# Perturbation analysis of bounded homogeneous generalized inverses on Banach
spaces

Jianbing Cao
Department of mathematics, Henan Institute of Science and Technology
Xinxiang, Henan, 453003, P.R. China
Department of Mathematics, East China Normal University,
Shanghai 200241, P.R. China
Email<EMAIL_ADDRESS>Yifeng Xue
Department of Mathematics, East China Normal University,
Shanghai 200241, P.R. China Email<EMAIL_ADDRESS>Corresponding
author

###### Abstract

Let $X,Y$ be Banach spaces and $T:X\to Y$ be a bounded linear operator. In
this paper, we initiate the study of the perturbation problems for bounded
homogeneous generalized inverse $T^{h}$ and quasi–linear projector generalized
inverse $T^{H}$ of $T$. Some applications to the representations and
perturbations of the Moore–Penrose metric generalized inverse $T^{M}$ of $T$
are also given. The obtained results in this paper extend some well–known
results for linear operator generalized inverses in this field.

2010 Mathematics Subject Classification: Primary 47A05; Secondary 46B20

Key words: homogeneous operator, stable perturbation, quasi–additivity,
generalized inverse.

##  1 Introduction

The expression and perturbation analysis of the generalized inverses (resp.
the Moore–Penrose inverses) of bounded linear operators on Banach spaces
(resp. Hilbert spaces) have been widely studied since Nashed’s book [18] was
published in 1976. Ten years ago, Chen and Xue proposed a notation so–called
the stable perturbation of a bounded operator instead of the rank–preserving
perturbation of a matrix in [8]. Using this new notation, they established the
perturbation analyses for the Moore–Penrose inverse and the least square
problem on Hilbert spaces in [6, 9, 26]. Meanwhile, Castro–González and Koliha
established the perturbation analysis for Drazin inverse by using of the
gap–function in [4, 5, 14]. Later, some of their results were generalized by
Chen and Xue in [27, 28] in terms of stable perturbation.

Throughout this paper, $X,Y$ are always Banach spaces over real field
$\mathbb{R}$ and $B(X,Y)$ is the Banach space consisting of bounded linear
operators from $X$ to $Y$. For $T\in B(X,Y)$, let $\mathcal{N}(T)$ (resp.
$\mathcal{R}(T)$) denote the null space (resp. range) of $T$. It is well–known
that if $\mathcal{N}(T)$ and $\mathcal{R}(T)$ are topologically complemented
in the spaces $X$ and $Y$, respectively, then there exists a (projector)
generalized inverse $T^{+}\in B(Y,X)$ of $T$ such that

$TT^{+}T=T,\quad T^{+}TT^{+}=T^{+},\quad T^{+}T=I_{X}-P_{\mathcal{N}(T)},\quad
TT^{+}=Q_{\mathcal{R}(T)},$

where $P_{\mathcal{N}(T)}$ and $Q_{\mathcal{R}(T)}$ are the bounded linear
projectors from $X$ and $Y$ onto $\mathcal{N}(T)$ and $\mathcal{R}(T)$,
respectively (cf. [6, 18, 25]). But, in general, not every closed subspace in
a Banach space is complemented. Thus the linear generalized inverse $T^{+}$ of
$T$ may not exist. In this case, we may seek other types of generalized
inverses for $T$. Motivated by the ideas of linear generalized inverses and
metric generalized inverses (cf. [18, 20]), by using the so–called homogeneous
(resp. quasi–linear) projector in Banach space, Wang and Li defined the
homogeneous (resp. quasi–linear) generalized inverse in [22]. Then, some
further study on these types of generalized inverses in Banach space was given
in [1, 17]. More important, from the results in [17, 20], we know that, in
some reflexive Banach spaces $X$ and $Y$, for an operator $T\in B(X,Y)$, there
may exists a bounded quasi–linear (projector) generalized inverse of $T$,
which is generally neither linear nor metric generalized inverse of $T$. So,
from this point of view, it is important and necessary to study homogeneous
and quasi–linear (projector) generalized inverses in Banach spaces.

Since the homogeneous (or quasi–linear) projector in Banach space are no
longer linear, the linear projector generalized inverse and the homogeneous
(or quasi–linear) projector generalized inverse of linear operator in Banach
spaces are quite different. Motivated by the new perturbation results of
closed linear generalized inverses [12], in this paper, we initiate the study
of the following problems for bounded homogeneous (resp. quasi–linear
projector) generalized inverse: let $T\in B(X,Y)$ with a bounded homogeneous
(resp. quasi–linear projector) generalized inverse $T^{h}$ (resp. $T^{H}$),
what conditions on the small perturbation $\delta T$ can guarantee that the
bounded homogeneous (resp. quasi–linear projector) generalized inverse
$\bar{T}^{h}$ (resp. $\bar{T}^{H}$) of the perturbed operator
$\bar{T}=T+\delta T$ exists? Furthermore, if it exists, when does
$\bar{T}^{h}$ (resp. $\bar{T}^{H}$) have the simplest expression
$(I_{X}+T^{h}\delta T)^{-1}T^{h}$ (resp. $(I_{X}+T^{H}\delta T)^{-1}T^{H}$)?
With the concept of the quasi–additivity and the notation of stable
perturbation in [8], we will present some perturbation results on homogeneous
generalized inverses and quasi–linear projector generalized inverses in Banach
spaces. Explicit representation and perturbation for the Moore–Penrose metric
generalized inverse of the perturbed operator are also given.

##  2 Preliminaries

Let $T\in B(X,Y)\backslash\\{0\\}$. The reduced minimum module $\gamma(T)$ of
$T$ is given by

$\gamma(T)=\inf\\{\|Tx\|\,|\,x\in X,\mathrm{dist}(x,\mathcal{N}(T))=1\\},$
(2.1)

where
$\mathrm{dist}(x,\mathcal{N}(T))=\inf\\{\|x-z\|\,|\,z\in\mathcal{N}(T)\\}$. It
is well–known that $\mathcal{R}(T)$ is closed in $Y$ iff $\gamma(T)>0$ (cf.
[16, 28]). From (2.1), we can obtain useful inequality as follows:

$\|Tx\|\geq\gamma(T)\,\mathrm{dist}(x,\mathcal{N}(T)),\quad\forall\,x\in X.$

Recall from [1, 23] that a subset $D$ in $X$ is called to be homogeneous if
$\lambda\,x\in D$ whenever $x\in D$ and $\lambda\in\mathbb{R}$; a mapping
$T\colon X\rightarrow Y$ is called to be a bounded homogeneous operator if $T$
maps every bounded set in $X$ into a bounded set in $Y$ and
$T(\lambda\,x)=\lambda\,T(x)$ for every $x\in X$ and every
$\lambda\in\mathbb{R}$.

Let $H(X,Y)$ denote the set of all bounded homogeneous operators from $X$ to
$Y$. Equipped with the usual linear operations on $H(X,Y)$ and norm on $T\in
H(X,Y)$ defined by $\|T\|=\sup\\{\|Tx\|\,|\,\|x\|=1,x\in X\\}$, we can easily
prove that $(H(X,Y),\|\cdot\|)$ is a Banach space (cf. [20, 23]).

######  Definition 2.1.

Let $M$ be a subset of $X$ and $T\colon X\rightarrow Y$ be a mapping. We call
$T$ is quasi–additive on $M$ if $T$ satisfies

$T(x+z)=T(x)+T(z),\qquad\forall\;x\in X,\;\forall\;z\in M.$

Now we give the concept of quasi–linear projector in Banach spaces.

######  Definition 2.2 (cf. [17, 20]).

Let $P\in H(X,X)$. If $P^{2}=P$, we call $P$ is a homogeneous projector. In
addition, if $P$ is also quasi–additive on $\mathcal{R}(P)$, i.e., for any
$x\in X$ and any $z\in\mathcal{R}(P)$,

$P(x+z)=P(x)+P(z)=P(x)+z,$

then we call $P$ is a quasi–linear projector.

Clearly, from Definition 2.2, we see that the bounded linear projectors,
orthogonal projectors in Hilbert spaces are all quasi–linear projector.

Let $P\in H(X,X)$ be a quasi–linear projector. Then by [17, Lemma 2.5],
$\mathcal{R}(P)$ is a closed linear subspace of $X$ and
$\mathcal{R}(I-P)=\mathcal{N}(P)$. Thus, we can define “the quasi–linearly
complement” of a closed linear subspace as follows. Let $V$ be a closed
subspace of $X$. If there exists a bounded quasi–linear projector $P$ on $X$
such that $V=\mathcal{R}(P)$, then $V$ is said to be bounded quasi–linearly
complemented in $X$ and $\mathcal{N}(P)$ is the bounded quasi–linear
complement of $V$ in $X$. In this case, as usual, we may write
$X=V\dotplus\mathcal{N}(P)$, where $\mathcal{N}(P)$ is a homogeneous subset of
$X$ and “$\dotplus$” means that $V\cap\mathcal{N}(P)=\\{0\\}$ and
$X=V+\mathcal{N}(P)$.

######  Definition 2.3.

Let $T\in B(X,Y)$. If there is $T^{h}\in H(Y,X)$ such that

$TT^{h}T=T,\ \quad T^{h}TT^{h}=T^{h},$

then we call $T^{h}$ is a bounded homogeneous generalized inverse of $T$.
Furthermore, if $T^{h}$ is also quasi–additive on $\mathcal{R}(T)$, i.e., for
any $y\in Y$ and any $z\in\mathcal{R}(T)$, we have

$T^{h}(y+z)=T^{h}(y)+T^{h}(z),$

then we call $T^{h}$ is a bounded quasi–linear generalized inverse of $T$.

Obviously, the concept of bounded homogeneous (or quasi-linear) generalized
inverse is a generalization of bounded linear generalized inverse.

Definition 2.3 was first given in paper [1] for linear transformations and
bounded linear operators. The existence of a homogeneous generalized inverse
of $T\in B(X,Y)$ is also given in [1]. In the following, we will give a new
proof of the existence of a homogeneous generalized inverse of a bounded
linear operator.

######  Proposition 2.4.

Let $T\in B(X,Y)\backslash\\{0\\}$. Then $T$ has a homogeneous generalized
inverse $T^{h}\in H(Y,X)$ iff $\mathcal{R}(T)$ is closed and there exist a
bounded quasi–linear projector $P_{\mathcal{N}(T)}\colon X\to\mathcal{N}(T)$
and a bounded homogeneous projector $Q_{\mathcal{R}(T)}:Y\to\mathcal{R}(T)$.

