# Local spectral properties of typical contractions on $\ell_{p}\,$\- spaces

Sophie Grivaux  CNRS, Univ. Lille, UMR 8524 - Laboratoire Paul Painlevé,
F-59000 Lille, France<EMAIL_ADDRESS>Étienne Matheron
Laboratoire de Mathématiques de Lens
Université d’Artois
Rue Jean Souvraz SP 18
62307 Lens, France<EMAIL_ADDRESS>
###### Abstract.

We study some local spectral properties of contraction operators on
$\ell_{p}$, $1<p<\infty$ from a Baire category point of view, with respect to
the Strong∗ Operator Topology. In particular, we show that a typical
contraction on $\ell_{p}$ has Dunford’s Property (C) but neither Bishop’s
Property $(\beta)$ nor the Decomposition Property $(\delta)$, and is
completely indecomposable. We also obtain some results regarding the
asymptotic behavior of orbits of typical contractions on $\ell_{p}$.

###### keywords:

Polish topologies, $\ell_{p}\,$\- spaces, typical properties of operators,
local spectrum, orbits of operators

###### 1991 Mathematics Subject Classification:

47A15, 47A16, 54E52

This work was supported in part by the project FRONT of the French National
Research Agency (grant ANR-17-CE40-0021) and by the Labex CEMPI
(ANR-11-LABX-0007-01).

_To the memory of Jörg Eschmeier_

##  1\. Introduction

In this note, we shall be interested in the _typicality_ , in the sense of
Baire Category, of some natural properties of Banach space operators
pertaining to _local spectral theory_.

In the whole paper, the letter $X$ will denote an infinite-dimensional complex
(separable) Banach space with separable dual. Most of the time, $X$ will be in
fact $\ell_{p}:=\ell_{p}(\mathbb{Z}_{+})$ for some $1<p<\infty$. We denote by
${\mathcal{B}}(X)$ the algebra of bounded operators on $X$, and by
${\mathcal{B}}_{1}(X)$ the set of contraction operators on $X$:

${\mathcal{B}}_{1}(X)=\bigl{\\{}T\in{\mathcal{B}}(X)\;;\;||T||\leq
1\bigr{\\}}.$

There are several natural topologies on ${\mathcal{B}}_{1}(X)$ which may turn
it into a Polish (i.e. separable and completely metrizable) space, thus
allowing for a study of “large” sets of contractions on $X$ in the sense of
Baire Category. Some well-known such topologies are the Weak Operator Topology
(WOT), the Strong Operator Topology (SOT), and the Strong∗ Operator Topology
(SOT∗). In this note, we shall almost exclusively be concerned with the
topology SOT∗, which is defined as follows: a net
$(T_{i})\subseteq{\mathcal{B}}(X)$ converges to $T\in{\mathcal{B}}(X)$ with
respect to SOT∗ if and only if $T_{i}x$ tends to $Tx$ in norm for every $x\in
X$ and $T_{i}^{*}x^{*}$ tends to $T^{*}x^{*}$ in norm for every $x^{*}\in
X^{*}$. In other words, $T_{i}\to T$ for SOT∗ if and only if $T_{i}\to T$ and
$T_{i}^{*}\to T^{*}$ for SOT. Under our assumptions on $X$, it is well-known
that ${\mathcal{B}}_{1}(X)$ is a Polish space when endowed with SOT∗. In
addition to its Polishness, one pleasant feature of the topology SOT∗ is that
when $X$ is reflexive, the map $T\mapsto T^{*}$ is a homeomorphism from
${\mathcal{B}}_{1}(X)$ onto $\mathcal{B}_{1}(X^{*})$.

In this paper, the word _typical_ will always refer to the topology SOT∗,
unless the contrary is explicitly mentioned. Thus, we will say that a given
property (P) of operators $T\in{\mathcal{B}}_{1}(X)$ is typical, or that _a
typical $T\in{\mathcal{B}}_{1}(X)$ has Property _(P)__ , if the set
$\bigl{\\{}T\in{\mathcal{B}}_{1}(X)\;;\;T\ \hbox{has property (P)}\bigr{\\}}$
is comeager in $({\mathcal{B}}_{1}(X),\texttt{SOT}\mbox{${}^{*}$})$, i.e.
contains a dense $G_{\delta}$ subset of
$({\mathcal{B}}_{1}(X),\texttt{SOT}\mbox{${}^{*}$})$. In other words, a
property (P) of contractions on $X$ is typical if the set of all
$T\in{\mathcal{B}}_{1}(X)$ enjoying it is very large in the sense of Baire
Category.

Typical properties of _Hilbert_ space contractions with respect to the
topologies WOT, SOT and SOT∗ are investigated in depth in [Ei] and [EM]. The
main conclusions drawn from these works are that the generic situation is
rather well understood in the case of WOT (a typical contraction is unitary);
that it is “trivial” in the case of SOT (a typical contraction is unitarily
equivalent to the backward shift with infinite multiplicity, so there is
generically _just one_ Hilbert space contraction); and that things are more
complicated for the topology SOT∗. This line of thought is pursued in [GMM1]
and [GMM2]. The setting of [GMM1] is mostly Hilbertian, although many of the
proofs work as well on $\ell_{p}\,$\- spaces; and for that reason, emphasis is
mostly put on the topology SOT∗. In [GMM2], the setting is explicitly that of
$\ell_{p}\,$\- spaces, and the typicality of certain properties of operators
pertaining to the study of the Invariant Subspace Problem is considered for
the topologies SOT and SOT∗.

This note may be viewed as an addendum to [GMM2]. Our aim will be to determine
whether certain properties of operators related to local spectral theory are
typical in $\mathcal{B}_{1}(\ell_{p})$, $1<p<\infty$. As will be explained
below, our study was in part motivated by results from [Br2], [EP1], and [EP2]
showing the existence of non-trivial invariant subspaces under suitable
spectral assumptions.

Throughout the paper, we denote by $\mathbb{D}$ the open unit disk in
$\mathbb{C}$, and by $\overline{\,\mathbb{D}}$ the closed unit disk. Also, we
denote by $\sigma(T)$ the spectrum of an operator $T\in{\mathcal{B}}(X)$.
Thus, we have $\sigma(T)\subseteq\overline{\,\mathbb{D}}$ for every
$T\in{\mathcal{B}}_{1}(X)$.

##  2\. Background from local spectral theory

In order to be as self-contained as possible, we recall in this section some
basic definitions as well as some important results from local spectral
theory. We follow the terminology and notations of [LN].

###  2a. Decomposable operators

A large part of local spectral theory is motivated by the study of so-called
_decomposable operators_. An operator $T\in{\mathcal{B}}(X)$ is decomposable
if for any covering $(U_{1},U_{2})$ of $\mathbb{C}$ by two open subsets, there
exist two closed $T$\- invariant subspaces $E_{1}$ and $E_{2}$ such that
$\sigma(T_{|E_{1}})\subseteq U_{1}$, $\sigma(T_{|E_{2}})\subseteq U_{2}$, and
$X=E_{1}+E_{2}$. For instance, it follows from the Riesz Decomposition Theorem
that any operator with totally disconnected spectrum is decomposable. Also,
any surjective isometry is decomposable (see [LN, Proposition 1.6.7]). Two
related notions introduced and studied in [AtzSod] are _strong
indecomposability_ , and _complete indecomposability_ : an operator
$T\in{\mathcal{B}}(X)$ is strongly indecomposable if its spectrum $\sigma(T)$
is not a singleton and is included in the spectrum of the restriction of $T$
to any of its (closed) non-zero invariant subspaces; and $T$ is completely
indecomposable if both $T$ and $T^{*}$ are strongly indecomposable.

###  2b. Local spectra

The _local resolvent set_ of an operator $T\in{\mathcal{B}}(X)$ at a vector
$x\in X$ is the open subset $\rho_{x}(T)$ of $\mathbb{C}$ defined in the
following way: a complex number $\lambda$ belongs to $\rho_{x}(T)$ if there
exists a holomorphic function $f:V\to X$ defined on some neighborhood $V$ of
$\lambda$ such that $(T-z)f(z)=x$ for every $z\in V$. Note that the usual
resolvent set $\rho(T):=\mathbb{C}\,\setminus\,\sigma(T)$ is contained in
$\rho_{x}(T)$: for any $\lambda\in\rho(T)$, we may take $V:=\rho(T)$ and we
_must_ take $f(z):=(T-z)^{-1}x$ for every $z\in\rho(T)$. The _local spectrum_
of $T$ at $x$ is the compact set
$\sigma_{x}(T)=\mathbb{C}\,\setminus\,\rho_{x}(T)$, which is contained in
$\sigma(T)$. Obviously $\sigma_{x}(T)=\emptyset$ if $x=0$; but it may happen
that $\sigma_{x}(T)=\emptyset$ for some non-zero vector $x$.

###  2c. Eigenvector fields, and SVEP

Let $T\in{\mathcal{B}}(X)$. A _holomorphic $T\,$\- eigenvector field_ is a
holomorphic function $E:V\to X$ defined on some open set
$V\subseteq\mathbb{C}$, such that $(T-z)E(z)=0$ for every $z\in V$. (The
terminology is slightly inaccurate since $E(z)$ is allowed to be $0$.) The
operator $T$ has the _Single Valued Extension Property_ (SVEP) if for any open
set $V\subseteq\mathbb{C}$, the only holomorphic $T\,$\- eigenvector field
$E:V\to X$ is $E=0$. It is rather clear that if $T$ has SVEP then, for any
$x\in X$, there is a unique holomorphic function $f:\rho_{x}(T)\to X$ solving
the equation $(T-\lambda)f(\lambda)\equiv x$ on $\rho_{x}(T)$, which is called
the _local resolvent function for $T$ at $x$_. Moreover, it can be shown that
$T$ has SVEP if and only if $\sigma_{x}(T)\neq\emptyset$ for every $x\neq 0$
(see [LN, Proposition 1.2.16]). Obvious examples of operators with SVEP are
all operators whose set of eigenvalues has empty interior. In particular, any
isometry has SVEP. On the other hand, the usual backward shift $B$ on
$\ell_{p}$ lacks SVEP in a strong way, since it has a _spanning_ holomorphic
eigenvector field defined in the unit disk $\mathbb{D}$, i.e. a holomorphic
eigenvector field $E:\mathbb{D}\to\ell_{p}$ such that $\overline{\rm
span}\,\\{E(z);\;z\in\mathbb{D}\\}=\ell_{p}$; namely,
$E(z):=\sum_{j=0}^{\infty}z^{j}e_{j}$, where $(e_{j})_{j\geq 0}$ is the
canonical basis of $\ell_{p}$. More generally, by a classical result of Finch
[F] (see also [LN, Proposition 1.2.10]), any surjective but non-invertible
operator lacks SVEP. At the other extreme, the usual _forward_ shift $S$ on
$\ell_{p}$ has “maximal local spectra”, i.e. it satisfies
$\sigma_{x}(S)=\overline{\,\mathbb{D}}$ for every $x\neq 0$. In fact, the
Remark after [Vr, Theorem 1.5] shows the following: if $T\in\mathcal{B}(X)$ is
such that its Banach space adjoint $T^{*}\in\mathcal{B}(X^{*})$ admits a
spanning holomorphic eigenvector field $E$ defined on some connected open set
$\Omega\subseteq\mathbb{C}$, then $\sigma_{x}(T)$ contains $\overline{\Omega}$
for every $x\neq 0$.

###  2d. Local spectral radii

If $T\in{\mathcal{B}}(X)$ and $x\in X$, the _local spectral radius_ of $T$ at
$x$ is the number

$r_{x}(T):=\limsup_{n\to\infty}\,||T^{n}x||^{1/n}.$

It is not difficult to see that
$r_{x}(T)\geq\sup\\{|\lambda|\;;\;\lambda\in\sigma_{x}(T)\\}$; and a littler
harder to show that if $T$ has SVEP, then
$r_{x}(T)=\sup\\{|\lambda|\;;\;\lambda\in\sigma_{x}(T)\\}$ for every $x\neq
0$. This is the _local spectral radius_ formula, see [LN, Proposition 3.3.13].

###  2e. Spectral subspaces

Let $T\in{\mathcal{B}}(X)$. For any subset $F$ of $\mathbb{C}$, let

$X_{T}(F):=\\{x\in X\;;\;\sigma_{x}(T)\subseteq F\\}.$

With this notation, we see from 2c that the operator $T$ has SVEP if and only
if $X_{T}(\emptyset)=\\{0\\}$; and it turns out that this holds if and only if
$X_{T}(\emptyset)$ is closed in $X$. Also, it can be shown that $X_{T}(F)$ is
always a $T\,$\- hyperinvariant linear subspace of $X$. See [LN, Proposition
1.2.16]. Moreover, if $T$ has SVEP and if $F$ is a closed set such that
$X_{T}(F)$ is closed, then
$\sigma\bigl{(}T_{|X_{T}(F)}\bigr{)}\subseteq\sigma(T)\cap F$ (see [LN,
Proposition 1.2.20]).

The subspaces $X_{T}(F)$ may be called _local spectral subspaces_ for $T$,
since $x\in X_{T}(F)$ means that for any $\lambda\in\mathbb{C}\setminus F$,
there is a holomorphic solution of the equation $(T-z)f(z)\equiv x$ defined in
a neighborhood of $\lambda$. When $F$ is closed, one can also define a
_global_ spectral subspace, denoted by $\mathcal{X}_{T}(F)$: a vector $x$
belongs to $\mathcal{X}_{T}(F)$ if there is a holomorphic solution of
$(T-z)f(z)\equiv x$ globally defined on $\mathbb{C}\setminus F$. Obviously
$\mathcal{X}_{T}(F)\subseteq X_{T}(F)$; and if $T$ has SVEP then
$\mathcal{X}_{T}(F)=X_{T}(F)$ for any closed set $F\subseteq\mathbb{C}$.

