# Mach’s principle and dark matter

João Magueijo<EMAIL_ADDRESS>Theoretical Physics Group, The
Blackett Laboratory, Imperial College, Prince Consort Rd., London, SW7 2BZ,
United Kingdom

###### Abstract

In this paper we entertain a Machian setting where local physics is non-
locally affected by the whole Universe, taking the liberty to identify the
local (“Newton’s bucket”) with our visible Universe, and the whole Universe
(Mach’s “fixed stars”) with the global Universe beyond our horizon. Crucially,
we allow for the two to have different properties, so that we are beyond the
traditional FRW setting. For definiteness we focus on theories where non-
locality arises from evolution in the laws of physics in terms of spatially
global time variables dual to the constants of Nature. Since non-local
theories are foliation-dependent, the local (but not the global) Hamiltonian
constraint is lost. This is true not only while non-locality is taking place,
but also after it ceases: the local Hamiltonian constraint is only recovered
up to a constant in time, keeping a memory of the integrated past non-
locality. We show that this integration constant is equivalent to preserving
the local Hamiltonian constraint and adding an extra fluid with the same
cosmological properties as conventional pressureless dark matter. The
equivalence breaks down in terms of clustering properties, with the new
component attracting other matter, but not budging from its location. This is
the ultimate “painted-on” dark matter, attracting but not being attracted, and
nailing down a preferred frame.

##  I Introduction

In this paper we seek to relate the enigma of dark matter and an issue at the
core of the foundations of physics: the awkward interaction between General
Relativity (GR) in its final form and the Machian principle that provided some
of its inspiration MachBook ; MachReview . In trying to extend relativity of
motion to non-inertial frames, Mach implied that Newton’s rotating bucket
would not feel inertial forces if a presiding Universe were not present. But
by making inertial forces the result of relative acceleration between the
local and the global Universe, the door opened up to “pantheistic” direct
interactions between the whole Universe and its parts. Ironically this
introduces a preferred frame, albeit one determined by matter (the “fixed
stars”), as well as non-local interactions (action at a distance with infinite
speed). Hence Mach’s views, when interpreted in this fashion (see MachReview
for alternatives), are in contradiction with the GR it inspired, within which
the metric field is part of reality and suffices to locally explain the
inertial forces. In this paper we revisit the possibility of global actions on
local physics, thereby “returning to Mach” in a sense.

Mach never elevated his principle to a theory (although the interactions he
envisaged were manifestly gravitational). What follows is decoupled from the
issue of inertial forces, but preserves the idea of a theory with global
interactions. Furthermore, although most of this paper applies to any theory
where there are global variables interacting with local physics, for
concreteness we focus on theories where the global variables are canonical
pairs made up of the “constants” of Nature and their dual clocks JoaoPaper . A
prototype for this construction is the Henneaux-Teitelboim formulation unimod
of unimodular gravity unimod1 ; UnimodLee1 ; alan ; daughton ; sorkin1 ;
sorkin2 , where the cosmological constant $\Lambda$ is the target constant to
be demoted to an on-shell constant, with a canonical dual which on-shell is
the 4-volume to the past, or unimodular time Bombelli ; UnimodLee2 ;
JoaoLetter ; JoaoPaper . Other targets can be chosen pad ; pad1 ; vikman ;
vikman1 ; vikaxion ; JoaoPaper ; evolution . Usually the “times” ticked by
these clocks do not appear in the action, presenting as Lagrange multipliers
enforcing the on-shell constancy of the constants. But we can envisage
evolution in the laws of physics in terms of such time variables evolution ,
leading to concomitant variations in the constants of Nature to which they are
tied. As in Mach’s vision, in such theories there are global cosmic variables
which interfere with the local physics, a situation we have explored in the
context of black hole formation in BHevolution .

All such non-local theories peg down a global foliation of space-time into
space and time. This implies that the lapse function must be a spatially
global variable, so as to preserve the foliation. As a result there is
fundamentally no local Hamiltonian constraint, but only a global Hamiltonian
constraint. Furthermore, as proved in Sections II–IV, whenever global
interactions are switched off, the local Hamiltonian constraint is not
restored, but, rather, one obtains the weaker result that the local
Hamiltonian is conserved. Hence, it becomes a generally non-vanishing constant
in time (which can vary in space), representing the integrated memory of all
past evolution or any other global interactions. We can absorb this constant
into a redefined Hamiltonian satisfying a Hamiltonian constraint, containing a
new matter component, which we will call Machian matter. The fact that such
matter mimics some, but not all aspects of cold dark matter, is the point of
this paper. Specifically it reproduces all the gravitational effects of dark
matter as a homogeneous and isotropic background component (Sections IV and
V). In terms of inhomogeneities, however, it is strikingly different (Sections
VI and VII), providing grounds for testability and phenomenology.

##  II Local and global variables and the Hamiltonian constraint

Although what follows distils lessons learnt in BHevolution , we start by
presenting the argument in general, abstracting from concrete theories and
from the Machian setting. We consider a set of regions indexed by $I$, with
coordinate volume $V_{I}$ and $V_{\infty}=\sum_{I}V_{I}$. In general these
regions could be infinitesimal volumes around any point (with
$V_{I}\rightarrow d^{3}x$ and the summation replaced by an integral) but in
simplified models $I$ indexes regions where a symmetry reduction has taken
place. For example, in BHevolution  it indexes an inner collapsing FRW model,
a Schwarzchild bubble and an outer FRW Universe (see also OS ). We consider
$V_{I}$ to be coordinate volumes because the Lagrangian is assumed to be a
density, so that with this definition the action is a scalar. The $V_{I}$ are
also then constant.

We define global variables $\\{\alpha,T_{\alpha}\\}$, as opposed to local
variables $\\{q_{I},p_{I}\\}$, from the structure of the action:

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
dt\,[V_{\infty}\dot{\alpha}T_{\alpha}$ (1)
$\displaystyle+\sum_{I}V_{I}(\dot{q}_{I}p_{I}-NH_{I}(q_{I},p_{I};\alpha,T_{\alpha}))+...].$

Here $N$ and $H_{I}$ are the lapse function and Hamiltonian, and the ellipse
denotes other constraints. A foliation is implicit in this 3+1 splitting. In
this foliation the global variables do not depend on $I$, i.e. they are
spatially uniform. This can be a generic assumption for global variables, but
as we will see in Section IV.1, in unimodular-like theories it follows from
solving for the Lagrange multiplier ${\cal T}^{i}$, to eliminate the gauge-
dependent modes, with gauge-invariance present to enforce no new local degrees
of freedom. The volume factors appear because the variables are defined as
intensive densities (as is norm for local variables). As a result, the non-
vanishing Poisson brackets carry the respective volume factors:

$\displaystyle\\{q_{I},p_{I}\\}$ $\displaystyle=$
$\displaystyle\frac{1}{V_{I}}$ (2) $\displaystyle\\{\alpha,T_{\alpha}\\}$
$\displaystyle=$ $\displaystyle\frac{1}{V_{\infty}}.$ (3)

The volume factors are usually neglected (except upon quantization hbarrow ;
Afshordi ; realkod ; HHpacks ; Vilpacks ; laurent ) because in local theories
they cancel out in Hamilton’s equations, as they do here in the equations for
$q_{I}$ and $p_{i}$. However, they do matter for the global variables’
equations of motion:

$\displaystyle\dot{\alpha}$ $\displaystyle=$ $\displaystyle
N\sum_{I}\frac{V_{I}}{V_{\infty}}\frac{\partial H_{I}}{\partial T_{\alpha}}$
(4) $\displaystyle\dot{T}_{\alpha}$ $\displaystyle=$
$\displaystyle-N\sum_{I}\frac{V_{I}}{V_{\infty}}\frac{\partial
H_{I}}{\partial\alpha}.$ (5)

As we see, the larger the $V_{I}$ the more the region $I$ contributes to the
dynamics of the global variables. We can define a global Hamiltonian density:

$H=\frac{1}{V_{\infty}}\sum_{I}V_{I}H_{I},$ (6)

to repackage these equations as

$\displaystyle\dot{\alpha}$ $\displaystyle=$ $\displaystyle N\frac{\partial
H}{\partial T_{\alpha}}$ (7) $\displaystyle\dot{T}_{\alpha}$ $\displaystyle=$
$\displaystyle-N\frac{\partial H}{\partial\alpha}.$ (8)

Usually the lapse function $N$ can depend on $I$, but given our definition of
global variables and their appearance in the local action, $N$ should be
foliation preserving, that is $N_{I}(t)\equiv N(t)$. This is a general feature
of theories which break foliation invariance (Horava-Lifschitz theory
providing an example HL ). Hence, we lose the local Hamiltonian constraint; by
varying with respect to $N$ we still find a global Hamiltonian constraint:

$H=0,$ (9)

however, no $H_{I}=0$ constraint exists a priori. We can diagnose the loss of
the local Hamiltonian constraint through the loss of the secondary constraint
it would have generated. Since $H_{I}=H_{I}(q_{I},p_{I};\alpha,T_{\alpha})$ we
have:

$\displaystyle\dot{H}_{I}$ $\displaystyle=$
$\displaystyle\frac{\partial{H}_{I}}{\partial
q_{I}}\dot{q}_{I}+\frac{\partial{H}_{I}}{\partial
p_{I}}\dot{p}_{I}+\frac{\partial{H}_{I}}{\partial{\bm{\alpha}}}\dot{\bm{\alpha}}+\frac{\partial{H}_{I}}{\partial{\bm{T}}_{\bm{\alpha}}}\dot{\bm{T}}_{\bm{\alpha}}$
(10) $\displaystyle=$
$\displaystyle\frac{\partial{H}_{I}}{\partial{\bm{\alpha}}}\frac{\partial
H}{\partial{\bm{T}}_{\bm{\alpha}}}-\frac{\partial{H}_{I}}{\partial{\bm{T}}_{\bm{\alpha}}}\frac{\partial
H}{\partial{\bm{\alpha}}},$

where we have used Hamilton’s equations for the local and global variables.
The first two terms cancel, but in general the last two do not111The obvious
exceptions are the case where $\dot{\alpha}=0$ and $H_{I}$ does not depend on
$T_{\alpha}$, or else the case where there is only one $I$, such as the FRW
approximation studied in evolution ., so:

$\dot{H}_{I}\neq 0$ (11)

invalidating the possibility of a local Hamiltonian constraint. Summing over
$I$, we still find:

$\dot{H}=\frac{1}{V_{\infty}}\sum_{I}V_{I}\dot{H}_{I}=0$ (12)

as it should, since $H=0$.