###### Proof.

Suppose that there is $T^{h}\in H(Y,X)$ such that $TT^{h}T=T$ and
$T^{h}TT^{h}=T^{h}$. Put $P_{\mathcal{N}(T)}=I_{X}-T^{h}T$ and
$Q_{\mathcal{R}(T)}=TT^{h}$. Then $P_{\mathcal{N}(T)}\in H(X,X)$,
$Q_{\mathcal{R}(T)}\in H(Y,Y)$ and

$\displaystyle P_{\mathcal{N}(T)}^{2}$
$\displaystyle=(I_{X}-T^{h}T)(I_{X}-T^{h}T)=I_{X}-T^{h}T-T^{h}T(I_{X}-T^{h}T)=P_{\mathcal{N}(T)},$
$\displaystyle Q_{\mathcal{R}(T)}^{2}$
$\displaystyle=TT^{h}TT^{h}=TT^{h}=Q_{\mathcal{R}(T)}.$

From $TT^{h}T=T$ and $T^{h}TT^{h}=T^{h}$, we can get that
$\mathcal{N}(T)=\mathcal{R}(P_{\mathcal{N}(T)})$ and
$\mathcal{R}(T)=\mathcal{R}(Q_{\mathcal{R}(T)})$. Since for any $x\in X$ and
any $z\in\mathcal{N}(T)$,

$\displaystyle P_{\mathcal{N}(T)}(x+z)$
$\displaystyle=x+z-T^{h}T(x+z)=x+z-T^{h}Tx$
$\displaystyle=P_{\mathcal{N}(T)}x+z=P_{\mathcal{N}(T)}x+P_{\mathcal{N}(T)}z,$

it follows that $P_{\mathcal{N}(T)}$ is quasi–linear. Obviously, we see that
$Q_{\mathcal{R}(T)}:Y\to\mathcal{R}(T)$ is a bounded homogeneous projector.

Now for any $x\in X$,

$\mathrm{dist}(x,\mathcal{N}(T))\leq\|x-P_{\mathcal{N}(T)}x\|=\|T^{h}Tx\|\leq\|T^{h}\|\|Tx\|.$

Thus, $\gamma(T)\geq\dfrac{1}{\|T^{h}\|}>0$ and hence $\mathcal{R}(T)$ is
closed in $Y$.

Conversely, for $x\in X$, let $[x]$ stand for equivalence class of $x$ in
$X/\mathcal{N}(T)$. Define mappings
$\phi\colon\mathcal{R}(I-P_{\mathcal{N}(T)})\rightarrow X/\mathcal{N}(T)$ and
$\hat{T}\colon X/\mathcal{N}(T)\rightarrow\mathcal{R}(T)$ respectively, by

$\phi(x)=[x],\quad\forall\,x\in\mathcal{R}(I-P_{\mathcal{N}(T)})\ \text{and}\
\hat{T}([z])=Tz,\quad\forall\,z\in X.$

Clearly, $\hat{T}$ is bijective. Noting that the quotient space
$X/\mathcal{N}(T)$ with the norm $\|[x]\|=\mathrm{dist}(x,\mathcal{N}(T))$,
$\forall\,x\in X$, is a Banach space (cf. [25]) and
$\|Tx\|\geq\gamma(T)\,\mathrm{dist}(x,\mathcal{N}(T))$ with $\gamma(T)>0$,
$\forall\,x\in X$, we have $\|\hat{T}[x]\|\geq\gamma(T)\|[x]\|$,
$\forall\,x\in X$. Therefore,
$\|\hat{T}^{-1}y\|\leq\dfrac{1}{\gamma(T)}\|y\|$,
$\forall\,y\in\mathcal{R}(T)$.

Since $P_{\mathcal{N}(T)}$ is a quasi–linear projector, it follows that $\phi$
is bijective and $\phi^{-1}([x])=(I-P_{\mathcal{N}(T)})x$, $\forall\,x\in X$.
Obviously, $\phi^{-1}$ is homogeneous and for any $z\in\mathcal{N}(T)$,

$\|\phi^{-1}([x])\|=\|(I-P_{\mathcal{N}(T)})(x-z)\|\leq(1+\|P_{\mathcal{N}(T)}\|)\|x-z\|$

which implies that $\|\phi^{-1}\|\leq 1+\|P_{\mathcal{N}(T)}\|$. Put
$T_{0}=\hat{T}\circ\phi\colon\mathcal{R}(I-P_{\mathcal{N}(T)})\rightarrow\mathcal{R}(T)$.
Then
$T_{0}^{-1}=\phi^{-1}\circ\hat{T}^{-1}\colon\mathcal{R}(T)\rightarrow\mathcal{R}(I-P_{\mathcal{N}(T)})$
is homogeneous and bounded with
$\|T_{0}^{-1}\|\leq\gamma(T)^{-1}(1+\|P_{\mathcal{N}(T)}\|)$. Set
$T^{h}=(I-P_{\mathcal{N}(T)})T_{0}^{-1}Q_{\mathcal{R}(T)}$. Then $T^{h}\in
H(Y,X)$ and

$TT^{h}T=T,\ T^{h}TT^{h}=T^{h},\ TT^{h}=Q_{\mathcal{R}(T)},\
T^{h}T=I_{X}-P_{\mathcal{N}(T)}.$

This finishes the proof. ∎

Recall that a closed subspace $V$ in $X$ is Chebyshev if for any $x\in X$,
there is a unique $x_{0}\in V$ such that $\|x-x_{0}\|=\mathrm{dist}(x,V)$.
Thus, for the closed Chebyshev space $V$, we can define a mapping
$\pi_{V}\colon X\rightarrow V$ by $\pi_{V}(x)=x_{0}$. $\pi_{V}$ is called to
be the metric projector from $X$ onto $V$. From [20], we know that $\pi_{V}$
is a quasi–linear projector with $\|\pi_{V}\|\leq 2$. Then by Proposition 2.4,
we have

######  Corollary 2.5 ([19, 20]).

Let $T\in B(X,Y)\backslash\\{0\\}$ with $\mathcal{R}(T)$ closed. Assume that
$\mathcal{N}(T)$ and $\mathcal{R}(T)$ are Chebyshev subspaces in $X$ and $Y$,
respectively. Then there is $T^{h}\in H(Y,X)$ such that

$TT^{h}T=T,\ T^{h}TT^{h}=T^{h},\ TT^{h}=\pi_{\mathcal{R}(T)},\
T^{h}T=I_{X}-\pi_{\mathcal{N}(T)}.$ (2.2)

The bounded homogeneous generalized inverse $T^{h}$ in (2.2) is called to be
the Moore–Penrose metric generalized inverse of $T$. Such $T^{h}$ in (2.2) is
unique and is denoted by $T^{M}$ (cf. [20]).

######  Corollary 2.6.

Let $T\in B(X,Y)\backslash\\{0\\}$ such that the bounded homogeneous
generalized inverse $T^{h}$ exists. Assume that $\mathcal{N}(T)$ and
$\mathcal{R}(T)$ are Chebyshev subspaces in $X$ and $Y$, respectively. Then
$T^{M}=(I_{X}-\pi_{\mathcal{N}(T)})T^{h}\pi_{\mathcal{R}(T)}$.

###### Proof.

Since $\mathcal{N}(T)$ and $\mathcal{R}(T)$ are Chebyshev subspaces, it
follows from Corollary 2.5 that $T$ has the unique Moore–Penrose metric
generalized inverse $T^{M}$ which satisfy

$TT^{M}T=T,\ T^{M}TT^{M}=T^{M},\ TT^{M}=\pi_{\mathcal{R}(T)},\
T^{M}T=I_{X}-\pi_{\mathcal{N}(T)}.$

Set $T^{\natural}=(I_{X}-\pi_{\mathcal{N}(T)})T^{h}\pi_{\mathcal{R}(T)}$. Then
$T^{\natural}=T^{M}TT^{h}TT^{M}=T^{M}TT^{M}=T^{M}.$ ∎

##  3 Perturbations for bounded homogeneous generalized inverse

In this section, we extend some perturbation results of linear generalized
inverses to bounded homogeneous generalized inverses. We start our
investigation with some lemmas, which are prepared for the proof of our main
results. The following result is well–known for bounded linear operators, we
generalize it to the bounded homogeneous operators in the following form.

######  Lemma 3.1.

Let $T\in H(X,Y)$ and $S\in H(Y,X)$ such that $T$ is quasi–additive on
$\mathcal{R}(S)$ and $S$ is quasi–additive on $\mathcal{R}(T)$, then
$I_{Y}+TS$ is invertible in $H(Y,Y)$ if and only if $I_{X}+ST$ is invertible
in $H(X,X)$.

###### Proof.

If there is a $\Phi\in H(Y,Y)$ be such that
$(I_{Y}+TS)\Phi=\Phi(I_{Y}+TS)=I_{Y}$, then

$\displaystyle I_{X}$ $\displaystyle=I_{X}+ST-ST=I_{X}+ST-S((I_{Y}+TS)\Phi)T$
$\displaystyle=I_{X}+ST-((S+STS)\Phi)T\quad(S\ \text{quasi-additive on}\
\mathcal{R}(T))$ $\displaystyle=I_{X}+ST-((I_{X}+ST)S\Phi)T$
$\displaystyle=(I_{X}+ST)(1_{X}-S\Phi T)\quad(T\ \text{quasi--additive on}\
\mathcal{R}(S)).$

Similarly, we also have $I_{X}=(I_{X}-S\Phi T)(I_{X}+ST)$. Thus, $I_{X}+ST$ is
invertible on $X$ with $(I_{X}+ST)^{-1}=(1_{X}-S\Phi T)\in H(X,X)$.

The converse can also be proved by using the same way as above. ∎

######  Lemma 3.2.

Let $T\in B(X,Y)$ such that $T^{h}\in H(Y,X)$ exists and let $\delta T\in
B(X,Y)$ such that $T^{h}$ is quasi–additive on $\mathcal{R}(\delta T)$ and
$(I_{X}+T^{h}\delta T)$ is invertible in $B(X,X)$. Then $I_{Y}+\delta
TT^{h}:Y\rightarrow Y$ is invertible in $H(Y,Y)$ and

$\Phi=T^{h}(I_{Y}+\delta TT^{h})^{-1}=(I_{X}+T^{h}\delta T)^{-1}T^{h}$ (3.1)

is a bounded homogeneous operator with $\mathcal{R}(\Phi)=\mathcal{R}(T^{h})$
and $\mathcal{N}(\Phi)=\mathcal{N}(T^{h})$.

###### Proof.

By Lemma 3.1, $I_{Y}+\delta TT^{h}:Y\rightarrow Y$ is invertible in $H(Y,Y)$.

