# Asymptotic Müntz-Szász Theorems

Jim Agler and John E. McCarthy Partially supported by National Science
Foundation Grant DMS 2054199

###### Abstract

We define a monomial space to be a subspace of ${\rm L}^{2}([0,1])$ that can
be approximated by spaces that are spanned by monomial functions. We describe
the structure of monomial spaces.

##  1 Introduction

What sorts of subspaces in ${\rm L}^{2}([0,1])$ can be limits of spans of
monomials? Specifically, let

$\mathbb{S}=\\{s\in\mathbb{C}\,|\,\operatorname{Re}s>-\frac{1}{2}\\},$

so that $s\in\mathbb{S}$ if and only if $x^{s}\in{\rm L}^{2}([0,1])$. For $S$
a finite subset of $\mathbb{S}$ we let $\mathcal{M}(S)$ denote the span in
${\rm L}^{2}([0,1])$ of the monomials whose exponents lie in $S$, i.e.,

$\mathcal{M}(S)=\\{\sum_{s\in S}a(s)x^{s}\,|\,a:S\to\mathbb{C}\\}.$

We refer to sets in ${\rm L}^{2}([0,1])$ that have the form $\mathcal{M}(S)$
for some finite subset $S$ of $\mathbb{S}$ as _finite monomial spaces_. We are
interested in what the limits of such spaces are.

######  Definition 1.1.

If $\mathcal{M}$ is a subspace of a Hilbert space $\mathcal{H}$ and
$\\{\mathcal{M}_{n}\\}$ is a sequence of closed subspaces, we say that
_$\\{\mathcal{M}_{n}\\}$ tends to $\mathcal{M}$_ and write

$\mathcal{M}_{n}\to\mathcal{M}\ \text{ as }\ n\to\infty$

if

$\mathcal{M}=\\{f\in\mathcal{H}\,|\,\lim_{n\to\infty}\operatorname{dist}(f,\mathcal{M}_{n})=0\\}.$

There are alternative ways to frame this definition; see Proposition 4.1.

######  Definition 1.2.

We say that a subspace $\mathcal{M}$ of ${\rm L}^{2}([0,1])$ is a _monomial
space_ if there exists a sequence $\\{\mathcal{M}_{n}\\}$ of finite monomial
spaces such that $\mathcal{M}_{n}\to\mathcal{M}$.

The goal of this paper is to study monomial spaces, which have a rich
structure and are intimately related to the Müntz-Szasz theorem and
generalizations thereof.

###  1.1 Monotone Monomial Spaces

In this subsection we recall several classical results that can be interpreted
as facts about monomial spaces. In each example we consider there is a limit
$\mathcal{M}_{n}\to\mathcal{M}$ of finite monomial spaces that is monotone,
i.e.,

$\mathcal{M}_{i}\subseteq\mathcal{M}_{j}\ \ \text{ whenever }\ \ i\leq j.$

For a detailed account of the results in this section, see [4] and [7].

######  Example 1.3.

The Weierstrass Approximation Theorem Let $S_{n}=\\{0,1,2,\ldots,n\\}$. The
Weierstrass Theorem, which implies that the polynomials are dense in ${\rm
L}^{2}([0,1])$ implies that

$\mathcal{M}(S_{n})\to{\rm L}^{2}([0,1]).$

In particular, ${\rm L}^{2}([0,1])$ is a monomial space.

######  Example 1.4.

Classical Müntz-Szász Theorem This was proved in [23, 27]. Fix a strictly
increasing sequence of nonnegative integers $s_{0},s_{1},s_{2},\ldots,$ and
let

$S_{n}=\\{s_{0},s_{1},\ldots,s_{n}\\}.$

Then there exists a space $\mathcal{M}$ such that

$S(\mathcal{M}_{n})\to\mathcal{M}.$

Furthermore,

$\mathcal{M}={\rm L}^{2}([0,1])\ \ \ \text{ if and only if }\ \ \
\sum_{k=1}^{\infty}\frac{1}{s_{k}}=\infty.$

######  Example 1.5.

Szász’s Theorem (real case), (a.k.a Full Müntz-Szász Theorem in ${\bf{\rm
L}^{2}([0,1])}$). This was proved in [28]. Fix a sequence of distinct real
numbers $s_{0},s_{1},s_{2},\ldots,$ in $\mathbb{S}$ and let

