# On the fine spectrum of the forward difference operator on the Hahn space

Medi̇ne Yeşi̇lkayagi̇l and Murat Ki̇ri̇şci̇  Department of Mathematics, Uşak
University, 1 Eylül Campus, 64200 - Uşak, Turkey
<EMAIL_ADDRESS>Department of Mathematical Education, Hasan
Ali Yucel Education Faculty, İstanbul University, Vefa, 34470–İstanbul, Turkey
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The main purpose of this paper is to determine the fine spectrum with respect
to Goldberg’s classification of the difference operator over the sequence
space $h$. As a new development, we give the approximate point spectrum,
defect spectrum and compression spectrum of the difference operator on the
sequence space $h$.

###### Key words and phrases:

Spectrum of an operator, spectral mapping theorem, the Hahn sequence space,
Goldberg’s classification.

###### 2010 Mathematics Subject Classification:

47A10, 47B37.

**Corresponding author.

##  1\. Introduction

An important branch of mathematics due to its application in other branches of
science is a Spectral Theory. It has been proved to be a very useful tool
because of its convenient and easy applicability of the different fields. In
numerical analysis, the spectral values may determine whether a discretization
of a differential equation will get the right answer or how fast a conjugate
gradient iteration will converge. In aeronautics, the spectral values may
determine whether the flow over a wing is laminar or turbulent. In electrical
engineering, it may determine the frequency response of an amplifier or the
reliability of a power system. In quantum mechanics, it may determine atomic
energy levels and thus, the frequency of a laser or the spectral signature of
a star. In structural mechanics, it may determine whether an automobile is too
noisy or whether a building will collapse in an earthquake. In ecology, the
spectral values may determine whether a food web will settle into a steady
equilibrium. In probability theory, they may determine the rate of convergence
of a Markov process.

There are several studies about the spectrum of the linear operators defined
by some triangle matrices over certain sequence spaces. This long-time
behavior was intensively studied over many years, starting with the work by
Wenger [29], who established the fine spectrum of the integer power of the
Cesàro operator in c. The generalization of [29] to the weighted mean methods
is due to Rhoades [26]. The study of the fine spectrum of the operator on the
sequence space $\ell_{p}$, $(1<p<\infty)$ was initiated by Gonzàlez [14]. The
method of the spectrum of the Cesàro operator prepared by Reade [25], Akhmedov
and Başar [1], and Okutoyi [21], respectively, whose established to this idea
on the sequence spaces $c_{0}$ and $bv$.In [31], the fine spectrum of the
Rhaly operators on the sequence spaces $c_{0}$ and $c$ was given. The spectrum
and fine spectrum for p-Cesàro operator acting on the space $c_{0}$ was
studied by Coşkun [9]. The investigation of the spectrum and the fine spectrum
of the difference operator on the sequence spaces $s_{r}$ and $c_{0}$, $c$ was
made by Malafosse [20] and Altay and Başar [4], respectively, where $s_{r}$
denotes the Banach space of all sequences $x=(x_{k})$ normed by
$\|x\|_{s_{r}}=\underset{k\in\mathbb{N}}{\sup}|x_{k}|/r^{k}$, $(r>0)$. The
idea of the fine spectrum applied to the Zweier matrix which is a band matrix
as an operator over the sequence spaces $\ell_{1}$ and $bv$ by Altay and
Karakus [5]. Let $\Delta_{\nu}$ is double sequential band matrix on $\ell_{1}$
such that $(\Delta_{\nu})_{nn}=\nu_{n}$ and $(\Delta_{\nu})_{n+1,n}=-\nu_{n}$
for all $n\in\mathbb{N}$, under certain conditions on the sequence
$\nu=(\nu_{k})$. The spectra and the fine spectra of matrix $\Delta_{\nu}$
were determined by Srivastava and Kumar [27]. Afterwards, these results of the
double sequential band matrix $\Delta_{\nu}$ generalized to the double
sequential band matrix $\Delta_{uv}$ such that defined by
$\Delta_{uv}x=(u_{n}x_{n}+v_{n-1}x_{n-1})_{n\in\mathbb{N}}$ for all
$n\in\mathbb{N}$ (see [28]). In [6], the fine spectra of the Toeplitz
operators represented by an upper and lower triangular $n$-band infinite
matrices, over the sequence spaces $c_{0}$ and $c$ was computed. The fine
spectra of upper triangular double-band matrices over the sequence spaces
$c_{0}$ and $c$ was obtained by Karakaya and Altun [16]. Let $\Delta_{a,b}$ is
a double band matrix with the convergent sequences $\widetilde{a}=(a_{k})$ and
$\widetilde{b}=(b_{k})$ having certain properties, over the sequence space
$c$. The fine spectrum of the matrix $\Delta_{a,b}$ was examined by Akhmedov
and El-Shabrawy [3]. The approach to the fine spectrum with respect to
Goldberg’s classification studied with of the operator $B(r,s,t)$ defined by a
triple band matrix over the sequence spaces $\ell_{p}$ and $bv_{p}$,
$(1<p<\infty)$ by Furkan et al. [11]. Quite recently, the fine spectrum with
respect to Goldberg’s classification of the operator defined by the lambda
matrix over the sequence spaces $c_{0}$ and $c$ computed by Yeşilkayagil and
Başar [30].

Hahn sequence space is defined as $x=(x_{k})$ such that
$\sum_{k=1}^{\infty}k|x_{k}-x_{k+1}|$ converges and $x_{k}$ is a null sequence
and is denoted by $h$. Initially, this space was defined and studied to some
general properties by Hahn [15]. It was examined different properties of this
space by Goes and Goes [12] and Rao [22], [23], [24]. Quite recently, the
studies on Hahn sequence space has been compiled by Kirisci [17]. Also in
[18], it has been defined a new Hahn sequence space by Cesàro mean.

In the present paper, our propose is to investigate the fine spectrum of the
difference operator $\Delta$ on the sequence space $h$. And also, we define
the approximate point spectrum, defect spectrum and compression spectrum of
the difference operator on the sequence space $h$, as a new approach.

##  2\. Preliminaries and Definition

Let $X$ and $Y$ be Banach spaces, and also let $T:X\rightarrow Y$ be a bounded
linear operator. The range of the operator $T$ defined by

$\displaystyle R(T)=\\{y\in Y:y=Tx,\text{ }x\in X\\}.$

The set of all bounded linear operators on $X$ into itself denoted by $B(X)$.

We choose any Banach space $X$ and let $T\in B(X)$. Then we can define the
adjoint $T^{\ast}$ of $T$ is a bounded linear operator on the dual $X^{\ast}$
of $X$ such that $\left(T^{\ast}f\right)(x)=f(Tx)$ for all $f\in Y^{\ast}$ and
$x\in X$.

Let $X\neq\\{\theta\\}$ be a non-trivial complex normed space. A linear
operator $T$ defined by $T:D(T)\rightarrow X$ on a subspace $D(T)\subseteq X$.
We do not assume that $D(T)$ is dense in $X$ or that T has closed graph
$\\{(x,Tx):x\in D(T)\\}\subseteq X\times X$. We mean by the expression  ”$T$
is invertible” that there exists a bounded linear operator $S:R(T)\rightarrow
X$ for which $ST=I$ on $D(T)$ and $\overline{R(T)}=X$; such that $S=T^{-1}$ is
necessarily uniquely determined, and linear; the boundedness of $S$ means that
$T$ must be bounded below, in the sense that there is $k>0$ for which
$\|Tx\|\geq k\|x\|$ for all $x\in D(T)$. The perturbed operator defined on the
same domain $D(T)$ as $T$ as follows:

$\displaystyle T_{\alpha}=\alpha I-T$

such that associated with each complex number $\alpha$. The spectrum
$\sigma(T,X)$ consists of those $\alpha\in\mathbb{C}$ for which $T_{\alpha}$
is not invertible, and the resolvent is the mapping from the complement
$\sigma(T,X)$ of the spectrum into the algebra of bounded linear operators on
X defined by $\alpha\mapsto T_{\alpha}^{-1}$.

Let $\omega$ is the space of all complex valued sequences and $\phi$ the set
of all infinitely nonzero sequences. A linear subspace of $\omega$ which
contain $\phi$ said a sequence space. We write $\ell_{\infty}$, $c$, $c_{0}$
and $bv$ for the spaces of all bounded, convergent, null and bounded variation
sequences which are the Banach spaces with the sup-norm
$\|x\|_{\infty}=\underset{k\in\mathbb{N}}{\sup}|x_{k}|$ and
$\|x\|_{bv}=\stackrel{{\scriptstyle\infty}}{{\underset{k=0}{\sum}}}|x_{k}-x_{k+1}|$,
respectively, while $\phi$ is not a Banach space with respect to any norm,
where $\mathbb{N}=\\{0,1,2,\ldots\\}$. Also by $\ell_{p}$, we denote the space
of all $p$–absolutely summable sequences which is a Banach space with the norm
$\|x\|_{p}=\left(\stackrel{{\scriptstyle\infty}}{{\underset{k=0}{\sum}}}|x_{k}|^{p}\right)^{1/p}$,
where $1\leq p<\infty$.