###  2f. Property (C)

An operator $T\in{\mathcal{B}}(X)$ is said to have _Dunford’s Property
$(\emph{C})$_ if $X_{T}(F)$ is closed in $X$ for every closed set
$F\subseteq\mathbb{C}$. By what has been said just after the definition of
$X_{T}(F)$, Property (C) implies SVEP. Moreover, it can be shown that
decomposable operators have Property (C), and in fact that an operator
$T\in{\mathcal{B}}(X)$ is decomposable if and only if (i) $T$ has Property (C)
and (ii) for every open covering $(U_{1},U_{2})$ of $\mathbb{C}$, it holds
that $X=X_{T}(\overline{\,U_{1}})+X_{T}(\overline{\,U_{2}})$. See [LN, Theorem
1.2.23].

###  2g. Property $(\delta)$

The global spectral subspaces $\mathcal{X}_{T}(F)$ are involved in the
definition of the important _Decomposition Property_ $(\delta)$: an operator
$T\in{\mathcal{B}}(X)$ has Property $(\delta)$ if for any open covering
$(U_{1},U_{2})$ of $\mathbb{C}$, we have
$\mathcal{X}_{T}(\overline{\,U_{1}})+\mathcal{X}_{T}(\overline{\,U_{2}})=X$.
Every decomposable operator has property $(\delta)$; and more precisely, an
operator is decomposable if and only if it has both properties (C) and
$(\delta)$. See [LN, Proposition 1.2.29].

###  2h. Property $(\beta)$

An operator $T\in{\mathcal{B}}(X)$ has _Bishop’s Property $(\beta)$_ if the
following holds true: for any open set $V\subseteq\mathbb{C}$ and any sequence
of holomorphic functions $\phi_{n}:V\to X$, if
$(T-\lambda)\phi_{n}(\lambda)\to 0$ uniformly on compact subsets of $V$, then
$\phi_{n}(\lambda)\to 0$ uniformly on compact sets. Obviously, Property
$(\beta)$ implies SVEP; and in fact, Property $(\beta)$ implies Property (C)
(see [LN, Proposition 1.2.19]). That the converse is not true is a classical
result due to Miller and Miller [MiMi] (see also [LN, Theorem 1.6.16]). On the
other hand, decomposable operators have Property $(\beta)$ (see [LN, Theorem
1.2.7]); and the link between decomposability and Property $(\beta)$ is
actually much deeper: by a famous result of Albrecht and Eschmeier [AE] (see
also [LN, Theorem 2.4.4]), an operator $T\in{\mathcal{B}}(X)$ has Property
$(\beta)$ if and only if it is similar to the restriction of some decomposable
operator to one of its invariant subspaces.

Since property $(\beta)$ implies property (C), and since $T$ is decomposable
if and only if it has both properties (C) and $(\delta)$, it follows that $T$
is decomposable if and only if it has both properties $(\beta)$ and
$(\delta)$.

Also, since invertible isometries are decomposable and since every (not
necessarily invertible) isometry is the restriction of some invertible
isometry on a larger Banach space by a classical result of Douglas [D] (see
also [LN, Proposition 1.6.6]), we see that every isometry has Property
$(\beta)$, and hence Property (C). (Incidentally, this shows that an operator
$T$ may have Property (C) whereas $T^{*}$ does not and even lacks SVEP:
consider the usual forward shift $S$ on $\ell_{p}$, for which $X_{T}(F)=X$ if
$F\supseteq\overline{\,\mathbb{D}}$ and $X_{T}(F)=\\{0\\}$ otherwise.)

Finally, there is a quite remarkable duality between properties $(\beta)$ and
$(\delta)$, due to Albrecht-Eschmeier [AE] (see also [LN, Theorem 2.5.18]): an
operator $T$ has Property $(\beta)$ if and only if $T^{*}$ has Property
$(\delta)$, and $T$ has Property $(\delta)$ if and only if $T^{*}$ has
Property $(\beta)$. In particular, $T$ is decomposable if and only if $T^{*}$
is decomposable.

###  2i. Invariant subspaces

Local spectral theory provides sophisticated and extremely efficient tools for
extending to the Banach space setting some Hilbertian invariant subspaces
results concerning operators with _thick spectrum_. Recall that a compact
subset $K$ of $\mathbb{C}$ is said to be _thick_ if there exists a non-empty
bounded open set $U\subseteq\mathbb{C}$ in which $K$ is _dominating_ , which
means that $\sup_{z\in U}|f(z)|=\sup_{z\in U\cap K}|f(z)|$ for every bounded
holomorphic function $f$ on $U$.

A well-known result of Brown [Br2] states that any hyponormal operator with
thick spectrum on a Hilbert space admits a non-trivial invariant closed
subspace. This result admits several extensions to the Banach space setting,
perhaps the ultimate one being due to Eschmeier and Prunaru [EP2]. It involves
the notion of _localizable spectrum_ of an operator. If
$T\in{\mathcal{B}}(X)$, the localizable spectrum $\sigma_{loc}(T)$ of $T$ is
the closed subset of $\mathbb{C}$ defined as

$\sigma_{loc}(T):=\bigl{\\{}\lambda\in\mathbb{C}\;;\;\mathcal{X}_{T}(\overline{V})\neq\\{0\\}\
\textrm{for any neighborhood}\ V\ \textrm{of}\ \lambda\bigr{\\}}.$

It is not difficult to see, using Liouville’s Theorem, that
$\sigma_{loc}(T)\subseteq\sigma(T)$; and it is shown in [MiMiN, Theorem 2]
that equality holds if $T$ has Property $(\delta)$. Regarding the existence of
invariant subspaces, the main result of [EP2] runs as follows: If
$T\in{\mathcal{B}}(X)$ is such that either $\sigma_{loc}(T)$ or
$\sigma_{loc}(T^{*})$ is thick, then $T$ has a non-trivial closed invariant
subspace. Since $\sigma_{loc}(T)=\sigma(T)$ if $T$ has Property $(\delta)$, it
follows that any operator with Property $(\delta)$ or Property $(\beta)$ and
with thick spectrum has a non-trivial invariant subspace ([EP1], see also [LN,
Theorem 2.6.12]). As any hyponormal operator on a Hilbert space has property
$(\beta)$, this extends indeed the Hilbertian result from [Br2].

This result from [EP2] was the initial motivation for this work. Indeed,
[GMM2] left open the following basic question: does a typical operator on
$\ell_{p}$ for $1<p\neq 2<\infty$ has a non-trivial invariant subspace? As a
typical operator $T$ on $\ell_{p}$ satisfies
$\sigma(T)=\overline{\,\mathbb{D}}$ by [GMM2, Proposition 7.2], and hence has
thick spectrum, it is natural, in view of [EP2] to wonder whether a typical
$T\in\mathcal{B}_{1}(\ell_{p})$ has Property $(\delta)$ or Property $(\beta)$,
or has thick localizable spectrum. We will unfortunately see that none of this
happens. Thus, it seems that methods from local spectral theory are not likely
to help much for showing that a typical contraction on $\ell_{p}$, $p\neq 2$,
has a non-trivial invariant subspace. For $p=2$, it follows from the Brown-
Chevreau-Pearcy Theorem [BCP] and from the fact that a typical $T$ is such
that $\sigma(T)=\overline{\,\mathbb{D}}$, that a typical
$T\in\mathcal{B}_{1}(\ell_{2})$ does have a non-trivial invariant subspace.

###  2j. Results

The rest of the paper is organized as follows. In Section 3, we show that
typical operators on $\ell_{p}$ have maximal local spectra, and we draw
several consequences from this. Most notably, we show that a typical
$T\in{\mathcal{B}}_{1}(X)$ has Property (C) but has neither Property
$(\delta)$ nor Property $(\beta)$, is completely indecomposable and has empty
localizable spectrum. In Section 4, we present some results regarding the
asymptotic behavior of orbits of typical operators on $\ell_{p}$. The general
idea is that, for a typical $T\in\mathcal{B}_{1}(\ell_{p})$, no non-zero orbit
can be “too small”, yet most orbits are “partly small”. One consequence of
these results is that for any $M>1$, a typical $T\in\mathcal{B}_{M}(\ell_{2})$
satisfies a strong form of _distributional chaos_. In Section 5, we deviate a
little bit from our main topic: we state a simple condition ensuring that an
operator lacks SVEP in a strong way, and we use this to give examples of
operators admitting a mixing Gaussian measure with full support. Finally, we
gather in Section 6 some open questions and remarks.

###  2k. Duality

The following fact will be used several times in the paper. Each time we will
write “by duality”, this will refer implicitly to this fact.

######  Fact 2.1.

Let (P) be a property of Banach space operators. If, for any $1<p<\infty$, a
typical $T\in\mathcal{B}_{1}(\ell_{p})$ satisfies (P), then it is also true
that for any $1<p<\infty$, a typical $T\in\mathcal{B}_{1}(\ell_{p})$ is such
that $T^{*}$ satisfies (P).

###### Proof.

This is clear since the map $T\mapsto T^{*}$ is a homeomorphism from
$\mathcal{B}_{1}(\ell_{p})$ onto $\mathcal{B}_{1}(\ell_{q})$, where $q$ is the
conjugate exponent of $p$. Of course, this argument works only because we are
using the topology SOT∗ (it would break down for the topologies WOT and SOT,
for instance). ∎

##  3\. Operators with maximal local spectra

It is proved in [GMM2, Proposition 3.9] that a typical
$T\in(\mathcal{B}_{1}(\ell_{p}),\texttt{SOT})$, $1<p<\infty$ is such that
$\sigma(T)=\overline{\,\mathbb{D}}$; and the proof given there works for the
topology SOT∗ as well. Thus, typical contractions on $\ell_{p}$ have “maximal”
spectrum. The following result strengthens this statement and shows that
typical contractions on $\ell_{p}$ have maximal local spectra. This is to be
compared with [Vr, Theorem 1.5], where it is shown that for any Banach space
$X$ and any $T\in{\mathcal{B}}(X)$ with SVEP, the set $\\{x\in
X;\;\sigma_{x}(T)=\sigma(T)\\}$ is residual in $X$.

######  Theorem 3.1.

Let $X=\ell_{p}$, $1<p<\infty$. A typical $T\in{\mathcal{B}}_{1}(X)$ is such
that $\sigma_{x}(T)=\overline{\,\mathbb{D}}$ for every $x\neq 0$.
Equivalently, a typical $T\in{\mathcal{B}}_{1}(X)$ has the following property:
for any closed, $T\,$\- invariant subspace $Z\subseteq X$, it holds that
$\sigma_{x}\bigl{(}T_{|Z}\bigr{)}=\overline{\,\mathbb{D}}$ for every $x\neq 0$
in $Z$.

The equivalence of the two statements is easy to check. Indeed, it follows
directly from the definition of the local spectra that if
$T\in{\mathcal{B}}(X)$ and if $Z$ is a closed $T\,$\- invariant subspace of
$X$, then $\sigma_{x}\bigl{(}T_{|Z}\bigr{)}\supseteq\sigma_{x}(T)$ for every
$x\in Z$. Hence, if $\|T\|\leq 1$ and $\sigma_{x}(T)=\overline{\,\mathbb{D}}$
for some $x\in Z$, then a fortiori
$\sigma_{x}\bigl{(}T_{|Z}\bigr{)}=\overline{\,\mathbb{D}}$.

Before giving the proof of Theorem 3.1, we present some consequences.

######  Corollary 3.2.

Let $X=\ell_{p}$, $1<p<\infty$. A typical $T\in{\mathcal{B}}_{1}(X)$ has the
following properties : $T-\lambda$ is one-to-one with dense range for every
$\lambda\in\mathbb{C}$, and $T-\lambda$ does not have closed range for any
$\lambda\in\overline{\,\mathbb{D}}$. In particular, the essential spectrum of
a typical $T\in{\mathcal{B}}_{1}(X)$ is equal to $\overline{\,\mathbb{D}}$.

###### Proof.

These results are known; see [EM, Proposition 6.3], [GMM1, Proposition 2.24
and Remark 2.30] and [GMM2, Proposition 7.1]. However, they can also be
deduced from Theorem 3.1. Indeed, since $\sigma_{x}(T)\subseteq\\{\lambda\\}$
if $Tx=\lambda x$, it follows from Theorem 3.1 that a typical
$T\in{\mathcal{B}}_{1}(X)$ has no eigenvalue. By duality (see Fact 2.1), this
implies that a typical $T\in{\mathcal{B}}_{1}(X)$ is such that $T-\lambda$ has
dense range for every $\lambda\in\mathbb{C}$. Moreover, it also follows from
Theorem 3.1 that a typical $T\in{\mathcal{B}}_{1}(X)$ is such that
$\sigma(T)=\overline{\,\mathbb{D}}$; and so for a typical
$T\in{\mathcal{B}}_{1}(X)$, the operator $T-\lambda$ cannot have closed range
for any $\lambda\in\overline{\,\mathbb{D}}$. ∎

######  Corollary 3.3.

Let $X=\ell_{p}$, $1<p<\infty$. A typical $T\in{\mathcal{B}}_{1}(X)$ is
completely indecomposable.

###### Proof.

Since $\sigma_{x}\bigl{(}T_{|Z}\bigr{)}\subseteq\sigma\bigl{(}T_{|Z}\bigr{)}$
for any $T\,$\- invariant subspace $Z\subseteq X$ and every $x\in Z$, it is
clear from Theorem 3.1 that a typical $T\in{\mathcal{B}}_{1}(X)$ is strongly
indecomposable. By duality, it follows that a typical
$T\in{\mathcal{B}}_{1}(X)$ is such that $T^{*}$ is strongly indecomposable as
well. ∎

As another consequence of Theorem 3.1, we are able to determine whether the
local spectral properties presented in Section 2 hold, or not, for a typical
$T\in\mathcal{B}_{1}(\ell_{p})$:

######  Corollary 3.4.

Let $X=\ell_{p}$, $1<p<\infty$. A typical $T\in{\mathcal{B}}_{1}(X)$ has the
following properties.

  1. (i)

$r_{x}(T)=\sup\,\\{|\lambda|;\;\lambda\in\sigma_{x}(T)\\}=1$ for every $x\neq
0$.