$S_{n}=\\{s_{0},s_{1},\ldots,s_{n}\\}.$

Then there exists a space $\mathcal{M}$ such that

$\mathcal{M}(S_{n})\to\mathcal{M}.$

Furthermore,

$\mathcal{M}={\rm L}^{2}([0,1])\ \ \ \text{ if and only if }\ \ \
\sum_{k=0}^{\infty}\frac{2s_{k}+1}{(2s_{k}+1)^{2}+1}=\infty.$

######  Example 1.6.

Szász’s Theorem (complex case) Any of the proofs known to the authors of the
previous example, including Szász’s original proof, can be adapted to show
that if $s_{0},s_{1},s_{2},\ldots,$ is a sequence of distinct points in
$\mathbb{S}$ and

$S_{n}=\\{s_{0},s_{1},\ldots,s_{n}\\},$

then there exists a space $\mathcal{M}$ such that

$\mathcal{M}(S_{n})\to\mathcal{M},$

and where,

$\mathcal{M}={\rm L}^{2}([0,1])\ \ \ \text{ if and only if }\ \ \
\sum_{k=1}^{\infty}\frac{2\operatorname{Re}s_{k}+1}{|s_{k}+1|^{2}}=\infty.$

What happens in the above examples when $\mathcal{M}\not={\rm L}^{2}([0,1])$?

######  Example 1.7.

The Clarkson-Erdős Theorem. With the setup of Example 1.4, assume that
$\mathcal{M}\not={\rm L}^{2}([0,1])$. Then Clarkson and Erdős proved in [12]
that the elements of $\mathcal{M}$ extend to be analytic on $\mathbb{D}$!
Furthermore, if $f\in\mathcal{M}$, then $f$ has a power series representation
of the form

$f(z)=\sum_{k=0}^{\infty}a_{k}z^{s_{k}},\qquad z\in\mathbb{D}.$

This result was generalized to arbitrary real powers in
$(-\frac{1}{2},\infty)$ by Erdélyi and Johnson [16], who showed that if
$\mathcal{M}\not={\rm L}^{2}([0,1])$, then every $f$ in $\mathcal{M}$ is
analytic in ${\mathbb{D}}\setminus(-1,0]$

In honor of this remarkable theorem we introduce the following definition.

######  Definition 1.8.

We say that _$\mathcal{M}$ is a Clarkson-Erdős space_ if there exist a
sequence $\\{s_{0},s_{1},\ldots\\}$ in $\mathbb{S}$ of distinct points such
that

$\mathcal{M}(\\{s_{0},s_{1},\ldots,s_{n}\\})\to\mathcal{M}$

where $\mathcal{M}\not={\rm L}^{2}([0,1])$.

We want to allow for multiplicities. If an entry $s$ is repeated in a
sequence, this corresponds to multiplicity in the following way. The first
occurrence of $s$ in $S_{n}$ gives the function $x^{s}$ in $\mathcal{M}_{n}$.
The second occurrence gives $\frac{\partial}{\partial s}x^{s}=x^{s}\log x$. If
$s$ occurs $k$ times, then $\mathcal{M}_{n}$ contains the functions
$x^{s},x^{s}\log x,\dots,x^{s}(\log x)^{k-1}$. This leads to the following
generalization of a Clarkson-Erdős space.

######  Definition 1.9.

We say that _$\mathcal{M}$ is an Erdélyi-Johnson space_ if there exist a
sequence $\\{s_{0},s_{1},\ldots\\}$ in $\mathbb{S}$, with multiplicities
allowed, such that

$\mathcal{M}(\\{s_{0},s_{1},\ldots,s_{n}\\})\to\mathcal{M}$

where $\mathcal{M}\not={\rm L}^{2}([0,1])$.

###  1.2 A Non-monotone Monomial Space

In the study of monomial spaces it is natural to consider the class of
_monomial operators_ , i.e., the class of bounded operators $T$ acting on
${\rm L}^{2}([0,1])$ that take monomials to monomials, i.e.,

$\forall_{s\in\mathbb{S}}\ \ \exists_{\tau\in\mathbb{S}}\ \
\exists_{c\in\mathbb{C}}\ \ Tx^{s}=cx^{\tau}.$ (1.10)

In [2] the authors studied the special case wherein it is assumed that there
exists a fixed number $m$ such that in (1.10) $\tau$ can be chosen to equal
$s+m$ for all $s$. We call these flat monomial operators. In the course of
proving that flat monomial operators leave ${\rm L}^{2}([a,1])$ invariant for
each $a\in[0,1]$, the authors discovered the following example.