Clearly, $I_{X}+T^{h}\delta T$ is linear bounded operator and $I_{Y}+\delta
TT^{h}\in H(Y,Y)$. From the equation

$(I_{X}+T^{h}\delta T)T^{h}=T^{h}(I_{Y}+\delta TT^{h})$

and $T^{h}\in H(Y,X)$, we get that $\Phi$ is a bounded homogeneous operator.
Finally, from (3.1), we can obtain that $\mathcal{R}(\Phi)=\mathcal{R}(T^{h})$
and $\mathcal{N}(\Phi)=\mathcal{N}(T^{h})$. ∎

Recall from [8] that for $T\in B(X,Y)$ with bounded linear generalized inverse
$T^{+}\in B(Y,X)$, we say that $\bar{T}=T+\delta T\in B(X,Y)$ is a stable
perturbation of $T$ if $\mathcal{R}(\bar{T})\cap\mathcal{N}(T^{+})=\\{0\\}$.
Now for $T\in B(X,Y)$ with $T^{h}\in H(Y,X)$, we also say that
$\bar{T}=T+\delta T\in B(X,Y)$ is a stable perturbation of $T$ if
$\mathcal{R}(\bar{T})\cap\mathcal{N}(T^{h})=\\{0\\}$.

######  Lemma 3.3.

Let $T\in B(X,Y)$ such that $T^{h}\in H(Y,X)$ exists. Suppose that $\delta
T\in B(X,Y)$ such that $T^{h}$ is quasi–additive on $\mathcal{R}(\delta T)$
and $I_{X}+T^{h}\delta T$ is invertible in $B(X,X)$ Put $\bar{T}=T+\delta T$.
If $\mathcal{R}(\bar{T})\cap\mathcal{N}(T^{h})=\\{0\\}$, then

$\mathcal{N}(\bar{T})=(I_{X}+T^{h}\delta T)^{-1}\mathcal{N}(T)\ \text{and}\
\mathcal{R}(\bar{T})=(I_{Y}+\delta TT^{h})\mathcal{R}(T).$

###### Proof.

Set $P=(I_{X}+T^{h}\delta T)^{-1}(I_{X}-T^{h}T)$. We first show that $P^{2}=P$
and $\mathcal{R}(P)=\mathcal{N}(\bar{T})$. Since $T^{h}TT^{h}=T^{h}$, we get
$(I_{X}-T^{h}T)T^{h}\delta T=0$ and then

$\displaystyle(I_{X}-T^{h}T)(I_{X}+T^{h}\delta T)=I_{X}-T^{h}T$ (3.2)

and so that

$\displaystyle I_{X}-T^{h}T=(I_{X}-T^{h}T)(I_{X}+T^{h}\delta T)^{-1}.$ (3.3)

Now, by using (3.2) and (3.3), it is easy to get $P^{2}=P$.

Since $T^{h}$ is quasi–additive on $\mathcal{R}(\delta T)$, we see
$I_{X}-T^{h}T=(I_{X}+T^{h}\delta T)-T^{h}\bar{T}$. Then for any $x\in X$, we
have

$\displaystyle Px$ $\displaystyle=(I_{X}+T^{h}\delta T)^{-1}(I_{X}-T^{h}T)x$
$\displaystyle=(I_{X}+T^{h}\delta T)^{-1}[(I_{X}+T^{h}\delta
T)-T^{h}\bar{T}]x$ $\displaystyle=x-(I_{X}+T^{h}\delta T)^{-1}T^{h}\bar{T}x.$
(3.4)

From (3), we get that if $x\in\mathcal{N}(\bar{T})$, then
$x\in\mathcal{R}(P)$. Thus, $\mathcal{N}(\bar{T})\subset\mathcal{R}(P)$.

Conversely, let $z\in\mathcal{R}(P)$, then $z=Pz$. From (3), we get
$(I_{X}+T^{h}\delta T)^{-1}T^{h}\bar{T}x=0$. Therefore, we have
$\bar{T}x\in\mathcal{R}(\bar{T})\cap\mathcal{N}(T^{h})=\\{0\\}$. Thus,
$x\in\mathcal{N}(\bar{T})$ and then $\mathcal{R}(P)=\mathcal{N}(\bar{T})$.

From the Definition of $T^{h}$, we have
$\mathcal{N}(T)=\mathcal{R}(I_{X}-T^{h}T)$. Thus,

$(I_{X}+T^{h}\delta T)^{-1}\mathcal{N}(T)=(I_{X}+T^{h}\delta
T)^{-1}\mathcal{R}(I_{X}-T^{h}T)=\mathcal{R}(P)=\mathcal{N}(\bar{T}).$

Now, we prove that $\mathcal{R}(\bar{T})=(I_{Y}+\delta TT^{h})\mathcal{R}(T)$.
From $(I_{Y}+\delta TT^{h})T=\bar{T}T^{h}T$, we get that $(I_{Y}+\delta
TT^{h})\mathcal{R}(T)\subset\mathcal{R}(\bar{T})$. On the other hand, since
$T^{h}$ is quasi–additive on $\mathcal{R}(\delta T)$ and
$\mathcal{R}(P)=\mathcal{N}(\bar{T})$, we have for any $x\in X$,

$\displaystyle 0$ $\displaystyle=\bar{T}Px=\bar{T}(I_{X}+T^{h}\delta
T)^{-1}(I_{X}-T^{h}T)x$ $\displaystyle=\bar{T}x-\bar{T}(I_{X}+T^{h}\delta
T)^{-1}(T^{h}\delta Tx+T^{h}{T}x)$
$\displaystyle=\bar{T}x-\bar{T}(I_{X}+T^{h}\delta
T)^{-1}T^{h}\bar{T}x=\bar{T}x-\bar{T}T^{h}(I_{Y}+\delta TT^{h})^{-1}\bar{T}x$
$\displaystyle=\bar{T}x-(I_{Y}+\delta TT^{h}-I_{Y}+TT^{h})(I_{Y}+\delta
TT^{h})^{-1}\bar{T}x$ $\displaystyle=(I_{Y}-TT^{h})(I_{Y}+\delta
TT^{h})^{-1}\bar{T}x.$ (3.5)

Since $\mathcal{N}(I_{Y}-TT^{h})=\mathcal{R}(T)$, it follows (3) that
$(I_{Y}+\delta TT^{h})^{-1}\mathcal{R}(\bar{T})\subset\mathcal{R}(T)$, that
is, $\mathcal{R}(\bar{T})\subset(I_{Y}+\delta TT^{h})\mathcal{R}(T)$.
Consequently, $\mathcal{R}(\bar{T})=(I_{Y}+\delta TT^{h})\mathcal{R}(T)$. ∎

Now we can present the main perturbation result for bounded homogeneous
generalized inverse on Banach spaces.

######  Theorem 3.4.

Let $T\in B(X,Y)$ such that $T^{h}\in H(Y,X)$ exists. Suppose that $\delta
T\in B(X,Y)$ such that $T^{h}$ is quasi–additive on $\mathcal{R}(\delta T)$
and $I_{X}+T^{h}\delta T$ is invertible in $B(X,X)$. Put $\bar{T}=T+\delta T$.
Then the following statements are equivalent:

  1. $(1)$

$\Phi=T^{h}(I_{Y}+\delta TT^{h})^{-1}$ is a bounded homogeneous generalized
inverse of $\bar{T}$;

  2. $(2)$

$\mathcal{R}(\bar{T})\cap\mathcal{N}(T^{h})=\\{0\\}$;

  3. $(3)$

$\mathcal{R}(\bar{T})=(I_{Y}+\delta TT^{h})\mathcal{R}(T)$;

  4. $(4)$

$(I_{Y}+T^{h}\delta T)\mathcal{N}(\bar{T})=\mathcal{N}(T)$;

  5. $(5)$

$(I_{Y}+\delta TT^{h})^{-1}\bar{T}\mathcal{N}(T)\subset\mathcal{R}(T)$;

###### Proof.

We prove our theorem by showing that

$(3)\Rightarrow(5)\Rightarrow(4)\Rightarrow(2)\Rightarrow(3)\Rightarrow(1)\Rightarrow(3).$

$(3)\Rightarrow(5)$ This is obvious since $(I+\delta TT^{h})$ is invertible
and $\mathcal{N}(T)\subset X$.

$(5)\Rightarrow(4)$. Let $x\in\mathcal{N}(\bar{T})$, then we see
$(I_{X}+T^{h}\delta T)x=x-T^{h}Tx\in\mathcal{N}(T)$. Hence $(I_{X}+T^{h}\delta
T)\mathcal{N}(\bar{T})\subset\mathcal{N}(T)$. Now for any
$x\in\mathcal{N}(T)$, then by (5), there exists some $z\in X$ such that
$\bar{T}x=(I_{Y}+\delta TT^{h})Tz=\bar{T}T^{h}Tz.$ So
$x-T^{h}Tz\in\mathcal{N}(\bar{T})$ and hence

$(I_{X}+T^{h}\delta T)(x-T^{h}Tz)=(I_{X}-T^{h}T)(x-T^{h}Tz)=x.$

Consequently, $(I_{X}+T^{h}\delta T)\mathcal{N}(\bar{T})=\mathcal{N}(T)$.

$(4)\Rightarrow(2)$. Let $y\in R(\overline{T})\cap N(T^{h})$, then there
exists an $x\in X$ such that $y=\bar{T}x$ and $T^{h}\bar{T}x=0$. We can check
that

$T(I_{X}+T^{h}\delta T)x=Tx+TT^{h}\delta Tx=Tx+TT^{h}\bar{T}x-TT^{h}Tx=0.$

Thus, $(I_{X}+T^{h}\delta T)x\in\mathcal{N}(T)$. By (4),
$x\in\mathcal{N}(\bar{T})$ and so that $y=\bar{T}x=0$.

$(2)\Rightarrow(3)$ follows from Lemma 3.3.