  2. (ii)

For any $F\subseteq\mathbb{C}$, either $X_{T}(F)=\\{0\\}$ or $X_{T}(F)=X$;
more precisely: $X_{T}(F)=X$ if $F$ contains $\overline{\,\mathbb{D}}$, and
$X_{T}(F)=\\{0\\}$ otherwise.

  3. (iii)

$T$ and $T^{*}$ have Property _(C)_ , and hence _SVEP_.

  4. (iv)

$T$ and $T^{*}$ have neither Property $(\beta)$, nor Property $(\delta)$.

  5. (v)

$T$ has empty localizable spectrum: $\sigma_{loc}(T)=\emptyset$.

###### Proof.

Note that if $\sigma_{x}(T)=\overline{\,\mathbb{D}}$ for all $x\neq 0$, then
in particular $T$ has SVEP and hence
$r_{x}(T)=\sup\,\bigl{\\{}|\lambda|;\;\lambda\in\sigma_{x}(T)\bigr{\\}}$ for
every $x\neq 0$. This proves (i).

Part (ii) is clear.

From (ii), we deduce that a typical $T\in{\mathcal{B}}_{1}(X)$ has Property
(C) and does not have Property $(\delta)$. Indeed, this is clear for (C). As
for $(\delta)$, assume that $T$ satisfies (ii). Setting $U_{1}:=D(0,2/3)$ and
$U_{2}:=\mathbb{C}\setminus\overline{D}(0,1/3)$, we get an open covering
$(U_{1},U_{2})$ of $\mathbb{C}$ such that neither $\overline{\,U_{1}}$ nor
$\overline{\,U_{2}}$ contains $\overline{\,\mathbb{D}}$; so we have
$X_{T}(\overline{\,U_{1}})=\\{0\\}=X_{T}(\overline{\,U_{2}})$, which shows
that $T$ does not have Property $(\delta)$.

Thus, for any $1<p<\infty$, a typical $T\in\mathcal{B}(\ell_{p})$ has property
(C) and does not have Property $(\delta)$. By duality, and since an operator
$T$ has Property $(\beta)$ if and only if $T^{*}$ has Property $(\delta)$,
this proves (iii) and (iv). Part (v) follows immediately from (ii). ∎

The next corollary shows that the _algebraic core_ and the _analytic core_ of
a typical $T\in{\mathcal{B}}_{1}(X)$ are far from being equal. Recall that if
$T\in{\mathcal{B}}(X)$, the algebraic core of $T$, denoted by $C(T)$, is the
set of all $x\in X$ admitting a backward $T\,$\- orbit, i.e. a sequence
$(x_{n})_{n\geq 0}$ such that $x_{0}=x$ and $Tx_{n}=x_{n-1}$ for all $n\geq
1$; and that the analytic core of $T$, denoted by $K(T)$, is the set of all
$x\in X$ admitting a “controlled” backward $T\,$\- orbit, i.e. a backward
$T\,$\- orbit $(x_{n})$ such that $\|x_{n}\|\leq\delta^{n}$ for some constant
$\delta$ and all $n\geq 0$. Obviously, $K(T)\subseteq
C(T)\subseteq\bigcap_{n\in\mathbb{N}}T^{n}(X)$.

######  Corollary 3.5.

Let $X=\ell_{p}$, $1<p<\infty$. A typical $T\in{\mathcal{B}}_{1}(X)$ is such
that $K(T)=\\{0\\}$ whereas $C(T)$ is dense in $X$.

###### Proof.

It is known that for any $T\in{\mathcal{B}}(X)$, we have $K(T)=\\{x\in
X;\;0\not\in\sigma_{x}(T)\\}$, see e.g. [Mostafa, Proposition 1.3]. So, by the
previous corollary, a typical $T\in{\mathcal{B}}_{1}(X)$ is such that
$K(T)=\\{0\\}$. On the other hand, by Corollary 3.2, a typical
$T\in{\mathcal{B}}_{1}(X)$ is one-to-one with dense range. Now, if $T$ is one-
to-one then it is clear that $C(T)=\bigcap_{n\in\mathbb{N}}T^{n}(X)$; and if
$T$ has dense range then, by the Bourbaki-Mittag-Leffler Theorem (see e.g.
[Runde]), $\bigcap_{n\in\mathbb{N}}T^{n}(X)$ is dense in $X$. This concludes
the proof.∎

Recall that if $T\in{\mathcal{B}}(X)$, then a _spectral maximal subspace_ for
$T$ is a closed $T\,$\- invariant subspace $Z\subseteq X$ with the following
property: for any closed, $T\,$\- invariant subspace $Z^{\prime}\subseteq X$
such that
$\sigma\bigl{(}T_{|Z^{\prime}}\bigr{)}\subseteq\sigma\bigl{(}T_{|Z}\bigr{)}$,
it holds that $Z^{\prime}\subseteq Z$. (See [CF] for more on this notion.)
Obviously, $\\{0\\}$ and $X$ are spectral maximal subspaces for $T$.

######  Corollary 3.6.

Let $X=\ell_{p}$, $1<p<\infty$. A typical $T\in{\mathcal{B}}_{1}(X)$ is such
that $\sigma\bigl{(}T_{|Z}\bigr{)}=\overline{\,\mathbb{D}}$ for every closed
$T\,$\- invariant subspace $Z\neq\\{0\\}$. In particular, the only spectral
maximal subspaces of a typical $T\in{\mathcal{B}}_{1}(X)$ are $\\{0\\}$ and
$X$.

###### Proof.

The first part of the corollary follows immediately from Theorem 3.1 since
$\sigma\bigl{(}T_{|Z}\bigr{)}\supseteq\sigma_{x}\bigl{(}T_{|Z}\bigr{)}$ for
any $T\,$\- invariant subspace $Z$ and every $x\in Z$. For the second part,
apply the definition with a maximal spectral subspace $Z\neq\\{0\\}$ and
$Z^{\prime}:=X$. ∎

When $X=\ell_{2}$, we have at our disposal a very strong result for proving
the existence of invariant subspaces, namely the Brown-Chevreau-Pearcy Theorem
[BCP]. It states that any contraction on a Hilbert space whose spectrum
contains the unit circle $\mathbb{T}$ has a non-trivial invariant subspace.
Combining it with Corollary 3.6 above, we obtain:

######  Corollary 3.7.

A typical $T\in\mathcal{B}_{1}(\ell_{2})$ has no minimal invariant subspace
except $\\{0\\}$: for any closed $T\,$\- invariant subspace $Z\neq\\{0\\}$,
the operator $T_{|Z}$ has a non-trivial invariant subspace.

We now move over to the proof of Theorem 3.1. For any Banach space $X$, we
introduce the set

$\mathcal{G}:=\bigl{\\{}T\in{\mathcal{B}}_{1}(X)\;;\;\forall\,x\neq 0,\
\textrm{the set}\ \\{\mu\in\mathbb{D}\;;\;x\not\in\textrm{Ran}(T-\mu)\\}\
\textrm{is dense in}\ \mathbb{D}\bigr{\\}}.$

We start with two general lemmas:

######  Lemma 3.8.

Assume that $X$ is reflexive. Then, $\mathcal{G}$ is a $G_{\delta}$ subset of
$({\mathcal{B}}_{1}(X),\emph{{SOT}\mbox{${}^{*}$}})$.

This relies on the following fact.

######  Fact 3.9.

For any $\mu\in\mathbb{C}$ and any closed ball $B\subseteq X$, the set

$\mathcal{C}_{\mu,B}:=\\{(T,x)\in{\mathcal{B}}_{1}(X)\times
X;\;x\in(T-\mu)(B)\\}$

is closed in
$({\mathcal{B}}_{1}(X),{\texttt{SOT}\mbox{${}^{*}$}})\times(X,w)$.

###### Proof of Fact 3.9.

The set $\mathcal{C}_{\mu,B}$ is the projection along $B$ of

$\mathbf{C}_{\mu,B}:=\bigl{\\{}(T,y,x)\in{\mathcal{B}}_{1}(X)\times B\times
X;\;(T-\mu)y=x\bigr{\\}}.$

Therefore, since the ball $B$ is weakly compact, it is enough to check that
$\mathbf{C}_{\mu,B}$ is closed in
$({\mathcal{B}}_{1}(X),\texttt{SOT}\mbox{${}^{*}$})\times(B,w)\times(X,w)$.
Now, the map $(T,y)\mapsto(T-\mu)y$ is continuous from
$({\mathcal{B}}_{1}(X),\texttt{SOT}\mbox{${}^{*}$})\times(B,w)$ into $(X,w)$,
because we have ${\langle
x^{*},(T-\mu)y\rangle}={\langle(T^{*}-\mu)x^{*},y\rangle}$ for any $x^{*}\in
X^{*}$ and the map $(y^{*},y)\mapsto{\langle y^{*},y\rangle}$ is continuous on
$(K,\|\,\cdot\,\|)\times(B,w)$ for any bounded set $K\subseteq X^{*}$. So
$\mathbf{C}_{\mu,B}$ is indeed closed in
$({\mathcal{B}}_{1}(X),\texttt{SOT}\mbox{${}^{*}$})\times(X,w)\times(B,w)$. ∎

###### Proof of Lemma 3.8.

For any $N\in\mathbb{N}$, let us denote by $B_{N}$ the closed ball
$\overline{B}(0,N)\subseteq X$. Let also $(V_{i})_{i\in\mathbb{N}}$ be a
countable basis of (non-empty) open sets for $\mathbb{D}$. Then, the following
equivalence holds true for any $T\in{\mathcal{B}}_{1}(X)$:

$\displaystyle T\not\in\mathcal{G}\iff\exists x\in X\setminus\\{0\\}\;\exists
N\in\mathbb{N}\;$ $\displaystyle\exists i\in\mathbb{N}\;:\;$
$\displaystyle\Bigl{(}\forall\mu\in
V_{i}\;:\;(T,x)\in\mathcal{C}_{\mu,B_{N}}\Bigr{)}.$

Indeed, given $x\in X\setminus\\{0\\}$, each set
$F_{N}:=\\{\mu\in\mathbb{D}\,;\,x\in(T-\mu)B_{N}\\}$ is easily seen to be
closed in $\mathbb{D}$, so that $\Omega_{N}:=\mathbb{D}\setminus
F_{N}=\\{\mu\in\mathbb{D}\,;\,x\not\in(T-\mu)B_{N}\\}$ is open in
$\mathbb{D}$. Since

$\bigcap_{N\geq
1}\Omega_{N}=\\{\mu\in\mathbb{D}\,;\,x\in\textrm{Ran}(T-\mu)\\},$

the Baire Category Theorem implies that $\bigcap_{N\geq 1}\Omega_{N}$ is dense
in $\mathbb{D}$ if and only if $\Omega_{N}$ is dense in $\mathbb{D}$ for each
$N\geq 1$. Hence

$\displaystyle T\not\in\mathcal{G}\iff\exists x\in X\setminus\\{0\\}\;\exists
N\in\mathbb{N}\;;\;\\{\mu\in\mathbb{D}\,;\,x\not\in(T-\mu)B_{N}\\}\textrm{ is
not dense in }\mathbb{D};$

from which it follows that

$\displaystyle T\not\in\mathcal{G}\iff\exists x\in X\setminus\\{0\\}\;\exists
N\in\mathbb{N}\;$ $\displaystyle\exists i\in\mathbb{N}\;:\;$
$\displaystyle\Bigl{(}\forall\mu\in
V_{i}\;:\;(T,x)\in\mathcal{C}_{\mu,B_{N}}\Bigr{)}.$

By Fact 3.9, the condition under brackets defines a closed subset of
$({\mathcal{B}}_{1}(X),\texttt{SOT}\mbox{${}^{*}$})\times(X,w)$. Since
$X\setminus\\{0\\}$ is a $K_{\sigma}$ subset of $(X,w)$, it follows that
${\mathcal{B}}_{1}(X)\setminus\mathcal{G}$ is $F_{\sigma}$ in
$({\mathcal{B}}_{1}(X),\texttt{SOT}\mbox{${}^{*}$})$. ∎

Our second lemma provides a large class of operators belonging to the set
$\mathcal{G}$.

######  Lemma 3.10.

Let $T\in{\mathcal{B}}_{1}(X)$. Then $T$ belongs to $G$ provided the following
holds true: for every open set $V\neq\emptyset$ in $\mathbb{D}$,

$\overline{\rm span}^{\,w^{*}}\left(\bigcup_{\mu\in
V}\ker(T-\mu)^{*}\right)=X^{*}.$

###### Proof.

This is essentially obvious from the definition of $\mathcal{G}$. Indeed, an
operator $T\in{\mathcal{B}}_{1}(X)$ does not belong to $\mathcal{G}$ if and
only if one can find an open set $V\neq\emptyset$ in $\mathbb{D}$ such that
$\bigcap_{\mu\in V}{\rm Ran}(T-\mu)\neq\\{0\\}$; and if this holds then
$\overline{\rm span}^{\,w^{*}}\left(\bigcup_{\mu\in
V}\ker(T-\mu)^{*}\right)\neq X^{*}$ since $\overline{\rm
span}^{\,w^{*}}\left(\bigcup_{\mu\in
V}\ker(T-\mu)^{*}\right)\subseteq\left(\bigcap_{\mu\in V}{\rm
Ran}(T-\mu)\right)^{\perp}$. ∎

######  Remark 3.11.

The assumption of Lemma 3.10 is satisfied as soon as the operator $T^{*}$
admits a $w^{*}$\- spanning family of holomorphic eigenvector fields defined
on the unit disk $\mathbb{D}$, i.e. there exists a family $(E_{i})_{i\in I}$
of holomorphic eigenvector fields for $T^{*}$ defined on $\mathbb{D}$ such
that $\overline{\rm span}^{\,w^{*}}\,\\{E_{i}(\lambda);\;i\in
I,\;\lambda\in\mathbb{D}\\}=X^{*}$. This follows from the Hahn-Banach Theorem
and the identity principle for holomorphic functions.

We are now ready for the proof of Theorem 3.1.