######  Example 1.11.

Fix $\rho\in[0,1]$ and choose an increasing sequence of integers
$N_{1},N_{2},\ldots$ such that

$\lim_{n\to\infty}\frac{n}{n+N_{n}}=\sqrt{\rho}.$

If

$S_{n}=\\{n+1,n+2,\ldots,n+N_{n}\\},$

then

$\mathcal{M}(S_{n})\to{\rm L}^{2}([\rho,1]).$

In particular, ${\rm L}^{2}([\rho,1])$ is a monomial space for each
$\rho\in[0,1]$, where here and afterwards we identify ${\rm L}^{2}([\rho,1])$
with the subspace of ${\rm L}^{2}([0,1])$ consisting of functions that vanish
a.e. on $[0,\rho]$.

###  1.3 Characterization of monomial spaces

The Hardy operator $H:{\rm L}^{2}([0,1])\to{\rm L}^{2}([0,1])$ is defined by

$Hf(x)\ =\ \frac{1}{x}\int_{0}^{x}f(t)dt.$ (1.12)

This was introduced by Hardy in [20], where he proved it was bounded. As
$Hx^{s}=\frac{1}{s+1}x^{s}$ for all $s\in\mathbb{S}$, the Hardy operator
leaves invariant every monomial space. The converse is true.

######  Theorem 1.13.

A closed subspace of ${\rm L}^{2}([0,1])$ is a monomial space if and only if
it is invariant for $H$.

A proof of 1.13 that uses real analysis techniques is given in [1].

######  Corollary 1.14.

A bounded operator $T$ on ${\rm L}^{2}([0,1])$ is a monomial operator if and
only if for every $\mathcal{M}\in{\rm Lat}(H)$, the space $T\mathcal{M}$ is in
${\rm Lat}(H)$.

###  1.4 A Decomposition Theorem

######  Definition 1.15.

We say a space $\mathcal{M}$ in ${\rm L}^{2}([0,1])$ is a _singular space_ if
$\mathcal{M}$ is a monomial space that does not contain any Clarkson-Erdős
space.

######  Theorem 1.16.

Every monomial space $\mathcal{M}$ has a unique decomposition,

$\mathcal{M}=\overline{\mathcal{M}_{0}+\mathcal{M}_{1}},$

where $\mathcal{M}_{0}$ is an Erdélyi-Johnson space and $\mathcal{M}_{1}$ is a
singular space.

###  1.5 Atomic Spaces

Unitary monomial operators can be characterized using a theorem of Bourdon and
Narayan [8]. Their description is equivalent to the following reformulation,
from [3].

######  Theorem 1.17.

The operator

$T:x^{s}\mapsto c(s)x^{\tau(s)}$

is a unitary map from ${\rm L}^{2}([0,1])$ to ${\rm L}^{2}([0,1])$ if and only
if $\tau$ is a holomorphic automorphism of $\mathbb{S}$ and $c$ is given by

$c(s)\ =\ c_{0}\frac{1+\overline{\tau(0)}+\tau(s)}{1+s},$

where $c_{0}$ is a constant satisfying

$|c_{0}|\ =\ \frac{1}{\sqrt{1+2{\mathrm{R}e\,}\tau(0)}}.$

######  Definition 1.18.

We say a space $\mathcal{M}$ in ${\rm L}^{2}([0,1])$ is _atomic_ if there
exist $\rho\in(0,1)$ and a unitary monomial operator $T$ such that
$\mathcal{M}=T{\rm L}^{2}([\rho,1])$.

We can describe all atomic spaces in the following way.

######  Theorem 1.19.

The functions

$e_{n}(x)\ =\ \sum_{k=0}^{n}\binom{n}{k}\frac{(\ln x)^{k}}{k!},\qquad n\geq
0.$ (1.20)

form an orthonormal basis for ${\rm L}^{2}([0,1])$. Furthermore, the operator
$J$ defined on ${\rm L}^{2}([0,1])$ by requiring

$J(e_{n})=\left\\{\begin{array}[]{ll}e_{n}&\mbox{if }n\text{ is even}\\\
-e_{n}&\mbox{if }n\text{ is odd}\end{array}\right.$

is a unitary monomial operator, corresponding to the choice

$\tau(s)\ =\ \frac{-s}{1+2s},\qquad c(s)\ =\ \frac{1}{1+2s}$ (1.21)

in Theorem 1.17.