The central question now is, do we recover the local Hamiltonian constraint
(and full diffeomorphism invariance) once global interactions switch off? For
example, if the global variables are the constants of nature and their
conjugate times, the dependence of $H_{I}$ on such times may be confined to
restricted epochs (say $t_{i}<t<t_{f}$). What happens for $t>t_{f}$, once
Machian interactions switch off? The answer is that, if there ever were
Machian interactions (e.g. evolution), the local Hamiltonian constraint is
never fully recovered. Eq. (10) implies $\dot{H}_{I}=0$ when $H_{I}$ does not
depend on at least one of the variables in each global canonical pair, but
this integrates to:

$\displaystyle H_{I}(t)$ $\displaystyle\equiv$ $\displaystyle-m_{I}$ (13)
$\displaystyle=$
$\displaystyle\int^{t_{f}}_{t_{i}}dt\,\left(\frac{\partial{H}_{I}}{\partial{\bm{\alpha}}}\dot{\bm{\alpha}}+\frac{\partial{H}_{I}}{\partial{\bm{T}}_{\bm{\alpha}}}\dot{\bm{T}}_{\bm{\alpha}}\right)+H_{I}(t_{i})$

for $t>t_{i}$. An integration constant, therefore, keeps memory of past
interactions.

We can redefine the Hamiltonian as:

$H_{I}\rightarrow\tilde{H}_{I}=H_{I}+m_{I}=0$ (14)

and enforce $\tilde{H}_{I}=0$ by promoting the global lapse function to a
local one, $N=N_{I}(t)$. We thus recover the local Hamiltonian constraint and
full diffeomorphism invariance. However this is done at the expense of
introducing a new term in the action and Hamiltonian, which is intrinsically
tied to a preferred frame. The fact that this integration constant is akin to
some aspects of dark matter is the point to be made in this paper.

##  III The Machian setting

The discussion in the previous Section was kept general, but in a Machian
setting we can further group the local variables and Hamiltonian terms
labelled by $I$ into those pertaining to the region $L$ we want to study (the
Mach/Newton “bucket”, a collapsing star OS ; BHevolution , etc) and those
composing the presiding Universe $U$ (the “fixed stars”, the outer FRW model
in BHevolution , etc). The separation is conceptual and part of the
modelization (the Machian Universe being, after all, made up of a large number
of coarse-grained local regions), with the only condition being $V_{L}\ll
V_{U}\approx V_{\infty}$. This leads to a corresponding split in the local
terms in action and Hamiltonian, so that the total Hamiltonian density defined
in (6) can be approximated as:

$H=\frac{V_{U}}{V_{\infty}}H_{U}+\frac{V_{L}}{V_{\infty}}H_{L}\approx H_{U}$
(15)

with $H_{U}$ and $H_{L}$ collecting all the $I$ contained in the corresponding
sets. This has two important implications. First, (7) and (8) can be
approximated by:

$\displaystyle\dot{\bm{\alpha}}$ $\displaystyle\approx$ $\displaystyle
N(t)\frac{\partial{H}_{U}}{\partial{\bm{T}}_{\bm{\alpha}}}$ (16)
$\displaystyle\dot{\bm{T}}_{\bm{\alpha}}$ $\displaystyle\approx$
$\displaystyle-N(t)\frac{\partial{H}_{U}}{\partial{\bm{\alpha}}}.$ (17)

As pointed out before, the importance of a given local region in determining
the global variables is weighted by its volume. Hence, in a Machian setting,
the presiding Universe wins by volume and it is to a very good approximation
the sole driver of the global variables (which, thus, could have been added to
group $U$ for all practical purposes). This implies that in the study of a
local region the global variables can be seen as external, i.e. time dependent
variables fixed by a dynamics alien to the local variables under study.

A second remarkable implication is that, within this approximation, if we
insert (15) into (9) we find that the Machian Universe is subject to an
approximate Hamiltonian constraint:

$H_{U}\approx 0.$ (18)

The fact that we always have a global Hamiltonian constraint implies that the
presiding Universe is still approximately subject to one too. This is because
the “global” is effectively identified with the presiding Universe.

These are general features of theories with global variables in a Machian
setting. As explained, the separation between $L$ and $U$ is conceptual. In
this paper we make a very special choice for $L$. We note that unlike Mach’s
fixed stars, the presiding Universe does not need to be the observable
Universe. Indeed, if there is a cosmological horizon effect, it is not. By
Occam’s razor we should assume that whatever lies outside our cosmological
horizon, and dominates the global variables’ dynamics, has the same average
properties we observe in a smoothed version of our horizon, but this is not
necessary. We toy with this possibility in this paper. We will take our
observable FRW patch for local region $L$, “the bucket” as it were. By
shifting the presiding Universe outside our horizon and allowing it to have a
separate dynamics, we are releasing the visible Universe from the Hamiltonian
constraint.

Note that this is necessary for non-trivial local effects. None of the points
made in the previous Section are felt in a “minisuperspace scenario”, or in
any scenario with spatial homogeneity built-in. In that case, there is only
one region $I$, so that local and global are confused and none of the effects
derived are felt (this is also the case for the models investigated in
evolution ).

##  IV Theories, models and scenarios

Having kept the discussion general we now provide examples of theories, models
and scenarios fitting into the framework of the last two Sections.

###  IV.1 Theories

For concrete theories with global variables, we choose the “deconstantization”
procedure in JoaoPaper , itself a generalization of unimod . Hence, for global
variables w$\bm{\alpha}$ we take a set of “constants” of Nature. The procedure
consists in adding to a base action $S_{0}$ (which can be GR) a new term:

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\,(\partial_{\mu}{\bm{\alpha}})\cdot{\cal T}_{\bm{\alpha}}^{\mu}+S_{0},$
(19)

where ${\cal T}_{\bm{\alpha}}$ is a density. This density is then used unimod
; Bombelli ; sorkin1 ; sorkin2 ; UnimodLee2 ; JoaoLetter ; JoaoPaper  to
produce a foliation-dependent time variable as follows. Performing a 3+1
spacetime split, and solving for the Lagrange mutiplier ${\cal T}^{i}$, we
first obtain spatial constancy of $\bm{\alpha}$ on the leaves $\Sigma_{t}$
defined by the split. For each entry in the vector $\bm{\alpha}$, the theory
has gauge-invariance under

${\cal T}_{\bm{\alpha}}^{\mu}\rightarrow{\cal
T}^{\mu}_{\bm{\alpha}}+\epsilon^{\mu}_{\bm{\alpha}}\quad{\rm
with}\quad\partial_{\mu}\epsilon_{\bm{\alpha}}^{\mu}=0,$ (20)

a feature essential in making sure that the theory has no new local degrees of
freedom (see unimod  for details), and hence it is purely a theory of global
variables. Retaining only the gauge invariant zero-mode of ${\cal T}^{0}$ we
define a time variable:

${\bm{T}}_{\bm{\alpha}}(t)=\frac{1}{V_{\infty}}\int_{\Sigma_{t}}d^{3}x\,{\cal
T}^{0}_{\bm{\alpha}}.$ (21)

The first term in (19) then reduces to the first term in (1). As explained
before, in the definition (21) we divide by the spatial coordinate volume
$V_{\infty}$ of each leaf so that time is intensive. For example, in the
unimodular case, for which the target $\alpha$ is $\rho_{\Lambda}$ (the vacuum
energy), the time $T_{\Lambda}$ is 4-volume per unit of coordinate (comoving,
in the case of FRW) 3-volume.

For the rest of this paper we will take for $S_{0}$ the Einstein-Cartan theory
with a mixture of perfect fluids (including a cosmological constant) as matter
source. This is then adapted to have a lapse $N$ that is purely time
dependent. As targets for deconstantization we choose a selection of:

$\displaystyle{\bm{\alpha}}$ $\displaystyle=$
$\displaystyle\left(\rho_{\Lambda},\frac{3c_{P}^{2}}{8\pi G}\right),$ (22)

where $\rho_{\Lambda}$ is the vacuum energy, $c_{P}$ is the speed of light as
it appears in the gravitational commutation relations evolution ; JoaoPaper ;
JoaoLetter , and $G$ is the gravitational constant. This is similar to
sequestration padilla ; pad ; pad1 ; lomb  and leads to associated times:

$\displaystyle T_{1}(\Sigma_{t})\equiv T_{\Lambda}(\Sigma_{t})$
$\displaystyle=$
$\displaystyle-\frac{1}{V_{\infty}}\int^{\Sigma_{t}}_{\Sigma_{0}}d^{4}x\sqrt{-g},$
(23) $\displaystyle T_{2}(\Sigma_{t})\equiv T_{R}(\Sigma_{t})$
$\displaystyle=$
$\displaystyle\frac{1}{6V_{\infty}}\int^{\Sigma_{t}}_{\Sigma_{0}}d^{4}x\sqrt{-g}R,$
(24)

i.e.: unimodular and Ricci time. Here the 4-volume integration goes from some
reference zero-time leaf $\Sigma_{0}$ to leaf $\Sigma_{t}$.

It is then important that $S_{0}$, or equivalently $H$, depend on time(s)
$\bm{T}_{\bm{\alpha}}$. Otherwise (4) implies that the $\bm{\alpha}$ really
are spacetime constants, albeit on-shell only, and any leaf dependence
disappears. Properly constant $\bm{\alpha}$ are the quintessential global
variables (a matter discussed in padilla ; pad1  in terms of fluxes), but that
is not the point of this paper, which seeks a Machian interaction with global
variables which are only spatially constant.

We will therefore assume a $\bm{T}_{\bm{\alpha}}$ dependence in $H$, and
further that this happens via constants/parameters $\bm{\beta}$ other than
$\bm{\alpha}$. That the sets $\bm{\alpha}$ and $\bm{\beta}$ are disjoint is
assumed for simplicity (it avoids introducing a second class constraint). Thus
we consider functions $\bm{\beta}({\bm{T}}_{\bm{\alpha}})$, similar to
evolution potentials evolution . With these further assumptions, Eq.(13) can
be written as:

$\displaystyle\Delta H_{I}$ $\displaystyle=$
$\displaystyle\int^{t_{f}}_{t_{i}}dt\,\left(\frac{\partial{H}_{I}}{\partial{\bm{\alpha}}}\dot{\bm{\alpha}}+\frac{\partial{H}_{I}}{\partial{\bm{\beta}}}\dot{\bm{\beta}}\right).$
(25)

In what follows we choose for ${\bm{\beta}}$ a selection of:

$\displaystyle{\bm{\beta}}$ $\displaystyle=$
$\displaystyle\left(c_{g}^{2},c_{m}^{2}\right)$ (26)

the gravity and matter speed of light.

###  IV.2 Model

As already stressed, the separation of $U$ and $L$ is part of the model, and
we choose the setting explained at the end of Section III. Thus, $I$ only
takes 2 values, $I=L,U$, with $L$ the visible FRW model, and $U$ an outer FRW
model beyond our horizon, the two having different properties. There could be
other regions (indeed there must be at least a buffer region between these
two); however these do not affect the global variables (dominated by $U$) or
our local Universe, and so can be ignored.