###### Proof of Theorem 3.1.

By the definition of $\mathcal{G}$, it is rather clear that if
$T\in\mathcal{G}$ and $x\neq 0$, then $\sigma_{x}(T)$ contains $\mathbb{D}$,
and hence $\sigma_{x}(T)=\overline{\,\mathbb{D}}$ since $\sigma_{x}(T)$ is a
closed subset of $\mathbb{C}$ contained in $\overline{\,\mathbb{D}}$. Indeed,
let $\lambda\in\mathbb{D}$. Since $T\in\mathcal{G}$, we see that for every
open neighborhood $V$ of $\lambda$ there exists $\mu\in V$ such that
$x\not\in\textrm{Ran}(T-\mu)$. Hence there cannot exist _any_ function $f:V\to
X$ such that $(T-z)f(z)\equiv x$ on $V$, and thus $\lambda\not\in\rho_{x}(T)$.
So, by Lemma 3.8, it is enough to show that the set $\mathcal{G}$ is SOT∗\-
dense in ${\mathcal{B}}_{1}(X)$.

Let us denote by $(e_{j})_{j\geq 0}$ the canonical basis of $X=\ell_{p}$; and
for each $N\in\mathbb{Z}_{+}$, set $E_{N}:=[e_{0},\dots,e_{N}]$. To prove the
SOT∗-denseness of $\mathcal{G}$, we show that any finite-dimensional operator
$A\in\mathcal{B}(E_{N})$ such that $\|A\|<1$ (viewed as an operator on
$X=\ell_{p}$) can be approximated in the topology SOT∗ by operators belonging
to $\mathcal{G}$.

Let $M>N$ be a large integer, let $\delta$ be a small positive number, and let
$T\in\mathcal{B}(X)$ be defined as follows:

$Te_{j}:=\left\\{\begin{array}[]{ll}Ae_{j}+\delta e_{j+M+1}&\hbox{if $0\leq
j\leq M$},\\\ e_{j+M+1}&\hbox{if $j>M$}.\end{array}\right.$

Since $\|A\|<1$, we have $\|T\|<1$ if $\delta$ is small enough. Moreover, if
$M$ is large enough and $\delta$ small enough, then $T$ is as close as we wish
to $A$ in the topology SOT∗. So, by Remark 3.11, it is now enough to show that
the operator $T^{*}$ admits a spanning family of holomorphic eigenvector
fields defined on $\mathbb{D}$. But this follows from the proof of [GMM1,
Proposition 2.10]: indeed, with the notations of [GMM1, Proposition 2.10] and
identifying $A$ with $P_{M}AP_{M}$, we have $T^{*}=B_{\boldsymbol{\omega},A}$,
where the weight sequence
$\boldsymbol{\omega}=(\delta,\dots,\delta,1,1,1\dots)$ is such that
$\underline{\lim}_{k\to\infty}\,\omega_{kM+l}\cdots\omega_{M+l}\omega_{l}=1>\|A\|$
for every $0\leq l\leq M$. ∎

##  4\. Asymptotic behavior of orbits

The results in this section are concerned with the study of some properties of
orbits of typical contractions on $\ell_{p}\,$\- spaces. Recall that if
$T\in{\mathcal{B}}(X)$, the _orbit_ of a vector $x\in X$ under the action of
$T$ is the set $\\{T^{n}x\;;\;n\in\mathbb{Z}_{+}\\}$. The dynamics of typical
operators $T\in(\mathcal{B}_{1}(\ell_{2}),\texttt{SOT}\mbox{${}^{*}$})$ are
already studied in some detail in [GMM1], where emphasis is put mostly on
properties related to hypercyclicity. Here, we are interested in asymptotic
properties of orbits of typical operators. Our results are strongly motivated
by [Mu, Chapter V], where several striking results valid for _every_ operator
$T\in{\mathcal{B}}(X)$ are obtained.

###  4a. Not too small orbits

We begin this section with a somewhat abstract statement which will be later
on applied to several concrete situations.

######  Proposition 4.1.

Let $X=\ell_{p}$, $1<p<\infty$. Let also $\mathbf{G}$ be a $G_{\delta}$ subset
of $(0,\infty)^{\mathbb{N}}$. Assume that $\mathbf{G}$ contains all sequences
$\omega=(\omega_{n})_{n\in\mathbb{N}}\in(0,\infty)^{\mathbb{N}}$ such that
$\inf_{n\in\mathbb{N}}\omega_{n}>0$, and that $\mathbf{G}$ is _upward closed_
for the product ordering of $(0,\infty)^{\mathbb{N}}$, i.e.
($\omega\in\mathbf{G}$ and $\omega\leq\omega^{\prime}$) implies
$(\omega^{\prime}\in\mathbf{G})$. Then, a typical $T\in{\mathcal{B}}_{1}(X)$
is such that for every $x\neq 0$, the sequence
$\bigl{(}\|T^{n}x\|\bigr{)}_{n\in\mathbb{N}}$ belongs to $\mathbf{G}$.

The proof of Proposition 4.1 relies on the next two lemmas.

######  Lemma 4.2.

Let $X=\ell_{p}$, $1<p<\infty$. The set of all $T\in{\mathcal{B}}_{1}(X)$ such
that $\forall x\neq 0\;:\;\inf_{n\in\mathbb{N}}\,\|T^{n}x\|>0$ is _SOT ∗_\-
dense in ${\mathcal{B}}_{1}(X)$.

###### Proof.

Let us denote by $\mathcal{D}$ the considered set of operators. As in the
proof of Theorem 3.1, given a finite-dimensional $A\in\mathcal{B}(E_{N})$, we
consider the operator $T$ defined by

$Te_{j}:=\left\\{\begin{array}[]{ll}Ae_{j}+\delta e_{j+M+1}&\hbox{if $0\leq
j\leq M$},\\\ e_{j+M+1}&\hbox{if $j>M$},\end{array}\right.$

where $M>N$ and $\delta>0$. This operator $T$ belongs to
${\mathcal{B}}_{1}(X)$ and is SOT∗\- close to $A$ if $M$ is large enough and
if $\delta$ is small enough. So it is enough to show that $T$ belongs to
$\mathcal{D}$, i.e. that $\inf_{n\in\mathbb{N}}\,\|T^{n}x\|>0$ for every
$x\neq 0$.

Let $x\in X\setminus\\{0\\}$. If ${\langle e_{j_{0}}^{*},x\rangle}\neq 0$ for
some $j_{0}>M$, then, since ${\langle
e_{j_{0}+n(M+1)}^{*},T^{n}x\rangle}={\langle e_{j_{0}}^{*},x\rangle}$ for
every $n\in\mathbb{N}$ by the definition of $T$, we see that
$\inf_{n\in\mathbb{N}}\,\|T^{n}x\|>0$. If ${\langle e_{j}^{*},x\rangle}=0$ for
all $j>M$, i.e. if $x\in E_{M}$, then $x^{\prime}:=Tx$ satisfies ${\langle
e_{j_{0}}^{*},x^{\prime}\rangle}\neq 0$ for some $M<j_{0}\leq 2M+1$ by the
definition of $T$; so we have $\inf_{n\in\mathbb{N}}\,\|T^{n}x^{\prime}\|>0$,
i.e. $\inf_{n\geq 2}\,\|T^{n}x\|>0$, and hence
$\inf_{n\in\mathbb{N}}\,\|T^{n}x\|>0$ since $Tx=x^{\prime}\neq 0$. This
concludes the proof of the lemma. ∎

######  Lemma 4.3.

If $\mathbf{H}\subseteq(0,1]^{\mathbb{N}}$ is $G_{\delta}$ and upward closed,
then H can be written as $\mathbf{H}=\bigcap_{k\in\mathbb{N}}\mathbf{O}_{k}$,
where the sets $\mathbf{O}_{k}$ are open in $(0,1]^{\mathbb{N}}$ and also
upward closed.

###### Proof.

Write $\mathbf{H}=\bigcap_{k\in\mathbb{N}}\mathbf{U}_{k}$ for some open sets
$\mathbf{U}_{k}\subseteq(0,1]^{\mathbb{N}}$. For each $k\in\mathbb{N}$, set

$\mathbf{O}_{k}:=\bigl{\\{}\omega\in(0,1]^{\mathbb{N}};\;\forall\omega^{\prime}\geq\omega\;:\;\omega^{\prime}\in\mathbf{U}_{k}\bigr{\\}}.$

The sets $\mathbf{O}_{k}$ are obviously upward closed, and since $\mathbf{H}$
is upward closed, we have $\mathbf{H}=\bigcap_{k\in\mathbb{N}}\mathbf{O}_{k}$.
So we just need to check that each $\mathbf{O}_{k}$ is open in
$(0,1]^{\mathbb{N}}$. Now, since $[0,1]^{\mathbb{N}}$ is compact and
$(0,1]^{\mathbb{N}}$ is upward closed in $[0,1]^{\mathbb{N}}$, it is easily
checked that $(0,1]^{\mathbb{N}}\setminus\mathbf{O}_{k}$ is closed in
$(0,1]^{\mathbb{N}}$. ∎

We are now ready for the proof of Proposition 4.1.

###### Proof of Proposition 4.1.

By homogeneity and upward closedness of $\mathbf{G}$, it is enough to show
that a typical $T\in{\mathcal{B}}_{1}(X)$ is such that the sequence
$(\|T^{n}x\|)_{n\in\mathbb{N}}$ belongs to $\mathbf{G}$ for every $x\neq 0$
with $\|x\|\leq 1$. Moreover, if $T\in{\mathcal{B}}_{1}(X)$ and $\|x\|\leq 1$,
then $\|T^{n}x\|\leq 1$ for all $n\in\mathbb{N}$. So, setting
$\mathbf{H}:=\mathbf{G}\cap(0,1]^{\mathbb{N}}$ and denoting by $B_{X}$ the
closed unit ball of $X$, we have to show that the set

$\mathcal{H}:=\Bigl{\\{}T\in{\mathcal{B}}_{1}(X);\;\forall x\in
B_{X}\setminus\\{0\\}\;:\;\bigl{(}\|T^{n}x\|\bigr{)}_{n\in\mathbb{N}}\in\mathbf{H}\Bigr{\\}}$

is comeager in ${\mathcal{B}}_{1}(X)$.

By Lemma 4.2 and by assumption on $\mathbf{G}$, we already know that the set
$\mathcal{H}$ is dense in ${\mathcal{B}}_{1}(X)$. We show that $\mathcal{H}$
is also $G_{\delta}$.

By Lemma 4.3, we may write H as
$\mathbf{H}=\bigcap_{k\in\mathbb{N}}\mathbf{O}_{k}$, where the sets
$\mathbf{O}_{k}$ are open and upward closed in $(0,1]^{\mathbb{N}}$. So it is
enough to check that if $\mathbf{O}\subseteq(0,1]^{\mathbb{N}}$ is open and
upward closed, then the set

$\mathcal{O}:=\Bigl{\\{}T\in{\mathcal{B}}_{1}(X);\;\forall x\in
B_{X}\setminus\\{0\\}\;:\;\bigl{(}\|T^{n}x\|\bigr{)}_{n\in\mathbb{N}}\in\mathbf{O}\Bigr{\\}}$

is $G_{\delta}$ in ${\mathcal{B}}_{1}(X)$.

Let us denote by $\Sigma$ the set of all finite sequences from $(0,1]$; and
for any sequence $\sigma=(s_{1},\dots,s_{r})\in\Sigma$, let us set

$\mathbf{V}_{\sigma}:=\\{\omega\in(0,1]^{\mathbb{N}};\;\omega_{n}>s_{n}\;\hbox{for
$n=1,\dots,r$}\\}.$

Since $\mathbf{O}$ is upward closed and open in $(0,1]^{\mathbb{N}}$, one can
write

$\mathbf{O}=\bigcup_{\sigma\in I}\mathbf{V}_{\sigma},$

for some set $I\subseteq\Sigma$. Indeed, $\mathbf{O}$ is a union of basic open
sets of the form $\bigcap_{n=1}^{r}\mathbf{W}_{n,s_{n},t_{n}}$, where we set
$\mathbf{W}_{n,s,t}:=\\{\omega;\;s<\omega_{n}<t\\}$ for any $n\in\mathbb{N}$
and any real numbers $s,t$ such that $0<s\leq 1$ and $s<t$. Now, if
$t_{1},\ldots,t_{r}$ are given such that $t_{n}>s_{n}$ for every
$n=1,\ldots,r$, then, for every $\omega\in\mathbf{V}_{\sigma}$, one can find
$\omega^{\prime}\in\bigcap_{n=1}^{r}\mathbf{W}_{n,s_{n},t_{n}}$ such that
$\omega\geq\omega^{\prime}$. Hence, by upward closedness, we have that if
$\bigcap_{n=1}^{r}\mathbf{W}_{n,s_{n},t_{n}}\subseteq\mathbf{O}$, then in fact
$\mathbf{V}_{\sigma}\subseteq\mathbf{O}$; which proves the claim.

So, for any $T\in{\mathcal{B}}_{1}(X)$, we see that

$\displaystyle T\not\in\mathcal{O}\iff\exists x\in B_{X}$
$\displaystyle\setminus\\{0\\}$
$\displaystyle\forall\sigma=(s_{1},\dots,s_{n})\in I\;:\;\Bigl{(}\exists
n\in\llbracket 1,r\rrbracket\;:\;\|T^{n}x\|\leq s_{n}\Bigr{)}$

Now, observe that for any bounded set $B\subseteq X$, the map
$(A,x)\mapsto\|Ax\|$ is lower semi-continuous on
$({\mathcal{B}}_{1}(X),\texttt{SOT}\mbox{${}^{*}$})\times(B,w)$, because the
map $(A,x)\mapsto Ax$ is continuous from
$({\mathcal{B}}_{1}(X),\texttt{SOT}\mbox{${}^{*}$})\times(B,w)$ into $(X,w)$
and the norm is lower semi-continuous on $(X,w)$. Since
$B_{X}\setminus\\{0\\}$ is $K_{\sigma}$ in $(X,w)$, it follows easily that
${\mathcal{B}}_{1}(X)\setminus\mathcal{O}$ is $F_{\sigma}$ in
${\mathcal{B}}_{1}(X)$. ∎

A first consequence of Proposition 4.1 is that all non-zero orbits of a
typical $T\in\mathcal{B}_{1}(\ell_{p})$ are “not too small”.