Let $\mu$ and $\nu$ be two sequence spaces, and $A=(a_{nk})$ be an infinite
matrix of complex numbers $a_{nk}$, where $k,n\in\mathbb{N}$. Then, we say
that $A$ defines a matrix mapping from $\mu$ into $\nu$, and we denote it by
writing $A:\mu\rightarrow\nu$ if for every sequence $x=(x_{k})\in\mu$, the
$A$-transform $Ax=\\{(Ax)_{n}\\}$ of $x$ is in $\nu$; where

(2.1) $\displaystyle(Ax)_{n}=\sum_{k=0}^{\infty}a_{nk}x_{k}~{}\textrm{ for all
}~{}n\in\mathbb{N}.$

By $(\mu:\nu)$, we denote the class of all matrices $A$ such that
$A:\mu\rightarrow\nu$. Thus, $A\in(\mu:\nu)$ if and only if the series on the
right side of (2.1) converges for each $n\in\mathbb{N}$ and each $x\in\mu$,
and we have $Ax=\\{(Ax)_{n}\\}_{n\in\mathbb{N}}\in\nu$ for all $x\in\mu$.

The $BK-$space $h$ of all sequences $x=(x_{k})$ such that

$\displaystyle h=\left\\{x:\sum_{k=1}^{\infty}k|\Delta
x_{k}|<\infty~{}\textrm{ and }~{}\lim_{k\rightarrow\infty}x_{k}=0\right\\}$

was defined by Hahn [15]. Here and after $\Delta$ denotes the forward
difference operator, that is, $\Delta x_{k}=x_{k}-x_{k+1}$, for all
$k\in\mathbb{N}$. The following norm

$\displaystyle\|x\|_{h}=\sum_{k}k|\Delta x_{k}|+\sup_{k}|x_{k}|$

was given on the space $h$ by Hahn [15] (and also [12]). Rao [22, Proposition
2.1] defined a new norm on $h$ as $\|x\|=\sum_{k}k|\Delta x_{k}|.$

Hahn proved following properties of the space $h$:

######  Lemma 2.1.

The following statements hold:

  * (i)

$h$ is a Banach space.

  * (ii)

$h\subset\ell_{1}\cap\int c_{0}$.

  * (iii)

$h^{\beta}=\rho_{\infty}$,

where $\int\lambda=\big{\\{}x=(x_{k})\in\omega:(kx_{k})\in\lambda\big{\\}}$
and
$\rho_{\infty}=\big{\\{}x=(x_{k})\in\omega:\sup_{n}n^{-1}\big{|}\sum_{k=1}^{n}x_{k}\big{|}<\infty\big{\\}}$.

Functional analytic properties of the $BK-$space $bv_{0}\cap d\ell_{1}$ was
studied by Goes and Goes [12], where
$d\ell_{1}=\\{x=(x_{k})\in\omega:\sum_{k=1}^{\infty}\frac{1}{k}|x_{k}|<\infty\\}$.
Also, in [12], the arithmetic means of sequences in $bv_{0}$ and $bv_{0}\cap
d\ell_{1}$ were considered, and used the fact that the Cesàro transform
$(n^{-1}\sum_{k=1}^{n}x_{k})$ of order one $x\in bv_{0}$ is a quasiconvex null
sequence.

Rao [22] studied some geometric properties of Hahn sequence space and gave the
characterizations of some classes of matrix transformations. Also, in [23] and
[24], Rao examined the different properties of Hahn sequence space.

Balasubramanian and Pandiarani [8] defined the new sequence space $h(F)$
called the Hahn sequence space of fuzzy numbers and proved that $\beta-$ and
$\gamma-$duals of $h(F)$ is the Cesàro space of the set of all fuzzy bounded
sequences.

Until the new studies of Kirişçi [17, 18], there has not been any work
containing the Hahn sequence space.

##  3\. Subdivision of the Spectrum

In this section, we define the parts called point spectrum, continuous
spectrum, residual spectrum, approximate point spectrum, defect spectrum and
compression spectrum of the spectrum. There are many different ways to
subdivide the spectrum of a bounded linear operator. Some of them are
motivated by applications to physics, in particular, quantum mechanics.

###  3.1. The Point Spectrum, Continuous Spectrum and Residual Spectrum

The name resolvent is appropriate since $T_{\alpha}^{-1}$ helps to solve the
equation $T_{\alpha}x=y$. Thus, $x=T_{\alpha}^{-1}y$ provided that
$T_{\alpha}^{-1}$ exists. More importantly, the investigation of properties of
$T_{\alpha}^{-1}$ will be basic for an understanding of the operator $T$
itself. Naturally, many properties of $T_{\alpha}$ and $T_{\alpha}^{-1}$
depend on $\alpha$, and the spectral theory is concerned with those
properties. For instance, we shall be interested in the set of all $\alpha$’s
in the complex plane such that $T_{\alpha}^{-1}$ exists. Boundedness of
$T_{\alpha}^{-1}$ is another property that will be essential. We shall also
ask for what $\alpha$’s the domain of $T_{\alpha}^{-1}$ is dense in X, to name
just a few aspects. A regular value $\alpha$ of $T$ is a complex number such
that $T_{\alpha}^{-1}$ exists and is bounded whose domain is dense in X. For
our investigation of T, $T_{\alpha}$ and $T_{\alpha}^{-1}$, we need some basic
concepts in the spectral theory which are given, as follows (see Kreyszig [19,
pp. 370-371]):

The resolvent set $\rho(T,X)$ of $T$ is the set of all regular values $\alpha$
of $T$. Furthermore, the spectrum $\sigma(T,X)$ is partitioned into the
following three disjoint sets:

The point (discrete) spectrum $\sigma_{p}(T,X)$ is the set such that
$T_{\alpha}^{-1}$ does not exist. An $\alpha\in\sigma_{p}(T,X)$ is called an
eigenvalue of $T$.

The continuous spectrum $\sigma_{c}(T,X)$ is the set such that
$T_{\alpha}^{-1}$ exists and is unbounded, and the domain of $T_{\alpha}^{-1}$
is dense in $X$.

The residual spectrum $\sigma_{r}(T,X)$ is the set such that $T_{\alpha}^{-1}$
exists (and may be bounded or not) but the domain of $T_{\alpha}^{-1}$ is not
dense in $X$.

Therefore, these three subspectra form a disjoint subdivision such that

(3.1)
$\displaystyle\sigma(T,X)=\sigma_{p}(T,X)\cup\sigma_{c}(T,X)\cup\sigma_{r}(T,X).$

To avoid trivial misunderstandings, let us say that some of the sets defined
above may be empty. This is an existence problem which we shall have to
discuss. Indeed, it is well-known that
$\sigma_{c}(T,X)=\sigma_{r}(T,X)=\emptyset$ and the spectrum $\sigma(T,X)$
consists of only the set $\sigma_{p}(T,X)$ in the finite-dimensional case.

###  3.2. The Approximate Point Spectrum, Defect Spectrum and Compression
Spectrum

In this subsection, three more subdivision of the spectrum called the
approximate point spectrum, defect spectrum and compression spectrum have been
defined as in Appell et al. [7].

Let $X$ is a Banach space and $T$ is a bounded linear operator. A $(x_{k})\in
X$ Weyl sequence for $T$ defined by $\left\|x_{k}\right\|=1$ and
$\left\|Tx_{k}\right\|\rightarrow 0$, as $k\rightarrow\infty$.

In what follows, we call the set

(3.2) $\displaystyle\sigma_{ap}(T,X):=\\{\alpha\in\mathbb{C}:\text{there
exists a Weyl sequence for }\alpha I-T\\}$

the approximate point spectrum of $T$. Moreover, the subspectrum

(3.3) $\displaystyle\sigma_{\delta}(T,X):=\\{\alpha\in\mathbb{C}:\alpha
I-T\text{ is not surjective}\\}$

is called defect spectrum of $T$.

The two subspectra given by (3.2) and (3.3) form a (not necessarily disjoint)
subdivision

$\displaystyle\sigma(T,X)=\sigma_{ap}(T,X)\cup\sigma_{\delta}(T,X)$

of the spectrum. There is another subspectrum,

$\displaystyle\sigma_{co}(T,X)=\\{\alpha\in\mathbb{C}:\overline{R(\alpha
I-T)}\neq X\\}$

which is often called compression spectrum in the literature. The compression
spectrum gives rise to another (not necessarily disjoint) decomposition

$\displaystyle\sigma(T,X)=\sigma_{ap}(T,X)\cup\sigma_{co}(T,X)$

of the spectrum. Clearly, $\sigma_{p}(T,X)\subseteq\sigma_{ap}(T,X)$ and
$\sigma_{co}(T,X)\subseteq\sigma_{\delta}(T,X)$. Moreover, comparing these
subspectra with those in (3.1) we note that

$\displaystyle\sigma_{r}(T,X)$ $\displaystyle=$
$\displaystyle\sigma_{co}(T,X)\backslash\sigma_{p}(T,X),$
$\displaystyle\sigma_{c}(T,X)$ $\displaystyle=$
$\displaystyle\sigma(T,X)\backslash[\sigma_{p}(T,X)\cup\sigma_{co}(T,X)].$

Sometimes it is useful to relate the spectrum of a bounded linear operator to
that of its adjoint. Building on classical existence and uniqueness results
for linear operator equations in Banach spaces and their adjoints are also
useful.