The symmetry reduced actions for $I=L,U$ are:

$\displaystyle S_{L}$ $\displaystyle=$ $\displaystyle V_{L}\int
dt\,[\alpha_{2}\dot{b}a^{2}+\sum_{k}\dot{m}_{k}T_{mk}-NH_{L}],$ (27)
$\displaystyle H_{L}$ $\displaystyle=$ $\displaystyle
a\left(-\alpha_{2}(b^{2}+kc_{g}^{2})+\sum_{k}\frac{m_{k}}{a^{1+3w_{k}}}\right),$
(28) $\displaystyle S_{U}$ $\displaystyle=$ $\displaystyle V_{U}\int
dt\,[\alpha_{2}\dot{\bar{b}}\bar{a}^{2}+\sum_{k}\dot{\bar{m}}_{k}\bar{T}_{mk}-NH_{U}],$
(29) $\displaystyle H_{U}$ $\displaystyle=$
$\displaystyle\bar{a}\left(-\alpha_{2}(\bar{b}^{2}+\bar{k}c_{g}^{2})+\sum_{k}\frac{{\bar{m}}_{k}}{{\bar{a}}^{1+3w_{k}}}\right)$
(30)

(see e.g evolution  and references therein). Here $a$ is the expansion factor,
$b$ the FRW connection variable (which in the standard theory is $\dot{a}$ on-
shell, that is the comoving inverse Hubble length), and the fluids indexed by
$k$ have been represented following brown ; gielen ; gielen1 ; evolution ,
with $w_{k}$ the equation of state. We use a bar for the $U$ variables and no
bar for the $L$ variables. This defines the Poisson brackets for all
variables, as well as the Hamiltonian.

By comparing (13) (with $I$ identified with $L$) with the fluids term in
$H_{L}$ (cf.(28)), we see that any past evolution in the laws of physics (or
more generally any non-locality) leaves a memory that can be identified with a
pressureless matter component. Indeed any integration constant in $H_{L}$ is
exactly matched by a $w_{k}=0$ component. This is because a constant in the
Hamiltonian is exactly identical at this level of symmetry with a pressureless
component. This should not be confused with any pressureless fluids already
present, such as baryonic or cold dark matter (BDM or CDM). The phenomenon we
are describing is not a source term for these species but, rather, it is
equivalent to adding a new matter component which we will call Machian matter
and label with an ingredient label $k\equiv M$. This $k=M$ component in the
fluids term of (28) is characterized by:

$\displaystyle w_{M}$ $\displaystyle=$ $\displaystyle 0$ (31) $\displaystyle
m_{M}$ $\displaystyle=$ $\displaystyle-\int
dt\,\left(\frac{\partial{H}_{L}}{\partial{\bm{\alpha}}}\dot{\bm{\alpha}}+\frac{\partial{H}_{L}}{\partial{\bm{\beta}}}\dot{\bm{\beta}}\right).$
(32)

There is nothing a priori forcing $m_{M}>0$, but since this is not a real
matter component, a negative “mass” need not signal an instability.
Furthermore, as we will see, the equivalence with dark matter only works at
the FRW level. When we examine inhomogeneities and clustering properties later
in this paper, some differences will be found between this effective
pressureless fluid and any conventional form of dark matter, be it BDM, CDM or
any other variant.

###  IV.3 Scenarios

We finally need to settle on possible “scenarios”. For the particular theories
described in IV.1, these follow from choices of $\bm{\beta}(T_{\bm{\alpha}})$.
For the Machian setting chosen in IV.2, the outer Universe $U$ then defines
functions $\bm{\alpha}(t)$ $\bm{T}_{\bm{\alpha}}(t)$ and $\bm{\beta}(t)$, and
these are external inputs for the $L$ dynamics. It is these functions that we
wish to classify here. Although there are exceptions (e.g., power-laws, such
as in Barrow1 ; Barrow2 ) many scenarios for varying constants are “phase
transitions” VSL0 ; VSL , presumably “first order”, since they were modelled
as step functions.

Along these lines we introduce the classification (illustrated with parameters
$\bm{\beta}$, but it applies to other variables):

  * •

Pulses or bleeps:

${\bm{\beta}}=\delta_{\bm{\beta}}\,\delta(t-t_{\star})+{\bm{\beta}}_{0},$ (33)

where ${\bm{\beta}}_{0}$ is the baseline value of the variable.

  * •

Steps or jumps:

${\bm{\beta}}={\bm{\beta}}_{-}+\Delta{\bm{\beta}}\,H(t-t_{\star}).$ (34)

where the variable jumps from one constant value to another (here from
${\bm{\beta}}_{-}$ to ${\bm{\beta}}_{+}={\bm{\beta}}_{-}+\Delta{\bm{\beta}}$).

  * •

Cusps: a discontinuity in the derivative of a variable.

Since these profiles are related by differentiation and integration, all
variables in a given scenario will usually fall into one of these categories,
or a combination thereof.

##  V A simple illustration

As a simple illustration we consider a theory where the only $\bm{\alpha}$ in
(22) is the Planck mass $\alpha_{2}$ (and so the only clock is $T_{R}$) and
the only $\bm{\beta}$ in (26) is $c_{g}^{2}$. The non-locality is therefore
encapsulated in the evolution potential $c_{g}^{2}=c_{g}^{2}(T_{R})$.
Furthermore, we assume that the Machian presiding Universe is spatially flat
($\bar{k}=0$), so that equations (16) and (17) reduce to:

$\displaystyle\dot{\alpha}_{2}$ $\displaystyle\approx$
$\displaystyle-N\alpha_{2}\frac{dc_{g}^{2}}{dT_{R}}\bar{a}\bar{k}=0$ (35)
$\displaystyle\dot{T}_{R}$ $\displaystyle\approx$
$\displaystyle\frac{N}{\alpha_{2}}{\bar{a}^{3}}\frac{\bar{\rho}-3\bar{p}}{2}.$
(36)

Hence, in this model the time-dependence in the Hamiltonian induced by
$c_{g}^{2}=c_{g}^{2}(T_{R})$ does not entail corresponding variations in
$\alpha_{2}$, the dual to the clock. This simplifies the calculations (see
evolution ; BHevolution  for the technical complications to be found
otherwise).

The local Hamiltonian therefore is:

$\displaystyle H_{L}$ $\displaystyle=$ $\displaystyle
Na\left(-\alpha_{2}(b^{2}+kc_{g}^{2})+\frac{m}{a}\right)$ (37)

where $\alpha_{2}$ does not vary and $c_{g}^{2}(t)$ results from composing the
functions $c_{g}^{2}(T_{R})$ and (36) and can be seen as an input external to
the $L$ dynamics. The Hamilton equations for the local Universe are:

$\displaystyle\dot{a}$ $\displaystyle=$ $\displaystyle\\{a,H_{L}\\}=Nb$ (38)
$\displaystyle\dot{b}$ $\displaystyle=$
$\displaystyle\\{b,H_{L}\\}=-\frac{N}{2a}(b^{2}+kc_{g}^{2})$ (39)
$\displaystyle\dot{m}$ $\displaystyle=$ $\displaystyle\\{m,H_{L}\\}=0,$ (40)

where we have not used $H_{L}=0$. Thus, unlike in standard cosmology, we do
not have a Friedman equation (equivalent to $H_{L}=0$). In addition, combining
the first and second equation leads to the Raychaudhuri equation stripped of
its standard simplification using the Friedman equation. The last equation is
a conservation equation, valid in this case (since $\rho$ does not depend on
any $\bm{\alpha}$ or $\bm{\beta}$). Note that because $H_{L}=0$ cannot be
imposed we lose the ability to redefine $N$ as a function of $a$ or $b$ (the
Poisson bracket would generate a non-vanishing term in the derivatives of $N$
multiplied by $H_{L}$ leading to new terms in these equations). In what
follows we assume $N=1$, that is we work with a global lapse function which
reduces to 1 in the local FRW patch.

Let us now assume a scenario with a pulse in $c_{g}^{2}$ at $t=t_{\star}$:

$c_{g}^{2}=c^{2}_{g0}+\delta_{c_{g}^{2}}\,\delta(t-t_{\star}),$ (41)

(so that $[\delta_{c_{g}^{2}}]=L^{2}/T$). Eqs. (38) and (39) imply a cusp in
$a$ (at $a(t_{\star})=a_{\star}$) and a step in $b$ with:

$\Delta b=-\frac{k\delta_{c_{g}^{2}}}{2a_{\star}}.$ (42)

Hence, in addition to suffering a bleep (that will be forgotten) $H_{L}$ feels
a step at $t_{\star}$ (that will persist) given by:

$\Delta H_{L}=-a_{\star}\alpha_{2}\Delta
b^{2}=-a_{\star}\alpha_{2}(2b_{-}\Delta b+(\Delta b)^{2})$ (43)

where $b_{-}$ is the value of $b$ just before the jump. This is consistent
with (25), which gives after an integration by parts (and use of $\int
dx\,\delta(x)H(x)=1/2$):

$\Delta H_{L}=\alpha_{2}\int
dt\,kbc_{g}^{2}=\alpha_{2}k\left(b_{-}+\frac{\Delta
b}{2}\right)\delta_{c_{g}^{2}}$ (44)

identical to (43). As explained before, this shift in the Hamiltonian presents
problems for enforcing a Hamiltonian constrain after the bleep, assuming it
was valid before. But this shift is equivalent to keeping the Hamiltonian
constraint after the pulse, and introducing a new pressureless fluid with with
$m_{M}=-\Delta H_{L}$. This is the simplest scenario we found for production
of Machian matter.

Another reasonably simple alternative results from a phase transition in
$c_{g}^{2}$. Then, analysis of our equations shows that $a$ and $b$ must be
continuous (with a cusp in $b$), and we find:

$\Delta H_{L}=-m_{M}=-a_{\star}k\alpha_{2}\Delta c_{g}^{2}.$ (45)

Other theories and scenarios can be investigated, but they all lead to similar
results: generically a pressureless component is created by past non-
localilty, according to expression (32).

Parenthetically we note that the simple model presented in this Section could
be used to solve the flatness and homogeneity problems. Since the production
of Machian matter is coupled to the local $k$ it could be used to fine tune
its flatness. If we see fluctuations around a $k=0$ Universe as small open and
closed Universes, this push to flatness would also homogenize the Universe
(albeit leaving behind an isocurvature fluctuation involving conventional
matter and Machian matter). However, this property is not generic, and is
specific of the concrete theory used in this Section.

##  VI Compare and contrast

The equivalent Machian matter component is not merely an addition to the zero
point Hamiltonian. Within the FRW approximation (homogeneity and isotropy) it
gravitates as dark matter for all practical purposes, just as $\Lambda$ does
but with different pressure and dilution rate. For example, the Raychaudhury
equation contains it, since:

$\ddot{a}=-\frac{N}{2a}(b^{2}+kc_{g}^{2})=-\frac{a}{2\alpha_{2}}(\rho+3p+\rho_{M})$
(46)

where the first identity is Hamilton’s equaton and the second uses the
effective Hamiltonian constraint containing $m_{M}$. But even at the FRW level
the equivalence breaks down beyond gravitation. For example, Machian matter
cannot replace BDM in its role in Big Bang Nucleosynthesis.

Nevertheless, at the FRW level the equivalence with CDM is complete, since
gravitation is its only manifestation. CDM, however, is more than a required
FRW background component: its clustering properties are also needed to explain
the phenomenology. It is at this level that the effective new component shows
some striking differences, providing opportunities for testability.