######  Corollary 4.4.

Let $X=\ell_{p}$, $1<p<\infty$.

  1. (i)

Given a sequence of positive real numbers $(a_{n})$ such that $a_{n}\to 0$, a
typical $T\in{\mathcal{B}}_{1}(X)$ is such that, for every $x\neq 0$, one has
$\|T^{n}x\|>a_{n}$ for infinitely many $n\in\mathbb{N}$.

  2. (ii)

Given a sequence $(\phi_{n})$ of increasing continuous functions from
$\mathbb{R}^{+}$ into itself such that
$\sum_{n=0}^{\infty}\phi_{n}(\varepsilon)=\infty$ for every $\varepsilon>0$, a
typical $T\in{\mathcal{B}}_{1}(X)$ is such that

$\sum_{n=0}^{\infty}\phi_{n}\bigl{(}\|T^{n}x\|\bigr{)}=\infty\quad\textrm{for
every }x\neq 0.$

###### Proof.

For (i), take
$\mathbf{G}_{1}:=\bigl{\\{}\omega\in(0,\infty)^{\mathbb{N}};\;\omega_{n}>a_{n}\;\hbox{for
infinitely many $n$}\bigr{\\}}$, which is obviously $G_{\delta}$ and upward
closed. For (ii), consider
$\mathbf{G}_{2}:=\bigl{\\{}\omega\in(0,\infty)^{\mathbb{N}};\;\sum_{n=0}^{\infty}\phi(\omega_{n})=\infty\bigr{\\}}$,
which is $G_{\delta}$ because $\omega\in\mathbf{G}_{2}\iff\forall
M\in\mathbb{N}\;\exists N\;:\;\sum_{n=0}^{N}\phi_{n}(\omega_{n})>M$, and
upward closed because the functions $\phi_{n}$ are increasing. ∎

######  Remark 4.5.

Taking e.g. $a_{n}:=\frac{1}{n}$ in (i), one gets another proof that a typical
operator $T\in{\mathcal{B}}_{1}(X)$ is such that $r_{x}(T)=1$ for every $x\neq
0$.

###  4b. Nilpotent operators

We now turn to a result that, sadly enough, we can prove only for
$X=\ell_{2}$. Let us denote by $\mathcal{N}(X)$ the set of all nilpotent
operators $T\in{\mathcal{B}}(X)$, and set
$\mathcal{N}_{1}(X):=\mathcal{N}(X)\cap{\mathcal{B}}_{1}(X)$.

######  Proposition 4.6.

If $X=\ell_{2}$, then $\mathcal{N}_{1}(X)$ is _SOT ∗_\- dense in
${\mathcal{B}}_{1}(X)$.

Before giving the proof of Proposition 4.6, we explore some of its
consequences. First, it implies that a typical $T\in\mathcal{B}(\ell_{2})$ has
lots of “partly small” orbits (see Proposition 4.14 and Proposition 4.15 below
for concrete examples):

######  Corollary 4.7.

Let $X=\ell_{2}$. Let also $\mathbf{G}$ be a $G_{\delta}$ subset of
$[0,\infty)^{\mathbb{N}}$. Assume that $\mathbf{G}$ contains all sequences
$\omega=(\omega_{n})\in[0,\infty)^{\mathbb{N}}$ which are eventually $0$.
Then, a typical $T\in{\mathcal{B}}_{1}(X)$ has the following property: for a
comeager set of vectors $x\in X$, the sequence
$\bigl{(}\|T^{n}x\|\bigr{)}_{n\in\mathbb{N}}$ belongs to $\mathbf{G}$.

###### Proof.

Let $D$ be a countable dense subset of $X$, and define

$\mathcal{G}:=\bigl{\\{}T\in{\mathcal{B}}_{1}(X);\;\forall z\in
D\;:\;(\|T^{n}z\|)_{n\in\mathbb{N}}\in\mathbf{G}\bigr{\\}}.$

Since $\mathbf{G}$ is $G_{\delta}$, it is easily checked that $\mathcal{G}$ is
SOT -$\,G_{\delta}$, hence SOT∗-$\,G_{\delta}$. Moreover, $\mathcal{G}$
contains all nilpotent operators in ${\mathcal{B}}_{1}(X)$ by assumption on
$\mathbf{G}$, and hence $\mathcal{G}$ is SOT∗\- dense in
${\mathcal{B}}_{1}(X)$ by Proposition 4.6. Thus, we see that a typical
$T\in{\mathcal{B}}_{1}(X)$ is such that the set $D_{T}:=\bigl{\\{}x\in
X;\;(\|T^{n}x\|)_{n\in\mathbb{N}}\in\mathbf{G}\bigr{\\}}$ contains $D$, and
hence is dense in $X$. But for any fixed $T\in{\mathcal{B}}_{1}(X)$, the set
$D_{T}$ is clearly $G_{\delta}$; so the proof is complete. ∎

######  Remark 4.8.

By the Kuratowski-Ulam Theorem, it follows from Corollary 4.7 that the set
$\bigl{\\{}(T,x)\in{\mathcal{B}}_{1}(X)\times X;\;(\|T^{n}x\|)_{n\in
N}\in\mathbf{G}\bigr{\\}}$ is comeager in
$\bigl{(}{\mathcal{B}}_{1}(X),\texttt{SOT}\mbox{${}^{*}$}\bigr{)}\times X$,
and also in $\bigl{(}{\mathcal{B}}_{1}(X),\texttt{SOT}\bigr{)}\times X$.
However this set has no reason for being $G_{\delta}$.

Let us point out another consequence of Proposition 4.6.

######  Corollary 4.9.

Let $X=\ell_{2}$. For any $a\in\mathbb{D}$, the set of all
$T\in{\mathcal{B}}_{1}(X)$ such that $\sigma(T)=\\{a\\}$ is _SOT ∗_\- dense in
${\mathcal{B}}_{1}(X)$.

###### Proof.

Let us denote by $\varphi_{a}$ the Möbius transformation associated with $a$,

$\varphi_{a}(z)=\frac{a-z}{1-\bar{a}z}\cdot$

Since $\varphi_{a}$ is holomorphic in a neighborhood of
$\overline{\,\mathbb{D}}$, one can define $\varphi_{a}(T)$ for any
$T\in{\mathcal{B}}_{1}(X)$. Moreover, the map $T\mapsto\varphi_{a}(T)$ from
${\mathcal{B}}_{1}(X)$ into itself is SOT∗\- continuous because the power
series of $\varphi_{a}$ is absolutely convergent on $\overline{\,\mathbb{D}}$;
and since $\varphi_{a}\circ\varphi_{a}(z)\equiv z$ in a neighborhood of
$\overline{\,\mathbb{D}}$, we have
$\varphi_{a}\bigl{(}\varphi_{a}(T)\bigr{)}=T$ for every
$T\in{\mathcal{B}}_{1}(X)$. Finally, we also have that
$\varphi_{a}(T)\in{\mathcal{B}}_{1}(X)$ for any $T\in{\mathcal{B}}_{1}(X)$, by
von Neumann’s inequality. So we may conclude that the map
$T\mapsto\varphi_{a}(T)$ is an involutive homeomorphism of
${\mathcal{B}}_{1}(X)$ onto itself. Since $\sigma(T)=\\{a\\}$ if and only if
$\sigma\bigl{(}\varphi_{a}(T)\bigr{)}=\\{0\\}$ and since the set
$\bigl{\\{}S\in{\mathcal{B}}_{1}(X);\;\sigma(S)=\\{0\\}\bigr{\\}}$ is SOT∗\-
dense in ${\mathcal{B}}_{1}(X)$ by Proposition 4.6, the result follows. ∎

The proof of Proposition 4.6 relies on the next two lemmas.

######  Lemma 4.10.

Let $T$ be a bounded operator on a Banach space $X$. If $E:\Omega\to X$ is a
holomorphic eigenvector field for $T$ defined on some connected open set
$\Omega\subseteq\mathbb{C}$ containing $0$, then

$\overline{\rm
span}\,\bigl{\\{}E(\lambda);\;\lambda\in\Omega\bigr{\\}}\subseteq\overline{\bigcup_{k\in\mathbb{N}}\ker(T^{k})}.$

###### Proof.

Note first that since $\Omega$ is connected, we have

$\overline{\rm
span}\,\bigl{\\{}E(\lambda);\;\lambda\in\Omega\bigr{\\}}=\overline{\rm
span}\,\bigl{\\{}E^{(n)}(0);\;n\geq 0\bigr{\\}}.$

Indeed, the inclusion $\supseteq$ is clear; and the other inclusion follows
from the Hahn-Banach Theorem and the identity principle for holomorphic
functions. Next, differentiating the identity $TE(\lambda)\equiv\lambda
E(\lambda)$, we get that $TE^{(n)}(\lambda)=\lambda
E^{(n)}(\lambda)+nE^{(n-1)}(\lambda)$ for every $\lambda\in\Omega$ and for all
$n\geq 1$. In particular, $TE^{(n)}(0)=nE^{(n-1)}(0)$ for all $n\geq 1$. Since
$TE(0)=0$, it follows that $T^{n+1}E^{(n)}(0)=0$ for all $n\geq 0$. So we see
that ${\rm span}\,\bigl{\\{}E^{(n)}(0);\;n\geq
0\bigr{\\}}\subseteq\bigcup_{k\in\mathbb{N}}\ker(T^{k})$, which concludes the
proof of the lemma. ∎

As an immediate corollary of Lemma 4.10, we obtain

######  Corollary 4.11.

If an operator $T\in{\mathcal{B}}(X)$ admits a spanning family of holomorphic
eigenvector fields $(E_{i})_{i\in I}$ defined on some connected open set
$\Omega\subseteq\mathbb{C}$ containing $0$, i.e. $\overline{\rm
span}\,\\{E_{i}(\lambda);\;i\in I,\lambda\in\Omega\\}=X$, then
$\bigcup_{k\in\mathbb{N}}\ker(T^{k})$ is dense in $X$.

Our second lemma is specific to the Hilbertian setting.

######  Lemma 4.12.

Let $X=\ell_{2}$. If $T\in{\mathcal{B}}_{1}(X)$ is such that
$\bigcup_{k\in\mathbb{N}}\ker(T^{k})$ is dense in $X$, then $T$ belongs to the
_SOT ∗_\- closure of $\mathcal{N}_{1}(X)$.

###### Proof.

For each $k\in\mathbb{N}$, let us denote by $Q_{k}\in{\mathcal{B}}(X)$ the
orthogonal projection of $X$ onto $\ker(T^{k})$. Then $\|Q_{k}\|\leq 1$ (we
use specifically here the fact that $X=\ell_{2}$), and since the sequence of
subspaces $\bigl{(}\ker(T^{k})\bigr{)}_{k\in\mathbb{N}}$ is increasing with
$\overline{\bigcup_{k\in\mathbb{N}}\ker(T^{k})}=X$, we see that
$Q_{k}\xrightarrow{\texttt{SOT}}I$. Moreover, the projections $Q_{k}$ are
self-adjoint, so in fact $Q_{k}\xrightarrow{\texttt{SOT}\mbox{${}^{*}$}}I$.
Hence, if we set $T_{k}:=TQ_{k}$, then $T_{k}\in{\mathcal{B}}_{1}(X)$ for all
$k\in\mathbb{N}$ and $T_{k}\xrightarrow{\texttt{SOT}\mbox{${}^{*}$}}T$.
Moreover, each operator $T_{k}$ is nilpotent, in fact $T_{k}^{k}=0$: indeed,
since $T({\rm Ran}(Q_{k}))=T(\ker(T^{k}))\subseteq\ker(T^{k})={\rm
Ran}(Q_{k})$, we have $Q_{k}TQ_{k}=TQ_{k}$, so $T_{k}^{n}=T^{n}Q_{k}$ for all
$n\in\mathbb{N}$ and hence $T_{k}^{k}=0$ since ${\rm Ran}(Q_{k})=\ker(T^{k})$.
So we have found a sequence $(T_{k})\subseteq\mathcal{N}_{1}(X)$ such that
$T_{k}\xrightarrow{\texttt{SOT}\mbox{${}^{*}$}}T$. ∎

Proposition 4.6 can now be readily deduced from Lemmas 4.10 and 4.12.

###### Proof of Proposition 4.6.

By Lemmas 4.10 and 4.12, it is enough to show that the set of all
$T\in{\mathcal{B}}_{1}(X)$ admitting a spanning family of holomorphic
eigenvector fields defined on $\mathbb{D}$ is SOT∗\- dense in
${\mathcal{B}}_{1}(X)$. But this follows from the proof of [GMM1, Corollary
2.12]. Indeed, this proof shows that for any $M>1$, the set of all
$T\in\mathcal{B}_{M}(X)$ admitting a spanning family of holomorphic
eigenvector fields defined on the open disk $D(0,M)$ is SOT∗\- dense in
$\mathcal{B}_{M}(X)$; so we get the required result by homogeneity. ∎

######  Remark 4.13.

Here is a completely different and highly non-elementary proof of Proposition
4.6. We prove in fact the following stronger result: _A typical
$T\in{\mathcal{B}}_{1}(X)$ belongs to the norm-closure of
$\mathcal{N}_{1}(X)$_.

By a deep result of Apostol-Foias-Voiculescu [AFV], an operator
$T\in{\mathcal{B}}(X)$ belongs to the norm-closure of nilpotent operators if
and only if it has the following properties:

  *     1. (i)

$\sigma(T)$ and $\sigma_{e}(T)$ are connected with $0\in\sigma_{e}(T)$.

    2. (ii)

${\rm ind}(T-\lambda)=0$ for any $\lambda\in\mathbb{C}$ such that $T-\lambda$
is a semi-Fredholm operator.