The polynomials

$p_{n}(t)\ =\ \sum_{k=0}^{n}\binom{n}{k}\frac{(t)^{k}}{k!},\qquad n\geq 0,$

are the Laguerre polynomials. They are the orthogonal polynomials on
$[0,\infty)$ for the measure $e^{-t}dt$. Under the change of variables
$x=e^{-t}$ they become the functions $e_{n}$ in (1.20). Their connection to
the Hardy operator was shwon in [22].

If $c$ is real, the multiplication operator $M_{x^{ic}}$ is also a unitary
monomial operator. These two operators can be used to build the general atomic
space.

  1. 1.

For any $w>0$, define

$\mathcal{A}_{1,w}\ :=\ L^{2}([e^{-2w},1]).$

  2. 2.

Define

$\mathcal{A}_{-1,w}\ :=\ J\,\mathcal{A}_{1,w}$

  3. 3.

For any $\tau\in{\mathbb{T}}\setminus\\{1\\}$, define

$\mathcal{A}_{\tau,w}\ :=\ M_{x^{ic}}\,\mathcal{A}_{-1,w^{\prime}},\qquad{\rm
where\ }2ic=\frac{\tau+1}{\tau-1},\ w^{\prime}=(1+4c^{2})w.$

(The reason for the strange scaling is to simplify the formulas in Section 5).
Atomic spaces are all of the form $\mathcal{A}_{\tau,w}$.

######  Theorem 1.22.

Every atomic space is equal to $\mathcal{A}_{\tau,w}$ for exactly one pair
$(\tau,w)\in{\mathbb{T}}\times(0,\infty)$.

###  1.6 The Structure of Singular Spaces

Say $\mathcal{M}$ is _finitely atomic_ if $\mathcal{M}$ is a finite sum of
atomic spaces. If

$\mu=\sum_{k=1}^{n}\ w_{k}\ \delta_{\tau_{k}}$

is a finitely atomic measure on $\mathbb{T}$, with distinct atoms $\tau_{k}$,
we define a finitely atomic space in ${\rm L}^{2}([0,1])$ by the formula

$\mathcal{M}(\mu)=\sum_{k=1}^{n}\mathcal{A}_{\tau_{k},w_{k}}$

######  Theorem 1.23.

The assignment $\mu\mapsto\mathcal{M}(\mu)$ extends by weak-* sequential
continuity to a map from the positive singular Borel measures on $\mathbb{T}$
into closed subspaces of ${\rm L}^{2}([0,1])$. When extended,

$\mu_{n}\to\mu\text{ weak-* }\implies\mathcal{M}(\mu_{n})\to\mathcal{M}(\mu).$

###  1.7 The main idea

The Hardy operator is unitarily equivalent to $1-S^{*}$, where $S$ is the
unilateral shift, via a unitary $U:{\rm L}^{2}([0,1])\to H^{2}$ that we call
the Sarason transform, described in Section 2. It follows that the invariant
subspaces of $H$ can be described by Beurling’s theorem [6] in terms of model
spaces, the invariant subspaces for the backward shift described for example
in [18]. However, all the theorems above have been stated in terms that are
intrinsic to ${\rm L}^{2}([0,1])$. We believe that finding proofs that are
also intrinsic to ${\rm L}^{2}([0,1])$ will illuminate this space with a new
light. So far, the authors have only succeeded in doing this for some of these
results.

######  Open Problem 1.24.

Find real analysis proofs to Theorems 1.16 and 1.23.

##  2 The Sarason Transform

###  2.1 The Definition

We let $k_{\alpha}$ denote the Szegö kernel function for ${\rm H}^{2}$, the
classical Hardy space of square integrable functions on $\mathbb{D}$, i.e.,

$k_{\alpha}(z)=\frac{1}{1-\bar{\alpha}z},\qquad z\in\mathbb{D}.$

As the monomials are linearly independant in ${\rm L}^{2}([0,1])$, there is a
well defined map $L$ defined on polynomials in ${\rm L}^{2}([0,1])$ into ${\rm
H}^{2}$ defined by the formula

