11institutetext: 🖂 Jeevan Manavalan 22institutetext:
<EMAIL_ADDRESS>33institutetext:
1 King’s College London, Bush House, Strand Campus, 30, Aldwych, London WC2B
4BG - United Kingdom

2 University of Edinburgh, 1.17. Bayes Centre,47 Potterrow, Edinburgh, EH8 9BT
- United Kingdom

# A Library for Constraint Consistent Learning

Yuchen Zhao1 Jeevan Manavalan1 Prabhakar Ray1 Hsiu-Chin Lin2 Matthew Howard1

###### Abstract

This paper introduces the first, open source software library for Constraint
Consistent Learning (CCL). It implements a family of data-driven methods that
are capable of  (i) learning state-independent and -dependent constraints,
(ii) decomposing the behaviour of redundant systems into task- and null-space
parts, and  (iii) uncovering the underlying null space control policy . It is
a tool to analyse and decompose many everyday tasks, such as wiping, reaching
and drawing. The library also includes several tutorials that demonstrate its
use with both simulated and real world data in a systematic way. This paper
documents the implementation of the library, tutorials and associated helper
methods. The software is made freely available to the community, to enable
code reuse and allow users to gain in-depth experience in statistical learning
in this area.

###### Keywords:

Software Library Constraints Learning from Demonstration Model Learning

##  1 Introduction

Constraint Consistent Learning (Constraint Consistent LearningCCLConstraint
Consistent Learning) is a family of methods for learning different parts of
the equations of motion of redundant and constrained systems in a data-driven
fashion Howard2009b ; Towell2010 ; Lin2015 . It is able to learn
representations of self-imposed or environmental constraints Lin2015 ;
LinHumanoid ; Armesto2017 , decompose the movement of redundant systems into
task- and null space parts LinTaskConstraint ; Towell2010 , and uncover the
underlying null space control policy Howard2009 ; Howard2008a . Constraint
Consistent Learning enables:

  1. 1.

Learning the constraints encountered in many everyday tasks through various
representations.

  2. 2.

Learning the underlying control behaviour from movements observed under
different constraints.

In contrast to many standard learning approaches that incorporate _the whole
of the observed motions into a single control policy estimate_
schaal2003computational , Constraint Consistent Learning separates the problem
into learning  (i) the constraint representations and  (ii) the underlying
control policy . This provides more flexibility to the robot in the
reproduction of the behaviour, for example, in face of demonstration data of
one behaviour recorded under different constraints, Constraint Consistent
Learning can learn a single policy that generalises across the constraints
Howard2008a .

In terms of 1), the type of constraints may fall into the categories of _state
independent_ or _state dependent_ constraints. For example, for state-space
represented as end-effector coordinates, when wiping a table (see Figure
1(b)), the flat surface acts as a hard restriction on the actions available
(motions perpendicular to the surface will be eliminated by the constraint),
regardless of where the effector is located on the surface, so represents a
state independent constraint. When wiping or stirring soup in a bowl (see
Figure 1(c)), the curved surface introduces a state dependency in the
constraint, since the restriction of motion is dependent on the location of
the effector. The ability to predict how the constraints can influence the
outcomes of control actions can speed-up learning of the new skills and
enhance safety (e.g., the learned constraints can prevent exploration actions
that cause excessive force). Furthermore, the availability of constraint
knowledge can reduce the dimension of the search space when optimising
behaviours Bitzer2010 .

Figure 1: Typical examples in CCL.  (a) Three different tasks: moving the
finger to an $x,y,z$, position, liquid pouring and wiping a surface. In each
case, redundancy is resolved in the same way.  (b) A table wiping task is
subject to a state-independent flat environment constraint using a robot arm.
(c) A bowl wiping task subject to a state-dependent curvature constraint.

In terms of 2), the observable movements may contain partial information about
the control policy masked by the constraint, or higher priory task objectives
Towell2010 . For example, in a reaching task (see Figure 1(a)), humans move
their arms towards the target (primary task) while minimising effort, for
instance, by keeping the elbow low (secondary objective). The imperative of
executing the primary task represents a _self-imposed constraint_ , that
restricts the secondary objective. Interaction with _environmental
constraints_ , can also mask the actions applied by the demonstrator. For
example, when grasping an object on a table such as a pen, the fingers slide
along the table surface resulting in a discrepancy between the observed motion
and that predicted by the applied actions were the surface different in shape
or orientation. Constraint Consistent Learning can help uncover these
components, enabling the intended actions to be reconstructed and applied to
new situations LinEtAl2018 .

Figure 2: A system diagram of the Constraint Consistent Learning library. The
complete dependency tree can be found in the package documentation. The
library has been written under three categories: learn A, learn null-space
component and learn policy. The arrows links the dependencies of the
implementations to the main functions.

This paper introduces the _Constraint Consistent Learning library_ as a
community resource for learning in the face of kinematic constraints,
alongside tutorials to guide users who are not familiar with learning
constraint representations and underlying policies. As its key feature,
Constraint Consistent Learning is capable of decomposing many everyday tasks
(e.g., wiping Lin2015 , reaching Towell2010  and gait Lin2014a ) and learn
subcomponents (e.g., null space components Towell2010 , constraints Lin2015
and unconstrained control policy Howard2008a ) individually. The methods
implemented in the library provide means by which these quantities can be
extracted from the data, based on a collection of algorithms developed since
Constraint Consistent Learning was first conceived in 2007 Howard2007 . It
presents the _first unified _Application Programmers Interface (Application
Programmers InterfaceAPIApplication Programmers Interface)_ implementing and
enabling use of these algorithms in a common framework_.

##  2 Application Domain

The Constraint Consistent Learning library addresses the problem of learning
from motion subject to various constraints for redundant systems. The
behaviour of constrained and redundant systems is well-understood from an
analytical perspective Udwadia1996 , and a number of closely related
approaches have been developed for managing redundancy at the levels of
dynamics Peters2008a , kinematics Liegeois1977 ; Whitney1969 , and actuation
Howard2011 ; Tahara2009 . However, a common factor in these approaches is the
assumption of prior knowledge of the constraints, the control policies, or
both. Constraint Consistent Learning addresses the case where there are
uncertainties in these quantities and provides methods for their estimation.

###  2.1 Constraint Formalism

The Constraint Consistent Learning library assumes training data to come from
systems subject to the following formalism.

The system is considered to be subject to a set of $\mathcal{{S}}$-dimensional
($\mathcal{{S}}\leq\mathcal{{Q}}$) constraints with the general form

$\mathbf{A}(\mathbf{x})\mathbf{u}(\mathbf{x})=\mathbf{b}(\mathbf{x})$ (1)

where $\mathbf{x}\in\mathbb{R}^{\mathcal{{P}}}$ represents state and
$\mathbf{u}\in\mathbb{R}^{\mathcal{{Q}}}$ represents the action.
$\mathbf{A}(\mathbf{x})\in\mathbb{R}^{\mathcal{{S}}\times\mathcal{{Q}}}$ is
the constraint matrix which projects the task space policy onto the relevant
part of the control space. The vector term
$\mathbf{b}\in\mathbb{R}^{\mathcal{{S}}}$, if present, represents constraint-
imposed motion (in redundant systems, it is the policy in task space).

Inverting (1), results in the relation

$\mathbf{u}(\mathbf{x})=\underbrace{{\mathbf{A}(\mathbf{x})}^{\dagger}\mathbf{b}(\mathbf{x})}_{\mathbf{v}}+\underbrace{\mathbf{N}(\mathbf{x})\boldsymbol{\pi}(\mathbf{x})}_{\mathbf{w}}$
(2)

where ${\mathbf{A}}^{\dagger}$ denotes the unique Moore-Penrose pseudo-inverse
of the matrix $\mathbf{A}$,
$\mathbf{N}(\mathbf{x})=(\mathbf{I}-{\mathbf{A}(\mathbf{x})}^{\dagger}\mathbf{A}(\mathbf{x}))\in\mathbb{R}^{\mathcal{{Q}}\times\mathcal{{Q}}}$
is a projection matrix that projects the policy $\boldsymbol{\pi}(\mathbf{x})$
onto the null space of the constraint. Note that, the constraint can be state
dependent ($\mathbf{A}(\mathbf{x})$) or state independent ($\mathbf{A}$).
$\mathbf{v}(\mathbf{x})\equiv{\mathbf{A}(\mathbf{x})}^{\dagger}\mathbf{b}(\mathbf{x})$
and
$\mathbf{w}(\mathbf{x})\equiv\mathbf{N}(\mathbf{x})\boldsymbol{\pi}(\mathbf{x})$
are termed the task-space and the null-space component, respectively.
Typically, it is assumed that the only directly observable quantities are the
state-action pairs $(\mathbf{x}_{n},\mathbf{u}_{n})$ $n\in
1,\cdots,\mathcal{{N}}$ which contain an unknown combination of
$\mathbf{v}(\mathbf{x})$ and $\mathbf{w}(\mathbf{x})$, or at greater
granularity $\mathbf{A}(\mathbf{x})$, $\mathbf{b}(\mathbf{x})$,
$\mathbf{N}(\mathbf{x})$ and $\boldsymbol{\pi}(\mathbf{x})$. The Constraint
Consistent Learning library provides methods for estimating each of these
quantities—implementation details are provided in §3.

Note that, application of learning approaches klanke2008library  that do not
consider the composition of the data in terms of the constraints are prone to
poor performance and modelling errors. For example, applying direct regression
to learn policies where there are variations in the constraints can result in
model averaging effects that risk unstable behaviour Howard2009b . This is due
to factors such as  (i) the non-convexity of observations under different
constraints, and  (ii) degeneracyin the set of possible policies that could
have produced the movement under the constraint . Providing an open-source
collection of software tools suitable for application to this class of
learning problems, can help those working in the field to avoid these
pitfalls.

##  3 Implementation

A system diagram of the Constraint Consistent Learning library is shown in
Figure 2. The naming convention is following ccl_xxx_xxx indicating the
category and functionality of the implementations. The following provides
brief notes on the implementation languages and library installation, explains
the data structures used in toolbox.

###  3.1 Language and Installation Notes

The library is implemented in both C and Matlab and is available for download
from Github 111www.github.com/mhoward3210/ccl. The Constraint Consistent
Learning library is provided under the GNU General Public License v3.0, and
documentation is provided online222nms.kcl.ac.uk/rll/CCL_doc/index.html. To
use the Matlab package, the user can just add the Constraint Consistent
Learning library into the current path. Installation of the C package has been
tested on Ubuntu 14.04 Linux systems and is made easy through use of the
autoconf utility. For Windows systems, only the Matlab package is currently
tested and supported. The C package’s only dependency is on the 3rd-party,
open-source GNU Scientific Library (GNU Scientific LibraryGSLGNU Scientific
Library)333www.gnu.org/software/gsl/.

A detailed explanation of the functions follows, alongside tutorial examples
that are provided to enable the user to easily extend use of the library to
their chosen application (see §6).

###  3.2 Data Structures

The methods implemented in the toolbox work on data that is given as tuples
$\\{\mathbf{x}_{n},\mathbf{u}_{n}\\}_{n=1}^{\mathcal{{N}}}$ of observed states
and constrained actions. It is assumed that all the actions $\mathbf{u}$ are
generated using the same underlying policy $\boldsymbol{\pi}(\mathbf{x})$. In
particular444For brevity, here, and throughout the paper, the notation
$\mathbf{a}_{n}$ is used to denote the $n$th sample of the (matrix or vector)
quantity a. Where that quantity is a function of $\mathbf{x}$, the notation
$\mathbf{a}_{n}$ denotes the quantity calculated on the $n$th sample of
$\mathbf{x}$, i.e., $\mathbf{a}_{n}=\mathbf{a}(\mathbf{x}_{n})$.
$\mathbf{u}_{n}={\mathbf{A}_{n}}^{\dagger}\mathbf{b}_{n}+\mathbf{N}_{n}\boldsymbol{\pi}_{n}$,
where $\mathbf{A}_{n}$, $\mathbf{b}_{n}$ and $\boldsymbol{\pi}_{n}$ are not
explicitly known. The observations are assumed to be grouped into
$\mathcal{K}$ subsets of $\mathcal{{N}}$ data points, each (potentially)
recorded under different (task or environmental) constraints.