Unlike $\Lambda$, the new component can be inhomogeneous, and if so, the other
forms of matter feel its gravity. That an inhomogeneous $m_{M}(x)$ attracts
other matter can be probed, for example, with the spherical collapse model or
the Oppenheimer-Snyder model OS ; BHevolution  (the new component contributes
to the ADM mass as usual), but it can also be proved in general. Adapting
Section II to the generic continuous case without any symmetry reduction, we
have for the total Hamiltonian:

$H_{T}=V_{\infty}{\cal H}_{T}=\int_{\Sigma_{t}}d^{3}x\,{\cal H}(x)$ (47)

where ${\cal H}_{T}$ is the total Hamiltonian density and
$V_{\infty}=\int_{\Sigma_{t}}d^{3}x$ (either finite or with a limiting
procedure implied). This is defined so that the $V_{\infty}$ factors cancel
out in:

$\displaystyle\dot{\alpha}$ $\displaystyle=$
$\displaystyle\\{\alpha,NH_{T}\\}=N\frac{\partial{\cal H}_{T}}{\partial
T_{\alpha}}$ (48) $\displaystyle\dot{T}_{\alpha}$ $\displaystyle=$
$\displaystyle\\{T_{\alpha},NH_{T}\\}=-N\frac{\partial{\cal
H}_{T}}{\partial\alpha}.$ (49)

(where $N=N(t)$), just as in (7) and (8). Then, the expressions at the end of
Section II become in general:

$\displaystyle m_{M}(x)$ $\displaystyle=$ $\displaystyle-\int
dt\,\left(\frac{\partial{\cal
H}(x)}{\partial{\bm{\alpha}}}\dot{\bm{\alpha}}+\frac{\partial{\cal
H}(x)}{\partial{\bm{T}}_{\bm{\alpha}}}\dot{\bm{T}}_{\bm{\alpha}}\right)$
$\displaystyle=$ $\displaystyle-\int dt\,\left(\frac{\partial{\cal
H}(x)}{\partial{\bm{\alpha}}}\frac{\partial{\cal
H}_{T}}{\partial{\bm{T}}_{\bm{\alpha}}}-\frac{\partial{\cal
H}(x)}{\partial{\bm{T}}_{\bm{\alpha}}}\frac{\partial\bar{\cal
H}}{\partial{\bm{\alpha}}}\right)$

with:

$\tilde{\cal H}(x)={\cal H}(x)+m_{M}(x)=0.$ (50)

Hence, the redefined action acquires a Machian mass shift as in:

$\tilde{S}=S+S_{M}={S}-\int dt\,N(t)\int_{\Sigma_{t}}d^{3}x\,m_{M}(x).$ (51)

Evaluating the stress energy tensor by varying with respect to the metric
(here $g_{00}=-N^{2}$ only, using ADM variables adapted to $\Sigma_{t}$) we
find that the only non-vanishing component in these coordinates is:

$T^{00}_{M}=\frac{2}{N\sqrt{h}}\frac{\delta S_{M}}{\delta
N^{2}}=\frac{m_{M}(x)}{N^{2}\sqrt{h}}.$ (52)

Defining the velocity 4-vector as $u^{\mu}=(N,\mathbf{0})$ (not identical to
the normal to $\Sigma_{t}$, $n_{\mu}=(N,\mathbf{0})$, if the shift $N_{i}\neq
0$) we can write:

$T_{M}^{\mu\nu}=\rho_{M}(x)u^{\mu}u^{\nu}$ (53)

(with $\tilde{T}^{\mu\nu}=T^{\mu\nu}+T_{M}^{\mu\nu}$) where:

$\rho_{M}=\frac{m_{M}}{\sqrt{h}}.$ (54)

This is the stress-energy tensor of a dust fluid and variation of the full
action with respect to the metric shows that $T^{\mu\nu}_{M}$ appears in the
right hand side of the Einstein equations, so it has the usual active
gravitational properties, as claimed.

However the passive gravity is very different, as can be guessed from the fact
that $m_{M}(x)$ does not depend on time, regardless of whatever other matter
may be around to “pull it”. The density $\rho_{M}$ does depend on time but
only via the spatial determinant of the metric $h$, so no conservation law is
expected, and indeed this is verified by calculation. Recall that for
conventional dust one finds independently geodesic motion for $u^{\mu}$ and
constancy of the dust grains rest mass, $\nabla_{\mu}(\rho u^{\mu})=0$.
Instead, here $u^{\mu}$ is fixed by $\Sigma_{t}$ (and so signals the presence
of a preferred frame), and is generically non-geodesic:

$S^{\alpha}=P^{\alpha}_{\;\;\nu}\nabla_{\mu}T^{\mu\nu}=u^{\mu}\nabla_{\mu}u^{\alpha}\neq
0$ (55)

(with $P_{\mu\nu}=g_{\mu\nu}+u_{\mu}u_{\nu}$ the usual projector, and
$S_{\alpha}u^{\alpha}=0$). Independently from this, we find that:

$S=-u_{\nu}\nabla_{\mu}T^{\mu\nu}=\nabla_{\mu}(\rho
u^{\mu})=(m_{M}N^{i})_{,i}$ (56)

(where $N^{i}$ is defined as in the ADM formalism) is generically non-zero.
This should not be surprising. Machian matter does not correspond to any
grains of dust, so there is no reason why it should follow geodesics or its
“rest mass” be conserved.

We thus see that, once we go beyond homogeneity and isotropy, our Machian
matter becomes a form of “painted-on” dark matter. It is encrusted in the
preferred frame according to a fixed $m_{M}(x)$ and has zero passive
gravitational mass, refusing to budge from its locus regardless of the gravity
of other matter and itself. And yet, it does attract other forms of matter
(including visible matter). This opens it up to being experimentally
distinguished from standard dark matter, as we will briefly discuss in our
final Section.

Note that the fact that $T^{\mu\nu}_{M}$ is not conserved, but enters the
right hand side of the Einstein equations is not contradictory, because of the
implied frame dependence and consequent collapse of the Bianchi identities. We
should also stress that the non-conservation of $T^{\mu\nu}_{M}$ is a separate
effect from the non-conservation of the other $T^{\mu\nu}$ explored in
evolution . The latter happens while non-locality/evolution is switched on;
the former perdures even after it switches off. The latter can happens in the
FRW approximation; the former cannot.

##  VII Discussion

Dark matter is usually introduced as a putative new cosmic ingredient to
explain several observations. But could what we call dark matter be a memory
of past evolution in the laws of physics? In this paper we considered this
possibility within the context of evolution in terms of global time variables,
so that our conclusions apply more broadly to any theory with non-local
interactions. Hence, we probed a more general Machian idea, where the global
Universe can act directly on the local physics. We focused on the legacy
effect of past global interactions: their integrated effect spoils the
Hamiltonian constraint even after they switch off, but this can be rectified
with a redefined Hamiltonian containing a new form of matter, which for lack
of a better name we called Machian matter222Other possibilities include
“memorious matter” in a nod to Borges, and “mnematter” or “mnemomatter” in a
nod to the Greek language.. We went on to note that Machian matter mimics many
features of conventional dark matter.

“Dark matter” is a broad term serving several aspects of phenomenology. We
found that Machian matter reproduces all of its gravitational effects at the
level of an homogeneous and isotropic background. Beyond that the differences
are significant, and are two-fold:

  * •

Although Machian matter perfectly mimics the active gravitational pull of
inhomogeneous dark matter, it fails to cluster around other forms of matter or
feel its own gravity.

  * •

Machian matter remains stuck in position as defined by a fixed frame, thereby
physically pegging down a preferred frame.

Thus, Machian matter is the ultimate painted-on dark matter. Computer
simulations show that, in the non-linear regime, conventional dark matter
occasionally fails to stay put where it is needed for phenomenology, under the
pull of its own gravity and of other matter. A case in point is the dark
matter needed to explain galaxy rotation curves, and its ungainly tendency to
self-collapse into a cusp, in contradiction with the required profile. Fuzzy
dark matter fuzzy ; fuzzya ; fuzzy1  is an alternative proposed to skirt this
drawback. We do not have this problem: Machian matter stays where we put it.
Of course any suggestion that the seed for any galaxy lies in $m_{M}(x)$
stumbles into an initial condition issue; but is this more fine-tuned than
fuzzy dark matter? The question extends to clusters, lensing, the bullet
cluster, etc.

Clearly there are some positive aspects of conventional dark matter which
Machian matter cannot mimic: it is never meant to be a full replacement to
cold dark matter. But conversely, there are some problems with dark matter
which Machian matter can solve, at the expense of a designer $m_{M}(x)$.
Another aspect that is different is peculiar velocities. Machian matter
generates peculiar flows; yet it does not partake in them. All of this may be
seen as desirable, since it makes our proposal observationally distinguishable
from the more parochial forms of dark matter.

We close with some general comments on the nature of our construction. One
sometimes classifies cosmological setups into dark matter/energy models vs.
modified gravity theories, with some ambiguity on whether they can be made
equivalent. We have at hand something that does not quite fit either
description. At its roots our class of theories is best described as a
modified gravity, with a dark matter reinterpretion, but as Section VI showed
this interpretation has its limitations. Then, of course, Machian matter
breaks Lorentz invariance, something the usual dark matter does not. It pegs
down a preferred frame and breaks the equivalence principle, since it has
active but no passive gravitational mass. Once again Mach’s principle inspires
a construction that eventually conflicts with its best intentions.

##  VIII Acknowledgments

We thank S. Koushiappas for discussion, and P. Bassani and A. Lupi for
assistance with terminology. This work was partly supported by the STFC
Consolidated Grants ST/T000791/1 and ST/X00575/1.

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# Zero divisors and topological divisors of zero in certain Banach algebras

Anurag Kumar Patel  Department of Mathematics, Banaras Hindu University,
Varanasi-221005, India<EMAIL_ADDRESS>and  Harish Chandra  Department
of Mathematics, Banaras Hindu University, Varanasi-221005, India
<EMAIL_ADDRESS>
###### Abstract.

In this paper we prove that an element $f\in\mathcal{A}(\mathbb{D})$ is a
topological divisor of zero(TDZ) if and only if there exists
$z_{0}\in\mathbb{T}$ such that $f(z_{0})=0.$ We also give a characterization
of TDZ in the Banach algebra $L^{\infty}(\mu).$ Further, we prove that the
multiplication operator $M_{h}$ is a TDZ in $\mathcal{B}(L^{p}(\mu))~{}(1\leq
p\leq\infty)$ if and only if $h$ is a TDZ in $L^{\infty}(\mu).$ Subsequently,
we show that a composition operator $C_{\phi}$ is a TDZ in
$\mathcal{B}(L^{2}(\mu))$ if and only if $\frac{d\mu\phi^{-1}}{d\mu}$ is a TDZ
in $L^{\infty}(\mu).$ Lastly, we determine composition operators on the Hardy
spaces $\mathbb{H}^{p}(\mathbb{D})$ and $\ell^{p}$ spaces which are zero-
divisors.