$L(\sum_{n=0}^{N}a_{n}x^{n})=\sum_{n=0}^{N}a_{n}\frac{1}{n+1}k_{\frac{n}{n+1}}$

Noting that

$\langle\,L(x^{i})\,,L(x^{j})\,\rangle_{{\rm
H}^{2}}=\langle\,x^{i}\,,x^{j}\,\rangle_{{\rm L}^{2}([0,1])}$

for all nonnegative integers $i$ and $j$, it follows that

$\langle\,L(p)\,,L(q)\,\rangle_{{\rm H}^{2}}=\langle\,p\,,q\,\rangle_{{\rm
L}^{2}([0,1])}$

for all polynomials $p$ and $q$, i.e., $L$ is isometric. Hence, as the
polynomials are dense in ${\rm L}^{2}([0,1])$, $L$ has a unique extension to
an isometry $U$ defined on all of ${\rm L}^{2}([0,1])$. Finally, noting that

$\\{\frac{n}{n+1}\,|\,n\text{ is a nonegative integer}\\}$

is a set of uniqueness for ${\rm H}^{2}$, it follows that the range of $L$ is
dense in ${\rm H}^{2}$, which implies that $U$ is a unitary transformation
from ${\rm L}^{2}([0,1])$ onto ${\rm H}^{2}$.

######  Definition 2.1.

We let $U$ denote the unique unitary transformation from ${\rm L}^{2}([0,1])$
onto ${\rm H}^{2}$ that satisfies

$U(x^{n})=\frac{1}{n+1}k_{\frac{n}{n+1}}$

for all nonnegative integers $n$.

We call $U$ the Sarason transform, as it is similar to the transform from
$L^{2}([0,\infty))$ onto ${\rm H}^{2}$ used in [26].

###  2.2 Moments in ${\rm L}^{2}([0,1])$ and Interpolation in ${\rm H}^{2}$

As the monomials are dense in ${\rm L}^{2}([0,1])$, a function $f\in{\rm
L}^{2}([0,1])$ is uniquely determined by its moment sequence

$\int_{0}^{1}x^{n}\ f(x)dx,\qquad n=0,1,\ldots.$

Similarly, as the sequence $\\{1-\tfrac{1}{n+1}\,|\,n=0,1,\ldots\\}$ is a set
of uniqueness for ${\rm H}^{2}$, a function $h\in{\rm H}^{2}$ is the unique
solution $g$ in ${\rm H}^{2}$ to the interpolation problem

$g(\frac{n}{n+1})=h(\frac{n}{n+1}),\qquad n=0,1,\ldots$

The following proposition follows immediately from Definition 2.1.

######  Proposition 2.2.

Fix a sequence of complex numbers $w_{0},w_{1},w_{2},\ldots$. If $f$ in ${\rm
L}^{2}([0,1])$ solves the moment problem

$\int_{0}^{1}x^{n}\ f(x)\ dx=w_{n},\qquad n=0,1,\ldots,$

then $Uf\in{\rm H}^{2}$ and solves the interpolation problem

$Uf(\frac{n}{n+1})=(n+1)w_{n},\qquad n=0,1,\ldots\ .$

If $h\in{\rm H}^{2}$ solves the interpolation problem

$h(\frac{n}{n+1})=w_{n},\qquad n=0,1,\ldots,$

then $U^{*}h\in{\rm L}^{2}([0,1])$ and solves the moment problem

$\int_{0}^{1}x^{n}\ U^{*}h(x)\ dx=\frac{1}{n+1}w_{n},\qquad n=0,1,\ldots\ .$

The correspondence between moments and interpolation described in the
preceding proposition allows us to easily calculate the Sarason transform of
many common functions. We illustrate this with the following two lemmas.

######  Lemma 2.3.

If $\alpha\in\mathbb{D}$, then

$U^{*}(k_{\alpha})(x)=\frac{1}{1-\bar{\alpha}}x^{\tfrac{\bar{\alpha}}{1-\bar{\alpha}}},\qquad
x\in[0,1].$ (2.4)

If $Re\ \beta>-\frac{1}{2}$, then

$U(x^{\beta})\ =\ \frac{1}{\beta+1}k_{\frac{\bar{\beta}}{\bar{\beta}+1}}.$

###### Proof.

We note that the two assertions of the lemma are equivalent. Therefore it
suffices to prove (2.4). Since the left and right hand sides of (2.4) are in
${\rm L}^{2}([0,1])$, to show (2.4) it suffices to show that for each $n\geq
0$

$\langle\,x^{n}\,,U^{*}k_{\alpha}\,\rangle_{{\rm
L}^{2}([0,1])}=\langle\,x^{n}\,,\frac{1}{1-\bar{\alpha}}x^{\tfrac{\bar{\alpha}}{1-\bar{\alpha}}}\,\rangle_{{\rm
L}^{2}([0,1])}.$