######  Proposition 3.1.

[7, Proposition 1.3, p. 28] The following relations on the spectrum and
subspectrum of an operator $T\in B(X)$ and its adjoint $T^{\ast}\in
B(X^{\ast})$ hold:

  1. (a)

$\sigma(T^{\ast},X^{\ast})=\sigma(T,X)$.

  2. (b)

$\sigma_{c}(T^{\ast},X^{\ast})\subseteq\sigma_{ap}(T,X)$.

  3. (c)

$\sigma_{ap}(T^{\ast},X^{\ast})=\sigma_{\delta}(T,X)$.

  4. (d)

$\sigma_{\delta}(T^{\ast},X^{\ast})=\sigma_{ap}(T,X)$.

  5. (e)

$\sigma_{p}(T^{\ast},X^{\ast})=\sigma_{co}(T,X)$.

  6. (f)

$\sigma_{co}(T^{\ast},X^{\ast})\supseteq\sigma_{p}(T,X)$.

  7. (g)

$\sigma(T,X)=\sigma_{ap}(T,X)\cup\sigma_{p}(T^{\ast},X^{\ast})=\sigma_{p}(T,X)\cup\sigma_{ap}(T^{\ast},X^{\ast})$.

The relations (c)-(f) show that the approximate point spectrum is in a certain
sense dual to the defect spectrum and the point spectrum is dual to the
compression spectrum. The equality (g) implies, in particular, that
$\sigma(T,X)=\sigma_{ap}(T,X)$ if $X$ is a Hilbert space and $T$ is normal.
Roughly speaking, this shows that normal (in particular, self-adjoint)
operators on Hilbert spaces are most similar to matrices in finite dimensional
spaces (see Appell et al. [7]).

###  3.3. Goldberg’s Classification of Spectrum

If $X$ is a Banach space and $T\in B(X)$, then there are three possibilities
for $R(T)$:

  * (A)

$\quad R(T)=X$.

  * (B)

$\quad R(T)\neq\overline{R(T)}=X$.

  * (C)

$\quad\overline{R(T)}\neq X$.

and

  * (1)

$\quad T^{-1}$ exists and is continuous.

  * (2)

$\quad T^{-1}$ exists but is discontinuous.

  * (3)

$\quad T^{-1}$ does not exist.

If these possibilities are combined in all possible ways, nine different
states are created. These are labelled by: $A_{1}$, $A_{2}$, $A_{3}$, $B_{1}$,
$B_{2}$, $B_{3}$, $C_{1}$, $C_{2}$, $C_{3}$. If an operator is in state
$C_{2}$ for example, then $\overline{R(T)}\neq X$ and $T^{-1}$ exists but is
discontinuous (see Goldberg [13]).

$C_{3}$$C_{2}$$C_{1}$$B_{3}$$B_{2}$$B_{1}$$A_{3}$$A_{2}$$A_{1}$$C_{3}$$C_{2}$$C_{1}$$B_{3}$$B_{2}$$B_{1}$$A_{3}$$A_{2}$$A_{1}$$T$$T^{*}$Table
1.1: State diagram for $B(X)$ and $B(X^{\ast})$ for a non-reflective Banach
space $X$

If $\alpha$ is a complex number such that $T_{\alpha}\in A_{1}$ or
$T_{\alpha}\in B_{1}$, then $\alpha\in\rho(T,X)$. All scalar values of
$\alpha$ not in $\rho(T,X)$ comprise the spectrum of $T$. The further
classification of $\sigma(T,X)$ gives rise to the fine spectrum of $T$. That
is, $\sigma(T,X)$ can be divided into the subsets
$A_{2}\sigma(T,X)=\emptyset$, $A_{3}\sigma(T,X)$, $B_{2}\sigma(T,X)$,
$B_{3}\sigma(T,X)$, $C_{1}\sigma(T,X)$, $C_{2}\sigma(T,X)$,
$C_{3}\sigma(T,X)$. For example, if $T_{\alpha}$ is in a given state, $C_{2}$
(say), then we write $\alpha\in C_{2}\sigma(T,X)$.

By the definitions given above, we can illustrate the subdivision (3.1) in the
following table:

|  | 1 | 2 | 3
---|---|---|---|---
|  |  $T^{-1}_{\alpha}$ exists |  $T^{-1}_{\alpha}$ exists | $T^{-1}_{\alpha}$
|  | and is bounded | and is unbounded | does not exist
|  |  |  | $\alpha\in\sigma_{p}(T,X)$
A | $R(\alpha I-T)=X$ | $\alpha\in\rho(T,X)$ | – | $\alpha\in\sigma_{ap}(T,X)$
|  |  | $\alpha\in\sigma_{c}(T,X)$ | $\alpha\in\sigma_{p}(T,X)$
B | $\overline{R(\alpha I-T)}=X$ | $\alpha\in\rho(T,X)$ | $\alpha\in\sigma_{ap}(T,X)$ | $\alpha\in\sigma_{ap}(T,X)$
|  |  | $\alpha\in\sigma_{\delta}(T,X)$ | $\alpha\in\sigma_{\delta}(T,X)$
|  | $\alpha\in\sigma_{r}(T,X)$ | $\alpha\in\sigma_{r}(T,X)$ | $\alpha\in\sigma_{p}(T,X)$
C | $\overline{R(\alpha I-T)}\not=X$ | $\alpha\in\sigma_{\delta}(T,X)$ | $\alpha\in\sigma_{ap}(T,X)$ | $\alpha\in\sigma_{ap}(T,X)$
|  |  | $\alpha\in\sigma_{\delta}(T,X)$ | $\alpha\in\sigma_{\delta}(T,X)$
|  | $\alpha\in\sigma_{co}(T,X)$ | $\alpha\in\sigma_{co}(T,X)$ | $\alpha\in\sigma_{co}(T,X)$

Table 1.2: Subdivision of spectrum of a linear operator

One can observe by the closed graph theorem that in the case $A_{2}$ cannot
occur in a Banach space $X$. If we are not in the third column of Table 1.2,
i.e., if $\alpha$ is not an eigenvalue of $T$, we may always consider the
resolvent operator $T^{-1}_{\alpha}$ (on a possibly thin domain of definition)
as algebraic inverse of $\alpha I-T$.

The forward difference operator $\Delta$ is represented by the matrix

$\displaystyle\Delta=\left[\begin{array}[]{cccccc}1&-1&0&0&\ldots\\\
0&1&-1&0&\ldots\\\ 0&0&1&-1&\ldots\\\ 0&0&0&1&\ldots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right].$

######  Corollary 3.2.

$\Delta:h\rightarrow h$ is a bounded linear operator.

##  4\. On the fine spectrum of the forward difference operator on the Hahn
space

In this section, we determine the spectrum and fine spectrum of the forward
difference operator $\Delta$ on the Hahn space $h$ and calculate the norm of
the operator

######  Theorem 4.1.

$\sigma(\Delta,h)=\left\\{\alpha\in\mathbb{C}:\left|1-\alpha\right|\leq
1\right\\}$.

###### Proof.

Let $\left|1-\alpha\right|>1$. Since $\Delta-\alpha I$ is triangle,
$(\Delta-\alpha I)^{-1}$ exists and solving the matrix equation
$(\Delta-\alpha I)x=y$ for $x$ in terms of $y$ gives the matrix
$(\Delta-\alpha I)^{-1}=B=(b_{nk})$, where

$\displaystyle
b_{nk}=\left\\{\begin{array}[]{ccl}\frac{1}{(1-\alpha)^{n+1}}&,&0\leq k\leq
n,\\\ 0&,&k>n\end{array}\right.$

for all $k,n\in\mathbb{N}$. Thus, we observe that

$\displaystyle\|(\Delta-\alpha I)^{-1}\|_{(h:h)}$ $\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}n|b_{nk}-b_{n+1,k}|$ $\displaystyle\leq$
$\displaystyle\sum_{n=1}^{\infty}n|b_{nk}|+\sum_{n=1}^{\infty}n|b_{n+1,k}|$
$\displaystyle=$
$\displaystyle\sum_{n=1}^{\infty}\frac{n}{|1-\alpha|^{n+1}}+\sum_{n=1}^{\infty}\frac{n}{|1-\alpha|^{n+2}}.$

From the ratio test, we have

$\displaystyle\|(\Delta-\alpha I)^{-1}\|_{(h:h)}<\infty,$

that is, $(\Delta-\alpha I)^{-1}\in(h:h)$. But for $\left|1-\alpha\right|\leq
1$,

$\displaystyle\|(\Delta-\alpha I)^{-1}\|_{(h:h)}=\infty,$

that is, $(\Delta-\alpha I)^{-1}$ is not in $B(h)$. This completes the proof.
∎

######  Theorem 4.2.

$\sigma_{p}(\Delta,h)=\emptyset$.