The Constraint Consistent Learning library uniformly stores the data tuples in
a data structure reflecting this problem structure. The latter includes the
data fields incorporating samples of the input state ($\mathbf{x}$), action
($\mathbf{u}$), actions from unconstrained policy ($\boldsymbol{\pi}$), task
space component ($\mathbf{v}$) and null space component ($\mathbf{w}$). The
major data structure and data fields are listed in Table 1.

Table 1: Main data structures of the Constraint Consistent Learning library.
dim* indicates the data fields for definition of the data dimensionality.

Struct | Main Fields | Definition | Other Fields
---|---|---|---
data | X | $\mathbf{X}=(\mathbf{x}_{1},\cdots,\mathbf{x}_{\mathcal{{N}}})\in\mathbb{R}^{\mathcal{{P}}\times\mathcal{{N}}}$ |
| U | ${\mathbf{U}}=(\mathbf{u}_{1},\cdots,\mathbf{u}_{\mathcal{{N}}})\in\mathbb{R}^{\mathcal{{Q}}\times\mathcal{{N}}}$ |
| Pi | $\mathbf{\Pi}=(\boldsymbol{\pi}_{1},\cdots,\boldsymbol{\pi}_{\mathcal{{N}}})\in\mathbb{R}^{\mathcal{{Q}}\times\mathcal{{N}}}$ |
| Ts | ${\mathbf{V}}=(\mathbf{v}_{1},\cdots,\mathbf{v}_{\mathcal{{N}}})\in\mathbb{R}^{\mathcal{{Q}}\times\mathcal{{N}}}$ |
| Un | ${\mathbf{W}}=(\mathbf{w}_{1},\cdots,\mathbf{w}_{\mathcal{{N}}})\in\mathbb{R}^{\mathcal{{Q}}\times\mathcal{{N}}}$ |
model | M | Nullspace component Model weighting parameters $\mathbf{\omega}$. |
| c | $\mathbf{\mu}$ |
| s | $\mathbf{\Sigma}$ |
| phi | $\boldsymbol{\beta}$ |
|  |  | dim*
search | theta | $(\boldsymbol{\theta}_{1},\cdots,\boldsymbol{\theta}_{\mathcal{{N_{S}}}})$ |
|  |  | dim*
optimal | nmse | Normalised mean square error. |
| f_proj | Function handle for projection matrix. |
|  |  | dim*
options | TolFun | Tolerance for residual. |
| TolX | Tolerance for model parameters. |
| MaxIter | Maximum iterations for optimiser. |
|  |  | dim*
stats | nmse | Normalised mean square error. |
| mse | Mean squared error. |
| var | Variance. |

Learning is more effective when the data contains sufficiently rich variations
in one or more of the quantities defined in (2), since methods learn the
consistency by teasing out the variations. For instance, when learning
$\mathbf{v}$, variations in $\mathbf{w}$ are desirable Towell2010 . For
learning $\boldsymbol{\pi}$, observations subject to multiple constraints
(variation in $\mathbf{A}$) are necessary Howard2009b . For learning
constraint $\mathbf{A}$, variations in $\boldsymbol{\pi}$ are desirable
LinTaskConstraint .

##  4 Learning Functions

The Application Programmers Interface of the Constraint Consistent Learning
library provides functions for a series of algorithms for the estimation of
the quantities $\mathbf{A}$, $\mathbf{v}$, $\mathbf{w}$ and
$\boldsymbol{\pi}$. The following provides a summary of the methods provided,
including implementation notes and a brief summary555Due to space constraints,
the reader is referred to the original research literature for in-depth
details of the theoretical aspects. of the theoretical basis for each
algorithm.

###  4.1 Learning the Constraint Matrix

The library implements several methods for estimating the constraint matrix
$\mathbf{A}$. These assume that  (i) each observation contains a constraint
$\mathbf{A}\neq\boldsymbol{0}$,  (ii) the observed actions take the form
$\mathbf{u}=\mathbf{w}=\mathbf{N}\boldsymbol{\pi}$,  (iii) $\mathbf{u}$are
generated using the same null space policy $\boldsymbol{\pi}$, and  (iv)
$\mathbf{A}$is not explicitly known for any observation  (optionally, features
in form of candidate rows of $\mathbf{A}$, may be provided as prior
knowledge–see §4.1.2).

The methods for learning $\mathbf{A}$ under these assumptions, are based the
insight that the projection of $\mathbf{w}$ also lies in the same image space
Lin2015 , i.e.,

$(\mathbf{I}-{\mathbf{A}}^{\dagger}\mathbf{A})\mathbf{w}=\mathbf{N}\mathbf{w}=\mathbf{w}.$
(3)

If the estimated constraint matrix $\tilde{\mathbf{A}}$ is accurate, it should
also obey this equality. Hence, the estimate can be formed by minimising the
difference between the left and right hand sides of (3), as captured by the
error function

$E[\tilde{\mathbf{N}}]=\sum^{\mathcal{{N}}}_{n=1}||\mathbf{w}_{n}-\tilde{\mathbf{N}}_{n}\mathbf{w}_{n}||^{2}$
(4)

where
$\tilde{\mathbf{N}}_{n}=\mathbf{I}-{\tilde{\mathbf{A}}_{n}}^{\dagger}\tilde{\mathbf{A}}_{n}$.
This can be expressed in simplified form

$E[\tilde{\mathbf{A}}]=\sum^{\mathcal{{N}}}_{n=1}||\mathbf{w}_{n}^{\top}{\tilde{\mathbf{A}}_{n}}^{\dagger}\tilde{\mathbf{A}}_{n}\mathbf{w}_{n}||^{2}.$
(5)

Note that, in forming the estimate, _no prior knowledge of $\boldsymbol{\pi}$
is required_. However, an appropriate representation of the constraint matrix
is needed.666Note: While the pseudoinverse operation has conditioning problems
when close to a rank-deficient configuration, the constraint learning methods
deal with this during the inversion by ignoring all singular values below a
certain threshold. As outlined in §1, in the context of learning $\mathbf{A}$,
there are two distinct cases depending on whether the constraints are state-
dependent or not. The Constraint Consistent Learning library implements
methods for both scenarios.

####  4.1.1 Learning State Independent Constraints

The simplest situation occurs if the constraint is unvarying throughout the
state space. In this case, the constraint matrix can be represented as the
concatenation of a set of unit vectors

$\tilde{\mathbf{A}}=(\hat{\mathbf{a}}_{1}^{\top},\hat{\mathbf{a}}_{2}^{\top},\cdots,\hat{\mathbf{a}}_{\mathcal{{S}}}^{\top})^{\top}$
(6)

where the $s$th unit vector $\hat{\mathbf{a}}_{s}$ is constructed from the
parameter vector $\boldsymbol{\theta}\in\mathbb{R}^{\mathcal{{Q}}-1}$ that
represents the orientation of the constraint in action space (see Lin2015  for
details).

Figure 3:  Three one-dimensional unit vector constraints $\hat{\mathbf{a}}$
where  (a) $\theta=0^{\circ}$,  (b) $\theta=45^{\circ}$, and  (c)
$\theta=90^{\circ}$ (reproduced from Lin2015 ).

A visualisation of the simplest case of a one-dimensional constraint in a
system with two degrees of freedom where
$\mathbf{A}=\hat{\mathbf{a}}\in\mathbb{R}^{1\times 2}$ is shown in Figure 3.
It can be seen how $\boldsymbol{\theta}$ (in this case,
$\hat{\mathbf{a}}=(\cos{\theta},\sin{\theta})$) corresponds to different
estimates of the constraint its associated null-space.

The Constraint Consistent Learning library provides the function  optimal
=ccl_learna_nhat(Un )

to form the estimate $\tilde{\mathbf{A}}$ in this scenario, where Un is the
collated set of samples of $\mathbf{w}$ (see Table 1) and optimal is the
learnt model. The C counterpart777The C library implements all learning
functions, but for the remainder of the paper, Matlab functions will be used
throughout for consistency. of this function is  ⬇ void learn_nhat (const
double *Un,const int dim_u, const int dim_n,NHAT_Model *optimal) where the
input argument Un and output argument optimal in the C library is identical
with Matlab implementation. The arguments dim_u and dim_n are used for
defining the dimensionality of the array Un.

The function works by searching for rows of the estimated constraint matrix
$\tilde{\mathbf{A}}$ using the function  ⬇ [model, stats] = ccl_learna_sfa
(UnUn,Un,model,search)

the arguments model and search contain the initial model (untrained) model,
parameters controlling the search, respectively and UnUn is the matrix
${\mathbf{W}}^{\top}{\mathbf{W}}$ where ${\mathbf{W}}$ is a matrix containing
samples of $\mathbf{w}$. This is pre-calculated prior to the function call for
speed. The function returns the learnt model model and learning statistics
stats (see Table 1).

####  4.1.2 Learning State Dependent Constraint

Another proposed method for learning a state-dependent constraint, i.e., the
constraint $\mathbf{A}(\mathbf{x})$ depends on the current state of the robot
$\mathbf{x}$. For learning $\mathbf{A}(\mathbf{x})$, two scenarios may be
encountered where  (i) there is prior knowledge of the constraint in form of
candidate rows (see LinTaskConstraint ), or  (ii) no prior knowledge is
available .

In the case of (i),

$\mathbf{A}(\mathbf{x})=\tilde{\boldsymbol{\Lambda}}(\mathbf{x})\boldsymbol{\Phi}(\mathbf{x}).$
(7)

$\boldsymbol{\Phi}(\mathbf{x})\in\mathbb{R}^{R\times\mathcal{{Q}}}$ is the
feature matrix and
$\tilde{\boldsymbol{\Lambda}}(\mathbf{x})\in\mathbb{R}^{\mathcal{{S}}\times
R}$ is the selection matrix specifying which rows represent constraints. The
feature matrix can take the form
$\boldsymbol{\Phi}(\mathbf{x})=\mathbf{J}(\mathbf{x})$, where $\mathbf{J}$ is
the Jacobian mapping from the joint space to the end-effector task space.
Similar to (6), $\tilde{\boldsymbol{\Lambda}}(\mathbf{x})$ is described by a
set of $\mathcal{{S}}$ orthonormal vectors

$\tilde{\boldsymbol{\Lambda}}(\mathbf{x})=(\boldsymbol{\lambda}_{1}(\boldsymbol{\theta})^{\top},\boldsymbol{\lambda}_{2}(\boldsymbol{\theta})^{\top},\cdots,\boldsymbol{\lambda}_{\mathcal{{S}}}(\boldsymbol{\theta})^{\top})^{\top}$
(8)

where $\boldsymbol{\lambda}_{s}\in\mathbb{R}^{\mathcal{{S}}}$ is the $s$th
dimension in the task space and
$\boldsymbol{\lambda}_{i}\perp\boldsymbol{\lambda}_{j}$ for $i\neq j$.
Parameter $\boldsymbol{\theta}_{s}\in\mathbb{R}^{\mathcal{{S}}-1}$ is used for
representing the constraint vector $\boldsymbol{\lambda}$. Each
$\boldsymbol{\theta}_{s}$ is modelled as
$\boldsymbol{\theta}_{s}=\mathbf{\omega}_{s}\boldsymbol{\beta}(\mathbf{x})$
where $\mathbf{\omega}_{s}\in\mathbb{R}^{(\mathcal{{Q}}-1)\times G}$ is the
weight matrix, and $\boldsymbol{\beta}(\mathbf{x})\in\mathbb{R}^{G}$ is the
vector of $G$ basis functions that transform $\mathbf{x}$ into a set of
feature vectors in higher dimensional space. Substituting
$\tilde{\mathbf{A}}_{n}=\tilde{\boldsymbol{\Lambda}}_{n}\boldsymbol{\Phi}_{n}$,
(5) can be written as

$E[\tilde{\boldsymbol{\Lambda}}]=\sum^{\mathcal{{N}}}_{n=1}||\mathbf{u}_{n}^{\top}{(\tilde{\boldsymbol{\Lambda}}_{n}\boldsymbol{\Phi}_{n})}^{\dagger}\tilde{\boldsymbol{\Lambda}}_{n}\boldsymbol{\Phi}_{n}\mathbf{u}_{n}||^{2}.$
(9)

The optimal $\tilde{\boldsymbol{\Lambda}}$ can be formed by iteratively
searching the choice of $\boldsymbol{\lambda}_{s}$ that minimises (9).