Now, by Corollary 3.2, a typical $T\in{\mathcal{B}}_{1}(X)$ certainly
satisfies (i), and a typical $T\in{\mathcal{B}}_{1}(X)$ also satisfies (ii)
vacuously for $\lambda\in\overline{\,\mathbb{D}}$. Since $T-\lambda$ is
invertible if $|\lambda|>1$, the result follows.

###  4c. Power-regular operators

According to [Atz], an operator $T\in{\mathcal{B}}(X)$ is said to be _power-
regular_ if $\|T^{n}x\|^{1/n}\to r_{x}(T)$ as $n\to\infty$ for every $x\in X$.
It is shown in [Atz] that any decomposable operator is power-regular.
Combining Theorem 3.1 and Corollary 4.7, we obtain the following result.

######  Proposition 4.14.

Let $X=\ell_{2}$. A typical $T\in{\mathcal{B}}_{1}(X)$ is not power-regular.
More precisely, a typical $T\in{\mathcal{B}}_{1}(X)$ is such that $r_{x}(T)=1$
for all $x\neq 0$ yet $\underline{\lim}\,\|T^{n}x\|^{1/n}=0$ for a comeager
set of vectors $x\in X$.

###### Proof.

By Theorem 3.1 (or Remark 4.5), a typical $T\in{\mathcal{B}}_{1}(X)$ is such
that $r_{x}(T)=1$ for every $x\neq 0$. On the other hand, a typical
$T\in{\mathcal{B}}_{1}(X)$ is such that $\underline{\lim}\,\|T^{n}x\|^{1/n}=0$
for a comeager set of vectors $x\in X$: this follows from Corollary 4.7
applied to the set

$\mathbf{G}:=\bigl{\\{}\omega=(\omega_{n})\in[0,\infty)^{\mathbb{N}};\;\underline{\lim}\,\omega_{n}^{1/n}=0\bigr{\\}}.$

∎

###  4d. Distributionally chaotic operators

Let us recall that a vector $x\in X$ is said to be _distributionally
irregular_ for an operator $T\in{\mathcal{B}}(X)$ if there exist two sets
$A,B\subseteq\mathbb{N}$ both having upper density equal to $1$, such that
$\|T^{n}x\|\to 0$ as $n\to\infty$ along $A$ and $\|T^{n}x\|\to\infty$ as
$n\to\infty$ along $B$. The operator $T$ is said to be _densely
distributionally chaotic_ if it has a dense set of distributionally irregular
vectors. We refer to [BBMP] for more on these notions.

Note that for $T$ to have any distributionally irregular vector, it is
necessary that $\|T\|>1$. It is shown in [GMM1] that for any $M>1$, a typical
$T\in\mathcal{B}_{M}(\ell_{2})$ is densely distributionally chaotic. This
result can be slightly improved, as follows:

######  Proposition 4.15.

Let $X=\ell_{2}$. For any $M>1$, a typical $T\in{\mathcal{B}}_{M}(X)$ has the
following properties: for every $x\neq 0$, there is a set
$A\subseteq\mathbb{N}$ with $\overline{\rm dens}(A)=1$ such that
$\|T^{n}x\|\to\infty$ as $n\to\infty$ along $A$; and for a comeager set of
vectors $x\in X$, there is a set $B\subseteq\mathbb{N}$ with $\overline{\rm
dens}(B)=1$ such that $\|T^{n}x\|\to 0$ as $n\to\infty$ along $B$.

###### Proof.

Let us first show that a typical $T\in{\mathcal{B}}_{M}(X)$ is such that for
every $x\neq 0$, there exists a set $A\subseteq\mathbb{N}$ with $\overline{\rm
dens}(A)=1$ such that $\|T^{n}x\|\to\infty$ as $n\to\infty$ along $A$. Choose
$\alpha\in(0,1)$ such that $M\alpha>1$, and set

$\mathbf{G}:=\Bigl{\\{}\omega=(\omega_{n})\in(0,\infty)^{\mathbb{N}};\;\overline{\rm
dens}\,\bigl{(}\\{n\in\mathbb{N};\;\omega_{n}>\alpha^{n}\\}\bigr{)}=1\Bigr{\\}}.$

The set $\mathbf{G}$ is easily seen to be $G_{\delta}$ in
$(0,\infty)^{\mathbb{N}}$. Moreover, $\mathbf{G}$ is clearly upward closed,
and since $\alpha<1$ it contains all sequences $\omega$ such that
$\inf_{n}\omega_{n}>0$. By Proposition 4.1, it follows that a typical
$T\in{\mathcal{B}}_{1}(X)$ is such that for every $x\neq 0$, the set
$\\{n\in\mathbb{N};\;\|T^{n}x\|>\alpha^{n}\\}$ has upper density equal to $1$.
Hence, by homogeneity, a typical $T\in{\mathcal{B}}_{M}(X)$ is such that for
every $x\neq 0$, the set
$A_{x,T}:=\\{n\in\mathbb{N};\;\|T^{n}x\|>(M\alpha)^{n}\\}$ has upper density
equal to $1$; and clearly $\|T^{n}x\|\to\infty$ as $n\to\infty$ along
$A_{x,T}$.

Now, let us show that a typical $T\in{\mathcal{B}}_{M}(X)$ is such that for a
comeager set of vectors $x\in X$, there exists a set $B\subseteq\mathbb{N}$
with $\overline{\rm dens}(B)=1$ such that $\|T^{n}x\|\to 0$ as $n\to\infty$
along $B$. This follows in fact from [GMM1, Proposition 2.40], but the proof
we give here is rather different and more elementary. Choose $\beta>0$ such
that $\beta M<1$. Applying Corollary 4.7 to the set

$\mathbf{G}:=\Bigl{\\{}\omega=(\omega_{n})\in[0,\infty)^{\mathbb{N}};\;\overline{\rm
dens}\,\bigl{(}\\{n\in\mathbb{N};\;\omega_{n}<\beta^{n}\\}\bigr{)}=1\Bigr{\\}},$

we see that a typical $T\in{\mathcal{B}}_{1}(X)$ is such that $\overline{\rm
dens}\,\bigl{(}\\{n\in\mathbb{N};\;\|T^{n}x\|<\beta^{n}\\}\bigr{)}=1$ for a
comeager set of vectors $x\in X$. By homogeneity, it follows that a typical
$T\in{\mathcal{B}}_{M}(X)$ has the following property: for a comeager set of
vectors $x\in X$, the set

$B_{T,x}:=\\{n\in\mathbb{N};\;\|T^{n}x\|<(\beta M)^{n}\\}$

has upper density equal to $1$. Since obviously $\|T^{n}x\|\to 0$ as
$n\to\infty$ along $B_{T,x}$, this concludes the proof of Proposition 4.15. ∎

######  Remark 4.16.

A slight modification of the above proof gives the following result. Let
$(\alpha_{n})$ be any sequence of positive real numbers tending to $0$, and
let $(\beta_{n})$ be any sequence of positive real numbers. Then, a typical
$T\in{\mathcal{B}}_{M}(X)$ has the following properties: for every $x\neq 0$,
there is a set $A\subseteq\mathbb{N}$ with $\overline{\rm dens}(A)=1$ such
that $\|T^{n}x\|>\alpha_{n}M^{n}$ for all $n\in A$; and for a comeager set of
vectors $x\in X$, there is a set $B\subseteq\mathbb{N}$ with $\overline{\rm
dens}(B)=1$ such that $\|T^{n}x\|<\beta_{n}$ for all $n\in B$.

######  Remark 4.17.

It is _not_ true that given $M>1$, a typical $T\in{\mathcal{B}}_{M}(X)$ is
such that every $x\neq 0$ is a distributionally irregular vector for $T$.
Indeed, by [Mu, Corollary V.37.9], for any $T\in{\mathcal{B}}(X)$, there is a
dense set of vectors $x\in X$ such that $\|T^{n}x\|^{1/n}\to r(T)$ as
$n\to\infty$. Since a typical $T\in{\mathcal{B}}_{M}(X)$ is such that its
spectral radius is equal to $M$, it follows that for a typical
$T\in{\mathcal{B}}_{M}(X)$, there is a dense set of vectors $x\in X$ such that
$\|T^{n}x\|\to\infty$, and hence a dense set of vectors $x$ which are not
distributionally irregular for $T$.

##  5\. Generalized kernels and Gaussian mixing

As we have seen, sets of the form
$\ker^{*}(T):=\bigcup_{k\in\mathbb{N}}\ker(T^{k})$, where $T$ is a bounded
operator on a complex separable Banach space $X$, play a prominent role in the
proof of Proposition 4.6. The set $\ker^{*}(T)$ is usually called the
_generalized kernel_ of the operator $T$. Operators with a $1\,$\- dimensional
kernel and a dense generalized kernel are called _generalized backward shifts_
in [GS], where it is shown that operators commuting with generalized backward
shifts have remarkable dynamical properties. We would like to point out the
following kind of converse to Corollary 4.11, which is very much in the spirit
of [GS, Theorem 3.6]. The proof is essentially the same as that of [F, Theorem
2], but we give the details for convenience of the reader.

######  Proposition 5.1.

Let $X$ be a complex separable Banach space, and let $T\in{\mathcal{B}}(X)$.
Assume that $T$ is _onto_ and that there exists an operator $B$ with dense
generalized kernel such that $TB=BT$ and $\ker(B)\subseteq\ker(T)$. Then, $T$
admits a spanning family of holomorphic eigenvector fields defined on the disk
$D(0,c_{T})$, where $c_{T}:=\inf\,\\{\|T^{*}x^{*}\|;\;\|x^{*}\|=1\\}$.

###### Proof.

The quantity $c_{T}$ is positive because $T$ is onto; and we know that for any
$x\in X$ and any $c>c_{T}^{-1}$, one can find $x^{\prime}\in X$ such that
$\|x^{\prime}\|\leq c\,\|x\|$ and $Tx^{\prime}=x$.

Note also that the assumptions imply that $\ker(B^{k})\subseteq\ker(T^{k})$
for every $k\in\mathbb{N}$. So $T$ itself has a dense generalized kernel.

Let $Z:=\ker^{*}(T)$. For any $z\in Z$, let us denote by $k_{z}$ the smallest
integer $k\geq 1$ such that $T^{k}z=0$. By the definition of $c_{T}$, one can
find a sequence $(z_{j})_{j\geq k_{z}-1}$ of vectors of $Z$ with
$z_{k_{z}-1}=z$ such that $Tz_{j}=z_{j-1}$ and
$\|z_{j}\|\leq(1+2^{-j})c_{T}^{-1}\,\|z_{j-1}\|$ for all $j\geq k_{z}$. Set
also $z_{j}:=T^{k_{z}-1-j}z$ for $0\leq j<k_{z}-1$, so that $Tz_{j}=z_{j-1}$
for every $j>0$. Then, for any $\lambda\in D(0,c_{T})$, the series
$\sum_{j\geq 0}\lambda^{j}z_{j}$ is convergent; so the formula

$E_{z}(\lambda):=\sum_{j\geq 0}\lambda^{j}z_{j}$

defines a holomorphic function $E_{z}:D(0,c_{T})\to X$. Note that $Tz_{0}=0$
since $z_{0}=T^{k_{z}-1}z$. By the choice of the sequence $(z_{j})$, it
follows that $TE_{z}(\lambda)=\lambda E_{z}(\lambda)$ for every $\lambda\in
D(0,c_{T})$. So we have defined on $D(0,c_{T})$ a family of holomorphic
eigenvector fields $(E_{z})_{z\in Z}$ for $T$. Moreover, this family is
spanning. Indeed, if $x^{*}\in X^{*}$ is such that ${\langle
x^{*},E_{z}(\lambda)\rangle}=0$ for every $z\in Z$ and $\lambda\in
D(0,c_{T})$, then ${\langle x^{*},z\rangle}={\langle
x^{*},z_{k_{z}-1}\rangle}=0$ for every $z\in Z$, and hence $x^{*}=0$ since $Z$
is dense in $X$. ∎

######  Remark 5.2.

Combining Lemma 4.10 with the proof of Proposition 5.1, we obtain the
following result: if $T\in{\mathcal{B}}(X)$ is onto, then there exists a
family of holomorphic eigenvector fields $(E_{i})_{i\in I}$ defined on
$D(0,c_{T})$ such that $\overline{\rm span}\,\\{E_{i}(\lambda);\,i\in
I,\lambda\in D(0,c_{T})\\}=\overline{\ker^{*}(T)}$.

We now use Proposition 5.1 to provide a condition under which an operator
$T\in{\mathcal{B}}(X)$ is _mixing in the Gaussian sense_ , which means that
$T$ admits an invariant Gaussian probability measure with full support with
respect to which it is a strongly mixing transformation. (We refer to [BMbook,
Chapter 5] and [BG] for unexplained terminology and for more about the ergodic
theory of linear dynamical systems.) This result is essentially proved in
[Mu2, Corollary 9], but our formulation is slightly more general.

######  Corollary 5.3.

Let $X$ be a complex separable Banach space, and let $T\in{\mathcal{B}}(X)$.
Assume that $T$ has the form $T=f(B)$, where $B\in{\mathcal{B}}(X)$ is onto
with a dense generalized kernel, and $f$ is a function holomorphic on an open
set $\Omega\supseteq\sigma(B)$, not constant on any connected component of
$\Omega$, and such that $|f(z_{0})|=1$ for some $z_{0}\in\Omega\cap
D(0,c_{B})$. Then, the operator $T$ is mixing in the Gaussian sense.

###### Proof.

Let $V\subseteq\Omega\cap D(0,c_{B})$ be a connected open neighborhood of
$z_{0}$. By assumption, the function $f$ is not constant on $V$, so
$f(V)\cap\mathbb{T}$ is a non-empty open subset of $\mathbb{T}$. In particular
$f(V)\cap\mathbb{T}$ is uncountable, so $V\cap f^{-1}(\mathbb{T})$ is
uncountable. On the other hand, the set $\\{z\in V;\;f^{\prime}(z)=0\\}$ is
countable. So $W:=V\cap\\{z\in V;\;f^{\prime}(z)\neq 0\\}$ is a non-empty open
set such that $f(W)\cap\mathbb{T}\neq\emptyset$. By the implicit function
Theorem, it follows that one can find a non-trivial closed arc
$\Lambda\subseteq\mathbb{T}$ and a (one-to-one) Lipschitz map $\phi:\Lambda\to
V$ such that $f(\phi(\lambda))=\lambda$ for every $\lambda\in\Lambda$.