###### Proof.

Suppose that $\Delta x=\alpha x$ for $x\neq\theta$ in $h$. Then, by solving
the system of linear equations

$\displaystyle\begin{array}[]{rcl}x_{0}&=&\alpha x_{0},\\\
-x_{0}+x_{1}&=&\alpha x_{1}\\\ -x_{1}+x_{2}&=&\alpha x_{2}\\\ &\vdots&\\\
-x_{n-1}+x_{n}&=&\alpha x_{n}\\\ \\\ &\vdots&\end{array}$

we find that if $x_{n_{0}}$ is the first nonzero entry of the sequence
$x=(x_{n})$, then $\alpha=1$. From the equality $-x_{n_{0}}+x_{n_{0}+1}=\alpha
x_{n_{0}+1}$ we have $x_{n_{0}}$ is zero. This contradicts the fact that
$x_{n_{0}}\neq 0$, which completes the proof. ∎

######  Theorem 4.3.

$\sigma_{p}(\Delta^{\ast},h^{\ast})=\left\\{\alpha\in\mathbb{C}:\left|1-\alpha\right|<1\right\\}$.

###### Proof.

Suppose that $\Delta^{\ast}x=\alpha x$ for $x\neq\theta$ in
$h^{\ast}\cong\sigma_{\infty}$. Then, by solving the system of linear
equations

$\displaystyle\begin{array}[]{rcl}x_{0}-x_{1}&=&\alpha x_{0},\\\
x_{1}-x_{2}&=&\alpha x_{1},\\\ &\vdots&\\\ x_{n-1}-x_{n}&=&\alpha x_{n},\\\
&\vdots&\end{array}$

we observe that $x_{n}=(1-\alpha)^{n}x_{0}$. Therefore,
$\stackrel{{\scriptstyle}}{{\underset{n}{\sup}}}\frac{|x_{0}|}{n}\stackrel{{\scriptstyle
n}}{{\underset{k=1}{\sum}}}|1-\alpha|^{k}<\infty$ if and only if
$|1-\alpha|<1$. This step concludes the proof. ∎

If $T\in B(h)$ with the matrix $A$, then it is known that the adjoint operator
$T^{\ast}:h^{\ast}\rightarrow h^{\ast}$ is defined by the transpose $A^{t}$ of
the matrix $A$. It should be noted that the dual space $h^{\ast}$ of $h$ is
isometrically isomorphic to the Banach space $\sigma_{\infty}$ of absolutely
summable sequences normed by
$\|x\|=\stackrel{{\scriptstyle\infty}}{{\underset{k=0}{\sum}}}k|x_{k}-x_{k+1}|$.

######  Lemma 4.4.

[13, p. 59] $T$ has a dense range if and only if $T^{\ast}$ is one to one.

######  Theorem 4.5.

$\sigma_{r}(\Delta,h)=\sigma_{p}(\Delta^{\ast},h^{\ast})$.

###### Proof.

For $\left|1-\alpha\right|<1$, the operator $\Delta-\alpha I$ is triangle, so
has an inverse. But $\Delta^{\ast}-\alpha I$ is not one to one by Theorem 4.3.
Therefore by Lemma 4.4, $\overline{R(\Delta-\alpha I)}\neq h$ and this step
concludes the proof. ∎

######  Theorem 4.6.

$\sigma_{c}(\Delta,h)=\left\\{\alpha\in\mathbb{C}:\left|1-\alpha\right|=1\right\\}$.

###### Proof.

For $\left|1-\alpha\right|<1$, the operator $\Delta-\alpha I$ is triangle, so
has an inverse but is unbounded. Also $\Delta^{\ast}-\alpha I$ is one to one
by Theorem 4.3. By Lemma 4.4, $\overline{R(\Delta-\alpha I)}=h$. Thus, the
proof is completed. ∎

######  Theorem 4.7.

$A_{3}\sigma(\Delta,h)=B_{3}\sigma(\Delta,h)=C_{3}\sigma(\Delta,h)=\emptyset$.

###### Proof.

From Theorem 4.2 and Table 1.2.,
$A_{3}\sigma(\Delta,h)=B_{3}\sigma(\Delta,h)=C_{3}\sigma(\Delta,h)=\emptyset$
is observed. ∎

######  Theorem 4.8.

$C_{1}\sigma(\Delta,h)=\emptyset$ and $\alpha\in\sigma_{r}(\Delta,h)\cap
C_{2}\sigma(\Delta,h)$.

###### Proof.

We know $C_{1}\sigma(\Delta,h)\cup C_{2}\sigma(\Delta,h)=\sigma_{r}(\Delta,h)$
from Table 1.2. For $\alpha\in\sigma_{r}(\Delta,h)$, the operator
$(\Delta-\alpha I)^{-1}$ is unbounded by Theorem 4.1. So
$C_{1}\sigma(\Delta,h)=\emptyset$. This completes the proof. ∎

######  Theorem 4.9.

The following results hold:

  1. (a)

$\sigma_{ap}(\Delta,h)=\sigma(\Delta,h)$.

  2. (b)

$\sigma_{\delta}(\Delta,h)=\sigma(\Delta,h)$.

  3. (c)

$\sigma_{co}(\Delta,h)=\left\\{\alpha\in\mathbb{C}:\left|1-\alpha\right|<1\right\\}$.

###### Proof.

(a) Since $\sigma_{ap}(\Delta,h)=\sigma(\Delta,h)\backslash
C_{1}\sigma(\Delta,h)$ from Table 1.2. and $C_{1}\sigma(\Delta,h)=\emptyset$
by Theorem 4.8, we have $\sigma_{ap}(\Delta,h)=\sigma(\Delta,h)$.

(b) Since $\sigma_{\delta}(\Delta,h)=\sigma(\Delta,h)\backslash
A_{3}\sigma(\Delta,h)$ from Table 1.2 and $A_{3}\sigma(\Delta,h)=\emptyset$ by
Theorem 4.7, we have $\sigma_{\delta}(\Delta,h)=\sigma(\Delta,h)$.

(c) Since the equality $\sigma_{co}(\Delta,h)=C_{1}\sigma(\Delta,h)\cup
C_{2}\sigma(\Delta,h)\cup C_{3}\sigma(\Delta,h)$ holds from Table 1.2, we have
$\sigma_{co}(\Delta,h)=\left\\{\alpha\in\mathbb{C}:\left|1-\alpha\right|<1\right\\}$
by Theorems 4.8 and 4.9. ∎

The next corollary can be obtained from Proposition 2.1.

######  Corollary 4.10.

The following results hold:

  1. (a)

$\sigma_{ap}(\Delta^{\ast},\ell_{1})=\sigma(\Delta,h)$.

  2. (b)

$\sigma_{\delta}(\Delta^{\ast},\ell_{1})=\left\\{\alpha:\left|\alpha-(2-\delta)^{-1}\right|=(1-\delta)(2-\delta)\right\\}\cup
E$.

  3. (c)

$\sigma_{p}(\Delta^{\ast},\ell_{1})=\left\\{\alpha\in\mathbb{C}:\left|\alpha-(2-\delta)^{-1}\right|<(1-\delta)/(2-\delta)\right\\}\cup
S$.

##  5\. Conclusion

Hahn [15] defined the space $h$ and gave its some general properties. Goes and
Goes [12] studied the functional analytic properties of the space $h$. The
study on the Hahn sequence space was initiated by Rao [22] with certain
specific purpose in Banach space theory. Also Rao [22] emphasized on some
matrix transformations. Rao and Srinivasalu [23] introduced a new class of
sequence space called the semi replete space. Rao and Subramanian [24] defined
the semi Hahn space and proved that the intersection of all semi Hahn spaces
is Hahn space. Balasubramanian and Pandiarani [8] defined the new sequence
space $h(F)$ called the Hahn sequence space of fuzzy numbers and proved that
$\beta-$ and $\gamma-$duals of $h(F)$ is the Cesàro space of the set of all
fuzzy bounded sequences. The sequence space $h$ was introduced by Hahn [15]
and Goes and Goes [12], and Rao [22, 23, 24] investigated some properties of
the space $h$. Quite recently, Kirişci [17] has defined a new Hahn sequence
space by using Cesàro mean, in [18].

The difference matrix $\Delta$ was used for determining the spectrum or fine
spectrum acting as a linear operator on any of the classical sequence spaces
$c_{0}$ and $c$, $\ell_{1}$ and $bv$, $\ell_{p}$ for $(1\leq p<\infty)$,
respectively in [4], [10] and [2].

As a natural continuation of this paper, one can study the spectrum and fine
spectrum of the Cesàro operator, Weighted mean operator or another known
operators in the sequence space $h$.

## Acknowledgement

We would like to thank Professor Feyzi Başar, Fatih University, Büyükçekmece
Campus, 34500 - İstanbul, Turkey, for his careful reading and valuable
suggestions on the earlier version of this paper.