The objective function for (5) and ((9)) are embedded in function

⬇

[fun] = ccl_obj_AVn (model,W,BX,RnUnUnn)

where fun is the objective function handle, M is the model parameter
$\mathbf{\omega}$, BX is the feature vectors in high dimension space
$\boldsymbol{\beta}(\mathbf{x})$, and RnUnUnn is the pre-rotated $UnUn_{n}$.

The implementation of case (i) is:

⬇

[optimal] =

ccl_learn_lambda (Un,X,Phi,options)

Where Phi is the feature matrix $\boldsymbol{\Phi}$. The implementation of
case (ii) where

$\mathbf{A}(\mathbf{x})=(\mathbf{a}_{1}(\boldsymbol{\theta})^{\top},\mathbf{a}_{2}(\boldsymbol{\theta})^{\top},\cdots,\mathbf{a}_{\mathcal{{S}}}(\boldsymbol{\theta})^{\top})^{\top}$
(10)

is:

⬇

[optimal] =

ccl_learn_alpha (Un,X,options)

The non-linear parameter optimisation solver used for learning
$\boldsymbol{\theta}_{s}$ calculates the direction in which the parameter is
improved for this, the Levenberg-Marquardt (LM) solver is used. The latter is
embedded in the function

⬇

[xf,S,msg] = ccl_math_solve_lm (varargin)

where the inputs varargin are the objective function (fun), initial guess of
model parameters (xc, i.e., $\mathbf{\omega}$) and training options (options).

###  4.2 Learning the Task and Null Space Components

When the unconstrained control policy is subject to both constraint and some
task space policy ($\mathbf{b}(\mathbf{x})\neq\boldsymbol{0}$), it is often
useful to extract the task and null space components ($\mathbf{v}(\mathbf{x})$
and $\mathbf{w}(\mathbf{x})$, respectively) of the observed actions
$\mathbf{u}(\mathbf{x})$. The Constraint Consistent Learning library provides
the following methods to estimate these quantities. These assume that the
underlying null-space policy and the task constraint are consistent.

Assuming that the underlying null-space policy $\boldsymbol{\pi}(\mathbf{x})$
and the task constraint $\mathbf{A}(\mathbf{x})$ are consistent across the
data set, the null space component $\mathbf{w}(\mathbf{x})$ should satisfy the
condition
$\mathbf{N}(\mathbf{x})\mathbf{u}(\mathbf{x})=\mathbf{w}(\mathbf{x})$ as noted
in Towell2010 , in this case, $\mathbf{w}(\mathbf{x})$ by minimising

$E[\tilde{\mathbf{w}}]=\sum^{\mathcal{{N}}}_{n=1}||\tilde{\mathbf{P}}_{n}\mathbf{u}_{n}-\tilde{\mathbf{w}}(\mathbf{x}_{n})||^{2}$
(11)

with
$\tilde{\mathbf{P}}_{n}=\tilde{\mathbf{w}}_{n}\tilde{\mathbf{w}}_{n}^{\top}/||\tilde{\mathbf{w}}_{n}||^{2}$,
where $\tilde{\mathbf{w}}_{n}=\tilde{\mathbf{w}}(\mathbf{x}_{n})$ projects an
arbitrary vector onto the same space of $\mathbf{w}$, and $\mathbf{u}_{n}$ is
the $n$th data point. This error function eliminates the task-space components
$\mathbf{v}(\mathbf{x})$ by penalising models that are inconsistent with the
constraints, i.e., those where the difference between the model,
$\tilde{\mathbf{w}}(\mathbf{x})$, and the observations projected onto that
model is large (for details, see Towell2010 ). In the Constraint Consistent
Learning library, this functionality is implemented in the function

⬇

[fun,J] = ccl_obj_ncl (model,W,BX,U)

where U is the observed actions ${\mathbf{U}}$. The outputs are fun (function
handle) and J (analytical Jacobian of the objective function (11)).

Minimisation of (11) penalises models that are inconsistent with the
constraints, i.e., those where the difference between the model,
$\tilde{\mathbf{w}}_{k}(\mathbf{x})$, and the observations projected onto that
model is large (for details, see Towell2010 ). In the Constraint Consistent
Learning library, this functionality is implemented in the function

⬇

model = ccl_learnv_ncl(X,U,model)

where the inputs are X (input state), Y (observed actions combined with task
and null space components) and model (model parameters). The output is the
learnt model model.

###  4.3 Learning the Unconstrained Control Policy

The Constraint Consistent Learning library also implements methods for
estimating the underlying _unconstrained control policy_
($\boldsymbol{\pi}(\mathbf{x})$). These assume either (i) $\mathbf{b}=0$ no
additional task is involved or (ii) $\mathbf{w}$ is learnt using the method
proposed in §4.2.

As shown in Howard2008a , an estimate $\tilde{\boldsymbol{\pi}}(\mathbf{x})$
can be obtained by minimising the _inconsistency error_

$E_{\pi}=\sum^{\mathcal{{N}}}_{n=1}||\mathbf{u}_{n}-\mathbf{P}_{n}\tilde{\boldsymbol{\pi}}(\mathbf{x}_{n})||^{2};\quad\mathbf{P}_{n}=\frac{\mathbf{u}_{n}\mathbf{u}_{n}^{\top}}{||\mathbf{u}_{n}||^{2}}.$
(12)

The risk function (12) is compatible with many regression models. The
Constraint Consistent Learning library implements a parametric policy learning
scheme. Where
$\tilde{\boldsymbol{\pi}}(\mathbf{x})={\mathbf{W}}\boldsymbol{\beta}(\mathbf{x})$,
${\mathbf{W}}\in\mathbb{R}^{\mathcal{{Q}}\times M}$ is a matrix of weights and
$\boldsymbol{\beta}(\mathbf{x})\in\mathbb{R}^{M}$ is a vector of fixed basis
functions (e.g., linear features or Gaussian radial basis functions). A
locally-weighted linear policy learning is also implemented in the library to
improve the robustness of the policy learning (Details see Towell2010 ). A
reinforcement learning scheme (with possibly deep network structure) can also
be used to learn more complex policies Howard2013 ; Howard2013b ; Gu2016 .

The Constraint Consistent Learning library implements the learning of these
models through the functions

⬇

model = ccl_learnp_pi (X,U,model)

where inputs X and U are observed states and actions. The outputs are the
learnt model model.

##  5 Evaluation Criteria

For testing different aspects of the learning performance, a number of
evaluation criteria have been defined in the Constraint Consistent Learning
literature. These include metrics comparing estimation against the ground
truth (if known), as well as those that provide a pragmatic estimate of
performance based only on observable quantities. In the context of evaluating
the constraints, the Constraint Consistent Learning library provides
implementations of the following functions.

If the ground truth constraint matrix $\mathbf{A}_{n}$ and unconstrained
policy $\boldsymbol{\pi}_{n}$ are known, then the normalised projected policy
error (Normalised Constrained Policy Error) provides the best estimate of
learning performance Lin2015 . In the Constraint Consistent Learning library,
The Normalised Constrained Policy Error is computed through the functions

⬇

[nPPE,vPPE,uPPE] =

ccl_error_ppe (U_t,N_p,Pi)

where U_t is the true null space components ${\mathbf{W}}$, N_p is the learned
projection matrix $\mathbf{N}$, and Pi is the unconstrained control policy
$\boldsymbol{\pi}$. The outputs are the Normalised Constrained Policy Error,
variance and (non-normalised) mean-squared PPE, respectively.

In the absence of ground truth $\mathbf{A}_{n}$ and $\boldsymbol{\pi}_{n}$,
the normalised projected observation error (Normalised Constrained Policy
Error) must to be used Lin2015 . The functions implemented for computing the
Normalised Constrained Policy Error are

⬇

[nPOE,vPOE,uPOE] =

ccl_error_poe (U_t,N_p,Pi)

where the inputs and outputs conventions are similar to those of the functions
computing the Normalised Constrained Policy Error.

To evaluate the predictions of the null-space component model
$\tilde{\mathbf{w}}_{k}(\mathbf{x})$, the null space projection error (NPE)
can be used Towell2010 . This is implemented in the function

⬇

[umse,v,nmse] = ccl_error_npe (Un,Unp)

where Un and Unp are the true and predicted null space component. The return
values are the NPE, variance and (non-normalised) mean squared projection
error. Note that, use of this function assumes knowledge of the ground truth
$\mathbf{w}_{n}$.

Table 2: Key Functions in the CCL Library

Function Type | Name | Description
---|---|---
Learn Constraint | Learn Alpha | Learning state dependent projection N (9) + (8)
Learn Lambda | Learning state dependent selection matrix $\tilde{\boldsymbol{\Lambda}}$ (9) + (10)
Learn Nhat | Learning state independent constraint N (5)
Learn Nullspace Component | Learn ncl | Learn the nullspace component of u = v+w (11)
Learn unconstrained control Policy | Learn CCL Pi | Learn null space policy (12)
Evaluate Constraint error | NPPE |  | normalised projected policy error
---
NPOE |  | normalised projected observation error
---
Evaluate Nullspace component error | NPE | nullspace projection error
Evaluate unconstrained control policy | NUPE | normalised unconstrained policy error
NCPE | normalised constrained policy error

To evaluate the estimated unconstrained control policy model
$\tilde{\boldsymbol{\pi}}(\mathbf{x})$, the normalised unconstrained policy
error (NUPE) and normalised constrained policy error (NCPE) Howard2008a  are
used. The former assumes access to the ground truth unconstrained policy
$\boldsymbol{\pi}_{n}$, while the latter assumes the true $\mathbf{w}_{n}$ is
known. They are implemented in the functions

⬇

[umse,v,nmse] = ccl_error_nupe (F,Fp)

and

⬇

[umse,v,nmse] = ccl_error_ncpe(F,Fp,P)

where input F is the true unconstrained control policy commands, Fp is the
learned unconstrained control policy and P is the projection matrix. The
outputs are Normalised Unconstrained Policy Error (respectively, NCPE), the
sample variance, and the mean squared Unconstrained Policy Error
(respectively, Constrained Policy Error).

Table 2 shows an overview of all key functions as well as their respective
equations for learning and evaluation which are covered in this section.

##  6 Tutorials

The Constraint Consistent Learning library package provides a number of
tutorial scripts to illustrate its use. Multiple examples are provided which
demonstrate learning using both simulated and real world data, including,
learning in a simulated 2-link arm reaching task, and learning a wiping task.
The following describes the toy example in detail, alongside a brief
description of other included demonstrations which were tested with real world
systems. Further details of the latter are included in the documentation.

In the first, a simple, illustrative toy example is used, in which a two-
dimensional system is subject to a one-dimensional constraint. In this, the
learning of  (i) the null space components,  (ii) the constraints, and  (iii)
the null space policy is demonstrated.

The toy example code has been split into three main sections for learning
different parts of the equations. Detailed comments of the functions can be
found in the Matlab script. The procedure for generating data is dependant on
the part of the equation you wish to learn. Details can be found in the
documentations.

####  6.0.1 Learning Null Space Components $\mathbf{w}$

This section gives a guidance of how to use the library for learning
${\mathbf{W}}$. A sample Matlab snippet is shown in Table 3 and explained as
follows. Firstly, the user needs to generate training data given the
assumption that $\mathbf{w}$ is fixed but $\mathbf{v}$ is varying in your
demonstration. The unconstrained policy is a limit cycle policy LinHumanoid .
For learning, the centre of the Radial Basis Function s are chosen according
to K-means and the variance as the mean distance between the centres. A
parametric model with 16 Gaussian basis functions is used. Then the null-space
component model can be learnt through ccl_learnv_ncl. Finally, the evaluation
metrics NUPE and NPE are used to report the learning performance.

####  6.0.2 Learning Null Space Constraints $\mathbf{A}$

This section gives guidance of how to use the library to learn a state
independent $\mathbf{A}$ and state dependent $\mathbf{A}(\mathbf{x})$, where
$\mathbf{b}=0$. A sample Matlab script is provided in Table 4 and explained as
following. Firstly, it simulates a problem where the user faces is learning a
fixed constraint in which the systems null-space controller $\boldsymbol{\pi}$
varies. For learning a linear constraint problem, constant $\mathbf{A}$ is
used and ccl_learn_nhat is implemented. For learning a parabolic constraint, a
state-dependent constraint $\mathbf{A}(\mathbf{x})$ of the form
$\mathbf{A}(\mathbf{x})=[-2ax,1]$ is used. For this, ccl_learn_alpha is
implemented. Finally, nPPE and nPOE are used to evaluate the learning
performance.