###### Key words and phrases:

Banach algebra, Composition operators, Hardy spaces, Measurable functions

###### 2010 Mathematics Subject Classification:

43A99, 47B33, 30H10, 28A20

The first author is supported by the Council of Scientific and Industrial
Research (CSIR) NET-JRF, New Delhi, India, through grant
$09/013(0891)/2019-EMR-I$

##  1\. Introduction

The notion of zero-divisor[12, 22] is well known from ring theory. In a ring
$\mathfrak{R},$ an element $z\in\mathfrak{R}$ is called a left $(right)$ zero-
divisor in $\mathfrak{R}$ if $zx=0~{}(xz=0)$ for some $0\neq
x\in\mathfrak{R}.$ An element $z\in\mathfrak{R}$ is called zero-divisor if it
is either a left (or right) zero-divisor. The concept of topological divisor
of zero(TDZ)[7, 22] is an important generalization in a Banach algebra,
introduced by Shilov in [21], with significant later contributions by Zelazko,
Kaplansky et al. Let $\mathfrak{B}$ be a Banach algebra. An element
$x\in\mathfrak{B}$ is a TDZ in $\mathfrak{B}$ if

$xx_{n}\to 0\text{~{}or~{}}x_{n}x\to 0$

for some sequence $\\{x_{n}\\}_{n=1}^{\infty}$ of elements of $\mathfrak{B}$
satisfying

$\|x_{n}\|=1,\quad n=1,2,...$

Bhatt and Dedania in [4] gave a classification of Banach algebras in which
every element is a TDZ. The purpose of the present paper is to determine TDZ
in some Banach algebras which do not fall in the preceding category.
In this paper, $\mathbb{N}$ denotes the set of all positive integers,
$\mathbb{C}$ the set of complex numbers, $|A|$ the cardinality of a set $A$
and $\mathcal{R}(f)$ denotes the range of a function $f$. Let
$\mathbb{D}=\\{z\in\mathbb{C}:|z|<1\\},$
$\mathbb{\overline{D}}=\\{z\in\mathbb{C}:|z|\leq 1\\}$ and
$\mathbb{T}=\\{z\in\mathbb{C}:|z|=1\\}$ be the unit circle. Let
$\mathcal{A}(\mathbb{D})$ denote the Banach algebra of all continuous
functions on $\mathbb{\overline{D}}$ which are analytic in $\mathbb{D}$ under
the sup-norm and $L^{\infty}(\mu)$ be the Banach algebra of all essentially
bounded complex-valued measurable functions on $X$ under the essential
supremum norm. Further, $\sigma_{p}(M_{h}),~{}\sigma_{c}(M_{h})$ and
$\sigma(M_{h})$ denotes the point spectrum, the continuous spectrum and the
spectrum of $M_{h}$ respectively. Further, $\mathcal{B}(L^{p}(\mu))~{}(1\leq
p\leq\infty)$ denotes the Banach algebra of all bounded linear operators on
$L^{p}(\mu).$
The current article is aimed at finding zero-divisors and TDZ in
$\mathcal{A}(\mathbb{D})$ and $L^{\infty}(\mu)$ in its full generality. It
also determines the conditions under which the multiplication operator on
$L^{p}(\mu)~{}(1\leq p\leq\infty)$ and the composition operator on
$L^{2}(\mu)$ are TDZ. The paper is structured as follows.
In Section 2, we cite certain definitions and results which will be used in
this paper. Section 3.1 is devoted for the characterization of TDZ in
$\mathcal{A}(\mathbb{D}).$ In section 3.2, we characterize the zero-divisors
and TDZ in $L^{\infty}(\mu).$ Section 3.3 is devoted to find a necessary and
sufficient condition for a multiplication operator $M_{h}$ to be TDZ in
$\mathcal{B}(L^{p}(\mu)).$ In Section 3.4, we find a necessary and sufficient
condition for the composition operator $C_{\phi}$ on $L^{2}(\mu)$ to be a TDZ
in $\mathcal{B}(L^{2}(\mu)).$ In Section 3.5 we classify the composition
operators on $\mathbb{H}^{p}(\mathbb{D})$ which are zero-divisors in
$\mathcal{B}(\mathbb{H}^{p}(\mathbb{D})).$ Finally, in Section 3.6, we
determine the composition operators on $\ell^{p}$ which are zero-divisors and
TDZ in $\mathcal{B}(\ell^{p}).$

##  2\. Preliminaries

In this section, we state few definitions and some known results which will be
used later.

######  Definition 2.1.

[9, 22] Let $\mathfrak{B}$ denote a Banach algebra with identity $1$. An
element $x\in\mathfrak{B}$ is called regular if there exists
$y\in\mathfrak{B}$ such that $xy=yx=1.$ Otherwise, $x\in\mathfrak{B}$ is
called a singular element.

######  Theorem 2.2.

[25] In a Banach algebra, the set of all TDZ is a closed set.

######  Definition 2.3.

[2, 10, 13] For $0<p<\infty$ the Hardy space $\mathbb{H}^{p}(\mathbb{D})$ is
the set of functions analytic on the unit disk for which

$\sup_{r\in(0,1)}\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{i\theta})|^{p}d\theta<\infty.$

Let $\mathbb{H}^{\infty}(\mathbb{D})$ be the Banach algebra of all bounded
analytic functions in $\mathbb{D}$ under the sup-norm.

######  Definition 2.4.

[8, 11, 20] Let $\phi$ be a holomorphic function on $\mathbb{D}$ with
$\phi(\mathbb{D})\subset\mathbb{D}.$ Then the equation $C_{\phi}f=f\circ\phi$
defines a composition operator $C_{\phi}$ on the space of holomorphic
functions in $\mathbb{D}.$ The Littlewood’s subordination principle[10, 15]
says that $C_{\phi}$ is bounded linear operator on
$\mathbb{H}^{p}(\mathbb{D})~{}(1\leq p\leq\infty).$

######  Theorem 2.5.

[5] Let $\phi:\mathbb{D}\to\mathbb{D}$ be an analytic map which maps
$\mathbb{D}$ conformally on to a Carath$\acute{e}$odory domain, then the
$\mathcal{R}(C_{\phi})$ is dense in $\mathbb{H}^{p}(\mathbb{D}),~{}1\leq
p<\infty.$

######  Theorem 2.6.

([16, 19]) Let $\phi:\mathbb{D}\to\mathbb{D}$ be an analytic map and $\phi$ is
a weak* generator of $H^{\infty}$, then the $\mathcal{R}(C_{\phi})$ is dense
in $\mathbb{H}^{p}(\mathbb{D}),~{}0<p<\infty.$

######  Definition 2.7.

([26]) Let $(X,\Omega,\mu)$ be a measure space and $Y$ denote a topological
space. Let $f:X\to Y$ be a measurable function. An element $y\in Y$ is in the
essential range of $f$ if for any neighborhood $U$ of $y$, $\mu(f^{-1}(U))>0.$
The set of all such elements is called the essential range of $f$ and is
denoted by ess.range$(f).$

######  Definition 2.8.

[1, 3, 17] Suppose $(X,\Omega,\mu)$ is a $\sigma-$finite measure space and
$1\leq p<\infty.$ Let $L^{p}(\mu)=\\{f:X\to\mathbb{C}:f\textrm{ is measurable
and }\int_{X}|f|^{p}d\mu<\infty\\}$ and
$L^{\infty}(\mu)=\\{f:X\to\mathbb{C}:f\textrm{ is measurable and ess.sup.
}|f|<\infty\\}.$ Note that

  1. (i)

For $1\leq p<\infty,~{}L^{p}(\mu)$ is a Banach space under the $L^{p}-$norm
defined as

$||f||_{p}=(\int|f|^{p}d\mu)^{\frac{1}{p}}~{}\forall~{}f\in L^{p}(\mu).$

  2. (ii)

$L^{\infty}(\mu)$ is a Banach algebra with identity under the norm defined as

$\|f\|=\|f\|_{\infty}=\inf\\{K:\mu\\{t:|f(t)|>K\\}=0\\}~{}\forall~{}f\in
L^{\infty}(\mu).$

######  Lemma 2.9.

[3, 9, 26] Suppose $(X,\Omega,\mu)$ is a measure space and $h\in
L^{\infty}(\mu).$ For $1\leq p\leq\infty,$ define the multiplication operator
$M_{h}:L^{p}(\mu)\to L^{p}(\mu)$ by $M_{h}(f)(x)=h(x)f(x)~{}\forall~{}x\in X$.
Then $M_{h}\in\mathcal{B}(L^{p}(\mu))$ and

  1. (i)

$\textrm{ess.range}(h)=\sigma(M_{h})=\\{\lambda\in\mathbb{C}:\forall~{}\epsilon>0,\mu(\\{x\in
X:|h(x)-\lambda|<\epsilon\\})>0\\}$

  2. (ii)

$\sigma_{p}(M_{h})=\\{\lambda\in\mathbb{C}:\mu(\\{x\in
X:h(x)=\lambda\\})>0\\}.$

######  Lemma 2.10.

[6] Let $\mathfrak{B}$ be a commutative Banach algebra and $x\in\mathfrak{B}$
be a TDZ. Then for each $y\in\mathfrak{B},\ xy$ is also a TDZ.

######  Theorem 2.11.

[6] Let $\mathcal{A}=\mathcal{A}(\mathbb{D})$ be the disk algebra. Let
$f(z)=\left(\frac{z-z_{0}}{2}\right)$ for some $z_{0}\in\mathbb{C}$. Then $f$
is TDZ in $\mathcal{A}$ if and only if $|z_{0}|=1.$

######  Definition 2.12.

[24, 23] If $(X,\Omega,\mu)$ is a $\sigma-\text{finite}$ measure space and
$\phi$ is a non-singular measurable transformation on $X$ into itself, then
the composition transformation $C_{\phi}$ on $L^{p}(\mu)~{}(1\leq p<\infty)$
is defined by the following equation.

$C_{\phi}f=f\circ\phi~{}\text{for every}~{}f\in L^{p}(\mu).$

If $C_{\phi}\in\mathcal{B}(L^{p}(\mu)),$ the Banach algebra of all bounded
linear operators on $L^{p}(\mu),$ then it is called a composition operator
induced by $\phi.$

######  Theorem 2.13.

[24] A measurable transformation $\phi:X\to X$ defines a composition operator
on $L^{p}(\mu)~{}(1\leq p<\infty)$ if and only if $\mu\phi^{-1}\ll\mu$ and
$\frac{d\mu\phi^{-1}}{d\mu}\in L^{\infty}(\mu),$ where
$\frac{d\mu\phi^{-1}}{d\mu}$ is Radon-Nikodym derivative of $\mu\phi^{-1}$
with respect to $\mu.$ In this case

$\displaystyle\|C_{\phi}\|=\|\frac{d\mu\phi^{-1}}{d\mu}\|_{\infty}^{\frac{1}{p}}.$

When $X=\mathbb{N}$ and $\mu$ is the counting measure on $\mathbb{N},$ then
$L^{p}(\mu)$ becomes $\ell^{p}.$ In this setting we have the following result.

######  Theorem 2.14.

[23] Let $\phi:\mathbb{N}\to\mathbb{N}$ be a self map. Then the composition
transformation $C_{\phi}$ induced by $\phi$ on $\ell^{p}~{}(1\leq p<\infty)$
and defined as

$C_{\phi}(f)=\sum_{n=1}^{\infty}f(n)\raisebox{0.0pt}[4.30554pt][4.30554pt]{$\chi$}_{\phi^{-1}(n)}$

is a bounded linear operator if and only if the set
$\\{|\phi^{-1}(n)|:n\in\mathbb{N}\\}$ is bounded.