By Proposition 5.1, the operator $B$ admits a spanning family of holomorphic
eigenvector fields $(E_{i})_{i\in I}$ defined on $D(0,c_{B})$. If we set
$\widetilde{E_{i}}(\lambda):=E_{i}(\phi(\lambda))$, we obtain a family of
Lipschitz eigenvector fields for $T=f(B)$ defined on the arc $\Lambda$.
Moreover, the family $(\widetilde{E_{i}})_{i\in I}$ is spanning by the Hahn-
Banach Theorem and the identity principle for holomorphic functions, because
$\widetilde{E_{i}}(\Lambda)=E_{i}(\phi(\Lambda))$ for all $i\in I$ and
$\phi(\Lambda)$ is an infinite compact subset of $D(0,c_{B})$. By [BG, Theorem
3.4], it follows that $T$ is mixing in the Gaussian sense.∎

######  Example 5.4.

Let $X$ be a complex separable Banach space, and let $B\in{\mathcal{B}}(X)$ be
onto with a dense generalized kernel. For any $\lambda_{0}\in\mathbb{C}$ such
that $\operatorname{dist}(\lambda_{0},\mathbb{T})<c_{B}$ (in particular, for
any $\lambda_{0}\in\mathbb{T}$) the operator $T:=\lambda_{0}+B$ is mixing in
the Gaussian sense.

###### Proof.

Apply Corollary 5.3 with $f(z):=\lambda_{0}+z$. ∎

######  Remark 5.5.

If one only assumes that $\ker^{*}(B)$ is dense in $X$ and
$\ker^{*}(B)\subseteq{\rm Ran}(B)$, then one can conclude that $\lambda_{0}+B$
is _topologically_ mixing for any $\lambda_{0}\in\mathbb{T}$ (see [BMbook,
Corollary 2.3]). But these weaker assumptions do not entail mixing in the
Gaussian sense. For example, if $B$ is a compact weighted backward shift on
$c_{0}$ or $\ell_{p}$, then $\lambda_{0}+B$ is not even frequently hypercyclic
by [Shk, Theorem 1.2], since its spectrum is countable.

##  6\. Some further comments and questions

We collect in this last section some natural questions which arise in
connection with this work.

To begin with, observe that since the space $c_{0}:=c_{0}(\mathbb{Z}_{+})$ has
separable dual, the ball $\mathcal{B}_{1}(c_{0})$ endowed with the topology
SOT∗ is a Polish space. So it makes sense to ask whether the results from
Sections 3 and 4 can be extended to typical contractions of $c_{0}$.

######  Question 6.1.

Are Theorem 3.1 and Proposition 4.1 still valid for $X=c_{0}$?

Observe that the proofs of Theorem 3.1 and Proposition 4.1 use in a crucial
way the weak compactness of closed balls of $\ell_{p}$, $1<p<\infty$.

We are able to prove the SOT∗\- density of $\mathcal{N}_{1}(X)$ in
${\mathcal{B}}_{1}(X)$ only in the case where $X=\ell_{2}$, and, as a
consequence, Corollaries 4.9 and 4.11 as well as Propositions 4.14 and 4.15
are proved only in the Hilbertian setting. It would thus be interesting to be
able to answer the following:

######  Question 6.2.

Let $X=\ell_{p}$, $1<p\neq 2<\infty$. Is it true that $\mathcal{N}_{1}(X)$ is
SOT∗\- dense in ${\mathcal{B}}_{1}(X)$?

Corollary 4.9 and its proof motivate the next question:

######  Question 6.3.

Let $X=\ell_{2}$ (or even $X=\ell_{p}$, $1<p<\infty$). Let $a\in\mathbb{T}$.
Is it true that the set of all $T\in{\mathcal{B}}_{1}(X)$ such that
$\sigma(T)=\\{a\\}$ is SOT∗\- dense in $X$?

In relation to this question, it is shown in [GMM2, Proposition 3.10] that if
$X=\ell_{p}$ for some $1\leq p<\infty$, the set of all
$T\in{\mathcal{B}}_{1}(X)$ such that $\sigma(T)\subseteq\mathbb{T}$ is SOT\-
dense in ${\mathcal{B}}_{1}(X)$. The proof given there shows that for
$1<p<\infty$, this set of operators is SOT∗\- dense in ${\mathcal{B}}_{1}(X)$
as well.

The results of this note show that essentially no known criterion of a
spectral flavor can be brought to use to show that a typical
$T\in\mathcal{B}_{1}(\ell_{p})$, $1<p\neq 2<\infty$, has a non-trivial
invariant subspace. In this line of thought, let us mention here an open
question from [GMM2]. An important criterion for showing the existence of a
non-trivial invariant subspace for operators is given by the famous Lomonosov
Theorem, which states that if the commutant $\\{T\\}^{\prime}$ of an operator
$T$ on a Banach space $X$ contains an operator which is not a multiple of the
identity operator and which commutes with a non-zero compact operator, then
$T$ has a non-trivial invariant subspace. It is shown in [GMM2, Theorem 7.5]
that a typical $T\in\mathcal{B}_{1}(\ell_{2})$ does _not_ commute with any
non-zero compact operator.

######  Question 6.4.

Let $X=\ell_{p}$, $1<p<\infty$. Is it true that a typical
$T\in{\mathcal{B}}_{1}(X)$ does not satisfy the assumptions of the Lomonosov
Theorem? Is it at least true that a typical $T\in{\mathcal{B}}_{1}(X)$ does
not commute with a non-zero compact operator?

Lastly, we mention that if $p>2$, Theorem 3.1 and Proposition 4.1 hold true
also for a typical $T\in(\mathcal{B}_{1}(\ell_{p}),\texttt{SOT})$. Indeed, by
[GMM2, Theorem 5.12], any SOT∗\- comeager subset of
$\mathcal{B}_{1}(\ell_{p})$ is also SOT \- comeager.

######  Question 6.5.

Let $1<p<2$. Do the statements of Theorem 3.1 and Proposition 4.1 hold true
also for a typical $T\in(\mathcal{B}_{1}(\ell_{p}),\texttt{SOT})$?

Observe that the answer to Question 6.5 is negative if $p=1$ or $p=2$. Indeed,
it is proved in [GMM2] (resp. in [EM]) that a typical
$T\in(\mathcal{B}_{1}(\ell_{1}),\texttt{SOT})$ (resp. a typical
$T\in(\mathcal{B}_{1}(\ell_{2}),\texttt{SOT})$) is such that any
$\lambda\in\mathbb{D}$ is an eigenvalue of $T$, and that $T^{*}$ is a
surjective isometry. So $T^{*}$ has property $(\beta)$ (see [LN, Proposition
1.6.6]), and hence $T$ has property $(\delta)$.

## References


Comment on Quantum Massive Conformal Gravity

Yun Soo Myung***e-mail address<EMAIL_ADDRESS>
Institute of Basic Sciences and Department of Computer Simulation, Inje
University Gimhae 621-749, Korea

###### Abstract

In a recent paper in EPJC March 2016, Faria has shown that quantum massive
conformal gravity is renormalizable but has ghost states. We comment this
paper on the aspect of renormalizability.

Typeset Using LaTeX

Recently, a paper in EPJC March 2016 [1], Faria has insisted that quantum
massive conformal gravity is renormalizable but has ghost states. The proposed
action for this purpose is just the massive conformal gravity action(MCG) [2,
3]

$\displaystyle S_{\rm MCG}=-\frac{1}{12}\int
d^{4}x\sqrt{-g}\Big{[}\Big{(}\phi^{2}R+6\partial_{\mu}\phi\partial^{\mu}\phi\Big{)}-\frac{1}{m^{2}}C^{2}\Big{]},~{}~{}C^{2}=C^{\mu\nu\rho\sigma}C_{\mu\nu\rho\sigma}$
(1)

which is classically invariant under the conformal transformations as

$g_{\mu\nu}\to\Omega^{2}(x)g_{\mu\nu},~{}~{}\phi\to\frac{\phi}{\Omega}.$ (2)

Here $\Omega(x)$ is an arbitrary function of the spacetime coordinates. We
note here that the MCG action ($S_{\rm MCG}$) is composed of the conformal
dilation gravity ($S_{\rm CDG}$) and the conformal gravity ($S_{\rm CG}$).
Since the integral of the Euler density
($E=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}-4R_{\mu\nu}R^{\mu\nu}+R^{2}$) is
a topological invariant quantity, $S_{\rm CG}$ reduces to the Weyl gravity

$S_{\rm WG}=\frac{1}{6m^{2}}\int
d^{4}x\sqrt{-g}\Big{[}R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}R^{2}\Big{]}\equiv\frac{1}{6m^{2}}\int
d^{4}x\sqrt{-g}\tilde{C}^{2}.$ (3)

It was argued that the massive conformal gravity (1) is a renormalizable
quantum theory of gravity which has two massive ghost states. However, Faria’s
work is far from showing that (1) is a renormalizable quantum gravity because
it was based on performing the canonical quantization of the second-order
bilinear action. Concerning the renormalizability, Faria has mentioned that
the graviton propagator of $\Psi_{\mu\nu}=h_{\mu\nu}-\eta_{\mu\nu}h/2$,

$D^{\mu\nu,\alpha\beta}_{\Psi}=-\frac{i}{2}\Big{(}\eta^{\mu\alpha}\eta^{\nu\beta}+\eta^{\mu\beta}\eta^{\nu\alpha}-\eta^{\mu\nu}\eta^{\alpha\beta}\Big{)}\int\frac{d^{4}p}{(2\pi)^{4}}\frac{m^{2}e^{-ip\cdot(x-y)}}{(p^{2}-i\chi)(p^{2}+m^{2}-i\chi)}$
(4)

has a good behavior of $1/p^{4}$ at high momenta, making (1) power-counting
renormalizable in the Minkowski spacetime. Unless the massive pole is shown to
be unphysical, the MCG (1) is perturbatively meaningless because the graviton
propagator (4) has ghost state. We wish to point out that this was a clear
observation since the seminal work of Stelle released 40 years ago [4].

It was known that the MCG (1) is a promising quantum (gravity) model because
the conformal symmetry restricts the number of counter-terms arising from the
perturbative quantization of the scalar (dilaton) $\phi$, while keeping the
graviton fixed [5]. The inclusion of conformal symmetry is the reason for the
cancelations. Explicitly, the one-loop counter term of the conformal dilation
gravity ($S_{\rm CDG}$) is given by [6, 7]

$\Gamma_{\rm CDG}=-\frac{1}{960\pi^{2}(n-4)}\int
d^{n}x\sqrt{-g}\Big{[}R_{\mu\nu}R^{\mu\nu}-\frac{1}{3}R^{2}\Big{]},$ (5)

which is proportional to the $S_{\rm WG}$ (3), reflecting the conformal
symmetry. We don’t need any counter terms more. In the absence of conformally
coupled term ($\phi^{2}R$), however, quantizing a purely kinetic term requires
(5) as well as $R^{2}$-term (conformally non-invariant term). Hence (1) plays
the role of a proper quantization action when quantizing the dilation $\phi$
in the fixed curved background $g_{\mu\nu}$.

On the other hand, Stelle has proposed the conformally non-invariant
Lagrangian of $\sqrt{-g}(R+a\tilde{C}^{2}+bR^{2})$ to improve the perturbative
properties of Einstein gravity [4]. The first two terms could be derived from
(1) by making use of the conformal transformation (2). Also, one includes
$R^{2}$ term as a counter term of $g_{\mu\nu}$. If $ab\not=0$, the
renormalizability was achieved but the unitarity was violated, which shows
that the renormalizability is not compatible with the unitarity. Although the
Weyl-squared term ($\tilde{C}^{2}$) improves the ultraviolet divergence, it
induces ghost excitations which spoil the unitarity simultaneously. Here, the
price one has to pay for making the theory renormalizable is the loss of
unitarity.

Up to now, there is no obvious way to attain the renormalizability without
violating the unitarity in quantizing the gravity.

## References

  * [1] F. F. Faria, Eur. Phys. J. C 76, no. 4, 188 (2016) doi:10.1140/epjc/s10052-016-4037-5 [arXiv:1503.04355 [gr-qc]].
  * [2] F. F. Faria, Adv. High Energy Phys. 2014, 520259 (2014) doi:10.1155/2014/520259 [arXiv:1312.5553 [gr-qc]].
  * [3] Y. S. Myung, Phys. Lett. B 730, 130 (2014) [arXiv:1401.1890 [gr-qc]].
  * [4] K. S. Stelle, Phys. Rev. D 16, 953 (1977). doi:10.1103/PhysRevD.16.953
  * [5] G. ’t Hooft and M. J. G. Veltman, Annales Poincare Phys. Theor. A 20, 69 (1974).
  * [6] G. ’t Hooft, Found. Phys. 41, 1829 (2011) doi:10.1007/s10701-011-9586-8 [arXiv:1104.4543 [gr-qc]].
  * [7] E. Alvarez, M. Herrero-Valea and C. P. Martin, JHEP 1410, 115 (2014) doi:10.1007/JHEP10(2014)115 [arXiv:1404.0806 [hep-th]].


# Semi-Supervised GCN for learning Molecular Structure-Activity Relationships

Alessio Ragno 1, Dylan Savoia1, Roberto Capobianco1,2

1Dept. of Computer, Control and Management
Engineering “Antonio Ruberti”, Sapienza University of Rome
2Sony AI
Correspondence to<EMAIL_ADDRESS>
###### Abstract

Since the introduction of artificial intelligence in medicinal chemistry, the
necessity has emerged to analyse how molecular property variation is modulated
by either single atoms or chemical groups. In this paper, we propose to train
graph-to-graph neural network using semi-supervised learning for attributing
structure-property relationships. As initial case studies we apply the method
to solubility and molecular acidity while checking its consistency in
comparison with known experimental chemical data. As final goal, our approach
could represent a valuable tool to deal with problems such as activity cliffs,
lead optimization and de-novo drug design.