Table 3: Sample code snippet for learning null-space component from Matlab

Data = ccl_data_gen(settings) ;
---
model.c = ccl_math_gc (X, model.dim_basis) ;
model.s = mean(mean(sqrt(ccl_math_distances(model.c, model.c))))^2 ;
model.phi = @(x)ccl_basis_rbf ( x, model.c, model.s);
model = ccl_learnv_ncl (X, U, model) ;
f_ncl = @(x) ccl_learnv_pred_ncl ( model, x ) ;
NSp = f_ncl (X) ;
NUPE = ccl_error_nupe(Un, Unp) ;
NPE = ccl_error_npe (Un, Unp) ;

Table 4: To do: Sample code snippet for learning state independent and
dependent constraint from matlab

Data = ccl_data_gen(settings) ;
---
model = ccl_learn_nhat (Un) ; % for learning $\mathbf{A}$
model = ccl_learn_alpha (Un, X, settings) ; % for learning
$\mathbf{A}(\mathbf{x})$
nPPE = ccl_error_ppe(U, model.P, Pi) ;
nPOE = ccl_error_poe(U, model.P, Pi) ;

Table 5: Sample code snippet for learning null-space control policy from
Matlab

Data = ccl_data_gen(settings);
---
model.c = ccl_math_gc (X, model.dim_basis) ;
model.s = mean(mean(sqrt(ccl_math_distances(model.c, model.c))))^2 ;
model.phi = @(x)ccl_basis_rbf ( x, model.c, model.s);
model = ccl_learnp_pi (X, Un, model) ;
fp = @(x) ccl_learnp_pred_linear ( model, x ) ;
Pi = fp (X) ;
NUPE = ccl_error_nupe(Un, Unp) ;
NCPE = ccl_error_ncpe (Un, Unp) ;

####  6.0.3 Learning Null Space Policy $\pi$

This section will give guidance on learning an unconstrained control policy
$\boldsymbol{\pi}$. This applies to the use case where $\boldsymbol{\pi}$ is
consistent but $\mathbf{A}$ is varying. A sample Matlab script is provided in
Table 5 and explained as the following. For learning, 10 Radial Basis Function
s are used and learned using the same K-means algorithm. ccl_learnp_pi is then
implemented for training the model. Finally, NUPE and NCPE are calculated to
evaluate the model’s performance. In the library, a locally weighted policy
learning method is also implemented in both Matlab and C. For details please
refer to the documentation.

Other examples such as a 2-link arm and wiping examples are also implemented
which follow a similar procedure to the toy example but in a higher dimension
to aid novice users wanting to take advantage of the Constraint Consistent
Learning to learn the kinematic redundancy of a robot. Moreover, users can
easily adapt these learning methods, where the provided demonstrations such as
wiping and real data are setup to use the 7 degrees of freedomKUKA LWR3 and
Trakstar 3D electromagnetic tracking sensor, respectively, are reconfigured to
suit their own systems and requirements.

##  7 Conclusion

This paper introduces the Constraint Consistent Learning library, an open-
source collection of software tools for learning different components of
constrained movements and behaviours. The implementations of the key functions
have been explained throughout the paper, and interchangeable and expandable
examples have been demonstrated using both simulated and real world data for
their usage. For the first time, the library brings together a diverse
collection of algorithms developed over the past ten years into a unified
framework with a common interface for software developers in the robotics
community. In the future work, Matlab and python wrappers will be released by
taking advantage of the fast computation routine implemented in C.

## References

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  * (2) C. Towell, M. Howard, and S. Vijayakumar, “Learning null space policies,” in _IEEE. C. Intel. & Rob. Sys._, 2010.
  * (3) H.-C. Lin, M. Howard, and S. Vijayakumar, “Learning null space projections,” in _ICRA_ , 2015.
  * (4) V. Ortenzi, H. C. Lin, M. Azad, R. Stolkin, J. A. Kuo, and M. Mistry, “Kinematics-based estimation of contact constraints using only proprioception,” in _Humanoids_ , Nov 2016, pp. 1304–1311.
  * (5) L. Armesto, J. Moura, V. Ivan, A. Sala, and S. Vijayakumar, “Learning Constrained Generalizable Policies by Demonstration,” in _Robotics: Sci. & Sys._, 2017.
  * (6) H.-C. Lin, P. Ray, and M. Howard, “Learning task constraints in operational space formulation,” in _ICRA_ , 1 2017.
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# Spectral representation of absolutely minimum attaining unbounded normal
operators

S. H. Kulkarni  Department of Mathematics,
IIT Palakkad, Ahalia Integrated Campus,
Kozhippara, Palakkad, Kerala, India 678557<EMAIL_ADDRESS>and  G. Ramesh
Department of Mathematics,
IIT Hyderabad,
Telangana, India 502 284<EMAIL_ADDRESS>
###### Abstract.

Let $T:D(T)\rightarrow H_{2}$ be a densely defined closed operator with domain
$D(T)\subset H_{1}$. We say $T$ to be absolutely minimum attaining if for
every closed subspace $M$ of $H_{1}$, the restriction operator
$T|_{M}:D(T)\cap M\rightarrow H_{2}$ attains its minimum modulus $m(T|_{M})$.
That is, there exists $x\in D(T)\cap M$ with $\|x\|=1$ and
$\|T(x)\|=\inf\\{\|T(m)\|:m\in D(T)\cap M:\|m\|=1\\}$. In this article, we
prove several characterizations of this class of operators and show that every
operator in this class has a nontrivial hyperinvariant subspace.

We also prove a spectral theorem for unbounded normal operators of this class.
It turns out that every such operator has a compact resolvent.

###### Key words and phrases:

closed operator, minimum attaining, absolutely minimum attaining operator,
invariant subspace, bounded transform, essential spectrum.

###### 1991 Mathematics Subject Classification:

Primary 47A75,47A05 ; Secondary 47A10,47A15

##  1\. Introduction

The class of absolutely minimum attaining unbounded operators was introduced
in [12] where its basic properties and structure were described under some
additional assumptions. On the other hand, similar studies in the case of
bounded absolutely minimum operators were carried out in [8, 2]. There is a
significant difference between the bounded absolutely minimum attaining
operators and the unbounded ones. The results in the present article improve
results from [12]. It is interesting to note that this class contains densely
defined closed operators with finite-dimensional null space and a compact
generalized inverse. The complete characterization, structure, spectral
properties and hyperinvariant subspaces are studied in [8, 2, 3]. A class
larger than the absolutely norm attaining is explored in [19, 20]. Hence in
this article we exclusively study the class of absolutely minimum attaining
operators in the unbounded setting. Since the methods of the bounded case do
not work in this case, we adopt different methods for proving our results.

The class of minimum attaining unbounded operators has been recently studied
in [12, 13]. It is proved that this class is dense in the class of densely
defined closed operators with respect to the gap metric. This result can be
compared with the Lindenstrauss theorem of norm attaining operators. Moreover,
a quantitative version of the Lindenstrauss theorem for minimum attaining
operators is discussed in [1].

In this article we prove that a densely defined closed operator is absolutely
minimum attaining if and only if it has a finite-dimensional null space and
the Moore-Penrose inverse (generalized inverse) of the operator is compact.
Using this characterization we study some properties of this class of
operators. In particular, we prove a spectral theorem for absolutely minimum
attaining unbounded normal operators. The Spectral Theorem for unbounded
normal operators is already available in the literature, for example, in [22].
Since the class considered by us is a subclass of this class, we need to
emphasize that the spectral theorem proved by us is of a different type and
should be compared with the spectral theorem for compact normal operators. In
the end, we show that every such operator has a non-trivial hyperinvariant
subspace.

In the second section we provide basic results which will be used throughout
the article. In the third section, we prove main results of this paper.

##  2\. Preliminaries

In this section we give details of basic notations, definitions and results
which we need to prove our main results.

Throughout we work with infinite-dimensional complex Hilbert spaces denoted by
$H,H_{1},H_{2}$ etc. The inner product and the induced norm on these spaces
are denoted by $\langle\cdot\rangle$ and $\|.\|$, respectively. If $T$ is a
linear operator with domain $D(T)$, a subspace of $H_{1}$ and taking values in
$H_{2}$, then $T$ is said to be densely defined if $D(T)$ is dense in $H_{1}$.
The graph $G(T)$ of $T$ is defined by $G(T):={\\{(x,Tx):x\in D(T)}\\}\subseteq
H_{1}\oplus H_{2}$. We say $T$ to be a closed operator if $G(T)$ is closed.
Equivalently, we can say that $T$ is closed if and only if for every sequence
$(x_{n})$ in $D(T)$ such that $x_{n}\rightarrow x\in H_{1}$ and
$Tx_{n}\rightarrow y\in H_{2}$, then $x\in D(T)$ and $Tx=y$.

The closed graph Theorem [22] assert that an everywhere defined closed
operator is bounded. That is, the domain of an unbounded closed operator must
be a proper subspace of a Hilbert space.

The space of all bounded operators between $H_{1}$ and $H_{2}$ is denoted by
$\mathcal{B}(H_{1},H_{2})$ and the class of all closed operators between
$H_{1}$ and $H_{2}$ is denoted by $\mathcal{C}(H_{1},H_{2})$. We write
$\mathcal{B}(H)$ for $\mathcal{B}(H,H)$ and $\mathcal{C}(H)$ for
$\mathcal{C}(H,H)$. We denote the space of all compact operators between
$H_{1}$ and $H_{2}$ by $\mathcal{K}(H_{1},H_{2})$ and $\mathcal{K}(H,H)$ by
$\mathcal{K}(H)$. Let us denote by $\mathcal{F}(H_{1},H_{2})$, the space of
all finite rank bounded operators from $H_{1}$ into $H_{2}$ and by
$\mathcal{F}(H)$, the space of all finite rank bounded operators on $H$.

The unit sphere of a subspace $M$ of $H$ is defined by $S_{M}:={\\{x\in
M:\|x\|=1}\\}$. If $M$ is closed, then the othogonal projection of $H$ onto
$M$ is denoted by $P_{M}$.

The null space and the range space of $T\in\mathcal{C}(H_{1},H_{2})$ are
denoted by $N(T)$ and $R(T)$ respectively and the space $C(T):=D(T)\cap
N(T)^{\bot}$ is called the carrier of $T$. In fact,
$D(T)=N(T)\oplus^{\bot}C(T)$ [4, page 340].

If $S$ and $T$ are closed operators with the property that $D(T)\subseteq
D(S)$ and $Tx=Sx$ for all $x\in D(T)$, then $T$ is called the restriction of
$S$ denoted by $T\subseteq S$, and $S$ is called an extension of $T$, which is
denoted by $T\subseteq S$. Furthermore, $S=T$ if and only if $S\subseteq T$
and $T\subseteq S$.

If $T$ is a densely defined operator, then there exists a unique linear
operator (in fact, a closed operator) $T^{*}:D(T^{*})\rightarrow H_{1}$, with

$D(T^{*}):={\\{y\in H_{2}:x\rightarrow\langle Tx,y\rangle\,\text{for
all}\,x\in D(T)\,\text{is continuous}}\\}\subseteq H_{2}$

satisfying $\langle Tx,y\rangle=\langle x,T^{*}y\rangle$ for all $x\in D(T)$
and $y\in D(T^{*})$.

A densely defined operator $T\in\mathcal{C}(H)$ is said to be

  1. (1)

normal if $T^{*}T=TT^{*}$

  2. (2)

self-adjoint if $T=T^{*}$

  3. (3)

positive if $\langle Tx,x\rangle\geq 0$ for all $x\in D(T)$

  4. (4)

symmetric if $T\subseteq T^{*}$. In other words, $T$ is symmetric if $\langle
Tx,y\rangle=\langle x,Ty\rangle$ for every $x,y\in D(T)$.

If $T$ is positive, then there exists a unique positive operator $S$ such that
$T=S^{2}$. The operator $S$ is called the square root of $T$ and it is denoted
by $S=T^{\frac{1}{2}}$.