######  Theorem 2.15.

[24] Let $\phi:X\to X$ be a measurable transformation and let $C_{\phi}$ be
the composition operator induced by $\phi$ on $L^{2}(\mu).$ Then
$C_{\phi}^{*}C_{\phi}=M_{\frac{d\mu\phi^{-1}}{d\mu}},$ where
$\frac{d\mu\phi^{-1}}{d\mu}$ is Radon-Nikodym derivative of $\mu\phi^{-1}$
with respect to $\mu.$

######  Theorem 2.16.

[23, 24] Let $\phi$ be a self map on $\mathbb{N}$ and $C_{\phi}$ be the
composition operator induced by $\phi$ on $\ell^{p}~{}(1\leq p<\infty).$ Then,

  1. $(1)$

$C_{\phi}$ is one-one if and only if $\phi$ is onto;

  2. $(2)$

$C_{\phi}$ is onto if and only if $\phi$ is one-one;

  3. $(3)$

$C_{\phi}$ is invertible if and only if $\phi$ is invertible;

##  3\. Main results

###  3.1. TDZ in $\mathcal{A}(\mathbb{D})$

In the following theorem we characterize the TDZ in $\mathcal{A}(\mathbb{D}).$

######  Theorem 3.1.

An element $f\in\mathcal{A}(\mathbb{D})$ is a TDZ if and only if there exists
$z_{0}\in\mathbb{T}$ such that $f(z_{0})=0.$

###### Proof.

Let $f\in\mathcal{A}(\mathbb{D})$ be a TDZ. Then there exists a sequence
$\\{f_{n}\\}_{n=1}^{\infty}$ in $\mathcal{A}(\mathbb{D})$ such that
$\|f_{n}\|=1\ \forall\ n\in\mathbb{N}$, and $\|ff_{n}\|\to 0.$ Suppose that
$f(z)\neq 0$ for all $z\in\mathbb{T}.$ Then $|f|>0$ on $\mathbb{T}.$ Now the
continuity of $|f|$ on the compact set $\mathbb{T}$ implies that $|f|$ attains
its maximum and minimum values on $\mathbb{T}.$ Therefore, there exists a
$\delta>0$ such that

$|f(z)|>\delta~{}\forall z\in\mathbb{T}.$

Further, for each $n\in\mathbb{N}$

$\|ff_{n}\|=\max_{z\in\mathbb{T}}|f(z)f_{n}(z)|>\delta\cdot\max_{z\in\mathbb{T}}|f_{n}(z)|=\delta\cdot\|f_{n}\|=\delta>0.$

This contradicts the fact $\|ff_{n}\|\to 0.$ Hence there exists a point
$z_{0}\in\mathbb{T}$ such that $f(z_{0})=0.$
Conversely, assume that there exists $z_{0}\in\mathbb{T}$ such that
$f(z_{0})=0.$ Since $f$ is continuous on $\mathbb{\overline{D}}$ and analytic
on $\mathbb{D},$ by Mergelyan’s theorem[18], there exists a sequence of
polynomials $\\{p_{n}\\}_{n=1}^{\infty}$ that uniformly converges to $f.$ Let
$q_{n}(z)=p_{n}(z)-p_{n}(z_{0}).$ Then $\\{q_{n}\\}_{n=1}^{\infty}$ uniformly
converges to $f$ and $q_{n}(z_{0})=0.$ Hence
$q_{n}(z)=(z-z_{0})r_{n}(z)~{}\forall~{}n\geq 1.$ Now, since
$z_{0}\in\mathbb{T},$ it follows from the Theorem 2.11 that $h(z)=(z-z_{0})$
is a TDZ. Consequently, by Lemma 2.10, $q_{n}(z)$ is also a TDZ for each
$n\geq 1$. Since $\\{q_{n}\\}_{n=1}^{\infty}$ converges to $f,$ hence by
Theorem 2.2, $f$ is also a TDZ. ∎

######  Remark 3.2.

We realise that an alternative proof of the above Theorem 3.1 can be obtained
using the notion of Shilov boundary and applying the result [[14], Corollary
3.4.6, page 171].

###  3.2. TDZ in $L^{\infty}(\mu)$

The following theorem characterizes the zero-divisors in $L^{\infty}(\mu).$

######  Theorem 3.3.

Let $f\in L^{\infty}(\mu)$. Then $f$ is a zero-divisor if and only if
$\mu(E^{c})>0,$ where $E=\\{x\in X:f(x)\neq 0\\}.$

###### Proof.

Suppose $E=\\{x\in X:f(x)\neq 0\\}$ and $\mu(E^{c})>0.$ Let
$g=\raisebox{0.0pt}[4.30554pt][4.30554pt]{$\chi$}_{E^{c}}.$ Then, $g\neq 0$
and $f\cdot g=0.$ This implies that $f$ is a zero- divisor.
Conversely, suppose that $f$ is a zero-divisor. Then there exists a $0\neq
g\in L^{\infty}(\mu)$ such that $f\cdot g=0.$ Let $E=\\{x\in X:f(x)\neq 0\\}.$
Then $E$ is a measurable set. Also, $f\cdot g=0$ implies $E\subset\\{x\in
X:g(x)=0\\}$. This implies that $\\{x\in X:g(x)\neq 0\\}\subset E^{c}.$ Since
$g\neq 0,$ then $\mu(\\{x\in X:g(x)\neq 0\\})>0.$ Hence $\mu(E^{c})>0.$ ∎

The next theorem determines the topological divisors of zero in
$\L^{\infty}(\mu).$

######  Theorem 3.4.

Let $f\in L^{\infty}(\mu).$ Then $f$ is a TDZ if and only if there exists a
sequence $\\{E_{n}\\}_{n=1}^{\infty}$ of measurable sets in $X$ with
$\mu(E_{n})>0$ for each $n\geq 1$ and $\|f|_{E_{n}}\|\to 0.$

###### Proof.

Suppose there exists a sequence $\\{E_{n}\\}_{n=1}^{\infty}$ of measurable
sets with $\mu(E_{n})>0$ for each $n\geq 1$ and $\|f|_{E_{n}}\|\to 0$ as
$n\to\infty.$ Let
$g_{n}=\raisebox{0.0pt}[4.30554pt][4.30554pt]{$\chi$}_{E_{n}}.$ Then
$\|g_{n}\|=1$ for each $n\geq 1$ and $fg_{n}=f|_{E_{n}}.$ Hence
$\|fg_{n}\|=\|f|_{E_{n}}\|\to 0$ as $n\to\infty.$ This implies that $f$ is a
TDZ.
Conversely, suppose $f\in L^{\infty}(\mu)$ is a TDZ. Then there exists a
sequence $\\{g_{n}\\}_{n=1}^{\infty}$ in $L^{\infty}(\mu)$ with $\|g_{n}\|=1$
for each $n\geq 1$ and $\|fg_{n}\|\to 0.$
Let $E_{n}=\\{x\in X:|g_{n}(x)|>1-\frac{1}{n}\\}.$ Since $\|g_{n}\|=1,$ hence
$\mu(E_{n})>0.$ Clearly, $\|fg_{n}\|\to 0$ implies $\|fg_{n}|_{E_{n}}\|\to 0$
as $n\to\infty.$
Further, for $x\in E_{n}$

$|fg_{n}(x)|=|f(x)||g_{n}(x)|>|f(x)|(1-\frac{1}{n}).$

Consequently

$|f(x)|<\frac{1}{(1-\frac{1}{n})}|fg_{n}(x)|\ \forall\ x\in E_{n}.$

Hence

$\|f|_{E_{n}}\|\leq\frac{1}{(1-\frac{1}{n})}\|fg_{n}|_{E_{n}}\|\to 0.$

Thus

$\|f|_{E_{n}}\|\to 0.$

∎

The following Remark can be easily obtained.

######  Remark 3.5.

An element $f\in L^{\infty}(\mu)$ is regular if and only if there exists a
$\lambda_{0}>0$ such that $|f|\geq\lambda_{0}$ a.e. on $X.$

######  Corollary 1.

In $L^{\infty}(\mu),$ The collection of all TDZ is equivalent to the
collection of all singular elements.

###### Proof.

Clearly, a TDZ is a singular element.
Let $f\in L^{\infty}(\mu)$ be a singular element. Then $f$ is not regular in
$L^{\infty}(\mu).$ Therefore, for each $n\in\mathbb{N},$ there exists a
measurable set $E_{n}$ with $0<\mu(E_{n})<\infty$ such that
$|f|_{E_{n}}|<\frac{1}{n}.$ Hence $\|f|_{E_{n}}\|\leq\frac{1}{n}\ \forall\
n\geq 1.$ Therefore $\|f|_{E_{n}}\|\to 0$ as $n\to\infty.$
Consequently, by Theorem $\ref{anurag8},$ $f$ is a TDZ. ∎

######  Remark 3.6.

We note that the Corollary 1 also follows from the result that any non-
invertible element in a symmetric normed algebra without radicals is a TDZ.
(See [12], Corollary $2$, sec.$10.2$).

######  Theorem 3.7.

Let $h\in L^{\infty}(\mu)$ and $M_{h}$ be the multiplication operator on
$L^{\infty}(\mu)$ induced by $h.$ Then $h$ is a TDZ if and only if
$0\in\textrm{ess.range}(h)=\sigma(M_{h}).$

###### Proof.

Let $0\in
ess.range(h)=\sigma(M_{h})=\\{\lambda\in\mathbb{C}:\forall\epsilon>0,\mu(\\{x\in
X:|h(x)-\lambda|<\epsilon\\})>0\\}.$ Then for each $n\geq 1,$ $\mu(\\{x\in
X:|h(x)|<\frac{1}{n}\\})>0.$ Let $E_{n}=\\{x\in X:|h(x)|<\frac{1}{n}\\}.$ Thus
for each $n\geq 1,~{}\mu(E_{n})>0$ and $|h|_{E_{n}}|<\frac{1}{n}.$ This
implies that $\|h|_{E_{n}}\|\leq\frac{1}{n}\ \forall\ n\in\mathbb{N}.$
Therefore $\|h|_{E_{n}}\|\to 0$ as $n\to\infty.$ Hence $h$ is a TDZ.
Conversely, suppose $h$ is a TDZ. Then, by Corollary $\ref{anurag9},$ $h$ is
not a regular element. Therefore, for each $\epsilon>0,~{}\mu(\\{x\in
X:|h(x)-0|<\epsilon\\})>0.$ This implies that $0\in$
ess.range$(h)=\sigma(M_{h}).$ ∎

######  Remark 3.8.

Let $h\in L^{\infty}(\mu)$ be a TDZ. Then

  1. (1)

$0\in\sigma_{p}(M_{h})$ if and only if $\mu(\\{x\in X:h(x)=0\\})>0.$

  2. (2)

$0\in\sigma_{c}(M_{h})$ if and only if $\mu(\\{x\in X:h(x)=0\\})=0$ and
$\mu(\\{x\in X:|h(x)|<\epsilon\\})>0$ for each $\epsilon>0.$