##  1 Introduction

Drug discovery is a process that consists in searching, developing and
optimizing molecules in order to disclose new lead compounds for future new
drugs as medicaments for specific diseases.

One big issue in drug discovery consists in experimentally evaluating the
molecules, a part of the drug development flow that is time-consuming and
expensive.

In order to compensate this problem, medicinal chemists generally use
computational approaches to derive models for forecasting molecules’ bio-
activity and selecting the ones associated to higher predicted values, for
example, quantitative structure-activity relationships (QSAR) consist in the
application of machine learning for predicting the molecules’ activity from
their structure.

In this paper we propose a method to study these relationships calculating
atom attributions through their graph representation. It is, in fact, very
important for the chemist to individuate the most relevant substructures in
order to exploit them for the development of new molecules. Once a QSAR model
is built, we propose to train a graph-to-graph network (GGN) to attribute the
molecules’ atoms for their properties.

##  2 Related Work

While, with classic machine learning, molecules are represented using
fingerprints and descriptors, after the introduction of graph convolutional
networks (GCN) (Kipf and Welling, 2016) it became possible to represent their
structure using graphs. This allows us to learn over molecules taking
advantage of both chemical and structural information. When studying
molecules’ properties, it is not only interesting to be able to predict them
using different estimators, but, it is important to understand how they are
modulated by the different functional groups.

In order to study these structure-activity relationships, researches have been
exploiting explainable AI (XAI) methods for interpreting the GCN predictions.
This allows to visualize the atoms that are considered “mostly responsible”
for modulating the activity. Some approaches exploit integrated gradients (IG)
(Sundararajan et al., 2017) to generate input-attributions on the atoms
(Jiménez-Luna et al., 2021). Although XAI methods do a good job in attributing
specific atoms, it is difficult for them to refer to functional groups.
Moreover, many approaches (Ying et al., 2019; Luo et al., 2020; Schlichtkrull
et al., 2021) involve the perturbation of graphs by means of atom removal,
masking the graphs or edge cutting, but they do not take into account the fact
that, in chemistry, removing an atom or a bond from a molecule might generate
an impossible structure to realize. Other techniques (Huang et al., 2020)
focus, instead, on the perturbation of the node features, but again, this
could cause the creation of non-sense values.

##  3 Proposed Approach

In this paper, we propose to train a GGN model, using a semi-supervised
approach (Kipf and Welling, 2016), which is able to give scores to atoms and
functional groups of a molecule, depending on their influence on a QSAR
model’s prediction. To train the model, we need to set some ground-truth
output values for the atoms: we modify the molecules by adding or substituting
atoms and we evaluate the activity of the generated molecules, finally, we
compare them with the original one. We then train the GGN model on the
modified molecules, predicting the score of the nodes corresponding only to
the added atoms. In this way, the resulting model is able to give scores to
every atom of the molecule, depending on its influence on the property.

With this idea, we build a dataset by generating new molecules via atom
addition and set the ground-truth scores: given a molecule $m$, with predicted
activity $a$, and a second molecule $m^{\prime}$, with predicted activity
$a^{\prime}$, derived from $m$ through an addition of a group $g$, the ground-
truth scores for the atoms of $g$ are defined as
$attr(g_{i})=\frac{a^{\prime}-a}{|g|}$. We train the GGN to approximate
$attr(g_{i})$, with $g_{i}\in g$.

For instance, given a QSAR model and molecule $m$ with a predicted activity
$a=1.30$ (benzene in Figure 1) we can add a group $g$ of 3 atoms (carboxyl
group in Figure 1) generating a second molecule $m^{\prime}$ (benzoic acid in
Figure 1). Assuming the evaluation using the QSAR model of $m^{\prime}$ is
$a^{\prime}=1.00$, we know that the addition led an activity decrement of
$1.00-1.30=-0.30$, therefore we can assume that the contribution of the added
atoms would be $\frac{-0.30}{3}=-0.10$.

Figure 1: Ground-truth score generation example with benzene and benzoic acid.

In order to make modifications to the molecules, we need a set of functional
groups to add. Ertl (2017) proposes an algorithm to extract functional groups
from a set of molecule and applied it to $\approx~{}$483,000 molecules with
activity below 10 $\mu$M on any target retrieved from ChEMBL database (Gaulton
et al., 2012). The resulting database consists in a set of 768 groups that are
present in at least 10 ChEMBL molecules, provided in pseudo-SMILES notation.

Molecule | IG | DeepLIFT | GGN | logS
---|---|---|---|---
hexane | -0.1166 | -0.1196 | -1.8828 | -9.1150
pentane | -0.0932 | -0.0956 | -1.8627 | -7.5483
1-hexanol | -0.0657 | -0.0711 | -0.5852 | -2.8134
1-pentanol | -0.0404 | -0.0460 | -0.5696 | -1.3863
3-pentanol | -0.0096 | -0.0163 | -0.1369 | -0.5447
Correlation with logS | 0.96 | 0.95 | 0.98 | 1.0
Table 1: Molecular scores obtained with IG, DeepLIFT and the proposed approach
compared with the logS of the molecules

##  4 Experimental Results

We evaluate our method on two datasets:

  * •

ESOL Water Solubility Dataset (Delaney, 2004), which contains more than 1k
molecules with the associated logarithm of the solubility expressed in molar
concentration (logS);

  * •

A set of 160k molecules retrieved from ChEMBL (Gaulton et al., 2012) with the
associated ChemAxon predicted pKa.

We train two GCN-QSAR models on the datasets and use the functional groups
database to generate the dataset for the GGN model with the ground-truth
attribution scores. Finally for each initial dataset, a GGN model is trained.
The following subsections analyse the attributions to the GCN-QSAR predictions
obtained with some molecules for both the tasks comparing them with two XAI
methods: IG (Sundararajan et al., 2017) and DeepLIFT (Shrikumar et al., 2017).
The attributions are analysed through a mask of colors on the atoms on a scale
from blue to red, where blue is negative and red is positive. The opacity of
the color indicates the intensity of the scores. To produce a quantitative
evaluation we sum up all the atom scores for each molecule: we expect
molecules with a higher value of the property in exam to have a higher
molecular scores. Finally we analyse the correlation between molecular scores
and their properties.

###  4.1 Water Solubility

Water solubility is very important in medicinal chemistry, since most of the
drugs must be present in the form of solution at the site of absorption.
Moreover, solubility works as a benchmark since we know from chemical theory
that the presence of specific atoms or patterns modulates significantly the
property.

As first analysis the attributions for 5 different molecules external to the
dataset with incremental solubility are compared: hexane, pentane, 1-hexanol,
1-pentanol and 3-pentanol.

Molecule | IG | DeepLIFT | GGN | pKa
---|---|---|---|---
pentanoic acid | 0.4884 | 0.9821 | 1.6650 | 5.01
4-chloropentanoic acid | 0.1337 | 0.0765 | 0.5025 | 4.27
3-chloropentanoic acid | 0.1610 | 0.2180 | 0.5006 | 4.24
2-chloropentanoic acid | 0.0253 | -0.0779 | 0.4777 | 3.96
Correlation with pKa | 0.98 | 0.41 | 0.98 | 1.0
Table 2: Molecular scores obtained with IG, DeepLIFT and the proposed approach
compared with the pKa of the first pKa case study molecules

Chemical theory and experimental values for solubility of the molecules in
exam suggest that the hydroxyl (OH-) group generally increases the solubility
if introduced in a molecule and the more the it is at the center of the carbon
chain, the more the solubility increases. Hexane and pentane are only composed
of a chain of carbons, while 1-hexanol and 1-pentanol are generated adding an
hydroxyl group to the formers, respectively, this makes them more soluble than
their original molecules and in fact the hydroxyl group receives positive
scores from all the methods (Figure 2(a), rows 3 and 4). Using the same
reasoning, the 3-pentanol, is more soluble than the previous molecules, as its
hydroxyl group is placed at the center of the chain. Also in this case, the
attribution methods report coherent values with the chemical theory (Figure
2(a), row 4). The molecular scores obtained with the different methods report
values in good agreement with the expectation, in fact the molecular scores
have a high positive correlation with the logS (Table 1).

###  4.2 pKa

The acidity of the molecules is a property which influences the solubility of
drugs and their pharmacodynamics and pharmacokinetics (ADMET) properties. The
strength of an acid is generally measured through the acid dissociation
constant, $K_{a}$, or its linearized version pKa.

(a) Solubility attribution comparison varying hydroxyl group position

(b) pKa attribution comparison varying the chlorine position

Figure 2: Molecular attributions compared between IG, DeepLIFT and the
proposed approach

We take into exam five molecules: pentanoic acid, 4-chloropentanoic acid,
3-chloropentanoic acid and 2-chloropentanoic acid. In this case study, the
molecules in exam have a negative trend of pKa, due to the position of the
chlorine along the molecules: the closer to the carboxyl group, the more pKa
decreases (more acid). The carboxyl group in the pentanoic acid is correctly
attributed with the lowest score since it is the group that determines the
molecule’s acidity (Figure 2(b), row 1). Moreover, our method is the only one
which correctly identifies the carboxyl group as a whole. For the other
molecules, only our method and DeepLIFT are able to attribute the chlorine as
an element which influences negatively on the pKa (Figure 2(b), rows 2, 3 and
4). Although DeepLIFT seems to perform similarly to our approach, correlation
between the molecular scores and pKa values show that the our methods’s
results are more coherent with the experimental values (Table 2).

##  5 Conclusions

Neural networks in drug design may help reducing the costs of lead discovery
and, with GCNs, it is possible to exploit both chemical and structure
information of the molecules thanks to the graph representation. We only
performed the experiments using the vanilla GCN layer as the main goal was to
provide a general method. Other variants, such as Graph Attention Layers,
could be used as well, both for the QSAR model and the GGN.

XAI proved to be a key tool for developing future models in drug design.
Although novel techniques are able to individuate the most relevant atoms, it
is still difficult for them to focus on functional groups. The proposed
approach managed to overcome this issue and we think that it could be extended
in order to optimize molecules’ activities/properties by adding or
substituting atoms in a rational way.

As a drawback, the proposed method consists in building a GGN network. This
approach proved to be more “chemically oriented” but brings the overload of
training an additional model.

## References

  * Delaney [2004] John S. Delaney.  ESOL: estimating aqueous solubility directly from molecular structure.  _Journal of Chemical Information and Computer Sciences_ , 44(3):1000–1005, May 2004.  doi: 10.1021/ci034243x.  URL https://doi.org/10.1021/ci034243x.

  * Ertl [2017] Peter Ertl.  An algorithm to identify functional groups in organic molecules.  _Journal of Cheminformatics_ , 9(1):36, Jun 2017\.  ISSN 1758-2946.  doi: 10.1186/s13321-017-0225-z.  URL https://doi.org/10.1186/s13321-017-0225-z.
  * Gaulton et al. [2012] Anna Gaulton, Louisa J. Bellis, A. Patricia Bento, Jon Chambers, Mark Davies, Anne Hersey, Yvonne Light, Shaun McGlinchey, David Michalovich, Bissan Al-Lazikani, and John P. Overington.  Chembl: a large-scale bioactivity database for drug discovery.  _Nucleic acids research_ , 40(Database issue):D1100–D1107, Jan 2012.  ISSN 1362-4962.  doi: 10.1093/nar/gkr777.  URL https://pubmed.ncbi.nlm.nih.gov/21948594.  21948594[pmid].
  * Huang et al. [2020] Qiang Huang, Makoto Yamada, Yuan Tian, Dinesh Singh, Dawei Yin, and Yi Chang.  Graphlime: Local interpretable model explanations for graph neural networks.  _CoRR_ , abs/2001.06216, 2020.  URL https://arxiv.org/abs/2001.06216.
  * Jiménez-Luna et al. [2021] José Jiménez-Luna, Miha Skalic, Nils Weskamp, and Gisbert Schneider.  Coloring molecules with explainable artificial intelligence for preclinical relevance assessment.  61(3):1083–1094, February 2021.  doi: 10.1021/acs.jcim.0c01344.  URL https://doi.org/10.1021/acs.jcim.0c01344.
  * Kipf and Welling [2016] Thomas N. Kipf and Max Welling.  Semi-supervised classification with graph convolutional networks.  _CoRR_ , abs/1609.02907, 2016.  URL http://arxiv.org/abs/1609.02907.
  * Luo et al. [2020] Dongsheng Luo, Wei Cheng, Dongkuan Xu, Wenchao Yu, Bo Zong, Haifeng Chen, and Xiang Zhang.  Parameterized explainer for graph neural network.  _CoRR_ , abs/2011.04573, 2020.  URL https://arxiv.org/abs/2011.04573.
  * Schlichtkrull et al. [2021] Michael Sejr Schlichtkrull, Nicola De Cao, and Ivan Titov.  Interpreting graph neural networks for {nlp} with differentiable edge masking.  In _International Conference on Learning Representations_ , 2021.  URL https://openreview.net/forum?id=WznmQa42ZAx.
  * Shrikumar et al. [2017] Avanti Shrikumar, Peyton Greenside, and Anshul Kundaje.  Learning important features through propagating activation differences.  _CoRR_ , abs/1704.02685, 2017.  URL http://arxiv.org/abs/1704.02685.
  * Sundararajan et al. [2017] Mukund Sundararajan, Ankur Taly, and Qiqi Yan.  Axiomatic attribution for deep networks, 2017.
  * Ying et al. [2019] Rex Ying, Dylan Bourgeois, Jiaxuan You, Marinka Zitnik, and Jure Leskovec.  GNN explainer: A tool for post-hoc explanation of graph neural networks.  _CoRR_ , abs/1903.03894, 2019.  URL http://arxiv.org/abs/1903.03894.