A bounded operator $V\in\mathcal{B}(H_{1},H_{2})$ is said to be a partial
isometry if $\|Vx\|_{2}=\|x\|_{1}$ for all $x\in N(V)^{\bot}$. In this case
$N(V)^{\bot}$ is called the initial space of $V$ and $R(V)$ is called the
final space of $V$.

If $T\in\mathcal{C}(H_{1},H_{2})$ is densely defined, then the operator
$|T|:=(T^{*}T)^{\frac{1}{2}}$ is called the modulus of $T$. There exists a
unique partial isometry $V:H_{1}\rightarrow H_{2}$ with initial space
$\overline{R(T^{*})}$ and the final space $\overline{R(T)}$ such that
$T=V|T|$. This factorization of $T$ is called the polar factorization or the
polar decomposition of $T$.

It can be verified that $D(|T|)=D(T)$ and $N(|T|)=N(T)$ and
$\overline{R(|T|)}=\overline{R(T^{*})}$.

Let $T\in\mathcal{C}(H)$ be densely defined. The resolvent of $T$ is defined
by

$\rho(T):={\\{\lambda\in\mathbb{C}:(T-\lambda I)^{-1}\in\mathcal{B}(H)}\\}.$

The set $\sigma(T)=\mathbb{C}\setminus\rho(T)$ is called the spectrum of $T$.
The set

$\sigma_{p}(T)={\\{\lambda\in\mathbb{C}:T-\lambda I:D(T)\rightarrow
H\;\text{is not one-to-one}}\\}$

is called the point spectrum of $T$.

For a self-adjoint operator $T\in\mathcal{C}(H)$, the discrete spectrum
$\sigma_{d}(T)$ is defined as the set of all isolated eigenvalues of $T$ with
finite multiplicity. The set $\sigma_{ess}(T)=\sigma(T)\setminus\sigma_{d}(T)$
is called the essential spectrum of $T$. For more details of this concept we
refer [23, Definition 8.3, page 178].

We refer [25, 10, 22, 5, 23] for the above basics of unbounded operators.

Here we recall the definition and properties of the Moore-Penrose inverse (or
generalized inverse) of a densely defined closed operator that we need for our
purpose.

If $T\in\mathcal{C}(H_{1},H_{2})$ is densely defined, then there exists a
unique densely defined operator $T^{\dagger}\in\mathcal{C}(H_{2},H_{1})$ with
domain $D(T^{\dagger})=R(T)\oplus^{\bot}R(T)^{\bot}$ and has the following
properties:

  1. (1)

$TT^{\dagger}y=P_{\overline{R(T)}}~{}y,~{}\text{for all}~{}y\in
D(T^{\dagger})$

  2. (2)

$T^{\dagger}Tx=P_{N(T)^{\bot}}~{}x,~{}\text{for all}~{}x\in D(T)$

  3. (3)

$N(T^{\dagger})=R(T)^{\bot}$.

The operator $T^{\dagger}$ is called the Moore-Penrose inverse of $T$. An
alternative definition of $T^{\dagger}$ is given below.

For every $y\in D(T^{\dagger})$, let

$L(y):=\Big{\\{}x\in D(T):\|Tx-y\|\leq\|Tu-y\|\quad\text{for all}\quad u\in
D(T)\Big{\\}}.$

Here any $u\in L(y)$ is called a least square solution of the operator
equation $Tx=y$. The vector $x=T^{\dagger}y\in
L(y),\,\text{and}\;\|T^{\dagger}y\|\leq\|x\|\quad\text{for all}\quad u\in
L(y)$ and it is called the least square solution of minimal norm. A different
treatment of $T^{\dagger}$ is given in [4, Pages 314, 318-320], where the
authors call this as “the Maximal Tseng generalized Inverse”.

Next we define minimum attaining operators and the absolutely minimum
attaining operators.

######  Definition 2.1.

[4, 10] Let $T\in\mathcal{C}(H_{1},H_{2})$ be densely defined. Then

$m(T):=\inf{\\{\|Tx\|:x\in S_{D(T)}}\\}$

is called the minimum modulus of $T$. The operator $T$ is said to be bounded
below if and only if $m(T)>0$.

######  Remark 2.2.

If $T\in\mathcal{C}(H_{1},H_{2})$ is densely defined, then

  * (i)

$m(T)>0$ if and only if $R(T)$ is closed and $T$ is one-to-one

  * (ii)

Since $D(T)=D(|T|)$ and $\|Tx\|=\|T|x\|$ for all $x\in D(T)$, we can conclude
that $m(T)=m(|T|)$.

######  Definition 2.3.

Let $T\in\mathcal{C}(H_{1},H_{2})$ be densely defined. Then $T$ is said to be

  1. (1)

minimum attaining if there exists $x_{0}\in S_{D(T)}$ such that
$\|Tx_{0}\|=m(T)=\inf{\\{\|Tx\|:x\in S_{D(T)}}\\}$

  2. (2)

absolutely minimum attaining if for every closed subspace $M$ of $H_{1}$, the
operator $T|_{M}:M\cap D(T)\rightarrow H_{2}$ is minimum attaining.

We denote the set of all densely defined closed absolutely minimum attaining
operators between $H_{1}$ and $H_{2}$ by $\mathcal{AM}_{c}(H_{1},H_{2})$ and
the set of all densely defined minimum attaining closed operators by
$\mathcal{M}_{c}(H_{1},H_{2})$. We write $\mathcal{AM}_{c}(H,H)$ and
$\mathcal{M}_{c}(H,H)$ by $\mathcal{AM}_{c}(H)$ and $\mathcal{M}_{c}(H)$,
respectively.

In particular, if $T\in\mathcal{B}(H_{1},H_{2})$, then $T$ is called minimum
attaining if there exists $x\in H_{1}$ with $\|x\|=1$ such that $\|Tx\|=m(T)$.
We say $T$ to be absolutely minimum attaining if for every closed subspace $M$
of $H_{1}$, the restriction operator $T|_{M}:M\rightarrow H_{2}$ is minimum
attaining. This class is denoted by $\mathcal{AM}(H_{1},H_{2})$. If
$H_{1}=H_{2}=H$, then we denote $\mathcal{AM}(H_{1},H_{2})$ by
$\mathcal{AM}(H)$.

Let $M$ be a closed subspace of $H$ and $T\in\mathcal{C}(H)$ be densely
defined. Then $M$ is said to be invariant under $T$, if $T(M\cap
D(T))\subseteq M$.

Let $P:=P_{M}$. If $P(D(T))\subseteq D(T)$ and $(I-P)(D(T))\subseteq D(T)$,
then

$T=\left(\begin{array}[]{cc}T_{11}&T_{12}\\\ T_{21}&T_{22}\\\
\end{array}\right),$

where $T_{ij}=P_{i}TP_{j}|M_{j}\;(i,j=1,2)$, $M_{1}=P(D(T))$ and
$M_{2}=(I-P)(D(T))$. Here $P_{1}=P$ and $P_{2}=I-P$. It is known that $M$ is
invariant under $T$ if and only if $T_{21}=0$. Also, $M$ reduces $T$ if and
only if $T_{21}=0$ and $T_{12}=0$.

##  3\. Main Results

In this section we prove our main results. First we recall a few basic results
on absolutely minimum attaining and absolutely norm attaining operators, that
we use frequently. Recall that $T\in\mathcal{B}(H_{1},H_{2})$ is norm
attaining if there exists $x\in S_{H_{1}}$ such that $\|Tx\|=\|T\|$. We say
$T$ to be absolutely norm attaining if for every closed subspace $M$ of
$H_{1}$, the restriction operator $T|_{M}:M\rightarrow H_{2}$ is norm
attaining. We denote the set of all absolutely norm attaining operators by
$\mathcal{AN}(H_{1},H_{2})$. For more details of this class operators we refer
to [18].

######  Theorem 3.1.

[8, Theorem 5.14] Let $T\in\mathcal{B}(H_{1},H_{2})$. Then the following are
equivalent;

  1. (1)

$T\in\mathcal{AM}(H_{1},H_{2})$

  2. (2)

$T^{*}T\in\mathcal{AM}(H_{1})$.

######  Theorem 3.2.

[12, Theorem 4.6] Let $T\in\mathcal{C}(H)$ be densely defined and
$T^{-1}\in\mathcal{B}(H)$. Then $T\in\mathcal{AM}_{c}(H)$ if and only if
$T^{-1}\in\mathcal{AN}(H)$.

Let $T\in\mathcal{C}(H)$ be a densely defined closed operator. Then
$Z_{T}=T(I+T^{*}T)^{-\frac{1}{2}}$ is called the bounded transform of $T$.
Here we list a few important properties of $Z_{T}$.

######  Theorem 3.3.

[23, Page 90] Let $T\in\mathcal{C}(H)$ be densely defined. Then we have

  1. (1)

$\|Z_{T}\|\leq 1$.

  2. (2)

$(Z_{T})^{*}=Z_{T^{*}}$

  3. (3)

$Z_{T}^{*}Z_{T}=I-(I+T^{*}T)^{-1}$.

It is to be noted that in particular, $T\in\mathcal{B}(H)$ if and only
$\|Z_{T}\|<1$ (see [24, Corollary 2.1]). Also we can easily prove that
$N(Z_{T})=N(T)$ and $R(T)=R(Z_{T})$ (see [17]).

######  Theorem 3.4.

[21, theorem VIII.3] Let $S$ be a densely defined closed symmetric operator.
Then

(3.1) $\|(S+iI)x\|^{2}=\|Sx\|^{2}+\|x\|^{2},\;\text{for all}\;x\in D(S).$

######  Theorem 3.5.

Let $S$ be a densely defined closed symmetric operator. Then the following are
true;

  1. (1)

$m(S+iI)=\sqrt{1+m(S)^{2}}$

  2. (2)

$S\in\mathcal{M}_{c}(H)$ if and only if $S+iI\in\mathcal{M}_{c}(H)$

  3. (3)

$S\in\mathcal{AM}_{c}(H)$ if and only if $S+iI\in\mathcal{AM}_{c}(H)$.

###### Proof.

All the proofs directly follow by Equation 3.1 and the definitions of the
minimum modulus, minimum attaining property and the absolutely minimum
attaining property, respectively. ∎

######  Theorem 3.6.

Let $T$ be a positive and unbounded operator. Then the following are
equivalent;

  1. (1)

$T\in\mathcal{AM}_{c}(H)$

  2. (2)

$T^{2}+I\in\mathcal{AM}_{c}(H)$

  3. (3)

$Z_{T}\in\mathcal{AM}(H)$.

###### Proof.

Proof of (1) if and only (2): By Theorem 3.5, we know that
$T\in\mathcal{AM}_{c}(H)$ if and only $T+iI\in\mathcal{AM}_{c}(H)$. Since
$T+iI$ is one to one, by applying [12, Theorem 4.10], we can conclude that
$T+iI\in\mathcal{AM}_{c}(H)$ if and only if
$T^{2}+I=(T+iI)^{*}(T+iI)\in\mathcal{AM}_{c}(H)$.

To prove the equivalence of (2) and (3), first we observe that

(3.2) $Z_{T}^{*}Z_{T}=T^{2}(I+T^{2})^{-1}=I-(I+T^{2})^{-1}.$

Now, if $I+T^{2}\in\mathcal{AM}_{c}(H)$, then $(I+T^{2})^{-1}$ is absolutely
norm attaining, by Theorem 3.2. Hence by [26, Theorem 2.5], there exists a
compact positive operator $K$ and positive finite rank operator $F$ and
$\alpha\geq 0$ such that

(3.3) $(I+T^{2})^{-1}=\alpha I+K-F$

with $KF=0$ and $F\leq\alpha I$. Now, by Equation 3.2, we have that
$Z_{T}^{*}Z_{T}=(1-\alpha)I-K+F$. Our idea is to apply [8, Theorem 5.8] and
conclude $Z_{T}^{*}Z_{T}\in\mathcal{AM}(H)$. Then this will imply
$Z_{T}\in\mathcal{AM}(H)$. For this purpose, we need to prove that $\|K\|\leq
1-\alpha$. By post multiplying Equation 3.3 and using the facts that $KF=0$
and $K$ commute with $(I+T^{2})^{-1}$, we get

(3.4) $(I+T^{2})^{-1}K=(\alpha I+K)K.$

Since $\|(I+T^{2})^{-1}\|\leq 1$, we have $(I+T^{2})^{-1}K\leq K$ or $(\alpha
I+K)K\leq K$. This implies that $K^{2}\leq(1-\alpha)K\leq(1-\alpha)\|K\|$.
From this we can conclude that $\|K^{2}\|=\|K\|^{2}\leq(1-\alpha)\|K\|$ or
$\|K\|\leq 1-\alpha$.