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# Selective Activation of Aromatic C-H Bonds Catalyzed by Single Gold Atoms at
Room Temperature

Benjamin Lowe  School of Physics and Astronomy, Monash University, Clayton,
Victoria 3800, Australia  Jack Hellerstedt  School of Physics and Astronomy,
Monash University, Clayton, Victoria 3800, Australia  Adam Matěj  Institute of
Physics, Academy of Sciences of the Czech Republic, Cukrovarnická 10, 1862 53,
Prague, Czech Republic  Pingo Mutombo  Institute of Physics, Academy of
Sciences of the Czech Republic, Cukrovarnická 10, 1862 53, Prague, Czech
Republic  Dhaneesh Kumar  School of Physics and Astronomy, Monash University,
Clayton, Victoria 3800, Australia  Martin Ondráček  Institute of Physics,
Academy of Sciences of the Czech Republic, Cukrovarnická 10, 1862 53, Prague,
Czech Republic  Pavel Jelinek  Institute of Physics, Academy of Sciences of
the Czech Republic, Cukrovarnická 10, 1862 53, Prague, Czech Republic
<EMAIL_ADDRESS>Agustin Schiffrin  School of Physics and Astronomy, Monash
University, Clayton, Victoria 3800, Australia<EMAIL_ADDRESS>
###### Abstract

Selective activation and controlled functionalization of C-H bonds in organic
molecules is one of the most desirable processes in synthetic chemistry.
Despite progress in heterogeneous catalysis using metal surfaces, this goal
remains challenging due to the stability of C-H bonds and their ubiquity in
precursor molecules, hampering regioselectivity. Here, we examine the
interaction between 9,10-dicyanoanthracene (DCA) molecules and Au adatoms on
an Ag(111) surface at room temperature (RT). Characterization via low-
temperature scanning tunneling microscopy, spectroscopy, and noncontact atomic
force microscopy, supported by theoretical calculations, revealed the
formation of organometallic DCA-Au-DCA dimers, where C atoms at the ends of
the anthracene moieties are bonded covalently to single Au atoms. The
formation of this organometallic compound is initiated by a regioselective
cleaving of C-H bonds at RT. Hybrid quantum mechanics/molecular mechanics
calculations show that this regioselective C-H bond cleaving is enabled by an
intermediate metal-organic complex which significantly reduces the
dissociation barrier of a specific C-H bond. Harnessing the catalytic activity
of single metal atoms, this regioselective on-surface C-H activation reaction
at RT offers promising routes for future synthesis of functional organic and
organometallic materials.

ARC Centre for Excellence in Low-Energy Electronics Technologies, Monash
University, Clayton, Victoria 3800, Australia Contributed equally to this work
ARC Centre for Excellence in Low-Energy Electronics Technologies, Monash
University, Clayton, Victoria 3800, Australia Contributed equally to this work
Regional Centre of Advanced Technologies and Materials, Czech Advanced
Technology and Research Institute (CATRIN), Palacký University Olomouc, 779 00
Olomouc, Czech Republic Department of Physical Chemistry, Faculty of Science,
Palacký University Olomouc, 771 46 Olomouc, Czech Republic Contributed equally
to this work ARC Centre for Excellence in Low-Energy Electronics Technologies,
Monash University, Clayton, Victoria 3800, Australia Regional Centre of
Advanced Technologies and Materials, Czech Advanced Technology and Research
Institute (CATRIN), Palacký University Olomouc, 779 00 Olomouc, Czech Republic
ARC Centre for Excellence in Low-Energy Electronics Technologies, Monash
University, Clayton, Victoria 3800, Australia

##  1 Introduction

The selective activation and controlled functionalization of C-H bonds in
molecular precursors is one of the most important areas of synthetic organic
chemistry for the design of C-C and C-heteroatom bonds in new compounds.1, 2,
3 Controlled C-H activation reactions have enticing applications including
natural gas transport,2, 4, 5 polymer fabrication,6, 7, 8 and late-stage
modification of pharmaceutical products.9, 10, 11, 12, 13, 14 The high bond-
dissociation enthalpy and ubiquity of C-H bonds, however, make their selective
cleavage challenging.1, 15, 16, 17 In solution, enzymatic and transition metal
(homogeneous) catalysis have provided avenues for the selective activation of
C-H bonds.7, 18, 19, 20, 21

Solid metal surfaces have also been shown to exhibit (heterogeneous) catalytic
activity on adsorbed molecules.22, 23, 24, 25 In particular, controlled
activation of aromatic C-H bonds on a surface has enabled the synthesis of
low-dimensional organic nanomaterials with promising electronic and magnetic
properties, such as nanographenes,26, 27, 28, 29 graphene nanoribbons,30, 31,
32 organic polymers,33, 34 and covalent organic frameworks35, 36. However,
broader application of on-surface C-H activation for the synthesis of
functional materials remains limited due to high-bond energies and poor
regioselectivity.37, 38

A common reaction used for on-surface design of aromatic C-C and C-heteroatom
bonds is Ullmann-type coupling, where C-X bonds (X: halogen atom) in precursor
molecules undergo surface-catalyzed dehalogenation.39 This approach has been
successfully exploited for the fabrication of a range of different
materials,22, 23, 24, 25, 39 although the requirement for pre-halogenation of
targeted carbons in precursor molecules limits its applicability.

One promising approach for on-surface aromatic C-H activation consists of
leveraging the catalytic activity of single metal adatoms. Single-atom
catalysis yields promise for reducing the amount of precious metals used in
heterogeneous catalysis.40, 41, 42, 43 In particular, despite its relative
chemical inertness, gold has emerged as a promising single-atom catalyst
(e.g., for the oxidation of carbon monoxide).42, 43, 44 While Au surfaces have
been used to catalyze the cleavage of C-H and C-X bonds,45, 38, 46, 47
sometimes resulting in the formation of organometallic C-Au bonds,48, 49, 50
atomic-scale studies of single Au atom catalysis for C-H activation (on
surfaces other than Au) have not yet been conducted.

Herein, we report the regioselective cleavage of an aromatic C-H bond of a
cyano-functionalized anthracene derivative, 9,10-dicyanoanthracene (DCA;
Figure 1a), activated by the interaction with single Au atoms on a Ag(111)
surface at room temperature (RT). This selective activation of a specific C-H
bond (located at the end of the anthracene group - labelled position A in
Figure 1a) is mediated by the on-surface formation of an intermediate DCA-Au
metal-organic complex, which subsequently enables selective dehydrogenation of
the DCA molecule (Figure 1d). This results in a reactive site at the DCA
anthracene extremity, which can then covalently bond to a Au adatom and form
organometallic DCA-Au-DCA dimers via a C-Au-C motif (Figure 1e). We used low-
temperature scanning tunneling microscopy (STM) and spectroscopy (STS) as well
as non-contact atomic force microscopy (ncAFM) to characterize the
organometallic dimers at the atomic scale. The experimental findings are
supported by density functional theory (DFT), ncAFM simulations and quantum
mechanics/molecular mechanics (QM/MM) calculations to shed light on the atomic
structure and reaction mechanism. The low activation energy, indicated by the
reaction occurring at RT, and regioselectivity of the observed C-H activation
open the door to unexplored on-surface synthesis protocols based on single
atom catalysis for the development of functional and robust organic and
organometallic materials.

##  2 Results

We co-deposited DCA molecules and Au atoms from the gas phase in ultra-high
vacuum (UHV) onto a clean Ag(111) surface held at RT (see Methods for further
details). Figure 1f shows an STM image of a sample region following this
preparation. We observe two types of well-ordered, two-dimensional (2D)
periodic arrays of self-assembled adsorbates: (i) a purely organic domain
(cyan box, region labelled DCA) consisting of identical elliptical protrusions
(solid blue outline) corresponding to non-covalently bonded DCA molecules; 51
(ii) a domain consisting of parallel rows of pairs (red box, region labelled
DCA+Au) of similar elliptical features (inset: solid blue outline) linked by
circular protrusions (inset: red dashed circular outline). Based on their
appearance and size, we identify the elliptical features in the DCA+Au domain
as DCA molecules,51 arranged in pairs in either cis (red box in Figure 1f) or
trans configuration (see SI Figure S1). Approximately equal occurrences of
cis- and trans- configuration were observed, though a quantitative statistical
analysis is beyond the scope of this study.

The qualitative difference in Figure 1f between the previously reported51 DCA
domain and the DCA+Au domain consisting of ordered DCA pairs is a strong
suggestion that the pairing of DCA molecules in the latter is mediated by the
interaction between DCA and Au adatoms. Notably, these results are also
qualitatively different from previously studied surface-supported systems
where 3-fold rotationally symmetric metal-ligand coordination between DCA
cyano groups and noble metal adatoms [e.g., Cu adatoms on Cu(111),52, 53, 54
Ag(111),55 Ir(111)-supported graphene,56 and NbSe257; Au adatoms on Au(111)58]
give rise to porous, 2D honeycomb-kagome metal-organic frameworks (MOFs) as
shown in Figure 1b. This is a strong indication that the interaction between
DCA and Au adatoms on Ag(111) is fundamentally different than in these other
systems.