######  Theorem 3.9.

Let $h\in L^{\infty}(\mu)$ and $M_{h}:L^{\infty}(\mu)\to L^{\infty}(\mu)$ be
the multiplication operator. Then $h$ is a zero-divisor if and only if
$0\in\sigma_{p}(M_{h}).$

###### Proof.

Let $0\in\sigma_{p}(M_{h}).$ Then $\exists~{}0\neq f\in L^{\infty}(\mu)$ such
that $M_{h}(f)=0\cdot f=0.$ This implies that $h\cdot f=0.$ Hence $h$ is a
zero-divisor in $L^{\infty}(\mu)$ .
Conversely, suppose $h\in L^{\infty}(\mu)$ is a zero-divisor. Then
$\exists~{}0\neq f\in L^{\infty}(\mu)$ such that $h\cdot f=0=0\cdot f$ a.e..
This implies that $M_{h}(f)=0\cdot f.$ Hence $0\in\sigma_{p}(M_{h}).$ ∎

###  3.3. Multiplication operator as TDZ

In this section, we state and prove the necessary and sufficient condition for
the multiplication operator $M_{h}$ to be TDZ in $\mathcal{B}(L^{p}(\mu)).$

######  Theorem 3.10.

Let $h\in L^{\infty}(\mu)$ and $1\leq p\leq\infty.$ Define the multiplication
operator $M_{h}:L^{p}(\mu)\to L^{p}(\mu)$ by $M_{h}(f)=h\cdot f=hf$. Then
$M_{h}$ is a TDZ in $\mathcal{B}(L^{p}(\mu))$ if and only if $h$ is a TDZ in
$L^{\infty}(\mu).$

###### Proof.

Suppose $h\in L^{\infty}(\mu)$ be a TDZ. Let $\\{h_{n}\\}_{n=1}^{\infty}$ be a
sequence in $L^{\infty}(\mu)$ such that

$\|h_{n}\|=1\ \forall\ n\in\mathbb{N}~{}and~{}\|hh_{n}\|\to 0.$

For each $n\geq 1,$ define $M_{h_{n}}:L^{p}(\mu)\to L^{p}(\mu)$ as
$M_{h_{n}}(f)=h_{n}f.$ Then

$\|M_{h_{n}}\|=\|h_{n}\|=1\ \forall\
n\in\mathbb{N}~{}and~{}M_{h}M_{h_{n}}(f)=hh_{n}f.$

Hence $\|M_{h}M_{h_{n}}\|=\|hh_{n}\|\to 0.$ This shows that $M_{h}$ is a TDZ
in $\mathcal{B}(L^{p}(\mu))$.
For the converse, suppose $h\in L^{\infty}(\mu)$ is not a TDZ in
$L^{\infty}(\mu)$. Then by Corollary1, $h$ is a regular element in
$L^{\infty}(\mu)$. Therefore there exists a $g\in L^{\infty}(\mu)$ such that
$h\cdot g=g\cdot h=1.$ This implies that $M_{hg}=\text{I}.$ Hence
$M_{hg}=M_{h}M_{g}=M_{g}M_{h}=\text{I}.$ This shows that $M_{h}$ is regular
element in $\mathcal{B}(L^{p}(\mu))$. Hence $M_{h}$ can not be a TDZ.

∎

###  3.4. Composition operator on $L^{2}(\mu)$ as TDZ

######  Theorem 3.11.

Let $H$ be a Hilbert space and $T\in\mathcal{B}(H).$ Then $T$ is a TDZ in
$\mathcal{B}(H)$ if and only if $T^{*}T$ is a TDZ in $\mathcal{B}(H).$

###### Proof.

Let $T\in\mathcal{B}(H)$ be a TDZ. Hence there exists a sequence
$\\{T_{n}\\}_{n=1}^{\infty}$ in $\mathcal{B}(H)$ such that
$\|T_{n}\|=1~{}\forall~{}n\in\mathbb{N}$ and $\|TT_{n}\|\to 0.$ Since
$\|T_{n}\|=\|T^{*}_{n}\|=1~{}\forall~{}n\in\mathbb{N}$ and $\|T^{*}\|=\|T\|,$
consequently

$\|(T^{*}T)T_{n}\|\leq\|T^{*}\|\|TT_{n}\|\to 0,$

and

$\|T^{*}_{n}(T^{*}T)\|\leq\|T^{*}_{n}T^{*}\|\|T\|=\|TT_{n}\|\|T\|\to 0.$

Hence $T^{*}T$ is a TDZ.
Conversely, suppose $T^{*}T$ is a TDZ in $\mathcal{B}(H).$ Then there exists a
sequence $\\{T_{n}\\}_{n=1}^{\infty}$ in $\mathcal{B}(H)$ such that
$\|T_{n}\|=1~{}\forall~{}n\in\mathbb{N}$ and $\|T^{*}TT_{n}\|\to 0.$ Observe
that, for each $x\in H$

$\|TT_{n}x\|^{2}=\langle
T^{*}TT_{n}x,\;T_{n}x\rangle\leq\|T^{*}TT_{n}x\|\|T_{n}x\|.$

Hence

$\|TT_{n}\|^{2}\leq\|T^{*}TT_{n}\|\|T_{n}\|=\|T^{*}TT_{n}\|\to 0.$

Thus, $T$ is TDZ. ∎

The following Remark can be readily obtained.

######  Lemma 3.12.

Let $H$ be a Hilbert space and $T\in\mathcal{B}(H).$ Then $T$ is a left(right)
TDZ in $\mathcal{B}(H)$ if and only if $T^{*}$ is a right(left) TDZ in
$\mathcal{B}(H).$

###### Proof.

Let $T\in\mathcal{B}(H)$ be a left TDZ. Then there exists a sequence
$\\{T_{n}\\}_{n=1}^{\infty}$ in $\mathcal{B}(H)$ such that
$\|T_{n}\|=1~{}\forall~{}n\in\mathbb{N}$ and $\|TT_{n}\|\to 0.$ Since
$\|T_{n}\|=\|T^{*}_{n}\|=1~{}\forall~{}n\in\mathbb{N}$ and

$\|(T^{*}_{n}T^{*})\|=\|(TT_{n})^{*}\|=\|TT_{n}\|\to 0.$

Hence $T^{*}$ is a right TDZ.
Conversely, suppose $T\in\mathcal{B}(H)$ and let $T^{*}$ be a right TDZ. Then,
from the above argument, $(T^{*})^{*}=T$ is a left TDZ in $\mathcal{B}(H).$ ∎

######  Theorem 3.13.

Composition operator $C_{\phi}$ on $L^{2}(\mu)$ is a TDZ in
$\mathcal{B}(L^{2}(\mu))$ if and only if we can find a sequence
$\\{E_{n}\\}_{n=1}^{\infty}$ of measurable sets in $X$ such that
$\mu(E_{n})>0$ for each $n\geq 1$ and
$\|\frac{d\mu\phi^{-1}}{d\mu}|_{E_{n}}\|\to 0.$

###### Proof.

Let $C_{\phi}\in\mathcal{B}(L^{2}(\mu))$ be a TDZ. Then by Theorem 3.11,
$C_{\phi}^{*}C_{\phi}$ is also a TDZ. Further, by Theorem 2.15,

$C_{\phi}^{*}C_{\phi}=M_{\frac{d\mu\phi^{-1}}{d\mu}}.$

Hence, by Theorem 3.10, $\frac{d\mu\phi^{-1}}{d\mu}$ is a TDZ in
$L^{\infty}(\mu)$. Consequently, by Theorem 3.4, we can find a sequence
$\\{E_{n}\\}_{n=1}^{\infty}$ of measurable sets in $X$ with $\mu(E_{n})>0$ for
each $n\geq 1$ and $\|\frac{d\mu\phi^{-1}}{d\mu}|_{E_{n}}\|\to 0.$
Conversely, suppose that we have a sequence $\\{E_{n}\\}_{n=1}^{\infty}$ of
measurable sets in $X$ such that $\mu(E_{n})>0$ for each $n\geq 1$ and
$\|\frac{d\mu\phi^{-1}}{d\mu}|_{E_{n}}\|\to 0.$ Then, by Theorem 3.4,
$\frac{d\mu\phi^{-1}}{d\mu}$ is a TDZ in $L^{\infty}(\mu)$. Now, combining
Theorem 3.10 and Theorem 3.11, the desired conclusion follows. ∎

######  Example 3.14.

Let $\mu$ denote the counting measure on $\mathbb{N}$ and
$\phi:\mathbb{N}\to\mathbb{N}$ be defined as $\phi(n)=n+1.$ Let $C_{\phi}$
denote the composition operator on $\ell^{2}$ induced by $\phi.$ It is easy to
see that

$\frac{d\mu\phi^{-1}}{d\mu}(n)=\begin{cases}{0},&n=1,\\\ 1,&n>1.\end{cases}$

Hence, by above theorem, $C_{\phi}$ is a TDZ in $\mathcal{B}(\ell^{2}).$

###  3.5. Composition operator on $\mathbb{H}^{p}(\mathbb{D})~{}~{}(1\leq
p<\infty)$ as zero-divisor

In this section, we give a characterization of the composition operators on
the Hardy space $\mathbb{H}^{p}(\mathbb{D})$ which are zero-divisors in
$\mathcal{B}(\mathbb{H}^{p}(\mathbb{D})).$ We begin with the following
theorem.

######  Proposition 1.

Let $X$ be a Banach space and $T\in\mathcal{B}(X).$ Then $T$ is a right zero-
divisor if and only if $\mathcal{R}(T)$ is not dense in $X$.

###### Proof.

Suppose $T\in\mathcal{B}(X)$ and $\mathcal{R}(T)$ is not dense in $X.$ Then by
Hahn-Banach theorem, there exists a non-zero bounded linear functional $f$ on
$X$ such that $f(y)=0$ for all $y\in\overline{T(X)}.$ For any nonzero vector
$x_{0}\in X,$ define $S:=x_{0}\bigotimes f:X\to X$ as $S(x)=f(x)x_{0}.$ Then,
$S$ is a nonzero element of $\mathcal{B}(X)$ such that

$\displaystyle(ST)(x)$ $\displaystyle=f(T(x))x_{0}$
$\displaystyle=0~{}~{}\text{for all}~{}x\in X.$

Hence $T$ is a right zero-divisor.
Conversely, suppose $\mathcal{R}(T)$ is dense in $X.$ Let $S\in\mathcal{B}(X)$
be such that $S(Tx)=0~{}\forall~{}x\in X.$ Then $S\equiv 0$ on
$\mathcal{R}(T).$ Since $\mathcal{R}(T)$ is dense in $X$ and $S$ is
continuous, it follows that $S\equiv 0.$ Consequently, $T$ can not be a right
zero-divisor. ∎

######  Remark 3.15.

Proposition 1 implies that $C_{\phi}\in B(\mathbb{H}^{2}(\mathbb{D}))$ is a
right zero-divisor if and only if $\mathcal{R}(C_{\phi})$ is not dense in
$\mathbb{H}^{2}(\mathbb{D}).$

######  Theorem 3.16.

Let $\phi:\mathbb{D}\to\mathbb{D}$ be any analytic map which either maps
$\mathbb{D}$ conformally on to a Carath$\acute{e}$odory domain or $\phi$ is a
weak* generator of $\mathbb{H}^{\infty}(\mathbb{D}).$ Then $C_{\phi}$ is not a
right zero-divisor in $\mathfrak{B}$.