To prove the implication
$(\ref{boundedtransformAM})\Rightarrow(\ref{AMplusidentity})$, let
$Z_{T}\in\mathcal{AM}(H)$. Then $Z_{T}^{*}Z_{T}\in\mathcal{AM}(H)$. Hence
there exists $K\in\mathcal{K}(H)_{+},\;F\in\mathcal{F}(H)_{+}$ and $\beta\geq
0$ satisfying $KF=0$ and $K\leq\beta I$ such that $Z_{T}^{*}Z_{T}=\beta
I-K+F$. Hence by Equation 3.2, we have that

$(T^{2}+I)^{-1}=(1-\beta)I+K-F\in\mathcal{AN}(H),$

by [15, Theorem 5.2]. Now by Theorem 3.2, we can conclude that
$I+T^{2}\in\mathcal{AM}_{c}(H)$. ∎

Our next goal before proving the spectral theorem is to show that an
absolutely minimum attaining closed operator can either have a finite-
dimensional null space or finite-dimensional range space. This property is
helpful in deciding the spectrum of such an operator. To achieve this we prove
the following results.

######  Lemma 3.7.

Let $T\in\mathcal{C}(H)$ be densely defined. If $Z_{T}$ is norm attaining,
then $T\in\mathcal{B}(H)$.

###### Proof.

We know that $\|Z_{T}\|\leq 1$. If $\|Z_{T}\|<1$, then clearly
$T\in\mathcal{B}(H)$, by [Tharifigap, Corollary 2.1]. Next assume that
$\|Z_{T}\|=1$. Since $Z_{T}$ is norm attaining, there exists $x_{0}\in S_{H}$
such that $Z_{T}^{*}Z_{T}x_{0}=x_{0}$. That is,
$(I-(I+T^{*}T)^{-1})x_{0}=x_{0}$ or $(I+T^{*}T)^{-1}{x_{0}}=0$. This imply
that $x_{0}=0$, a contradiction. Hence, the assumption that $\|Z_{T}\|=1$ is
wrong. That is, $\|Z_{T}\|<1$. Now the conclusion follows by the earlier case.
∎

######  Proposition 3.8.

Let $T\in\mathcal{C}(H_{1},H_{2})$ be densely defined. If $T$ has finite rank,
then $T\in\mathcal{B}(H_{1},H_{2})$.

###### Proof.

Since $T$ is closed, $N(T)$ is closed and hence $D(T)=N(T)\oplus C(T)$, where
$C(T)=D(T)\cap N(T)^{\bot}$. Note that $T_{0}=T|_{C(T)}$ is a bijection from
$C(T)$ onto $R(T)$. This implies that $C(T)$ is finite-dimensional and hence
$T_{0}$ is bounded. Suppose its norm is $M$ and consider any $x$ in the domain
$D(T)$ of $T$. Then $x$ can be written uniquely as $x=u+v$ with $u\in N(T)$
and $v\in C(T)$. Since $D(T)$ is a subspace, $v=x-u$ is in $D(T)$. Now

$\|T(x)\|=\|T(v)\|\leq M\|v\|\leq M\|x\|.$

This shows that $T$ is bounded on $D(T)$. Since $D(T)$ is an orthogonal direct
sum of a closed subspace $N(T)$ and a finite-dimensional subspace $C(T)$, it
is closed. As $T$ is densely defined, we obtain that $D(T)=H$. ∎

######  Corollary 3.9.

Let $T\in\mathcal{C}(H_{1},H_{2})$ be densely defined and unbounded. Then
$R(T)$ must be infinite-dimensional.

To prove the converse of Lemma 3.7, we need the following result.

######  Theorem 3.10.

Let $T$ be an unbounded positive operator $T\in\mathcal{AM}_{c}(H)$. Then
$N(T)$ is finite-dimensional.

###### Proof.

It is clear that $R(T)$ is infinite dimensional by Corollary 3.9. Suppose that
$N(T)$ is infinite-dimensional. Then $N(Z_{T})$ is infinite-dimensional and
hence $Z_{T}\in\mathcal{F}(H)$ by [2, Remark 3.2]. Since $R(T)=R(Z_{T})$, $T$
is a finite rank operator. Since $Z_{T}$ is norm attaining, by Lemma 3.7, we
can conclude that $T$ is bounded. This completes the proof. ∎

######  Corollary 3.11.

Let $T\in\mathcal{AM}_{c}(H_{1},H_{2})$ be densely defined and unbounded. Then
$N(T)$ is finite-dimensional.

###### Proof.

Since $T\in\mathcal{AM}_{c}(H_{1},H_{2})$, we have
$|T|\in\mathcal{AM}_{c}(H_{1})$ and by Theorem 3.10, we can conclude that
$N(|T|)=N(T)$ is finite dimensional. ∎

######  Remark 3.12.

If $A\in\mathcal{AM}(H_{1},H_{2})$, then either $R(A)$ or $N(A)$ is finite-
dimensional (see [8, Proposition 5.19] for details). But Corollary 3.11 says
that it is not the case when the operator is unbounded. The reason for this
difference is that the finite rank operators are included in the class of
bounded absolutely minimum attaining operators.

Next, we describe the structure of unbounded positive absolutely minimum
attaining operators. This result generalizes that of [12, Theorem 4.8].

######  Theorem 3.13.

Let $T\in\mathcal{AM}_{c}(H)$ be positive and unbounded. There exists an
unbounded (increasing) sequence ${\\{\lambda_{n}}\\}$ of eigenvalues of $T$
with corresponding orthonormal eigenvectors ${\\{\phi_{n}}\\}$ such that

  1. (1)

$D(T)=\left\\{x\in H:\displaystyle\sum_{n=1}^{\infty}\lambda_{n}^{2}|\langle
x,\phi_{n}\rangle|^{2}<\infty\right\\}$ and

$Tx=\displaystyle\sum_{n=1}^{\infty}\lambda_{n}\langle
x,\phi_{n}\rangle\phi_{n},\;\text{for all}\;x\in D(T)$

  2. (2)

$\sigma(T)\subseteq{\\{\lambda_{n}:n\in\mathbb{N}}\\}\cup{\\{0}\\}=\sigma_{p}(T)\cup{\\{0}\\}$

  3. (3)

If $\mu\in\sigma_{p}(T)$, then it is an eigenvalue with finite multiplicity

  4. (4)

$\overline{\text{span}}{\\{\phi_{n}:n\in\mathbb{N}}\\}=N(T)^{\bot}$

  5. (5)

$T$ is diagonalizable. That is, there exists an orthonormal basis of $H$
consisting of eigenvectors of $T$

  6. (6)

For every subset $S$ of $\mathbb{N}$, we have

$\inf{\\{\lambda_{n}:n\in S}\\}=\min{\\{\lambda_{n}:n\in S}\\}.$

###### Proof.

If $T$ is one-to-one then all the statements (1) to (4) follow from [12,
Theorem 4.8]. If $T$ is not one-to-one, then by Corollary 3.11, $N(T)$ is
finite-dimensional. That is, $0\in\sigma_{p}(T)$. Since $T=0\oplus
T|_{N(T)^{\bot}}$, applying [12, Theorem 4.8] to $T|_{N(T)^{\bot}}$, the
conclusion follows.

The proof of (5) follows by taking an orthonormal basis of $N(T)$ and
adjoining it to the eigenbasis of $N(T)^{\bot}$ obtained in (4). To prove (6),
first note that by (5), $T$ is diagonalizable. First assume that
$N(T)\neq{\\{0}\\}$. Then $N(T)$ is finite-dimensional and hence
$0\in\sigma_{d}(T)$. In this case
$\sigma(T)=\sigma_{d}(T)={\\{\lambda_{n}:n\in\mathbb{N}}\\}\cup{\\{0}\\}$. We
write $T=0\oplus T_{0}$, where $T_{0}=T|_{N(T)^{\bot}}$. Since
$T_{0}\in\mathcal{AM}(N(T)^{\bot})$ and $R(T_{0})=R(T)$ is closed as
$T\in\mathcal{AM}_{c}(H)$, we conclude that $T_{0}$ has a bounded inverse and
by Theorem 3.2, we have $T^{-1}_{0}\in\mathcal{AN}(N(T)^{\bot})$. Hence
$\sigma(T_{0}^{-1})={\\{\lambda_{n}^{-1}:n\in\mathbb{N}}\\}$. Now by [15,
Theorem 3.8(i)], we have $\max{\\{\lambda_{k}:k\in
S\setminus{0}}\\}=\sup{\\{\lambda_{k}^{-1}:k\in S\setminus{0}}\\}$. From this
we get that $\inf{\\{\lambda_{k}:k\in
S\setminus{0}}\\}=\min{\\{\lambda_{k}:k\in S\setminus{0}}\\}$. Note that if
$0\in S$, then

$\inf{\\{\lambda_{k}:k\in S}\\}=0=\min{\\{\lambda_{k}:k\in S}\\}.$

This completes the proof. ∎

######  Corollary 3.14.

Let $T\in\mathcal{C}(H_{1},H_{2})$ be densely defined and unbounded. Then
$T\in\mathcal{AM}_{c}(H_{1},H_{2})$ if and only if $D(T)$ does not contain an
infinite-dimensional closed subspace.

###### Proof.

We have that $T\in\mathcal{AM}_{c}(H_{1},H_{2})$ if and only if
$|T|\in\mathcal{AM}_{c}(H_{1})$. By Theorem 3.13, this is equivalent to the
fact that $|T|$ has pure discrete spectrum. By [23, Page 113, Exercise 25],
this is equivalent to the fact that $D(|T|)=D(T)$ does not contain an
infinite-dimensional closed subspace. ∎

######  Proposition 3.15.

Let $T\in\mathcal{C}(H)$ be positive and unbounded. If
$T\in\mathcal{AM}_{c}(H)$, then $T^{2}\in\mathcal{AM}_{c}(H)$.

###### Proof.

Note that if $T$ is one-to-one, then the result holds true by [12, Theorem
4.10]. So let us assume that $N(T)\neq{\\{0}\\}$. Then $T=0\oplus T_{0}$,
where $T_{0}=T|_{N(T)^{\bot}}$. By Corollary 3.11, $N(T)$ is finite-
dimensional. Clearly, by definition, $T_{0}\in\mathcal{AM}_{c}(N(T)^{\bot})$
and consequently by [12, Theorem 4.10],
$T_{0}^{2}\in\mathcal{AM}_{c}(N(T)^{\bot})$. Hence by Theorem 3.6,
$Z_{T_{0}^{2}}\in\mathcal{AM}(N(T)^{\bot})$. Since
$N(Z_{T^{2}})=N(T^{2})=N(T)$ is finite-dimensional, by [2, Proposition 3.6],
we can conclude that $Z_{T^{2}}=0\oplus Z_{T_{0}^{2}}\in\mathcal{AM}(H)$. Now
the conclusion follows by Theorem 3.6. ∎

######  Proposition 3.16.

If $T\in\mathcal{AM}_{c}(H)$ is positive, unbounded and $\alpha>0$, then
$T+\alpha I\in\mathcal{AM}_{c}(H)$.

###### Proof.

First note that since $T\in\mathcal{AM}_{c}(H)$, it follows that $R(T)$ is
closed. Also, $N(T)$ is finite-dimensional by Theorem 3.10. Since $N(T)$ is a
reducing subspace for $T$, we have that $T=\left(\begin{array}[]{cc}0&0\\\
0&T_{0}\\\ \end{array}\right)$, where $T_{0}=T|_{N(T)^{\bot}}$. Since $T_{0}$
is bijective and $T_{0}\in\mathcal{AM}_{c}(N(T)^{\bot})$, it follows that
$T_{0}+\alpha I_{N(T)^{\bot}}\in\mathcal{AM}_{c}(N(T)^{\bot})$. Hence by
Theorem 3.2, we can conclude that $(T_{0}+\alpha
I_{N(T)^{\bot}})^{-1}\in\mathcal{K}(N(T)^{\bot})$. Hence we have $(T+\alpha
I)^{-1}=\left(\begin{array}[]{cc}\alpha I_{N(T)}&0\\\ 0&(T_{0}+\alpha
I_{N(T)^{\bot}})^{-1}\\\ \end{array}\right)\in\mathcal{K}(H)$. Now the
conclusion follows by Theorem 3.2. ∎

######  Corollary 3.17.

Let $T\in\mathcal{C}(H)$ be positive and unbounded. If
$T^{2}\in\mathcal{AM}_{c}(H)$, then $T\in\mathcal{AM}_{c}(H)$.