We used ncAFM with a carbon-monoxide (CO)-functionalized tip (see Methods) to
gain further insight into the intramolecular structure of the Au-mediated DCA
pairs. NcAFM imaging of the DCA+Au domain, Figure 1g, and the DCA domain,51
Figure 1h, are similar in that the submolecular structure of individual DCA
molecules is clearly resolved in agreement with the overlaid chemical
structure in Figure 1h, with non-covalent attractive N–H interactions (Figure
1 white arrows) between adjacent molecules (DCA) or molecule pairs (DCA+Au).59

In contrast with ncAFM imaging of the DCA domain in Figure 1h where the
frequency shift ($\Delta f$) across each of the DCA benzene rings is
relatively uniform, individual DCA molecules in the DCA+Au domain, Figure 1g,
exhibit reduced $\Delta f$ at the benzene rings closest to the DCA pair center
(near the location of the circular protrusion in STM imaging; red dashed
circle). Such ncAFM appearance of $\pi$-conjugated rings is indicative of
direct coordination between atoms of the aromatic group and a metal adatom,60,
61, 62, 63 further supporting the involvement of Au adatoms within the DCA+Au
domain.

We conducted lateral STM manipulation to further probe the nature of the
intra- and inter-DCA pair interactions within the DCA+Au assembly. Figure 2a
shows an STM image of a DCA+Au domain with cis-configuration DCA pairs.
Individual DCA pairs, consisting of two elliptical features connected via a
circular feature in STM images, could be reproducibly removed from the edge of
a DCA+Au domain and manipulated about the surface (Figure 2b-d). Upon
manipulation, the STM appearance of the DCA pairs is unchanged, suggesting
that their chemical structure is preserved.

In some instances, we observed switching from the cis- to the trans-
configuration upon manipulation (dashed green and purple pairs in Figure 2).
We did not, however, observe evidence of such switching under normal tunneling
conditions (through e.g. telegraph noise64); cis- and trans- configurations
were both otherwise stable during characterization.

Within the range of STM manipulation parameters tested, we were unable to
observe further breaking of the DCA pairs into their individual constituents
(i.e. STM elliptical features corresponding to DCA molecules and circular
features). From this, we conclude that while lateral manipulation can overcome
the strength of bonding between adjacent DCA pairs, it cannot overcome the
interaction connecting the two DCA molecules within such pairs. This suggests
the intra-pair interaction is significantly stronger than the inter-pair
interaction, and that the Au-mediated DCA pairs are highly stable.

From these STM manipulation observations, combined with the reduced frequency
shift of the benzene rings near the DCA pair center (Figure 1f, 3c), we
propose that the DCA pairs consist of an organometallic DCA-Au-DCA motif,
wherein a position A (Figure 1a) carbon atom of each of the two DCA molecules
is covalently bonded to a Au atom (Figure 1e). The observation of cis- and
trans- configurations can be attributed to the chemical equivalence of the two
carbon atoms at the anthracene ends of the DCA molecule.

We performed DFT calculations to rationalize the proposed organometallic
structure. The energetically favorable atomic structure of an organometallic
trans DCA-Au-DCA dimer on Ag(111), optimized via DFT (see Methods), is shown
in Figure 3a). The DFT calculations revealed a negligible energy difference
between the trans and cis configurations (see SI Figure S2), supporting the
observation of the two configurations in approximately equal measure, and the
cis-to-trans conversion by STM manipulation.

The simulated ncAFM image (Figure 3b; obtained via the probe-particle method
65) corresponding to the DFT-relaxed DCA-Au-DCA dimer in Figure 3a shows
strong qualitative agreement with the experimental ncAFM image (Figure 3c),
including the reduced $\Delta f$ of the benzene rings closest to the C-Au-C
bonding motif at the DCA pair center. The simulated and experimental image
also have a similar distance $d$ between the centers of the benzene rings
closest to the Au atom (5.72 $\pm$ 0.06 Å for the simulated images and 6.0
$\pm$ 0.2 Å for the experimental images). We rule out a DCA-DCA pair with a
direct covalent C-C bond between facing anthracene groups, since ncAFM
simulations of this structure would result in a distance $d$ (4.76 $\pm$ 0.06
Å) significantly shorter than that observed experimentally, and would not
reproduce the experimentally observed reduced $\Delta f$ in the benzene rings
closest to the DCA-DCA linkage (Figure S3).

We performed $\mathrm{d}I/\mathrm{d}V$ STS to gain insight into the electronic
structure of an individual DCA-Au-DCA dimer. The spectra in Figure 4a were
acquired on the Au (orange) and outer (i.e. furthest from Au site) anthracene
end (blue) sites of a trans DCA-Au-DCA dimer, removed from the 2D assembly via
manipulation. Both orange and blue spectra show a step-like feature at
$\sim$-66 mV, and a peak at $\sim$376 and $\sim$560 mV, respectively.

We attribute the step-like feature to the Shockley surface state of Ag(111),66
and the two peaks to intrinsic electronic states of the DCA-Au-DCA unit.
Figure 4b,c show $\mathrm{d}I/\mathrm{d}V$ maps associated with each of these
electronic states. Both maps show significant intensity at the anthracene ends
furthest from the Au site, and at the cyano groups. These features are
qualitatively similar to the spatial distribution of the DCA lowest unoccupied
molecular orbital (LUMO).51, 67 The map in Figure 4b also shows some
$\mathrm{d}I/\mathrm{d}V$ intensity at the Au site, significantly more so than
Figure 4c (emphasized in the difference between Figure 4b and Figure 4c,
displayed in Figure 4d).

Figure 4e,f shows the DFT-calculated spatial distribution of the LUMO and
LUMO+1 of the structurally optimized gas-phase DCA-Au-DCA dimer (see Methods
and Figure S4 in the SI). These DFT-calculated LUMO and LUMO+1 are separated
in energy by $\sim$100 meV, on the same order of magnitude as the $\sim$184
meV energy difference between experimental $\mathrm{d}I/\mathrm{d}V$ peaks in
Figure 4a. Both the calculated LUMO and LUMO+1 have predominantly DCA orbital
character (see Figure S4 and Figure S5), with intensity at the anthracene ends
furthest from the Au site, and at the cyano groups. The LUMO (Figure 4e) shows
intensity at the Au atom whereas the LUMO+1 (Figure 4f) exhibits a nodal plane
at this site perpendicular to the DCA-DCA axis. The calculated LUMO and LUMO+1
are in qualitative agreement with the experimental $\mathrm{d}I/\mathrm{d}V$
maps in Figure 4b and c, respectively. This agreement provides further
evidence that the DCA+Au self-assembled 2D domain in Figure 1f) consists of
organometallic DCA-Au-DCA dimers, in which the position A carbon atoms are
covalently bonded to a Au atom.

The organometallic DCA-Au-DCA structure is qualitatively different to
previously observed 2D MOFs consisting of DCA molecules coordinated with metal
atoms via their cyano groups.52, 53, 54, 56, 55, 57, 58 Moreover, the
formation of the organometallic DCA-Au-DCA motif requires highly
regioselective cleavage of the position A aromatic C-H bonds to enable the
C-Au-C covalent bonding. Such C-H bonds have large dissociation enthalpies68,
69, 60, $\Delta H$, meaning their cleavage seldom occurs at RT,70 and they
often suffer from poor regioselectivity.60, 69, 68, 38, 25. For a gas-phase
DCA molecule for instance, we calculated $\Delta H$ values of 112.1 and 113.2
kcal/mol (4.86 and 4.91 eV) via DFT for the aromatic C-H bonds at positions A
and B, respectively (see Figure 1a and SI Figure S7).

In the following, we rationalize the highly selective C-H activation reaction
(and subsequent DCA-Au-DCA formation) at RT via DFT and QM/MM calculations.
First, we found using DFT that Au adatoms on Ag(111) retain their neutral,
atomic-like character (SI Figure S6). This contrasts with the case of Au
adatoms on Au(111), where significant electronic hybridization between adatom
and the surface takes place. That is, Au adatoms are less coordinated and
arguably more reactive on Ag(111) than on Au(111), providing an explanation
for the differences between organometallic (covalent C-Au bonding) and metal-
organic58 (N–Au) coordination complexes (Figure 1).

We then conducted QM/MM calculations of Gibbs free energy differences, $\Delta
G$, on a system composed of a single DCA molecule with a single Au adatom in
proximity on Ag(111) (system 1; see Figures 5b and S7). We considered a C-H
bond dissociation mechanism mediated by the direct interaction between Au and
H at $T=300$ K ($\sim$RT to match our experimental conditions; see Figures S7
and S8 for calculations at $T=0$ K). The dashed blue curve in Figure 5a) shows
the calculated $\Delta G$ as a function of position A C-H distance,
$d_{\mathrm{C-H}}$, when system 1 evolves from an initial state IS1A to a
transition state TS1A then to an intermediate state IM1A (where H becomes
bonded to Au; Figure 5b). These calculations yielded an activation energy
$\Delta
G\left[\mathrm{IS}_{\mathrm{1A}}\rightarrow\mathrm{TS}_{\mathrm{1A}}\right]=$
27.7 kcal/mol (1.20 eV). We claim that this energy is too high to explain the
dissociation of an anthracene C-H bond at RT. Note that similar calculations
performed for the position B C-H bond in system 1 suggest an even higher
dissociation barrier (Figure S7).

