to
L^{2}((-\infty,0),e^{-2(e^{y}-1)}\,dy)$ is injective. Finally, by Banach’s
open mapping theorem the norm estimates in (2.11) and (2.12) show that it
cannot have a closed range.

The following proposition characterizes when $P_{-}N$ is a closed subspace of
$L^{2}((-\infty,0),e^{-2(e^{y}-1)}\,dy)$.

######  Proposition 2.11.

With the above notation, $P_{-}N$ is closed if and only if the (not
necessarily direct) sum of the two closed subspaces $N$ and
$L^{2}((0,\infty),e^{-2(e^{y}-1)}\,dy)$ is closed.

###### Proof.

It is well known that if $P:H\to K$ is a projection onto a subspace $K$ of a
Hilbert space $H$ then $K$ is automatically closed: since if $(x_{n})$ is a
sequence in $K$ tending to $x\in H$, we have $x_{n}=Px_{n}\to Px$, so $x=Px\in
K$.

Now if $H:=N+L^{2}((0,\infty),e^{-2(e^{y}-1)}\,dy)$ is closed, the projection
$P_{-}$ maps $H$ to itself and its image is $P_{-}N$. Conversely, if $P_{-}N$
is closed then $N+L^{2}((0,\infty),e^{-2(e^{y}-1)}\,dy)=P_{-}N\oplus
L^{2}((0,\infty),e^{-2(e^{y}-1)}\,dy)$, the orthogonal direct sum of two
closed subspaces, and is therefore closed. ∎

##  3\. Functions of the Cesàro operator

Suppose that $(T(t))_{t\geq 0}$ is a $C_{0}$ semigroup with $\|T(t)\|\leq
e^{mt}$ for some $m<1$ and infinitesimal generator $A$.

It is a standard fact that

$\int_{0}^{\infty}e^{-\lambda t}T(t)\,dt=(\lambda-A)^{-1}$

provided that $\mathop{\rm Re}\nolimits\lambda>m$.

Now consider the operator

$B=\frac{1}{\sqrt{\pi}}\int_{0}^{\infty}\frac{e^{-\lambda
t}}{\sqrt{t}}T(t)\,dt.$

Then

$\displaystyle B^{2}$
$\displaystyle=\frac{1}{\pi}\int_{0}^{\infty}\frac{e^{-\lambda
t}}{\sqrt{t}}T(t)\,dt\int_{0}^{\infty}\frac{e^{-\lambda u}}{\sqrt{u}}T(u)\,du$
$\displaystyle=\frac{1}{\pi}\int_{w=0}^{\infty}e^{-\lambda
w}T(w)\,dw\int_{t=0}^{w}\frac{1}{\sqrt{t}\sqrt{w-t}}\,dt$

with $w=t+u$. The second integral is $\pi$ (use the substitution
$t=w\sin^{2}\theta$) and so $B^{2}=(\lambda-A)^{-1}$. Similarly for
$(-B)^{2}$, of course.

This gives an alternative way of looking at a result in [28] on the square
roots of the Cesàro operator, by using the composition semigroup and the
observations in Section 2. Not surprisingly, it is linked with the fact that
the Laplace transform of $e^{at}$ is $1/(s-a)$ and the Laplace transform of
$e^{at}/\sqrt{t}$ is $\sqrt{\pi}/\sqrt{s-a}$. There is a more general
functional calculus available here, but this calculation at least can be done
directly.

Indeed the Phillips functional calculus [22, Rem. 3.3.3] allows us, given a
bounded semigroup $(T(t))_{t\geq 0}$ of operators on a Banach space $X$, to
associate an operator $f(A)$ to a function $f$ that is the Laplace transform
of a Borel measure $\mu$ on $[0,\infty)$ of bounded variation, by the formula

$f(A)x=\int_{[0,\infty)}T(t)x\,d\mu(t)\qquad(x\in X).$

Note that the convention in [22] is that the generator is $-A$, rather than
$A$, and we have allowed for that in the discussion below. In particular, we
have

(3.1) $\int_{0}^{\infty}\frac{e^{-\lambda
t}}{t^{1-\beta}}\,dt=\lambda^{-\beta}\Gamma(\beta)$

for $\mathop{\rm Re}\nolimits\lambda,\mathop{\rm Re}\nolimits\beta>0$, so that

$\int_{0}^{\infty}\frac{e^{-t}e^{at}}{t^{1-\beta}}\,dt=(1-a)^{-\beta}\Gamma(\beta)$

for $\mathop{\rm Re}\nolimits a<1$ and $\mathop{\rm Re}\nolimits\beta>0$, from
which we obtain

(3.2)
$\int_{0}^{\infty}\frac{e^{-t}T(t)}{t^{1-\beta}}\,dt=(I-A)^{-\beta}\Gamma(\beta)$

for $\mathop{\rm Re}\nolimits\beta>0$. A similar formula holds on replacing
$T(t)$ by $T(t)^{*}$ and $A$ by $A^{*}$.

Recalling that $\mathcal{C}^{*}=(I-A)^{-1}$, we have the following:

######  Theorem 3.1.

For $\mathop{\rm Re}\nolimits\beta>0$ let $M_{\beta}$ denote the matrix of
$\mathcal{C}^{\beta}$ (as defined using (3.2)) with respect to the standard
orthonormal basis $(z^{j})_{j=0}^{\infty}$. Then

(3.3) $(M_{\beta})_{i,j}=\begin{cases}0&\text{if }i<j,\\\
\displaystyle{i\choose j}\sum_{k=0}^{i-j}(-1)^{k}{i-j\choose
k}(j+k+1)^{-\beta}&\text{if }i\geq j.\end{cases}$

###### Proof.

The $(j,i)$ entry of the matrix for $(I-A)^{-\beta}$ is the coefficient of
$z^{j}$ in

$\frac{1}{\Gamma(\beta)}\int_{0}^{\infty}\frac{e^{-t}}{t^{1-\beta}}(e^{-t}z+(1-e^{-t}))^{i}\,dt,$

namely, $0$ for $j>i$ and otherwise

$\frac{1}{\Gamma(\beta)}\int_{0}^{\infty}\frac{e^{-t}}{t^{1-\beta}}{i\choose
j}e^{-jt}(1-e^{-t})^{i-j}\,dt=\frac{1}{\Gamma(\beta)}{i\choose
j}\int_{0}^{\infty}\frac{e^{(-1-j)t}}{t^{1-\beta}}\sum_{k=0}^{i-j}{i-j\choose
k}(-1)^{k}e^{-kt}\,dt.$

Now, using (3.1) we obtain (3.3). ∎

In the case $\beta=1/2$ this agrees with the formula in [28].

######  Remark 3.2.

It is clear from the functional calculus that every subspace for $\mathcal{C}$
is also an invariant subspace for $\mathcal{C}^{\beta}$ for $\mathop{\rm
Re}\nolimits\beta>0$. Since invariant subspaces for $\mathcal{C}^{1/n}$ are
clearly invariant subspaces for $\mathcal{C}$ for $n=1,2,\ldots$, we may
conclude that $\mathcal{C}$ and $\mathcal{C}^{1/n}$ have the same lattice of
invariant subspaces.

## Acknowledgements

The authors are grateful to a referee for carefully reading the manuscript and
providing some extremely helpful comments which improved its readability.

## References

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