$(3)\Rightarrow(1)$ Noting that by Lemma 3.2, we have

$\Phi=T^{h}(I_{Y}+\delta TT^{h})^{-1}=(I_{X}+T^{h}\delta T)^{-1}T^{h}$

is a bounded homogeneous operator with $\mathcal{R}(\Phi)=\mathcal{R}(T^{h})$
and $\mathcal{N}(\Phi)=\mathcal{N}(T^{h})$. We need to prove that
$\Phi\bar{T}\Phi=\Phi$ and $\bar{T}\Phi\bar{T}=\bar{T}$. Since $T^{h}$ is
quasi–additive on $\mathcal{R}(\delta T)$, we have
$T^{h}\bar{T}=T^{h}T+T^{h}\delta T$. Therefore,

$\displaystyle\Phi\bar{T}\Phi$ $\displaystyle=(I_{Y}+T^{h}\delta
T)^{-1}T^{h}\bar{T}(I_{Y}+T^{h}\delta T)^{-1}T^{h}$
$\displaystyle=(I_{Y}+T^{h}\delta T)^{-1}T^{h}[(I_{X}+T^{h}\delta
T)-(I_{X}-T^{h}T)](I_{X}+T^{h}\delta T)^{-1}T^{h}$
$\displaystyle=(I_{X}+T^{h}\delta T)^{-1}T^{h}-(I_{X}+T^{h}\delta
T)^{-1}(I_{X}-T^{h}T)(I_{X}+T^{h}\delta T)^{-1}T^{h}$
$\displaystyle=(I_{X}+T^{h}\delta T)^{-1}T^{h}-(I_{X}+T^{h}\delta
T)^{-1}(I_{X}-T^{h}T)T^{h}(I_{X}+\delta TT^{h})^{-1}$ $\displaystyle=\Phi.$

$\mathcal{R}(\bar{T})=(I_{Y}+\delta TT^{h})\mathcal{R}(T)$ means that
$(I_{Y}-TT^{h})(I_{Y}+\delta TT^{h})^{-1}\bar{T}=0$. So

$\displaystyle\bar{T}\Phi\bar{T}$ $\displaystyle=(T+\delta
T)T^{h}(I_{Y}+T^{h}\delta T)^{-1}\bar{T}$ $\displaystyle=(I_{Y}+\delta
TT^{h}+TT^{h}-I_{Y})(I_{Y}+T^{h}\delta T)^{-1}\bar{T}$
$\displaystyle=\bar{T}.$

$(1)\Rightarrow(3)$ From $\bar{T}\Phi\bar{T}=\bar{T}$, we have
$(I_{Y}-TT^{h})(I_{Y}+\delta TT^{h})^{-1}\bar{T}=0$ by the proof of
$(3)\Rightarrow(1)$. Thus, $(I_{Y}+\delta
TT^{h})^{-1}\mathcal{R}(\bar{T})\subset\mathcal{R}(T)$. From $(I_{Y}+\delta
TT^{h})T=\bar{T}T^{h}T$, we get that $(I_{Y}+\delta
TT^{h})\mathcal{R}(T)\subset\mathcal{R}(\bar{T})$. So $(I_{Y}+\delta
TT^{h})\mathcal{R}(T)=\mathcal{R}(\bar{T})$. ∎

######  Corollary 3.5.

Let $T\in B(X,Y)$ such that $T^{h}\in H(Y,X)$ exists. Suppose that $\delta
T\in B(X,Y)$ such that $T^{h}$ is quasi–additive on $\mathcal{R}(\delta T)$
and $\|T^{h}\delta T\|<1$. Put $\bar{T}=T+\delta T$. If
$\mathcal{N}(T)\subset\mathcal{N}(\delta T)$ or $\mathcal{R}(\delta
T)\subset\mathcal{R}(T)$, then $\bar{T}$ has a homogeneous bounded generalized
inverse

$\bar{T}^{h}=T^{h}(I_{Y}+\delta TT^{h})^{-1}=(I_{X}+T^{h}\delta T)^{-1}T^{h}.$

###### Proof.

If $N(T)\subset N(\delta T)$, then $N(T)\subset N(\bar{T})$. So Condition (5)
of Theorem 3.4 holds. If $\mathcal{R}(\delta T)\subset R(T)$, then
$R(\bar{T})\subset\mathcal{R}(T)$. So
$\mathcal{R}(\bar{T})\cap\mathcal{N}(T)\subset\mathcal{R}(T)\cap\mathcal{N}(T^{h})=\\{0\\}$
and consequently, $\bar{T}$ has the homogeneous bounded generalized inverse
$T^{h}(I_{Y}+\delta TT^{h})^{-1}=(I_{X}+T^{h}\delta T)^{-1}T^{h}$ by Theorem
3.4. ∎

######  Proposition 3.6.

Let $T\in B(X,Y)$ with $\mathcal{R}(T)$ closed. Assume that $\mathcal{N}(T)$
and $\mathcal{R}(T)$ are Chebyshev subspaces in $X$ and $Y$, respectively. Let
$\delta T\in B(X,Y)$ such that $T^{M}$ is quasi–additive on
$\mathcal{R}(\delta T)$ and $\|T^{M}\delta T\|<1$. Put $\bar{T}=T+\delta T$.
Suppose that $\mathcal{N}(\bar{T})$ and $\overline{\mathcal{R}(\bar{T})}$ are
Chebyshev subspaces in $X$ and $Y$, respectively. If
$\mathcal{R}(\bar{T})\cap\mathcal{N}(T^{M})=\\{0\\}$, then
$\mathcal{R}(\bar{T})$ is closed in $Y$ and $\bar{T}$ has the Moore–Penrose
metric generalized inverse

$\bar{T}^{M}=(I_{X}-\pi_{\mathcal{N}(\bar{T})})(I_{X}+T^{M}\delta
T)^{-1}T^{M}\pi_{\mathcal{R}(\bar{T})}$

with $\|\bar{T}^{M}\|\leq\dfrac{2\|T^{M}\|}{1-\|T^{M}\delta T\|}$.

###### Proof.

$T^{M}$ exists by Corollary 2.5. Since $T^{M}\delta T$ is $\mathbb{R}$–linear
and $\|T^{M}\delta T\|<1$, we have $I_{X}+T^{M}\delta T$ is invertible in
$B(X,X)$. By Theorem 3.4 and Proposition 2.4,
$\mathcal{R}(\bar{T})\cap\mathcal{N}(T^{M})=\\{0\\}$ implies that
$\mathcal{R}(\bar{T})$ is closed and $\bar{T}$ has a bounded homogeneous
generalized inverse $\bar{T}^{h}=(I_{X}+T^{M}\delta T)^{-1}T^{M}$. Then by
Corollary 2.6, $\bar{T}^{M}$ has the form

$\bar{T}^{M}=(I_{X}-\pi_{\mathcal{N}(\bar{T})})(I_{X}+T^{M}\delta
T)^{-1}T^{M}\pi_{\mathcal{R}(\bar{T})}.$

Note that
$\|x-\pi_{\mathcal{N}(\bar{T})}x\|=\mathrm{dist}(x,\mathcal{N}(\bar{T}))\leq\|x\|$,
$\forall\,x\in X$. So $\|I_{X}-\pi_{\mathcal{N}(\bar{T})}\|\leq 1$. Therefore,

$\|\bar{T}^{M}\|\leq\|I_{X}-\pi_{\mathcal{N}(\bar{T})}\|\|(I_{X}+T^{M}\delta
T)^{-1}T^{M}\|\|\pi_{\mathcal{R}(\bar{T})}\|\leq\frac{2\|T^{M}\|}{1-\|T^{M}\delta
T\|}.$

This completes the proof. ∎

##  4 Perturbation for quasi–linear projector generalized inverse

We have known that the range of a bounded qausi–linear projector on a Banach
space is closed(see [17, Lemma 2.5]). Thus, from Definition 2.3 and the proof
of Proposition 2.4, the following result is obvious.

######  Proposition 4.1.

Let $T\in B(X,Y)\backslash\\{0\\}$. Then $T$ has a bounded quasi–linear
generalized inverse $T^{h}\in H(Y,X)$ iff there exist a bounded linear
projector $P_{\mathcal{N}(T)}\colon X\to\mathcal{N}(T)$ and a bounded
quasi–linear projector $Q_{\mathcal{R}(T)}:Y\to\mathcal{R}(T)$.

Motivated by related results in papers [1, 17, 22] and the definition of the
oblique projection generalized inverses in Banach space(see [18, 25]), based
on Proposition 4.1, we can give the following definition of quasi–linear
projector generalized inverse of a bounded linear operator on Banach space.

######  Definition 4.2.

Let $T\in B(X,Y)$. Let $T^{H}\in H(Y,X)$ be a bounded homogeneous operator. If
there exist a bounded linear projector $P_{\mathcal{N}(T)}$ from $X$ onto
$\mathcal{N}(T)$ and a bounded quasi–linear projector $Q_{\mathcal{R}(T)}$
from $Y$ onto $\mathcal{R}(T)$, respectively, such that

$\displaystyle(1)\,TT^{H}T=T;\quad(2)\,T^{H}TT^{H}=T^{H};\quad(3)\,T^{H}T=I_{X}-P_{\mathcal{N}(T)};\quad(4)\,TT^{H}=Q_{\mathcal{R}(T)}.$

Then $T^{H}$ is called a quasi–linear projector generalized inverse of $T$.

For $T\in B(X,Y)$, if $T^{H}$ exists, then from Proposition 4.1 and Definition
2.3, we see that $\mathcal{R}(T)$ is closed and $T^{H}$ is quasi–additive on
$\mathcal{R}(T)$, in this case, we may call $T^{H}$ is a quasi–linear
operator. Choose $\delta T\in B(X,Y)$ such that $T^{H}$ is also quasi–additive
on $\mathcal{R}(\delta T)$, then $I_{X}+T^{H}\delta T$ is a bounded linear
operator and $I_{Y}+\delta TT^{H}$ is a bounded linear operator on
$\mathcal{R}(\bar{T})$.

######  Lemma 4.3.

Let $T\in B(X,Y)$ such that $T^{H}$ exists and let $\delta T\in B(X,Y)$ such
that $T^{H}$ is quasi–additive on $\mathcal{R}(\delta T)$. Put
$\bar{T}=T+\delta T$. Assumes that
$X=\mathcal{N}(\bar{T})\dotplus\mathcal{R}(T^{H})$ and
$Y=\mathcal{R}(\bar{T})\dotplus\mathcal{N}(T^{H})$. Then

  1. $(1)$

$I_{X}+T^{H}\delta T:X\rightarrow X$ is a invertible bounded linear operator;

  2. $(2)$

$I_{Y}+\delta TT^{H}:Y\rightarrow Y$ is a invertible quasi–linear operator;

  3. $(3)$

$\Upsilon=T^{H}(I_{Y}+\delta TT^{H})^{-1}=(I_{X}+T^{H}\delta T)^{-1}T^{H}$ is
a bounded homogeneous operator.

###### Proof.

Since $I_{X}+T^{H}\delta T\in B(X,X)$, we only need to show that
$\mathcal{N}(I_{X}+T^{H}\delta T)=\\{0\\}$ and $\mathcal{R}(I_{X}+T^{H}\delta
T)=X$ under the assumptions.