But

$\displaystyle\langle\,x^{n}\,,\frac{1}{1-\bar{\alpha}}x^{\tfrac{\bar{\alpha}}{1-\bar{\alpha}}}\,\rangle_{{\rm
L}^{2}([0,1])}$
$\displaystyle=\frac{1}{1-\alpha}\langle\,x^{n}\,,x^{\tfrac{\bar{\alpha}}{1-\bar{\alpha}}}\,\rangle_{{\rm
L}^{2}([0,1])}$ $\displaystyle=\frac{1}{1-\alpha}\int_{0}^{1}\ x^{n}\
\overline{x^{\tfrac{\bar{\alpha}}{1-\bar{\alpha}}}}\ dx$
$\displaystyle=\frac{1}{1-\alpha}\ \frac{1}{n+\tfrac{\alpha}{1-\alpha}+1}$
$\displaystyle=\frac{1}{n+1}\ \frac{1}{1-\frac{n}{n+1}\alpha}$
$\displaystyle=\frac{1}{n+1}k_{\frac{n}{n+1}}(\alpha)$
$\displaystyle=(Ux^{n})(\alpha)$
$\displaystyle=\langle\,Ux^{n}\,,k_{\alpha}\,\rangle_{{\rm H}^{2}}$
$\displaystyle=\langle\,x^{n}\,,U^{*}k_{\alpha}\,\rangle_{{\rm L}^{2}([0,1])}$

∎

For $S$ a measurable set in $[0,1]$ let $\chi_{S}$ denote the characteristic
function of $S$.

######  Lemma 2.5.

If $s\in[0,1]$, then

$U\chi_{[0,s]}(z)=\sqrt{s}\ e^{\tfrac{1}{2}\ln s\ \frac{1+z}{1-z}}$ (2.6)

###### Proof.

We first observe that

$\displaystyle\frac{s^{n+1}}{n+1}$ $\displaystyle=\int_{0}^{s}x^{n}\ dx$
$\displaystyle=\langle\,\chi_{[0,s]}\,,x^{n}\,\rangle_{{\rm L}^{2}([0,1])}$
$\displaystyle=\langle\,U\chi_{[0,s]}\,,Ux^{n}\,\rangle_{{\rm H}^{2}}$
$\displaystyle=\langle\,U\chi_{[0,s]}\,,\frac{1}{n+1}k_{\frac{n}{n+1}}\,\rangle_{{\rm
H}^{2}}$ $\displaystyle=\frac{1}{n+1}U\chi_{[0,s]}(\frac{n}{n+1}),$

so that

$U\chi_{[0,s]}(\frac{n}{n+1})=s^{n+1}$

for all $n\geq 0$. On the other hand if for $w>0$ we let $E_{w}$ denote the
singular inner function defined by

$E_{w}(z)=e^{-w\frac{1+z}{1-z}}=e^{w}e^{-\frac{2w}{1-z}},$

we have that

$\displaystyle E_{w}(\frac{n}{n+1})$
$\displaystyle=e^{w}e^{-2w(n+1)}=e^{w}(e^{-2w})^{n+1}$

for all $n\geq 0$. Hence if we choose $w=-\tfrac{1}{2}\ln s$,

$U\chi_{[0,s]}(\frac{n}{n+1})=e^{-w}E_{w}(\frac{n}{n+1})$

for all $n\geq 0$. Since $\\{1-\tfrac{1}{n+1}\,|\,n\geq 0\\}$ is a set of
uniqueness for ${\rm H}^{2}$, it follows that

$U\chi_{[0,s]}(z)=e^{-w}E_{w}(z)$

for all $z\in\mathbb{D}$, which implies (2.6). ∎

###  2.3 A Formula for the Sarason Transform

######  Proposition 2.7.

If $f\in{\rm L}^{2}([0,1])$, then

$Uf(z)=\frac{1}{1-z}\int_{0}^{1}f(x)x^{\frac{z}{1-z}}dx$

for all $z\in\mathbb{D}$.