###### Proof.

If $T^{2}\in\mathcal{AM}_{c}(H)$, then $T^{2}+I\in\mathcal{AM}_{c}(H)$ by
Proposition 3.16. But we have
$T^{2}+I=(T-iI)(T+iI)=(T+iI)^{*}(T+iI)\in\mathcal{AM}_{c}(H)$. Since $T+iI$ is
one-to-one, by [12, Theorem 4.11], we have $T+iI\in\mathcal{AM}_{c}(H)$. Next,
by Theorem 3.5, we can conclude that $T\in\mathcal{AM}_{c}(H)$. ∎

######  Corollary 3.18.

Let $T\in\mathcal{C}(H_{1},H_{2})$ be densely defined and unbounded. Then
$T\in\mathcal{AM}_{c}(H_{1},H_{2})$ if and only if
$T^{*}T\in\mathcal{AM}_{c}(H_{1})$.

###### Proof.

We know that $T\in\mathcal{AM}_{c}(H_{1},H_{2})$ if and only if
$|T|\in\mathcal{AM}_{c}(H_{1})$. By Corollary 3.17, this is equivalent to the
fact $T^{*}T=|T|^{2}\in\mathcal{AM}_{c}(H_{1})$. The converse follows by
Proposition 3.15. ∎

######  Theorem 3.19.

Let $T\in\mathcal{C}(H)$ be positive. Then $T\in\mathcal{K}(H)$ if and only if
$Z_{T}\in\mathcal{K}(H)$.

###### Proof.

Assume that $T\in\mathcal{K}(H)$. Since
$(I+T^{2})^{-\frac{1}{2}}\in\mathcal{B}(H)$, we get that
$Z_{T}=T(I+T^{2})^{-\frac{1}{2}}\in\mathcal{K}(H)$. Next assume that
$Z_{T}\in\mathcal{K}(H)$. That is $Z_{T}$ is norm attaining. Hence by Lemma
3.7, it is clear that $T\in\mathcal{B}(H)$. Hence $(I+T^{2})^{1/2}\in B(H)$.
Thus $T=Z_{T}(I+T^{2})^{1/2}\in\mathcal{K}(H)$. ∎

The following result is proved in [12, Theorem 4.16]. Here we give a simple
proof in a particular case.

######  Proposition 3.20.

Let $T\in\mathcal{C}(H)$ be positive and unbounded. If
$T\in\mathcal{AM}_{c}(H)$ then $N(T)$ is finite-dimensional and
$T^{\dagger}\in\mathcal{K}(H)$.

###### Proof.

Clearly $N(T)$ is finite-dimensional. If $T$ is one-to-one, then the
conclusion follows by [12, Theorem 4.10]. Next, assume that
$N(T)\neq{\\{0}\\}$. Then $T$ has the representation
$\left(\begin{array}[]{cc}0&0\\\ 0&T_{0}\\\ \end{array}\right)$, where
$T_{0}=T|_{N(T)^{\bot}}$. Clearly, $T_{0}\in\mathcal{AM}_{c}(N(T)^{\bot})$.
Hence $T_{0}^{-{1}}\in\mathcal{K}(N(T)^{\bot})$, by Theorem 3.2. Hence
$T^{\dagger}=0\oplus T_{0}^{-1}$ is compact.

∎

Next we illustrate the above result with an example.

######  Example 3.21.

Let $A$ be a linear operator with

$D(A)={\\{(x_{n})\in\ell^{2}:(nx_{n})\in\ell^{2}}\\}.$

Define $A$ by

$A(x_{1},x_{2},x_{3},\dots)=(x_{1},2x_{2},3x_{3},\dots),\;(x_{n})\in\ell^{2}.$

It can be shown that $A$ is a densely defined closed operator. In fact, $A$ is
positive and invertible with $A^{-1}:\ell^{2}\rightarrow\ell^{2}$ given by

$A^{-1}(y_{1},y_{2},y_{3},\dots)=(y_{1},\frac{y_{2}}{2},\frac{y_{3}}{3},\dots),\;(y_{n})\in\ell^{2}.$

Now define $T$ on $D:=\ell^{2}\oplus D(A)\subset\ell^{2}\oplus\ell^{2}$ by
$T=0\oplus A$. Then $T^{\dagger}=0\oplus A^{-1}$. It can be easily seen that
$N(T)=\ell^{2}\oplus{\\{0}\\}$ which is infinite-dimensional. Also,
$T^{\dagger}\in\mathcal{K}(H)$. We claim that $T\notin\mathcal{AM}_{c}(H)$. It
is clear that $N(T)$ is a reducing subspace for $T$. It can be easily shown
that $Z_{T}=0\oplus Z_{A}$. A routine computation shows that $Z_{A}$ is
defined on the standard orthonormal basis ${\\{e_{n}:n\in\mathbb{N}}\\}$ of
$\ell^{2}$ by

$Z_{A}(e_{n})=\dfrac{n}{\sqrt{1+n^{2}}}e_{n},n\in\mathbb{N}.$

Note that $1$ is the limit point of the point spectrum

$\sigma_{p}(Z_{A})={\\{\dfrac{n}{\sqrt{1+n^{2}}}:n\in\mathbb{N}}\\}.$

Hence $\sigma_{ess}(Z_{A})={\\{1}\\}$ and since $0$ is an eigenvalue of
$Z_{T}$ with infinite multiplicity, we can conclude that
$0\in\sigma_{ess}(T)$. So $\sigma_{ess}(Z_{T})={\\{0,1}\\}$. By [2, Theorem
3.10], $Z_{T}\notin\mathcal{AM}(H)$. Now, by Theorem 3.6, we can conclude that
$T\notin\mathcal{AM}_{c}(H)$.

######  Proposition 3.22.

Let $T\in\mathcal{AM}_{c}(H_{1},H_{2})$ be densely defined and unbounded. Then
$N(T)$ is finite-dimensional and $T^{\dagger}\in\mathcal{K}(H_{2},H_{1})$.

###### Proof.

It is clear that $N(T)$ is finite-dimensional by Corollary 3.11. As
$T\in\mathcal{AM}_{c}(H_{1},H_{2})$, we have $|T|\in\mathcal{AM}_{c}(H_{1})$.
This implies that $|T|^{\dagger}=|(T^{*})^{\dagger}|\in\mathcal{K}(H_{1})$.
From this we can conclude that $T^{\dagger}\in\mathcal{K}(H_{2},H_{1})$. ∎

The following question is asked in [12, Question 4.17].

######  Question 3.23.

Let $T\in\mathcal{C}(H)$ be densely defined, unbounded and
$T^{\dagger}\in\mathcal{K}(H)$. Is it true that $T\in\mathcal{AM}_{c}(H)$?.

The Example 3.21 shows that the answer to Question 3.23 need not be
affirmative. But we have an affirmative answer in a particular case.

######  Proposition 3.24.

Let $T\in\mathcal{C}(H)$ be positive, unbounded and $N(T)$ be finite-
dimensional. If $T^{\dagger}\in\mathcal{K(H)}$, then
$T\in\mathcal{AM}_{c}(H)$.

###### Proof.

Since $N(T)$ reduces $T$, with respect to the subspaces $N(T)$ and
$N(T)^{\bot}$, we can write $T$ by $T=0\oplus T_{0}$, where
$T_{0}=T|_{N(T)^{\bot}}$. As $N(T)=N(Z_{T})$, we have that $Z_{T}=0\oplus
Z_{T_{0}}$. As $T_{0}^{-1}\in\mathcal{K}(N(T)^{\bot})$, we can conclude that
$Z_{T_{0}}\in\mathcal{K}(N(T)^{\bot})$. As $N(T)$ is finite-dimensional, by
[2, Proposition 3.6], it follows that $Z_{T}\in\mathcal{AM}(H)$. Now by
Theorem 3.6, we can conclude that $T\in\mathcal{AM}_{c}(H)$. ∎

######  Proposition 3.25.

Let $T\in\mathcal{C}(H_{1},H_{2})$ be densely defined, unbounded and $N(T)$ be
finite-dimensional. If $T^{\dagger}\in\mathcal{K}(H_{2},H_{1})$, then
$T\in\mathcal{AM}(H_{1},H_{2})$.

###### Proof.

If $T^{\dagger}\in\mathcal{K}(H_{2},H_{1})$, then
$|T^{{\dagger}^{*}}|\in\mathcal{K}(H_{2})$. But by [12, Proposition 3.19], we
have $|T|^{\dagger}=|T^{{\dagger}^{*}}|$. Since $N(T)=N(|T|)$, by Proposition
3.24, we have $|T|\in\mathcal{AM}_{c}(H_{1})$. Hence
$T\in\mathcal{AM}_{c}(H_{1},H_{2})$. ∎

By combining Propositions 3.22 and 3.25, we can state the following theorem.

######  Theorem 3.26.

Let $T\in\mathcal{C}(H_{1},H_{2})$ be densely defined and unbounded. Then
$T\in\mathcal{AM}_{c}(H_{1},H_{2})$ if and only if $N(T)$ is finite-
dimensional and $T^{\dagger}\in\mathcal{K}(H_{2},H_{1})$.

Next we ask whether the $\mathcal{AM}$-property of an operator imply the
$\mathcal{AM}$-property of its adjoint. This question in the case of bounded
operators is answered in [2, Proposition 3.11]. The unbounded case is
discussed in [12] by assuming an extra condition, namely the invertibility of
the operator.

######  Theorem 3.27.

[12, Theorem 4.11] If $T\in\mathcal{C}(H)$ is densely defined, unbounded and
has a bounded inverse, then $T\in\mathcal{AM}_{c}(H)$ if and only if
$T^{*}\in\mathcal{AM}_{c}(H)$.

Here we improve the above result by dropping the invertibility condition.

######  Theorem 3.28.

Let $T\in\mathcal{C}(H_{1},H_{2})$ be densely defined. Assume that both $N(T)$
and $N(T^{*})$ are finite-dimensional. Then
$T\in\mathcal{AM}_{c}(H_{1},H_{2})$ if and only if
$T^{*}\in\mathcal{AM}_{c}(H_{2},H_{1})$.

###### Proof.

First assume that $T\in\mathcal{AM}_{c}(H_{1},H_{2})$. Then by Proposition
3.22, $T^{\dagger}\in\mathcal{K}(H_{2},H_{1})$. This implies that
$(T^{*})^{\dagger}=(T^{\dagger})^{*}\in\mathcal{K}(H_{1},H_{2})$. As
$N(T^{*})$ is finite-dimensional, by Proposition 3.24, we can conclude that
$T^{*}\in\mathcal{AM}(H_{2},H_{1})$. The converse follows easily by applying
the above argument to $T^{*}$ and using the hypothesis that $N(T)$ is finite-
dimensional, we can conclude that $T\in\mathcal{AM}_{c}(H_{1},H_{2})$. ∎

Next we prove the spectral theorem for unbounded self-adjoint absolutely
minimum attaining operators.

######  Theorem 3.29 (Spectral theorem for self-adjoint
$\mathcal{AM}$-operators).

Let $H$ be an infinite-dimensional Hilbert space and $T\in\mathcal{C}(H)$ be
self-adjoint. Assume that $T$ is unbounded. Then the following statements are
equivalent.