We first show that $\mathcal{N}(I_{X}+T^{H}\delta T)=\\{0\\}$. Let
$x\in\mathcal{N}(I_{X}+T^{H}\delta T)$, then

$(I_{X}+T^{H}\delta T)x=(I_{X}-T^{H}T)x+T^{H}\bar{T}x=0$

since $T^{H}$ is quasi–linear. Thus $(I_{X}-T^{H}T)x=0=T^{H}\bar{T}x$ and
hence $\bar{T}x\in\mathcal{R}(\bar{T})\cap\mathcal{N}(T^{H})$. Noting that
$Y=\mathcal{R}(\bar{T})\dotplus\mathcal{N}(T^{H})$, we have $\bar{T}x=0$ and
hence $x\in\mathcal{R}(T^{H})\cap\mathcal{N}(\bar{T})$. From
$X=\mathcal{N}(\bar{T})\dotplus\mathcal{R}(T^{H})$, we get that $x=0$.

Now, we prove that $\mathcal{R}(I_{X}+T^{H}\delta T)=X$. Let $x\in X$ and put
$x_{1}=(I_{X}-T^{H}T)x$, $x_{2}=T^{H}Tx$. Since
$Y=\mathcal{R}(\bar{T})\dotplus\mathcal{N}(T^{H})$, we have
$\mathcal{R}(T^{H})=T^{H}\mathcal{R}(\bar{T})$. Therefore, from
$X=\mathcal{N}(\bar{T})\dotplus\mathcal{R}(T^{H})$, we get that
$\mathcal{R}(T^{H})=T^{H}\mathcal{R}(\bar{T})=T^{H}\bar{T}\mathcal{R}(T^{H})$.
Consequently, there is $z\in Y$ such that
$T^{H}(Tx_{2}-\bar{T}x_{1})=T^{H}\bar{T}T^{H}z$. Set $y=x_{1}+T^{H}z\in X$.
Noting that $T^{H}$ is quasi–additive on $\mathcal{R}(T)$ and
$\mathcal{R}(\delta T)$, respectively. we have

$\displaystyle(I_{X}+T^{H}\delta T)y$
$\displaystyle=(I_{X}-T^{H}T+T^{H}\bar{T})(x_{1}+T^{H}z)$
$\displaystyle=x_{1}+T^{H}\bar{T}x_{1}+T^{H}\bar{T}T^{H}z$
$\displaystyle=x_{1}+T^{H}\bar{T}x_{1}+T^{H}(Tx_{2}-\bar{T}x_{1})$
$\displaystyle=x.$

Therefore, $X=\mathcal{R}(I_{Y}+T^{H}\delta T)$.

Similar to Lemma 3.2, we have $\Upsilon=T^{H}(I_{Y}+\delta
TT^{H})^{-1}=(I_{X}+T^{H}\delta T)^{-1}T^{H}$ is a bounded homogeneous
operator. ∎

######  Theorem 4.4.

Let $T\in B(X,Y)$ such that $T^{H}$ exists and let $\delta T\in B(X,Y)$ such
that $T^{H}$ is quasi–additive on $\mathcal{R}(\delta T)$. Put
$\bar{T}=T+\delta T$. Then the following statements are equivalent:

  1. $(1)$

$I_{X}+T^{H}\delta T$ is invertible in $B(X,X)$ and
$\mathcal{R}(\bar{T})\cap\mathcal{N}(T^{H})=\\{0\\};$

  2. $(2)$

$I_{X}+T^{H}\delta T$ is invertible in $B(X,X)$ and
$\Upsilon=T^{H}(I_{Y}+\delta TT^{H})^{-1}=(I_{X}+T^{H}\delta T)^{-1}T^{H}$ is
a quasi–linear projector generalized inverse of $\bar{T};$

  3. $(3)$

$X=\mathcal{N}(\bar{T})\dotplus\mathcal{R}(T^{H})$ and
$Y=\mathcal{R}(\bar{T})\dotplus\mathcal{N}(T^{H})$, i.e.,
$\mathcal{N}(\bar{T})$ is topological complemented in $X$ and
$\mathcal{R}(\bar{T})$ is quasi–linearly complemented in $Y$.

###### Proof.

$(1)\Rightarrow(2)$ By Theorem 3.4, $\Upsilon=T^{H}(I_{Y}+\delta
TT^{H})^{-1}=(I_{X}+T^{H}\delta T)^{-1}T^{H}$ is a bounded homogeneous
generalized inverse of $T$. Let $y\in Y$ and $z\in\mathcal{R}(\bar{T})$. Then
$z=Tx+\delta Tx$ for some $x\in X$. Since $T^{H}$ is quasi–additive on
$\mathcal{R}(T)$ and $\mathcal{R}(\delta T)$, it follows that

$T^{H}(y+z)=T^{H}(y+Tx+\delta Tx)=T^{H}(y)+T^{H}(Tx)+T^{H}(\delta
Tx)=T^{H}y+T^{H}z,$

i.e., $T^{H}$ is quasi–additive on $\mathcal{R}(\bar{T})$ and hence $\Upsilon$
is quasi–linear. Set

$\bar{P}=(I_{X}+T^{H}\delta
T)^{-1}(I_{X}-T^{H}T),\qquad\bar{Q}=\bar{T}(I_{X}+T^{H}\delta T)^{-1}T^{H}.$

Then, by the proof of Lemma 3.3, $\bar{P}\in H(X,X)$ is a projector with
$\mathcal{R}(\bar{P})=\mathcal{N}(\bar{T})$. Noting that $(I_{X}+T^{H}\delta
T)^{-1}$ and $I_{X}-T^{H}T$ are all linear. So $\bar{P}$ is linear.
Furthermore,

$\displaystyle\Upsilon\bar{T}\\!$ $\displaystyle=(I_{X}+T^{H}\delta
T)^{-1}T^{H}(T+\delta T)\\!$ $\displaystyle=(I_{X}+T^{H}\delta
T)^{-1}(I_{X}+T^{H}\delta T+T^{H}T-I_{X})\\!$ $\displaystyle=I_{X}-\bar{P}.$

Since $T^{H}$ is quasi–additive on $\mathcal{R}(\bar{T})$, it follow that
$\bar{Q}=\bar{T}(I+T^{H}\delta T)^{-1}T^{H}=\bar{T}\Upsilon$ is quasi–linear
and bounded with $\mathcal{R}(\bar{Q})\subset\mathcal{R}(\bar{T})$. Noting
that

$\displaystyle\bar{Q}$ $\displaystyle=\bar{T}T^{H}(I_{Y}+\delta
TT^{H})^{-1}=(I_{Y}+\delta TT^{H}+TT^{H}-I_{Y})(I_{Y}+\delta TT^{H})^{-1}$
$\displaystyle=I_{Y}-(I_{Y}-TT^{H})(I_{Y}+\delta TT^{H})^{-1}$

and $(I_{Y}+\delta TT^{H})^{-1}\mathcal{R}(\bar{T})=\mathcal{R}(T)$ by Lemma
3.3, we have
$\mathcal{R}(\bar{T})=\bar{Q}(\mathcal{R}(\bar{T}))\subset\mathcal{R}(\bar{Q})$.
Thus, $\mathcal{R}(\bar{Q})=\mathcal{R}(\bar{T})$. From
$\Upsilon\bar{T}=I_{X}-\bar{P}$ and
$\mathcal{R}(\bar{P})=\mathcal{N}(\bar{T})$, we see that
$\Upsilon\bar{T}\Upsilon=\Upsilon$, then we have

$\bar{Q}^{2}=\bar{T}(I_{X}+T^{H}\delta T)^{-1}T^{H}\bar{T}(I_{X}+T^{H}\delta
T)^{-1}T^{H}=\bar{T}\Upsilon\bar{T}\Upsilon=\bar{Q}.$

Therefore, by Definition 4.2, we get $\bar{T}^{H}=\Upsilon$.

$(2)\Rightarrow(3)$ From $\bar{T}^{H}=T^{H}(I_{Y}+\delta
TT^{h})^{-1}=(I_{X}+T^{H}\delta T)^{-1}T^{H}$, we obtain that
$\mathcal{R}(\bar{T}^{H})=\mathcal{R}(T^{H})$ and
$\mathcal{N}(\bar{T}^{H})=\mathcal{N}(T^{H})$. From
$\bar{T}\bar{T}^{H}\bar{T}=\bar{T}$,
$\bar{T}^{H}\bar{T}\bar{T}^{H}=\bar{T}^{H}$, we get that

$\mathcal{R}(I_{X}-\bar{T}^{H}\bar{T})=\mathcal{N}(\bar{T}),\
\mathcal{R}(\bar{T}^{H}\bar{T})=\mathcal{R}(\bar{T}^{H}),\
\mathcal{R}(\bar{T}\bar{T}^{H})=\mathcal{R}(\bar{T}),\
\mathcal{R}(I_{Y}-\bar{T}\bar{T}^{H})=\mathcal{N}(\bar{T}^{H})$

Thus $\mathcal{R}(\bar{T}^{H}\bar{)}=\mathcal{R}(T^{H})$ and
$\mathcal{R}(I_{Y}-\bar{T}\bar{T}^{H})=\mathcal{N}(T^{H})$. Therefore,

$\displaystyle X$
$\displaystyle=\mathcal{R}(I_{X}-\bar{T}^{H}\bar{T})\dotplus\mathcal{R}(\bar{T}^{H}\bar{T})=\mathcal{N}(\bar{T})\dotplus\mathcal{R}(T^{H}),$
$\displaystyle Y$
$\displaystyle=\mathcal{R}(\bar{T}\bar{T}^{H})\dotplus\mathcal{R}(I_{Y}-\bar{T}\bar{T}^{H})=\mathcal{R}(\bar{T})\dotplus\mathcal{N}(T^{H}).$

$(3)\Rightarrow(1)$ By Lemma 4.3, $I_{X}+T^{H}\delta T$ is invertible in
$H(X,X)$. Now from $Y=\mathcal{R}(\bar{T})\dotplus\mathcal{N}(T^{H})$, we get
that $\mathcal{R}(\bar{T})\cap\mathcal{N}(T^{H})=\\{0\\}$. ∎

######  Lemma 4.5 ([2]).

Let $A\in B(X,X)$. Suppose that there exist two constants
$\lambda_{1},\lambda_{2}\in[0,1)$ such that

$\|Ax\|\leq\lambda_{1}\|x\|+\lambda_{2}\|(I+A)x\|,\quad\quad(\forall\;x\in
X).$

Then $I+A\colon X\rightarrow X$ is bijective. Moreover, for any $x\in X$,

$\displaystyle\frac{1-\lambda_{1}}{1+\lambda_{2}}\|x\|\leq\|(I+A)x\|\leq\frac{1+\lambda_{1}}{1-\lambda_{2}}\|x\|,\quad\frac{1-\lambda_{2}}{1+\lambda_{1}}\|x\|\leq\|(I+A)^{-1}x\|\leq\frac{1+\lambda_{2}}{1-\lambda_{1}}\|x\|.$

Let $T\in B(X,Y)$ such that $T^{H}$ exists. Let $\delta T\in B(X,Y)$ such that
$T^{H}$ is quasi–additive on $\mathcal{R}(\delta T)$ and satisfies

$\displaystyle\|T^{H}\delta
Tx\|\leq\lambda_{1}\|x\|+\lambda_{2}\|(I+T^{H}\delta T)x\|\quad(\forall\;x\in
X),$ (4.1)

where $\lambda_{1},\lambda_{2}\in[0,1)$.