###### Proof.

$\displaystyle Uf(z)$ $\displaystyle=\langle\,Uf\,,k_{z}\,\rangle_{{\rm
H}^{2}}$ $\displaystyle=\langle\,f\,,U^{*}k_{z}\,\rangle_{{\rm L}^{2}([0,1])}$
$\displaystyle(Lemma\eqref{sa.lem.10})\qquad$
$\displaystyle=\langle\,f\,,\frac{1}{1-\bar{z}}x^{\tfrac{\bar{z}}{1-\bar{z}}}\,\rangle_{{\rm
L}^{2}([0,1])}$
$\displaystyle=\frac{1}{1-z}\int_{0}^{1}f(x)x^{\frac{z}{1-z}}dx$

∎

To obtain a nonrigorous, but highly interesting proof of the proposition, let
us define the Sarason Transform of a measure $\mu$ on $[0,1]$,
$\mathcal{S}\\{\mu\\}$, to be the holomorphic function

$\mathcal{S}\\{\mu\\}(z)=\frac{1}{1-z}\int
x^{\frac{z}{1-z}}d\mu(x),\qquad|z|<1.$

Note that

$\mathcal{S}(\chi_{[0,s]})=\sqrt{s}\ e^{\ln\sqrt{s}\
\frac{1+z}{1-z}}=U(\chi_{[0,s]}).$

We have

$\mathcal{S}\\{\delta_{s}\\}\ =\
\frac{1}{1-z}s^{\frac{1}{1-z}-1}=\frac{1}{1-z}s^{\frac{z}{1-z}}.$

So formally we get

$Uf=\mathcal{S}(\int_{0}^{1}f(x)\delta_{x}\
dx)=\int_{0}^{1}f(x)\mathcal{S}(\delta_{x})\
dx=\int_{0}^{1}f(x)\frac{1}{1-z}x^{\frac{z}{1-z}}dx,$

the formula in Proposition 2.17.

###  2.4 Transforms of $H,V$ and $X$

The Hardy operator $H$ was defined by (1.12). Let $X$ denote multiplication by
$x$ on ${\rm L}^{2}([0,1])$, and define the Volterra operator $V:{\rm
L}^{2}([0,1])\to{\rm L}^{2}([0,1])$ by $V=XH$, so

$Vf(x)\ =\ \int_{0}^{x}f(t)dt.$

All three of these are monomial operators, and have simple descriptions in
terms of monomials.

$\displaystyle H:x^{n}$ $\displaystyle\ \mapsto\ $
$\displaystyle\frac{1}{n+1}x^{n}$ $\displaystyle X:x^{n}$
$\displaystyle\mapsto$ $\displaystyle x^{n+1}$ $\displaystyle V:x^{n}$
$\displaystyle\mapsto$ $\displaystyle\frac{1}{n+1}x^{n+1}.$

If $T$ is a bounded operator on ${\rm L}^{2}([0,1])$, let us write
$\widehat{T}=UTU^{*}$ for the unitarily equivalent operator on ${\rm H}^{2}$.
Monomial operators then become operators that map kernel functions to
multiples of other kernel functions, which are adjoints of weighted
composition operators. For more about weighted composition operators, see e.g.
[17, 13, 21, 14, 8, 5, 11].

We shall let $M_{g}$ denote the operator of multiplication by $g$, and
$C_{\beta}$ denote composition with $\beta$. It was observed in [19] that it
is possible for the product $M_{g}C_{\beta}$ to be bounded even when $M_{g}$
is not. The following theorem is proved in [3].

######  Theorem 2.8.

The operator $T:{\rm L}^{2}([0,1])\to{\rm L}^{2}([0,1])$ is a monomial
operator if and only if $\widehat{T^{*}}:{\rm H}^{2}\to{\rm H}^{2}$ is a
bounded operator of the form $M_{g}C_{\beta}$ for some holomorphic
$\beta:{\mathbb{D}}\to{\mathbb{D}}$ and some $g\in{\rm H}^{2}$.

Let $\gamma(z)=\frac{1}{2-z}$. This maps ${\mathbb{D}}$ to ${\mathbb{D}}$, and
maps $\frac{n}{n+1}$ to $\frac{n+1}{n+2}$. Using Lemma 2.3 and the preceding
formulas, it is easy to verify the following. We shall let $S$ denote the
unilateral shift, the operator of multiplication by $z$ on ${\rm H}^{2}$.

######  Proposition 2.9.

We have

$\displaystyle\widehat{H}$ $\displaystyle\ =\ $ $\displaystyle 1-S^{*}$
$\displaystyle\widehat{X}$ $\displaystyle\ =\ $ $\displaystyle
S^{*}C_{\gamma}^{*}$ $\displaystyle\widehat{V}$ $\displaystyle=$
$\displaystyle(1-S^{*})C_{\gamma}^{*}.$

The fact that $1-H$ is unitarily equivalent to a backward shift operator was
first proved in [10], and a proof similar to ours is in [22].