Therefore, we also considered a system composed of two DCA molecules and a
single Au adatom on Ag(111) (system 2; Figures 5c and S8), with a
configuration similar to that at the edges of self-assembled DCA-only domains
(Figures 1a and S14).51 We suggest that Au atoms diffusing on the surface may
encounter supramolecular DCA islands rather than isolated molecules (Figure
S14 and S15). This is supported by the fact that, experimentally, we observed
DCA-Au-DCA dimers both when DCA and Au were deposited sequentially (DCA first)
or simultaneously (see Methods). Furthermore, we did not observe any DCA+Au
domains which were surrounded by DCA-only domains. Based on this, we
considered the possibility that N–Au coordination could give rise to a metal-
organic DCA(N)–Au complex (see initial states IS2A or IS2B in Figure 5c,d) at
the edges of DCA-only domains, and conducted QM/MM simulations to investigate
whether this geometry could facilitate C-H activation at an adjacent molecule.
The solid blue curve in Figure 5a) shows $\Delta G$($T$ = 300 K) as a function
of position A C-H distance, $d_{\mathrm{C-H}}$, for system 2 as it evolves
from IS2A to an intermediate state IM2A via a transition state TS2A (Figure
5c). Here, we found a position A C-H activation energy $\Delta
G\left[\mathrm{IS}_{\mathrm{2A}}\rightarrow\mathrm{TS}_{\mathrm{2A}}\right]=$
19.1 kcal/mol (0.83 eV), significantly smaller than $\Delta
G\left[\mathrm{IS}_{\mathrm{1A}}\rightarrow\mathrm{TS}_{\mathrm{1A}}\right]$.

Similarly, the red curve in Figure 5a) corresponds to $\Delta G$($T$ = 300 K)
as a function of $d_{\mathrm{C-H}}-d_{\mathrm{Au-H}}$ for the position B C-H
bond, for the same system 2, evolving from IS2B to TS2B to IM2B (Figure 5d).
Here, the activation barrier $\Delta
G\left[\mathrm{IS}_{\mathrm{2B}}\rightarrow\mathrm{TS}_{\mathrm{2B}}\right]$
for the position B C-H bond is 33.6 kcal/mol (1.46 eV), significantly larger
than $\Delta
G\left[\mathrm{IS}_{\mathrm{2A}}\rightarrow\mathrm{TS}_{\mathrm{2A}}\right]$.
Based on these QM/MM calculations, the dissociation barrier of the position A
C-H bonds is significantly lowered via interaction with a DCA(N)–Au metal-
organic complex, enabling highly regioselective C-H bond cleavage at RT.

##  3 Discussion

To gain more insight into the regioselective lowering of the activation energy
of the C-H cleavage, we analyzed the electronic structure of the DCA(N)–Au
complex (present in IS2A) using DFT (Figure S10b). These calculations indicate
a weak, purely electrostatic interaction between Au and cyano group, with no
evidence of strong hybridization, and with an interaction energy of 4.2
kcal/mol (0.18 eV) and a N–Au distance of 2.47 Å. This interaction with the
cyano group results in a local polarization around and charge redistribution
within the Au atom, with an electron depletion on the outside of the Au atom
along the Au-cyano axis (Figure S10d). This depleted electron density can
reduce the Pauli repulsion between Au and a H atom of an adjacent DCA,
facilitating the formation of a Au-H chemical bond and of a DCA(H)-Au–(N)DCA
complex (see IM2A in Figure 5c). This partially explains the significant
lowering of the C-H bond activation energy.

Our DFT calculations also show that, in IM2A (Figure 5), covalent Au-H bonding
is concomitant with hybridization of Au 6s and 5d orbitals, giving rise to two
hybrid sd orbitals along the N–Au-H axis (see SI Section S9, Figures S9-S13),
and with electron transfer from Au to H (Figure S12). The resulting partial
positive charge on Au strengthens the electrostatic interaction with the
partially negatively charged cyano group (Figure S11), causing an increase in
the Au–(N)DCA interaction energy of $\sim$22 kcal/mol ($\sim$0.95 eV) and a
reduction of the Au–N bond length by $\sim$0.27 Å.

To understand how this change in Au–(N)DCA interaction energy influences the
C-H cleavage activation barrier, it is pertinent to consider the transition
state TS2A (Figure 5). Here, DFT calculations show a Au–(N)DCA interaction
energy of 26.0 kcal/mol (1.13 eV) with a Au–N bond length of 2.17 Å -
indicating a much stronger interaction than in IS2A. This synergistic effect
therefore contributes further to the significant lowering of the activation
energy of the C-H cleavage at position A.

Moreover, we claim that, for the position B C-H bond (Figure 5d), this effect
is offset by steric hindrance and N-N repulsion between adjacent DCA
molecules, contributing to a significantly larger dissociation barrier $\Delta
G\left[\mathrm{IS}_{\mathrm{2B}}\rightarrow\mathrm{TS}_{\mathrm{2B}}\right]$.

For TS2A, we calculated via DFT a C-H stretching mode in the direction of the
C-H dissociation with an eigenfrequency of 611.5 cm-1. Using this value, we
calculated for position A (with a dissociation barrier of 19 kcal/mol) at RT
an Arrhenius 71 C-H dissociation rate constant of 0.23 s-1. By contrast, the
RT Arrhenius dissociation rate constant for the position B C-H bond (with a
dissociation barrier of 33.6 kcal/mol) is $1.4\times 10^{-12}$ s-1. We claim
that this significant difference between positions A and B explains the
experimentally observed regioselectivity of the C-H dissociation. The 19
kcal/mol barrier represents a significant reduction relative to comparable C-H
activation reactions with barriers of 30-40 kcal/mol (1.3-1.8 eV),
experimentally occurring in a temperature window of 400-700 K.60, 68, 69, 72
Furthermore, other on-surface reactions with barriers of $\sim$20 kcal/mol
($\sim$0.87 eV) have been observed at RT,71, 73 supporting the interpretation
of our experimental results as RT C-H activation.

Although our DFT and QM/MM calculations rationalize the regioselective
cleaving of position A C-H bonds at RT, they do not explain the pathway from
IM2A (Figure 5) to the experimentally observed organometallic DCA-Au-DCA motif
(where C bonds covalently to Au). In the following, we provide a tentative,
qualitative scenario for such a pathway. We propose that, at RT, the hydrogen
atom in DCA(N)–Au-H in IM2A (Figure 5c) is likely to detach, bind to Ag(111)
and then desorb (see SI Section S10), leaving organometallic DCA(C)-Au on
Ag(111) which can diffuse on the surface and potentially react with another
dehydrogenated DCA$\cdot$ radical, forming the experimentally observed DCA-Au-
DCA unit. Non-covalent N–H bonds between neighboring DCA-Au-DCA dimers may
contribute to stabilizing the DCA-Au-DCA dimer rows as seen in the DCA-Au
domain in Figure 1f. The relatively weak and reversible DCA(N)–Au interaction,
combined with the reactivity of the DCA$\cdot$ radical, might explain why we
did not experimentally observe any of the metal-organic DCA(N)–Au species such
as in IS2A or IM2A in Figure 5. A quantitative theoretical description of this
process would require the consideration of many possible reaction pathways
(including, for example, surface diffusion, dynamic equilibrium of different
phases) in a system composed of many DCA molecules, Au adatoms and DCA-Au
complexes; this is beyond the scope of this work.

At $T=$ 0 K, QM/MM calculations suggest that the energies involved in the
dissociation of a position A C-H bond in a DCA-Au-DCA dimer (by interacting
with a DCA(N)—Au-H complex; see SI Figure S16) are approximately twice as
large as that for a position A C-H bond in a single DCA molecule. This
explains the observation of discrete DCA-Au-DCA dimers and absence of 1D
-[DCA-Au-DCA-Au]- organometallic chains.

We claim that the key difference between the previously observed DCA-Cu
honeycomb-kagome MOF on Ag(111),55 and the present study on the organometallic
DCA-Au motif is the strength of the metal adatom-cyano interaction. For Cu–N,
DFT calculations revealed an interaction energy of $\sim$18.5 kcal/mol
($\sim$0.8 eV), significantly larger than the aforementioned value of 4.2
kcal/mol for Au–N in IS2A (Figure 5c). This favors three-fold Cu-ligand
coordination and Cu-based MOF formation, preventing the reaction pathway for
C-H bond activation and organometallic bonding as is the case for Au.