######  Corollary 4.6.

Let $T\in B(X,Y)$ such that $T^{H}$ exists. Suppose that $\delta T\in B(X,Y)$
such that $T^{H}$ is quasi–additive on $\mathcal{R}(\delta T)$ and satisfies
(4.1). Put $\bar{T}=T+\delta T$. Then $I_{X}+T^{H}\delta T$ is invertible in
$H(X,X)$ and $\bar{T}^{H}=(I_{X}+T^{H}\delta T)^{-1}T^{H}$ is well defined
with

$\dfrac{\|\bar{T}^{H}-T^{H}\|}{\|T^{H}\|}\leq\dfrac{(2+\lambda_{1})(1+\lambda_{2})}{(1-\lambda_{1})(1-\lambda_{2})}.$

###### Proof.

By using Lemma 4.5, we get that $I_{X}+T^{H}\delta T$ is invertible in
$H(X,X)$ and

$\displaystyle\|(I_{X}+T^{H}\delta
T)^{-1}\|\leq\dfrac{1+\lambda_{2}}{1-\lambda_{1}},\qquad\|I_{X}+T^{H}\delta
T\|\leq\dfrac{1+\lambda_{1}}{1-\lambda_{2}}.$ (4.2)

From Theorem 4.4, we see $\bar{T}^{H}=T^{H}(I_{Y}+\delta
TT^{H})^{-1}=(I_{X}+T^{H}\delta T)^{-1}T^{H}$ is well–defined. Now we can
compute

$\displaystyle\dfrac{\|\bar{T}^{H}-T^{H}\|}{\|T^{H}\|}$
$\displaystyle\leq\dfrac{\|(I_{X}+T^{H}\delta
T)^{-1}T^{H}-T^{H}\|}{\|T^{H}\|}$
$\displaystyle\leq\dfrac{\|(I_{X}+T^{H}\delta T)^{-1}[I_{X}-(I_{X}+T^{H}\delta
T)]T^{H}\|}{\|T^{H}\|}$ $\displaystyle\leq\|(I_{X}+T^{H}\delta
T)^{-1}\|\|T^{H}\delta T\|.$ (4.3)

Since $\lambda_{2}\in[0,1)$, then from the second inequality in (4.2), we get
that $\|T^{H}\delta T\|\leq\dfrac{2+\lambda_{1}}{1-\lambda_{2}}$. Now, by
using (4) and (4.2), we can obtain

$\dfrac{\|\bar{T}^{H}-T^{H}\|}{\|T^{H}\|}\leq\dfrac{(2+\lambda_{1})(1+\lambda_{2})}{(1-\lambda_{1})(1-\lambda_{2})}.$

This completes the proof. ∎

######  Corollary 4.7.

Let $T\in B(X,Y)$ with $\mathcal{R}(T)$ closed. Assume that $\mathcal{R}(T)$
and $\mathcal{N}(T)$ are Chebyshev subspaces in $Y$ and $X$, respectively. Let
$\delta T\in B(X,Y)$ such that $\mathcal{R}(\delta T)\subset\mathcal{R}(T)$,
$\mathcal{N}(T)\subset\mathcal{N}(\delta T)$ and $\|T^{M}\delta T\|<1$. Put
$\bar{T}=T+\delta T$. If $T^{M}$ is quasi–additive on $\mathcal{R}(T)$, then
$\bar{T}^{M}=T^{M}(I_{Y}+\delta TT^{M})^{-1}=(I_{X}+T^{M}\delta T)^{-1}T^{M}$
with

$\dfrac{\|\bar{T}^{M}-T^{M}\|}{\|T^{M}\|}\leq\dfrac{\|T^{M}\delta
T\|}{1-\|T^{M}\delta T\|}.$

###### Proof.

From $\mathcal{R}(\delta T)\subset\mathcal{R}(T)$ and
$\mathcal{N}(T)\subset\mathcal{N}(\delta T)$, we get that
$\pi_{\mathcal{R}(T)}\delta T=\delta T$ and $\delta T\pi_{\mathcal{N}(T)}=0$,
that is, $TT^{M}\delta T=\delta T=\delta TT^{M}T$. Consequently,

$\bar{T}=T+\delta T=T(I_{X}+T^{M}\delta T)=(I_{Y}+\delta TT^{M})T$ (4.4)

Since $T^{M}$ is quasi–additive on $\mathcal{R}(T)$ and $\|T^{M}\delta T\|<1$,
we get that $I_{X}+T^{M}\delta T$ and $I_{Y}+\delta TT^{M}$ are all invertible
in $H(X,X)$. So from (4.4), we have $\mathcal{R}(\bar{T})=\mathcal{R}(T)$ and
$\mathcal{N}(\bar{T})=\mathcal{N}(T)$ and hence
$\bar{T}^{H}=T^{M}(I_{Y}+\delta TT^{M})^{-1}=(I_{X}+T^{M}\delta T)^{-1}T^{M}$
by Theorem 4.4. Finally, by Corollary 2.6,

$\displaystyle\bar{T}^{M}$
$\displaystyle=(I_{X}-\pi_{\mathcal{N}(\bar{T})})\bar{T}^{H}\pi_{\mathcal{R}(\bar{T})}=(I_{X}-\pi_{\mathcal{N}(T)})T^{M}(I_{Y}+\delta
TT^{M})^{-1}\pi_{\mathcal{R}(T)}$ $\displaystyle=(I_{X}+T^{M}\delta
T)^{-1}T^{M}\pi_{\mathcal{R}(T)}=(I_{X}+T^{M}\delta
T)^{-1}T^{M}=T^{M}(I_{Y}+\delta TT^{M})^{-1}$

and then

$\|\bar{T}^{M}-T^{M}\|\leq\|(I_{X}-T^{M}\delta
T)^{-1}-I_{X}\|\|T^{M}\|\leq\frac{\|T^{M}\delta T\|\|T^{M}\|}{1-\|T^{M}\delta
T\|}.$

The proof is completed. ∎

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# Milky Way and M31 rotation curves: $\Lambda$CDM vs. MOND

De-Chang Dai1,3111communicating author: De-Chang Dai,
email<EMAIL_ADDRESS>Glenn Starkman3, Dejan Stojkovic2 1 Center for
Gravity and Cosmology, School of Physics Science and Technology, Yangzhou
University, 180 Siwangting Road, Yangzhou City, Jiangsu Province, P.R. China
225002  2 HEPCOS, Department of Physics, SUNY at Buffalo, Buffalo, NY
14260-1500  3 CERCA/Department of Physics/ISO, Case Western Reserve
University, Cleveland OH 44106-7079

###### Abstract

We analyze the existing rotation-curve data of the Milky Way and M31 galaxies
that extends to very large distances and low accelerations. We find a
systematic downward trend in the weak acceleration (large distances) segment
of the radial acceleration. A similar downward trend has been noticed in the
$\Lambda$CDM EAGLE simulation [1], while the deviation from the generic MOND
prediction would need to be ascribed to an external field effect, or possibly
a post facto selected acceleration function $\mu(x)$.

Galaxy rotation curves were the first widely accepted evidence for the
existence of dark matter [3]. Outside the maximum radius where the luminous
matter contributes significantly to the total mass of a galaxy, Newtonian
gravity (and General Relativity (GR)) predict that the centripetal
acceleration binding tracers in orbit around the galaxy, and hence the
rotational velocity, should fall, yet consistently among all galaxies it
plateaus [3, 4, 5, 6, 7, 8, 9]. The dark matter hypothesis interprets this
plateau as evidence for the presence of additional non-luminous “dark” matter.
A number of lines of argument suggest that most, if not all, of this dark
matter would need to be non-relativistic – what cosmologists refer to as cold
dark matter (CDM) [10]. CDM is at the core of the concordance $\Lambda$CDM
model of cosmology, and the success of $\Lambda$CDM in predicting the details
of the power spectrum of temperature and polarization fluctuations of the
cosmic microwave background radiation [11] can be regarded as a stringent test
that the dark matter hypothesis has passed. Meanwhile, considerable
theoretical, experimental and observational effort has been invested in the
search for the nature of this dark matter through non-gravitational signals
and distinctive gravitational signals [12].

A competing hypothesis for the plateau in galaxy rotation curves, Modified
Newtonian Dynamics (MOND) [13], is that Newton’s law (and hence General
Relativity) is incorrect at small accelerations. Specific implementations of
the MOND paradigm at the level of phenomenological alternatives to Newton’s
law or of Poisson’s law have long been considered. While these
phenomenological alternatives are limited in their applicability (weak field,
non-relativistic, isolated systems, often requiring spherical or axial
symmetry), for those galaxies to which they can be applied they fit the
rotation curves with only a single parameter per system, the mass-to-light
ratio of the luminous matter, and a single universal function to match
standard Newtonian behavior at large acceleration to the desired behavior at
low acceleration [14]. However, the applicability of MOND to generic systems,
including cosmology, has been hampered by the absence of a suitable
Relativistic MOND (RMOND) theory – a modification of GR that reduces to MOND
in appropriate limits and is consistent with key cosmological observables and
other tests of GR. Recent work on such Relativistic MOND (RMOND) theories [15,
16] may have removed that difficulty.

The discovery [17] of the Radial Acceleration Relation (RAR) – a tight
correlation between the observed acceleration of low-acceleration tracers in
galaxies and the acceleration expected just from the baryonic matter – was
expected in the context of MOND. On the other hand it is widely regarded as a
mystery in $\Lambda$CDM, and therefore a failure of that theory. However, [18,
29] and one of the authors [1] independently examined data from the EAGLE
simulation [19, 20, 21], one of the largest cosmological hydrodynamical
simulations, and found that it unexpectedly demonstrated such a tight
correlation. Its origin must lie in physics emergent from the complex
interactions of baryonic matter and the non-linear structures it forms (e.g.,
stars), which are not directly captured by the standard particle interactions
of an N-body simulation, but which the EAGLE Project attempts to model
phenomenologically. Unsurprisingly, the EAGLE correlation does not precisely
match the observed RAR – the underlying theory is too complicated to expect an
accurate prediction from a phenomenological model that was not calibrated to
it – however this should probably be regarded as a very unexpected success of
$\Lambda$CDM, though perhaps a less compelling one than for MOND.