  1. (1)

$T\in\mathcal{AM}_{c}(H)$

  2. (2)

$T^{\dagger}\in\mathcal{K}(H)$ and $N(T)$ is finite-dimensional

  3. (3)

there exists a sequence $(\lambda_{n})$ of real numbers and an orthonormal
subset ${\\{v_{n}:n\in\mathbb{N}}\\}$ of $H$ such that
$\displaystyle\lim_{n\rightarrow\infty}|\lambda_{n}|\rightarrow\infty$ and
$Tv_{n}=\lambda_{n}v_{n}$ for each $n\in\mathbb{N}$. In this case,
$D(T)=\left\\{x\in H:\displaystyle\sum_{n=1}^{\infty}\lambda_{n}^{2}|\langle
x,v_{n}\rangle|^{2}<\infty\right\\}$ and

$Tx=\displaystyle\sum_{n=1}^{\infty}\lambda_{n}\langle x,v_{n}\rangle
v_{n},\;\text{for all}\;x\in D(T)$

  4. (4)

$T$ is diagonalizable, that is, $H$ has an orthonormal basis consisting of
eigenvectors of $T$ and each eigenvalue of $T$ has finite multiplicity

  5. (5)

$\sigma(T)\subseteq\\{0\\}\cup\\{\lambda_{n}:n\in\mathbb{N}\\}$ and every
spectral value is an eigenvalue with finite multiplicity, that is,
$\sigma(T)=\sigma_{d}(T)$.

  6. (6)

The resolvent $R_{\lambda}(T)=(T-\lambda I)^{-1}$ is compact for one, hence
for all, $\lambda\in\rho(T)$.

  7. (7)

the embedding $J_{T}:(D(T),\|\cdot\|_{T})\rightarrow H$ is compact. Here
$\|x\|_{T}=(\|x\|^{2}+\|Tx\|^{2})^{\frac{1}{2}}$ for all $x\in D(T)$.

###### Proof.

If $N(T)={\\{0}\\}$, then the result follows from [12, Theorem 4.18]. Next
assume that $N(T)\neq{\\{0}\\}$. Then the equivalence of (1) and (2) follows
by Propositions 3.25 and 3.22. Since $N(T)$ is finite-dimensional and $N(T)$
reduces $T$, we have that $T=0\oplus T_{0}$, where $T_{0}=T_{N(T)^{\bot}}$,
equivalence of the other statements follows by applying [12, Theorem 4.18] to
$T_{0}$ since $T_{0}$ is bijective and
$T_{0}\in\mathcal{AM}_{c}(N(T)^{\bot})$.

To find $D(T)$ in (3), we argue as follows;
$x\in D(T)$ if and only if $Tx\in H$ if and only if

$\displaystyle Tx$ $\displaystyle=\displaystyle\sum_{n=1}^{\infty}\langle
Tx,v_{n}\rangle v_{n}$ $\displaystyle=\displaystyle\sum_{n=1}^{\infty}\langle
x,Tv_{n}\rangle v_{n}$
$\displaystyle=\displaystyle\sum_{n=1}^{\infty}\lambda_{n}\langle
x,v_{n}\rangle v_{n}.$

It is clear that $Tx\in H$ if and only if
$\displaystyle\sum_{n=1}^{\infty}\lambda_{n}^{2}|\langle
x,v_{n}\rangle|^{2}<\infty$. Hence

$D(T)=\left\\{x\in H:\displaystyle\sum_{n=1}^{\infty}\lambda_{n}^{2}|\langle
x,v_{n}\rangle|^{2}<\infty\right\\}$

and $Tx=\displaystyle\sum_{n=1}^{\infty}\lambda_{n}\langle x,v_{n}\rangle
v_{n},\;\text{for all}\;x\in D(T).$ ∎

It is natural to ask whether Theorem 3.29 holds true for normal operators.
Here we answer this question.

######  Theorem 3.30 (Spectral theorem for unbounded normal
$\mathcal{AM}$-operators).

Let $H$ be an infinite-dimensional Hilbert space and $T\in\mathcal{C}(H)$ be
normal. Assume that $T$ is unbounded. Then the following statements are
equivalent.

  1. (1)

$T\in\mathcal{AM}_{c}(H)$

  2. (2)

$T^{\dagger}\in\mathcal{K}(H)$ and $N(T)$ is finite-dimensional

  3. (3)

there exists a sequence $(\lambda_{n})$ of complex numbers and an orthonormal
subset ${\\{v_{n}:n\in\mathbb{N}}\\}$ of $H$ such that
$\lim_{n\rightarrow\infty}|\lambda_{n}|\rightarrow\infty$ and
$Tv_{n}=\lambda_{n}v_{n}$ for each $n\in\mathbb{N}$. In this case,
$D(T)=\left\\{x\in H:\displaystyle\sum_{n=1}^{\infty}\lambda_{n}^{2}|\langle
x,v_{n}\rangle|^{2}<\infty\right\\}$ and

$Tx=\displaystyle\sum_{n=1}^{\infty}\lambda_{n}\langle x,v_{n}\rangle
v_{n},\;\text{for all}\;x\in D(T)$

  4. (4)

$T$ is diagonalizable, that is, $H$ has an orthonormal basis consisting of
eigenvectors of $T$ and each eigenvalue of $T$ has finite multiplicity.

  5. (5)

$\sigma(T)\subseteq\\{0\\}\cup\\{\lambda_{n}:n\in\mathbb{N}\\}$ and every
spectral value is an eigenvalue with finite multiplicity, that is,
$\sigma(T)=\sigma_{d}(T)$.

  6. (6)

The resolvent $R_{\lambda}(T)=(T-\lambda I)^{-1}$ is compact for one, hence
for all, $\lambda\in\rho(T)$.

  7. (7)

the embedding $J_{T}:(D(T),\|\cdot\|_{T})\rightarrow H$ is compact. Here
$\|x\|_{T}=(\|x\|^{2}+\|Tx\|^{2})^{\frac{1}{2}}$ for all $x\in D(T)$.

###### Proof.

Equivalence of (1) and (2) follows by Theorem 3.26. Next assume that (2) is
true. They by the spectral theorem for compact normal operators, there exists
a sequence $(\mu_{n})$ of complex numbers and an orthonormal system
${\\{\phi_{n}}\\}$ such that

$T^{\dagger}y=\sum_{n=1}^{\infty}\mu_{n}\langle
y,\phi_{n}\rangle\phi_{n},\;\text{for all}\;x\in H.$

From the above equation it is clear that
$T^{\dagger}(\phi_{n})=\mu_{n}\phi_{n}$ for each $n\in\mathbb{N}$. That is,
$\phi_{n}\in R(T^{\dagger})=C(T)$. It is also clear that $T^{\dagger}(y)\in
C(T)$. Since $T$ is normal, it is easy to observe that
$N(T)=N(T^{*})=N(T^{\dagger})$. Note that
$\phi_{n}=P_{R(T)}(\phi_{n})=TT^{\dagger}(\phi_{n})$ for all $n\in\mathbb{N}$.
Hence we have $T\phi_{n}=\lambda_{n}\phi_{n}$ for all $n\in\mathbb{N}$, where

$\lambda_{n}=\begin{cases}0\quad\mbox{if}\;\mu_{n}=0,\\\
\mu_{n}^{-1}\quad\mbox{if}\;\lambda_{n}\neq 0.\end{cases}$

Since $T^{\dagger}y\in C(T)$, $TT^{\dagger}y$ is meaningful. Following the
similar steps as in (3) of Theorem 3.29, we can determine $D(T)$ easily. On
the other hand, following similar arguments about $T$ and $T^{\dagger}$, we
can easily prove the implication
$(\ref{normaldiagonal})\Rightarrow(\ref{normalMPinverse})$.

Proof of $(\ref{normaldiagonal})\Leftrightarrow(\ref{normalpurespectrum})$: By
$(\ref{normaldiagonal})$ it is clear that $T|_{R(T)}$ is diagonalizable.
Observe that if $0\in\sigma(T)$ and as it is an isolated point, it must be an
eigenvalue of $T$ and hence by , $R(T)$ must be closed. So we have
$H=R(T)\oplus R(T)^{\bot}=R(T)\oplus N(T)$. As $N(T)$ is finite dimesnsional,
we can take an orthonormal basis of it and adjoin it with an orthonormal basis
${\\{\phi_{n}:n\in\mathbb{N}}\\}$ of $R(T)$ so that we can obtain an
orthonormal basis of $H$ consisting of eigenvectors of $T$.

On the other hand, since $T$ is not bounded, we can conclude that $\sigma(T)$
is unbounded. It is easy to establish the representation of $T$ and determine
the domain easily.

Proof of $(\ref{normalpurespectrum})\Leftrightarrow(\ref{discretespectrum})$:
the implication
$(\ref{normalpurespectrum})\Rightarrow(\ref{discretespectrum})$ is easy to
prove. To prove the other way implication, let us assume that
$\sigma(T)=\sigma_{d}(T)$. Let
$\sigma(T)={\\{\lambda_{1},\lambda_{2},\lambda_{3},\dots}\\}$. Write
$G_{n}=N(T-\lambda_{n}I)$. It is easy to prove that each $G_{n}$ reduces $T$.
By using the Zorn’s Lemma, we can show that
$H=\displaystyle\bigoplus_{n=1}^{\infty}G_{n}$ from which we can conclude that
$T$ is diagonalizable.

Proof of $(\ref{discretespectrum})\Leftrightarrow(\ref{cptresolvent})$: Let us
assume that
$\sigma(T)=\sigma_{d}(T)={\\{\lambda_{1},\lambda_{2},\lambda_{3},\dots}\\}$.
Since $T$ is unbounded, $\sigma(T)$ is unbounded closed subset of
$\mathbb{C}$. Hence $Tx=\displaystyle\sum_{n=1}^{\infty}\lambda_{n}P_{n}x$,
where $P_{n}$ is an orthogonal projection onto $N(T-\lambda_{n}I)$ with
$D(T)={\\{x\in
H:\displaystyle\sum_{n=1}^{\infty}|\lambda_{n}|^{2}\|P_{n}x\|^{2}<\infty}\\}$.
Hence if $\lambda\in\rho(T)$, then

$(T-\lambda
I)^{-1}=\displaystyle\sum_{n=1}^{\infty}(\lambda_{n}-\lambda)^{-1}P_{n}.$

It is easy to see that $|(\lambda_{n}-\lambda)^{-1}|\rightarrow 0$ as
$n\rightarrow\infty$. From this it is easy to see that $(T-\lambda
I)^{-1}\in\mathcal{K}(H)$.

Proof of $(\ref{cptresolvent})\Leftrightarrow(\ref{normalimbedding})$: the
proof this follows as in the similar lines of the self-adjoint case. We refer
to [23, Theorem 5.12] for details. ∎

Next, we turn our attention to invariant and hyperinvariant subspaces. Recall
that a closed subspace $M$ of $H$ is said to be invariant under a densely
defined closed operator $T$ if $T(D(T)\cap M)\subseteq M$, and hyperinvariant
if $M$ is invariant under every $A\in\mathcal{B}(H)$ such that $AT\subseteq
TA$.

If $T\in\mathcal{AM}_{c}(H)$ is unbounded and has a bounded inverse, then $T$
has a nontrivial hyperinvariant subspace. Here we remove the invertibility
condition and prove the same result.

######  Proposition 3.31.

If $T\in\mathcal{AM}_{c}(H)$ is densely defined and unbounded, then $T$ has a
non trivial hyperinvariant subspace.

###### Proof.

If $T$ is bijective, then $T^{-1}\in\mathcal{B}(H)$. Hence by [12, Theorem
4.14], $T$ has a hyperinvariant subspace. So assume that $T$ is not bijective.
Note that $R(T)$ is closed by [12, Proposition 4.2]. If $T$ is one-to-one but
not onto, then $R(T)$ is a non trivial hyperinvariant subspace for $T$. If $T$
is onto but not one-to-one, then $N(T)$ is a hyperinvariant subspace for $T$.

∎

######  Corollary 3.32.

Assume $T\in\mathcal{C}(H)$ is densely defined, unbounded. If $R(T)$ is closed
and $T^{\dagger}\in\mathcal{K}(H)$, then $T$ has a hyperinvariant subspace.

###### Proof.

First note that if $N(T)\neq{\\{0}\\}$, then clearly $N(T)$ is a
hyperinvariant subspace for $T$. So assume that $N(T)={\\{0}\\}$. By Theorem
3.25, it follows that $T\in\mathcal{AM}_{c}(H)$. Hence by Theorem 3.31, it is
clear that $T$ has a hyperinvariant subspace. ∎

### Acknowledgments

The second author’s research is supported by SERB grant No. MTR/2019/001307,
Govt. Of India.

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