###### Proof.

Let $\phi:\mathbb{D}\to\mathbb{D}$ be analytic map which maps $\mathbb{D}$
conformally on to a Carath$\acute{e}$odory domain. Then by Theorem 2.5,
$\mathcal{R}(C_{\phi})$ is dense in $\mathbb{H}^{2}(\mathbb{D}).$ Now, by
Proposition 1, $C_{\phi}$ can not be a right zero-divisor in
$\mathfrak{B}(\mathbb{H}^{2}(\mathbb{D}))$.
Further, if $\phi:\mathbb{D}\to\mathbb{D}$ is analytic map which is a weak*
generator of $\mathbb{H}^{\infty}(\mathbb{D}),$ then, by Theorem 2.6,
$\mathcal{R}(C_{\phi})$ is dense in $\mathbb{H}^{2}(\mathbb{D}).$ Then again,
$C_{\phi}$ can not be a right zero-divisor in
$\mathfrak{B}(\mathbb{H}^{2}(\mathbb{D}))$. ∎

######  Theorem 3.17.

The composition operator $C_{\phi}\in\mathcal{B}(\mathbb{H}^{p}(\mathbb{D}))$
is left zero-divisor if and only if $\phi$ is a constant function.

###### Proof.

Let $\phi:\mathbb{D}\to\mathbb{D}$ be defined as
$\phi(z)=z_{0}~{}\forall~{}z\in\mathbb{D}.$ Let $g_{0}(z)=z-z_{0}$ and
$T:\mathbb{H}^{p}(\mathbb{D})\to\mathbb{H}^{p}(\mathbb{D})$ be defined as
$T(f)=g_{0}\cdot f.$ Now, for each $f\in\mathbb{H}^{p}(\mathbb{D})$ and
$z\in\mathbb{D}$

$\displaystyle C_{\phi}(T)(f)(z)$ $\displaystyle=(g_{0}\cdot f)(\phi)(z)$
$\displaystyle=(g_{0}\cdot f)(z_{0})$ $\displaystyle=0.$

Hence $C_{\phi}$ is a left zero-divisor.
Conversely, suppose $C_{\phi}$ is a left zero-divisor. Then there exists
$0\neq T\in\mathcal{B}(\mathbb{H}^{p}(\mathbb{D}))$ such that
$C_{\phi}(T)(f)=0~{}\forall f\in\mathbb{H}^{p}(\mathbb{D}).$ This implies that

$T(f)|_{\mathcal{R}(\phi)}=0~{}\forall f\in\mathbb{H}^{p}(\mathbb{D}).$

Hence $\mathcal{R}(\phi)$ must be a singleton set. ∎

######  Theorem 3.18.

If $\phi:\mathbb{D}\to\mathbb{D}$ is a constant function, then
$C_{\phi}\in\mathcal{B}(\mathbb{H}^{p}(\mathbb{D}))$ is right zero-divisor.

###### Proof.

Let $\phi:\mathbb{D}\to\mathbb{D}$ be defined as
$\phi(z)=z_{0}~{}\forall~{}z\in\mathbb{D}$ and
$f\in\mathbb{H}^{p}(\mathbb{D}).$ Then $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}.$
Now, define $T:\mathbb{H}^{p}(\mathbb{D})\to\mathbb{H}^{p}(\mathbb{D})$ as

$T(f)=\sum_{n=0}^{\infty}a_{n+1}z^{n}.$

Then for each $f\in\mathbb{H}^{p}(\mathbb{D}),$

$(T\circ C_{\phi})(f)(z)=T((f\circ\phi)(z))=0.$

Hence $C_{\phi}$ is a right zero-divisor. ∎

######  Remark 3.19.

The following lemma shows that the converse of the above Theorem 3.18 is not
true.

######  Lemma 3.20.

For each integer $k\geq 2,$ let $\phi:\mathbb{D}\to\mathbb{D}$ be defined as
$\phi(z)=z^{k}~{}\forall~{}z\in\mathbb{D}.$ Then $C_{\phi}$ is a right zero-
divisor.

###### Proof.

Let $\phi(z)=z^{k}$ and $f\in\mathbb{H}^{2}(\mathbb{D}).$ Then

$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n},~{}\text{where}~{}\sum_{n=0}^{\infty}|a_{n}|^{2}<\infty.$

Hence,

$(C_{\phi}f)(z)=f(z^{k})=\sum_{n=0}^{\infty}a_{n}z^{nk}.$

Now, for $g(z)=\sum_{n=0}^{\infty}b_{n}z^{n}\in\mathbb{H}^{2}(\mathbb{D}),$
define $T:\mathbb{H}^{2}(\mathbb{D})\to\mathbb{H}^{2}(\mathbb{D})$ as

$T(g)=\sum_{n=0}^{\infty}a_{nk+1}z^{nk+1}.$

Then $T(C_{\phi}f)=0~{}\forall f\in\mathbb{H}^{2}(\mathbb{D}).$ Hence
$C_{\phi}$ is a right zero-divisor. ∎

###  3.6. Composition operator on $\ell^{p}~{}(1\leq p<\infty)$ space as TDZ

In this section, we give a characterization of the composition operators on
$\ell^{p}$ which are zero-divisors and topological divisors of zero in
$\mathcal{B}(\ell^{p}).$

######  Theorem 3.21.

$C_{\phi}\in\mathcal{B}(\ell^{p})$ is a right zero-divisor if and only if
$\phi$ is not injective.

###### Proof.

Suppose $\phi$ is injective. Let $T\in\mathcal{B}(\ell^{p})$ be such that
$TC_{\phi}=0.$ Then

$TC_{\phi}(\raisebox{0.0pt}[4.30554pt][4.30554pt]{$\chi$}_{n})=T(\raisebox{0.0pt}[4.30554pt][4.30554pt]{$\chi$}_{\phi^{-1}(n)})=0\
\text{ for each }\ n\geq 1.$

Since $\phi$ is injective, therefore $|\phi^{-1}(n)|\leq 1\ \forall\
n\in\mathbb{N}$. Also

$\bigcup_{n=1}^{\infty}\phi^{-1}(n)=\mathbb{N}.$

Hence, it follows that
$T(\raisebox{0.0pt}[4.30554pt][4.30554pt]{$\chi$}_{n})=0~{}\forall~{}n\geq 1.$
This implies that $T=0.$ Hence $C_{\phi}$ can not be a right zero-divisor.
Conversely, suppose $\phi$ is not injective. Then $C_{\phi}$ is not
surjective. Hence there exists an
$f_{0}\in\ell^{p}\backslash\mathcal{R}(C_{\phi})$. Let
$\mathfrak{B_{1}}=\\{f_{\alpha}:\alpha\in\Delta_{1}\\}$ be a Hamel basis for
$\mathcal{R}(C_{\phi})$ and $\mathfrak{B}$ be a Hamel basis for $\ell^{p}$
containing $\mathfrak{B_{1}}$ and $f_{0}.$ Now, define $T:\ell^{p}\to\ell^{p}$
as

$T(f)=\begin{cases}f_{0},&\text{if}~{}f=f_{0}\\\
0,&\text{if}~{}f\in\mathfrak{B}\backslash\\{f_{0}\\}.\end{cases}$

Clearly, $T$ defines a bounded linear operator on $\ell^{p}$ and for each
$f\in\ell^{p},$

$TC_{\phi}(f)=T(C_{\phi}f)=0.$

This implies that $C_{\phi}$ is a right zero-divisor. ∎

######  Theorem 3.22.

$C_{\phi}\in\mathcal{B}(\ell^{p})$ is a left zero-divisor if and only if
$\phi$ is not surjective.

###### Proof.

Suppose $\phi$ is surjective. Then $C_{\phi}$ is injective. Let
$T\in\mathcal{B}(\ell^{p})$ be such that $C_{\phi}T=0.$ Then for each $n\geq
1,~{}C_{\phi}T(\raisebox{0.0pt}[4.30554pt][4.30554pt]{$\chi$}_{n})=0.$ This
implies that,
$T(\raisebox{0.0pt}[4.30554pt][4.30554pt]{$\chi$}_{n})=0~{}\forall~{}n\geq 1.$
Therefore $T=0$ and hence, $C_{\phi}$ can not be a left zero-divisor.
Conversely, suppose $\phi$ is not surjective. Then $C_{\phi}$ is not
injective. Hence there exists $0\neq f_{0}\in\ell^{p}$ such that
$C_{\phi}(f_{0})=0.$ Define $T:\ell^{p}\to\ell^{p}$ as,

$T(f)=f(1)f_{0}~{}~{}\forall~{}f\in\ell^{p}.$

Clearly, $T\neq 0$ since
$T(\raisebox{0.0pt}[4.30554pt][4.30554pt]{$\chi$}_{1})=f_{0}\neq 0.$ Also, for
each $f\in\ell^{p},$

$\|Tf\|=|f(1)|\|f_{0}\|\leq\|f\|\|f_{0}\|.$

Hence $T$ is bounded, and for each $f\in\ell^{p},$
$C_{\phi}T(f)=C_{\phi}(f(1)f_{0})=f(1)C_{\phi}(f_{0})=0.$ Hence $C_{\phi}T=0.$
Therefore $C_{\phi}$ is a left zero-divisor. ∎

######  Theorem 3.23.

$C_{\phi}\in\mathcal{B}(\ell^{p})$ is a TDZ if and only if $\phi$ is not
invertible.

###### Proof.

If $\phi$ is invertible then, by Theorem $\ref{anurag7},$ $C_{\phi}$ is also
invertible. Hence $C_{\phi}$ can not be TDZ.
Conversely, suppose $\phi$ is not invertible. If $\phi$ is not injective, then
by Theorem $\ref{Anurag12},$ $C_{\phi}$ is a right zero-divisor. Further, if
$\phi$ is not surjective, then by Theorem $\ref{anurag13},$ $C_{\phi}$ is a
left zero-divisor. Therefore $C_{\phi}$ is a TDZ. ∎

######  Remark 3.24.

Let $\phi_{1}$ and $\phi_{2}$ be self maps on $\mathbb{N}$ such that either
$\phi_{1}$ is not surjective or $\phi_{2}$ is not injective, then
$C_{\phi_{2}}\circ C_{\phi_{1}}$ is a TDZ in $\mathcal{B}(\ell^{p}).$

###### Proof.

Suppose either $\phi_{1}$ is not surjective or $\phi_{2}$ is not injective,
then $\phi_{1}\circ\phi_{2}$ is not invertible. Hence by above theorem,
$C_{\phi_{1}\circ\phi_{2}}$ is a TDZ. But,
$C_{\phi_{1}\circ\phi_{2}}=C_{\phi_{2}}\circ C_{\phi_{1}}.$ Therefore
$C_{\phi_{2}}\circ C_{\phi_{1}}$ is a TDZ. ∎

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