While the dark matter and MOND hypotheses may agree, if only by construction,
on galaxy rotation curves at radii outside the luminous matter, there is no
reason for them to continue to do so at larger radii where the centripetal
acceleration falls much lower. Consider a spiral galaxy. Within the
$\Lambda$CDM model, there are three main structures in the galaxy - a bulge, a
disk and a dark halo. The innermost part is the bulge, which is the
gravitationally dominant component within several $kpc$ from the center. The
stellar disk is then dominant out to distances of several tens of kpc, where
the CDM halo takes over. The presence of the dark matter halo is the primary
reason why the rotational velocity does not decrease dramatically after the
effects of ordinary matter are saturated. However, within $\Lambda$CDM, the
effects of dark matter also eventually saturate. At larger radii, there is no
additional component to maintain the rotational velocity at a constant value,
so it should decrease outside some characteristic radius of the halo. Within
MOND, no such transition is predicted for isolated galaxies.

Most radial velocity data in galaxies cover only accelerations larger than
$a_{min}\sim 10^{-11}m/s^{2}$, as pointed out in [18]. The region of lower
accelerations has not been fully explored yet. However, as we discuss below,
recent data on the rotation curves of the Milky Way (MW) and M31 [22, 23, 24]
extend to larger distances and hence lower accelerations. Intriguingly, as
expected in $\Lambda$CDM, after plateauing at intermediate distances (several
tens of kpc from the galactic center), the rotation curve does not remain flat
at larger radii. Rotation velocities decrease by tens of $km/s^{2}$, down to
values that are much smaller than the plateau value. In the MW, this happens
at the distances larger than several tens of kpc. Similar trends can also be
found in the galaxy M31 (Fig. 5 in [24]).

As we discuss below, this decrease in the rotation velocity at large radius is
anticipated quantitatively in the results of the EAGLE simulation. It
therefore appears to be consistent with the general expectations of
$\Lambda$CDM. On the other hand, this is not what is expected for an isolated
galaxy in MOND – there are strong indications that the rotation curves deviate
from the MOND prediction for accelerations below $10^{-11}m/s^{2}$[1]. Such
deviations are ascribed by [2] to the “external field effect” (EFE) in MOND –
a consequence of a galaxy’s environment upsetting the normal MOND expectation
(see also studies based on other modified gravity models [38, 39, 40, 41, 42,
43, 44]).

Below we present rotation curves for the Milky Way and M31, and compare them
to the results of the EAGLE simulation and to the predictions of MOND for
isolated galaxies.

##  I The Milky Way and M31 as spiral galaxies in $\Lambda$CDM

In the $\Lambda$CDM model, the rotation curve of a spiral galaxy can be well
approximated as arising due to the combined contributions from and
axisymmetric bulge, disk and dark halo. (Of course both real and simulated
galaxies are more complicated.) This allows us to represent the rotational
velocity squared as the sum of three components,

$v^{2}(r)\equiv v_{b}^{2}(r)+v_{d}^{2}(r)+v_{h}^{2}(r)\,,$ (1)

where $r$ is the distance from the center of the galaxy, while $v_{b}$,
$v_{d}$ and $v_{h}$ are the contributions to $v^{2}$ due to the bulge, disk
and dark halo respectively. Newtonian mechanics simply relates $v_{i}(r)$ (for
$i=b,d,h$) to the mass of that component within the radius r,

$v_{i}^{2}=\frac{GM_{i}(r)}{r}\,.$ (2)

A galactic bulge is taken to be spherically symmetric with a de Vaucouleur
profile [25]. Its surface mass density is

$\Sigma_{B}(r)=\Sigma_{be}e^{-\kappa\left((\frac{r}{a_{B}})^{1/4}-1\right)},$
(3)

where $\kappa=7.6695$, and $\Sigma_{be}$ is the surface mass density at the
half-mass radius $r=a_{B}$. The mass density of the bulge is then

$\rho_{b}(r)=-\frac{1}{\pi}\int_{r}^{\infty}\frac{d\Sigma_{B}(x)}{dx}\frac{1}{\sqrt{x^{2}-r^{2}}}dx\,,$
(4)

so that the bulge mass within the radius $r$ is

$M_{b}(r)=4\pi\int^{r}_{0}\rho_{b}(r)r^{2}dr\,$ (5)

and

$v^{2}_{b}(r)=\frac{4\pi G}{r}\int^{r}_{0}\rho_{b}(r)r^{2}dr\,.$ (6)

A galactic disk can be approximated by an exponential disk [26, 27] with
surface mass density

$\Sigma_{d}(r)=\Sigma_{0}\exp\Big(-\frac{r}{a_{d}}\Big{missing})\,.$ (7)

$\Sigma_{0}$ is the central value and $a_{d}$ is a scale radius. The rotation
velocity squared due to the disc [27] can be written explicitly,

$v_{d}^{2}=\pi
G\Sigma_{0}\frac{r^{2}}{a_{d}}\left[I_{0}\left(\frac{r}{2a_{d}}\right)K_{0}\left(\frac{r}{2a_{d}}\right)-I_{1}\left(\frac{r}{2a_{d}}\right)K_{1}\left(\frac{r}{2a_{d}}\right)\right]\,,$
(8)

where $I_{i}$ and $K_{i}$ are modified Bessel functions.

A dark halo is taken to follow the NFW profile [28],

$\displaystyle\rho_{h}(r)$ $\displaystyle=$
$\displaystyle\frac{\rho_{0}}{\frac{r}{h}\left(1+\frac{r^{2}}{h^{2}}\right)}\,,$
(9)

where $\rho_{0}$ and $h$ are the scale density and scale radius of the dark
halo respectively. This leads to

$M_{h}(r)=4\pi\rho_{0}h^{3}\left[\ln\left(1+\frac{r}{h}\right)-\frac{r}{r+h}\right]\,,$
(10)

and

$v_{h}^{2}=\frac{GM_{h}(r)}{r}$ (11)

The total acceleration is then

$a_{t}=\frac{v_{b}^{2}+v_{d}^{2}+v_{h}^{2}}{r}\,,$ (12)

and the expected rotation curve is

$v_{t}(r)=\sqrt{v_{b}^{2}+v_{d}^{2}+v_{h}^{2}}\,.$ (13)

Figure 1: The rotation curve of the Milky Way. The data (solid dark circles
with error bars) for $r<100$kpc come from [22], while for $r>100$kpc from
[23]. The solid, dashed and doted lines describe the contribution from the
bulge, stellar disk and dark matter halo respectively, within a $\Lambda$CDM
model of the galaxy. The dashed-dot line is the total contribution of all
three components. The parameters of each component are taken from [24]. For
comparison, the Milky way rotation curve from GAIA data releaese II is shown
in color. The red dots are data from [34], the blue upward-pointing triangles
are from [35], while the cyan downward-pointing triangles are from [36].
Figure 2: The rotation curve of the M31. The data (solid purple squares with
error bars) come from [24]. The solid, dashed and doted lines describe the
contribution from the bulge, stellar disk and dark matter halo respectively,
within a $\Lambda$CDM model of the galaxy. The dashed-dot line is the total
contribution of all three components. The parameters of each component are
taken from [24].

There are $6$ parameters ($a_{B}$, $a_{d}$, $h$, $\Sigma_{be}$, $\Sigma_{0}$
and $\rho_{0}$) that fully specify all three components of a rotationally
supported galaxy in $\Lambda$CDM. We use the values for the MW and M31 from
[24]. For MW they are $a_{B}=0.87\pm 0.07$kpc, $a_{d}=5.73\pm 1.23$kpc,
$h=10.7\pm 2.9$kpc, $\Sigma_{be}=0.25\pm 0.02\times 10^{11}M_{\odot}$,
$\Sigma_{0}=1.12\pm 0.4\times 10^{11}M_{\odot}$ and $\rho_{0}=18.2\pm
7.4\times 10^{-3}M_{\odot}pc^{-3}$. For M31 they are $a_{B}=1.35\pm 0.02$kpc,
$a_{d}=5.28\pm 0.25$kpc, $h=34.6\pm 2.1$kpc, $\Sigma_{be}=0.35\pm 0.004\times
10^{11}M_{\odot}$, $\Sigma_{0}=1.26\pm 0.08\times 10^{11}M_{\odot}$ and
$\rho_{0}=2.23\pm-.24\times 10^{-3}M_{\odot}pc^{-3}$. Figure 1 and 2 show the
rotation curve of the Milky Way and M31 respectively, and its resolution into
the bulge, disk and halo components. The data for $r<100$kpc come from [22],
while for $r>100$kpc from [23]. As expected of a low-dimensional axisymmetric
approximation of a spiral galaxy, the fit to the model reproduces the gross
features of the rotation curve reasonably well, but not the fine details.
Recent Milky way rotation curve from GAIA data release II are plotted in
figure 1 in color [34, 35, 36, 37]. The general trend is very similar to the
rotation curve based the older data [24]. We see that while the GAIA rotation
curve at modest galactic radii differs significantly from earlier data, so
far, there is no statistically significant change at larger radii where the
model suggests that dark matter halo dominates the kinematics. One could try
to model the galaxy with GAIA data in the regime where the data is available,
however the behavior at modest galactic radii will not have significant
influence at larger radii where our analysis is concentrated. We also note
that there is no such discrepancy in M31 galaxy data.

##  II Comparing to $\Lambda$CDM and MOND

To compare observations with $\Lambda$CDM and MOND predictions, the Newtonian
gravitational acceleration due to the ordinary matter must be separated from
the total acceleration. Presumably the acceleration due to baryons, $a_{B}$,
comes from the bulge and the disk (with no dark matter halo contribution),
i.e.

$a_{B}(r)=\frac{v_{b}^{2}(r)+v_{d}^{2}(r)}{r}$ (14)

The MOND-predicted acceleration is obtained from [13]

$\mu(a/a_{0})\vec{a}=\vec{a}_{B}\,,$ (15)

where $a_{0}$ is a critical acceleration constant, which we will set to
$a_{0}=1.2\times 10^{-10}m/s^{2}$, while $\mu$ is an empirical function which
is often modeled with

$\mu(x)=\frac{x}{1+x}\,.$ (16)

MOND then predicts the total acceleration to be

$a_{M}=\frac{a_{B}+\sqrt{a_{B}^{2}+4a_{0}a_{B}}}{2}\,,$ (17)

which is to be compared with (13) in $\Lambda$CDM.