##  4 Conclusion

We have experimentally observed the formation of organometallic DCA-Au-DCA
dimers on a Ag(111) surface at RT. In these dimers, C atoms located
specifically at the extremity of the DCA molecule anthracene moiety (position
A) bond covalently to a Au adatom. These observations, supported by DFT,
indicate the regioselective C-H bond activation triggered by Au adatoms at RT.
Our QM/MM calculations show that this reaction is mediated by the interaction
with an intermediate DCA(N)–Au-H metal-organic complex, which significantly
reduces the activation barrier of a specific C-H bond. Our work provides
atomic-scale insight into selective C-H bond activation, catalyzed by single
noble metal atoms and intermediate metal-organic complexes at RT. Our findings
have the potential to contribute to addressing the large demand for the
synthesis of fuel, fine chemicals and functional organic and organometallic
materials. Further investigations could explore such protocols on surfaces or
host materials beyond noble metals.

##  5 Acknowledgements

A.S. acknowledges funding support from the ARC Future Fellowship scheme
(FT150100426). B.L., D.K., and J.H. acknowledge funding support from the
Australian Research Council (ARC) Centre of Excellence in Future Low-Energy
Electronics Technologies (CE170100039). B.L. is supported through an
Australian Government Research Training Program (RTP) Scholarship. A.M.
acknowledges the support from internal student grant agency of Palacky
University in Olomouc (IGA_PrF_2022_019). P.M, A.M., M.O. and P.J. acknowledge
support of Praemium Academie of the Academy of Science of the Czech Republic
and Czech Science Foundation (project no. 20-13692X). We acknowledge
computational resources provided by the project “e-Infrastruktura CZ” (e-INFRA
CZ LM2018140) supported by the Ministry of Education, Youth and Sports of the
Czech Republic.

##  6 Supporting Information

Experimental and theoretical methods; structural characterization of DCA+Au
domains; DFT optimized structures; further ncAFM simulations; electronic
properties of DCA+Au dimers; Au adatom character on Ag(111); QM/MM
calculations at $T=0$ K; hybridization and charge distribution across reaction
pathway; tentative organometallic dimer formation pathway; secondary C-H
activation QM/MM calculations.

Figure 1: Interactions between noble metal adatoms and cyano-functionalized
aromatic molecules on noble metal (111) surfaces: metal-cyano coordination vs.
covalent metal-carbon bonding. a) Schematic of 9,10-dicyanoanthracene (DCA)
molecules and metal adatoms (Cu or Au) deposited on a (111) noble metal
surface (Ag, Cu or Au), with C-H positions A (anthracene end) and B (cyano
adjacent) labelled. b) The interaction between DCA and Cu adatoms on Ag(111)
or on Cu(111), or with Au adatoms on Au(111), leads to formation of a 2D
honeycomb-kagome metal-organic framework (MOF) via three-fold rotationally
symmetric metal-cyano coordination52, 53, 54, 55, 58. c)-e) The interaction
between DCA and Au adatoms on Ag(111) gives rise to organometallic DCA-Au-DCA
dimers via selective (position A) C-H activation and subsequent C-Au covalent
bonding (IS: initial state, TS: transition state, FS: final state). f)
Constant-current STM image of self-assembled organic DCA-only (DCA) and DCA+Au
domains featuring organometallic DCA-Au-DCA dimers on Ag(111)
($V_{\mathrm{b}}=-20$ mV, $I_{\mathrm{T}}=25$ pA). Inset: high-resolution STM
image (same tunneling parameters) of cis organometallic DCA-Au-DCA dimers
within DCA+Au self-assembly. Solid blue ellipse: individual DCA molecule.
Dashed red circle: Au adatom. g)-h) Constant-height ncAFM images (CO-tip; tip
30 pm closer to the sample with respect to STM setpoint $V_{\mathrm{b}}=15$
mV, $I_{\mathrm{t}}=100$ pA) of DCA domain (blue box in f) and of three cis
DCA-Au-DCA organometallic dimers in DCA+Au domain (red box in f). White arrows
indicate intermolecular N–H interactions.  Figure 2: Deconstruction of
organometallic DCA+Au self-assembly on Ag(111) via lateral STM manipulation.
a)-d) Constant-current STM images showing manipulation of single DCA-Au-DCA
units from cis DCA+Au domain [images: $V_{\mathrm{b}}=-20$ mV,
$I_{\mathrm{t}}=10$ pA; manipulation: $V_{\mathrm{b}}=-10$ mV,
$I_{\mathrm{t}}=14$ nA for removal of DCA dimers from 2D film,
$V_{\mathrm{b}}=-10$ mV, $I_{\mathrm{t}}=5$ nA for subsequent translation on
Ag(111)]. Blue arrow indicates STM tip path during manipulation. Dashed black
square in a) and d) indicates region in b) and c). Dashed outlines indicate
manipulated DCA-Au-DCA units. White dashed circles indicate defect adsorbate
displaced during manipulation. Ball-and-stick model of DCA-Au-DCA dimer
chemical structure overlaid in b). Figure 3: Experimental and theoretical
submolecular structure of a trans DCA-Au-DCA pair on Ag(111). a) Ball-and-
stick model of DFT-relaxed structure. b) Simulated ncAFM image of optimized
structure in a). c) Constant-height (CO-tip) ncAFM image (tip 30 pm closer to
the sample with respect to STM setpoint $V_{\mathrm{b}}=15$ mV,
$I_{\mathrm{t}}=100$ pA). Inset in white box is a higher resolution image
taken with the tip 40 pm closer to the surface with respect to the same STM
setpoint (see Methods). Red arrow indicates distance, $d$, between the centers
of the benzene rings closest to the Au atom of each DCA molecule involved in
an organometallic pair. Simulated image in b): $d_{\textrm{sim}}=5.72\pm 0.06$
Å. Experimental image in c): $d_{\textrm{exp}}=6.0\pm 0.2$ Å. Scale bars: 2.5
Å.  Figure 4: Electronic properties of DCA-Au-DCA organometallic unit. a)
$\mathrm{d}I/\mathrm{d}V$ scanning tunneling spectra obtained at sites
indicated in inset (setpoint: $V_{\mathrm{b}}=-200$ mV, $I_{\mathrm{t}}=20$
pA). Black dashed curves: fits of experimental data consisting of an error
function (step-like feature, light gray shaded area), a Gaussian peak (dark
gray). An exponential function accounting for tunneling transmission was also
included in the fit for the orange curve. Silver reference spectrum (gray
curve) scaled down by a factor of 2. Vertical dotted gray lines indicate bias
voltages at which $\mathrm{d}I/\mathrm{d}V$ maps in (b) and (c) were acquired.
Inset: constant-current STM image of DCA-Au-DCA unit ($V_{\mathrm{b}}=-20$ mV,
$I_{\mathrm{t}}=50$ pA). b), c) Constant-current $\mathrm{d}I/\mathrm{d}V$
maps of DCA-Au-DCA unit ($I_{\mathrm{t}}=500$ pA) at $V_{\mathrm{b}}=400$ and
$550$ mV. d) Subtraction between $\mathrm{d}I/\mathrm{d}V$ maps in b), c) for
region corresponding to DCA-Au-DCA unit. DCA-Au-DCA chemical structure
overlaid for reference in a) inset and b)-d). Scale bars: 5 Å. We isolated the
single DCA-Au-DCA unit from a DCA+Au self-assembled 2D organometallic domain
via lateral STM manipulation. e)-f) Wavefunction isosurfaces (0.005 Å-3/2;
green: positive; blue: negative) for LUMO and LUMO+1 of gas-phase trans DCA-
Au-DCA pair calculated by DFT.  Figure 5: Pathways for Au-induced C-H cleavage
of DCA molecule on Ag(111). a) Gibbs free energy differences, $\Delta G$,
calculated via QM/MM simulations at $T$ = 300 K, for different C-H bond
dissociation pathways in a system comprised of a single DCA molecule and a
single Au adatom (system 1 in b; dashed curve), and in a system comprised of 2
DCA molecules and a single Au adatom (system 2 in c,d; solid curves). For
position A (blue curves), we considered C-H bond length $d_{\mathrm{{C-H}}}$
(bottom axis) as reaction coordinate; for position B (red curve), we
considered the difference $(d_{\mathrm{{C-H}}}-d_{\mathrm{{Au-H}}})$ (top
axis) between C-H bond length and Au-H distance as reaction coordinate
(computationally necessary for this pathway). Activation energy [$\Delta G$
between transition (TS) and initial (IS) states] for C-H bond dissociation is
reduced for position A in system 2, $\Delta
G[\mathrm{IS}_{\mathrm{2A}}\rightarrow\mathrm{TS}_{\mathrm{2A}}]=19.1$
kcal/mol (0.83 eV). b-d) Ball-and-stick models of IS, TS and intermediate (IM)
states for C-H bond (positions A and B) dissociation pathways in systems 1 (b)
and 2 (c, d).

Deposition of Au atoms and 9,10-dicyanoanthracene (DCA) molecules on Ag(111)
at room temperature (300 K) results in selective C-H activation to form
organometallic DCA-Au-DCA dimers.

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