In fact, this prediction assumes that the galaxies being observed are
isolated. The MOND isolated galaxy (MIG) prediction would be expected to fail
where tidal accelerations due to other structures in the environment disrupt
the symmetries of the system. This external field effect (EFE) is a
longstanding prediction of MOND, but its details – what specific tidal field
modifies (17) and how – would depend on how (15) emerges from a specific
relativistic theory of broader applicability.

In $\Lambda$CDM, disks of galaxies usually form at the centers of dark matter
halos. They span a relatively narrow range of acceleration, and the
acceleration profiles are self-similar from $a_{0}\sim 10^{-10}m/s^{2}$ to
$a_{min}\sim 10^{-11}m/s^{2}$ [18]. $\Lambda$CDM predictions for the rotation
curve then clearly deviate from MOND predictions only for $a\lesssim
a_{min}$[1]. Because this is within the dark-matter-dominated region, it is
not easily accessible to observations. Since there are very few distant
galaxies in which we can explore accelerations below $a_{min}$, this deviation
frequently goes unnoticed. However, in nearby galaxies like our own Milky Way
and M31, we can can explore such low accelerations and the distinguishable
predictions of MOND and $\Lambda$CDM, as shown in Fig. 3.

Figure 3: The total acceleration, $a$ vs. the Newtonian acceleration due to
baryons, $a_{B}$, for data and models. The black circles with error bars
represent Milky Way data. The purple squares with error bars represent M31.
The $\Lambda$CDM fit to them is the short-dashed line. (The dash-dot line is
the $\Lambda$CDM fitting curve of M31.) The dotted line is the reference line
for $a=a_{B}$. The dashed and solid lines are predicted by MOND with
$a_{0}=1.2\times 10^{-10}m/s^{2}$ and $a_{0}=1.9\times 10^{-10}m/s^{2}$. The
gray dots are from the EAGLE simulation (data file RefL0025N0752 on EAGLE’s
website) of $\Lambda$CDM in [1]. The thick horizontal (vertical) line crossing
the a ($a_{B}$) axis marks the acceleration (baryonic acceleration) below
which very little data is available – other than for the MW and M31.
Accelerations in M31 and the MW are smaller than the MOND prediction at lower
values of $a_{B}$. A potentially observable discrepancy appears at the radius
of several tens of kpc for both galaxies. We plot the results out to $500kpc$.
The galaxy mass in EAGLE’s data is chosen to be between $5\times
10^{10}M_{\odot}$ to $5\times 10^{11}M_{\odot}$. For comparison, the Milky way
rotation curve from GAIA data release II are shown in color. The red dots are
data from [34], the blue triangles are from [35], while the cyan down
triangles are from [36].

Fig. 3 shows that for smaller radii, i.e. larger accelerations, MOND does not
significantly deviate from expected $\Lambda$CDM model galaxies, but that this
changes for lower accelerations where the difference is clear. This feature
may not be easy to observe for galaxies that are further away, because the
discrepancy becomes clear only in dark halo regions far away from the bulk of
the visible matter. It should be noted that the model rotation curve of M31 is
closer to the MOND predictions than that of the Milky May at lower
accelerations. This is because M31’s halo is more spread out. ($h=34.6kpc$ for
M31 compared to $h=10.7kpc$ for the Milky Way.) These lower accelerations are
also where tidal fields are more likely to induce an external field effect in
the MOND theory.

Fig. 3 also compares the total observed acceleration $a$ with the acceleration
$a_{B}$ caused by the baryons for the case of the MW (for which the low-
acceleration data is much less noisy than for M31). The observational data
(black squares with error bars) follow the MOND isolated galaxies (MIG)
prediction (dashed and solid lines) very well for larger values of
acceleration (down to a few $10^{-10}m/s^{2}$); however, for smaller
accelerations, the observed acceleration is lower than the MIG prediction.
Fig. 1 of [2] does not extend to the same low accelerations as does this plot
with Milky Way data, which might be the reason why the discrepancy is not
striking in [2]. Fig. 4 explicitly shows this difference by plotting
$a/a_{M}-1$: MOND predicts accelerations that are much larger than observed,
and the theory and data appear well-separated for $a<10^{-11}m/s^{2}$. This is
the effect that was found in data in [2] and ascribed to the expected external
field effect (EFE) – the fact that tidal fields due to nearby mass
concentrations (e.g. other galaxies) disrupt the MIG prediciton (17).

In contrast, the MW and M31 models, like the MW data, are all alike in falling
below the MOND prediction at low $a_{B}$. One might complain that it is not
surprising that the MW model match the data, however Fig. 3 also includes the
data from EAGLE galaxies [19, 20, 21] as presented in [1]. The turndown in $a$
at low $a_{B}$ matches the feature previously pointed out by one of us [1] in
the EAGLE simulation [19, 20, 21]. We see that for EAGLE $a$ vs. $a_{B}$
follows a trend similar to that in MOND for $a\gtrsim a_{min}$ with reasonably
low dispersion, but then $a$ falls faster at low $a_{B}$ in EAGLE than
predicted by MOND, and the dispersion in the simulated data increases. This
broad feature is consistent with the data, and with the fits to simple dark-
matter models.

We note here that the EAGLE simulation data points in figure Fig. 4 are a
combination of many independent galaxies. The rotation curves of galaxies with
less dark matter are closer to Newtonian physics. On the other hand, the
rotation curves of galaxies with more dark matter are well above the MOND
prediction. In general, MOND predicts some average behavior, instead of a
precise behavior for every single galaxy. This is the reason why one can see
single-galaxy deviations from MOND clearly, while the deviation is not so
clear if many galaxies are included.

It would be attractive to be able to compare the data to the EAGLE simulation
quantitatively at these low $a$ and $a_{B}$; however this is not necessarily
justified, and we certainly resist doing so here. The baryonic physics that,
in EAGLE, results in the MOND-like radial acceleration relation and in the
downturn and increased (fractional) dispersion at low $a_{B}$, is microphysics
that is modeled phenomenologically. Even if we were convinced that all and
only the correct relevant physics has been captured in EAGLE, the appropriate
Bayesian hierarchal model comparison needed to draw a statistically robust
conclusion would certainly require more sophisticated simulation or emulation
infrastructure than is currently extant. Nevertheless, we think the case is
clear that where ultra-low-acceleration galaxy-rotation-curve data exist they
display both the RAR-like correlation of $a$ and $a_{B}$ expected from EAGLE
at moderately low accelerations, and the stronger tail off expected at the
lowest accelerations.

Figure 4: Figure 3 recast as a comparison between the total acceleration, $a$,
and the MOND prediction, $a_{M}$, as a function of the acceleration due to
baryons $a_{B}$. The solid horizontal line is $a=a_{M}$. The circles and
squares with error bars represent the Milky Way and M31 data, while the gray
dots are from the EAGLE simulation of $\Lambda$CDM in [1]. For $a_{B}\gtrsim
10^{-10}m/s^{2}$ any difference between $a$ and $a_{M}$is unclear. However,
once $a_{B}$ drops well below $10^{-11}m/s^{2}$, the discrepancy emerges. The
short-dashed line is the $\Lambda$CDM fitting curve of the MW. The dash-dot
line is the $\Lambda$CDM fitting curve of M31. The mass range of galaxies in
EAGLE’s data is chosen to be between $5\times 10^{10}M_{\odot}$ to $5\times
10^{11}M_{\odot}$. For comparison, the Milky way rotation curve from GAIA data
release II is shown in color. The red dots are data from [34], the blue
triangles are from [35], while the cyan down triangles are from [36]. While
the EAGLE simulation does not match the data perfectly, these plots indicate
that it is much easier to accommodate a systematic downward trend with the
$\Lambda$CDM model than with MOND.

Similar indications come from other studies. The authors of [30] found that
$\Lambda$CDM reproduces the acceleration relation predicted by MOND, and
conjectured that this relation was a consequence of dissipative collapse of
baryons. Such baryonic physics is not directly obtained by individual baryonic
particles in the simulations colliding and emitting photons into the
simulation volume, rather it is modeled phenomenologically. However, their
study extended only to accelerations $\gtrsim 10^{-11.5}$ m/s2, where we do
not expect to see a clear deviation from RAR. In [31] the authors found that
the deviation from RAR becomes clear for radii larger than 100kpc, which
agrees with our findings.

Meanwhile, the RAR has been studied using weak-lensing data for released by
the Kilo-Degree Survey, first [32] using KiDS GGL on isolated foreground
galaxies from 180 deg2 of the Galaxy and Mass Assembly (GAMA) survey, and more
recently [33] using a selection of 1 million foreground galaxies from
KiDS-1000 to achieve a fivefold increase in survey area. The GAMA survey
results were broadly consistent with the MOND prediction for an extension of
the RAR down to $a\simeq 10^{-12.5}$ m/s2 with the observed stellar baryons
plus inferred cold gas; the KiDS-1000 yielded $a/a_{b}$ that were
systematically higher, matching the MOND prediction with the additional
inclusion of a hot gas component. All of them are well above the MW and M31
RARs at $10^{-12.5}-10^{-11}$m/s2. Only the mean RAR is reported. These
results appear to be in tension with our analysis here. However, as the
authors of [32] state, a fundamental limitation of their analysis is that the
additional diffuse gas surrounding galaxies is difficult to measure, and has
not been included in their study. The existence of gaseous halos not included
in their study would enhance acceleration from baryons (and thus $a_{M}$), and
the measured rotation curve will be well below what MOND predicts.

In summary, flat galactic rotation curves can be explained either by
introducing a dark matter component, like in the $\Lambda$CDM model, or by
modifying Newtonian dynamics, like in MOND theories. However, it has been
noticed that the acceleration systematically goes down at large distances,
i.e. low accelerations. This systematic downward trend in the weak gravity
segment of the galactic rotation curves was interpreted in [2] as an external
field effect within the framework of the MOND theories. However, this feature
was observed in the $\Lambda$CDM model without invoking any modification of
Newtonian gravity, as was first found by one of the authors in the EAGLE
simulation [1]. They showed that dark halos in $\Lambda$CDM generate an
acceleration feature like MOND predicts, but that in the even-larger-radius,
even-lower-acceleration regions, the dark halos are saturated and the
acceleration then decreases as Kepler’s Laws, i.e. Newtonian gravity, but not
MOND predict. We corroborated this finding here by analyzing the Milky Way and
M31 data. At very large distances, these data are easier to accommodate in the
$\Lambda$CDM model.

###### Acknowledgements.

D.C Dai is supported by the National Natural Science Foundation of China
(Grant No. 11775140). D.S. is partially supported by the US National Science
Foundation, under Grant No. PHY-2014021. GDS is supported by a Department of
Energy grant DE-SC0009946 to the particle astrophysics theory group at CWRU.

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