###  2.5 The Sarason Transform and ${\rm Lat}(V)$

The invariant subspaces of the Volterra operator were described by Brodskii
[9] and Donoghue [15].

######  Theorem 2.10.

The space $\mathcal{M}\subseteq{\rm L}^{2}([0,1])$ is a closed invariant
subspace for $V$ if and only if $\mathcal{M}={\rm L}^{2}([\rho,1])$ for some
$\rho\in[0,1]$.

How do these spaces transform under the Sarason Transform?

For $s\in(0,1]$ let $\Phi_{s}$ be the singular inner function defined by

$\Phi_{s}(z)=e^{\tfrac{1}{2}\ln s\frac{1+z}{1-z}},\qquad z\in\mathbb{D}.$

For $s\in[0,1]$, define orthogonal projections $P^{\pm}_{s}$ on ${\rm
L}^{2}([0,1])$ by the formulas

$P^{-}_{s}f=\chi_{[0,s]}f\ \ \text{ and }\ \ P^{+}_{s}f=\chi_{[s,1]}f,\qquad
f\in{\rm L}^{2}([0,1]).$

######  Lemma 2.11.

$U\operatorname{ran}P_{s}^{-}=\Phi_{s}{\rm H}^{2}\text{ and
}U\operatorname{ran}P^{+}_{s}=\Phi_{s}{{\rm H}^{2}}^{\perp}$

###### Proof.

As $\operatorname{ran}P_{s}^{-}$ is invariant for $H^{*}$, it follows from
Proposition 2.9 that $U\operatorname{ran}P_{s}^{-}$ is invariant for the shift
$S$, and is therefore of the form $u{\rm H}^{2}$ for some inner function $u$
by Berurling’s theorem [6]. Moreover, $u$ is a constant multiple of the
projection of $1$ onto the invariant subspace. By Lemma 2.5, the projection of
$1$ is $\sqrt{s}\Phi_{s}$, so $u=\Phi_{s}$. ∎

###  2.6 The Sarason Transform and the Laplace Transform

Recall that the Laplace Transform is defined by the formula

$\mathcal{L}\\{f\\}(s)=\int_{0}^{\infty}e^{-st}f(t)dt.$

Further, if for $f\in{\rm L}^{2}([0,1])$ we define $f^{\sim}$ by the formula

$f^{\sim}(t)=e^{-\frac{t}{2}}f(e^{-t}),\qquad t\in(o,\infty),$

then the assignment $f\mapsto f^{\sim}$ is a Hilbert space isomorphism from
${\rm L}^{2}([0,1])$ onto ${\rm L}^{2}(0,\infty)$. By making the substitution
$x=e^{-t}$ we find that

$\displaystyle\int_{0}^{1}f(x)x^{\frac{z}{1-z}}dx$
$\displaystyle=\int_{\infty}^{0}f(e^{-t})\ e^{-t\frac{z}{1-z}}\ (-e^{-t})\ dt$
$\displaystyle=\int^{\infty}_{0}e^{-\frac{t}{2}}f(e^{-t})\
e^{-t(\frac{z}{1-z}+\frac{1}{2})}\ dt$
$\displaystyle=\int^{\infty}_{0}f^{\sim}(t)\
e^{-t(\frac{1}{2}\frac{1+z}{1-z})}\ dt$
$\displaystyle=\mathcal{L}\\{f^{\sim}\\}(\frac{1}{2}\frac{1+z}{1-z}).$

Hence,

$Uf(z)=\frac{1}{1-z}\mathcal{L}\\{f^{\sim}\\}(\frac{1}{2}\frac{1+z}{1-z}).$

So after changes of variable from ${\rm L}^{2}([0,1])$ of the disc to ${\rm
L}^{2}(0,\infty)$ and from ${\rm H}^{2}$ to ${\rm H}^{2}$ of the right half
plane, the Sarason Transform is simply the Laplace Transform.

##  3 The Inverse Sarason Transform

We know from Proposition 2.9 that $1-H^{*}$ is unitarily equivalent to the
unilateral shift. Let us find what the orthonormal basis $z^{n}$ in $H^{2}$
corresponds to. This was first done in [22]. In this section we shall use the
notation $\phi\sim g$ to mean that the function $f\in{\rm L}^{2}([0,1])$ is
mapped to $\phi\in H^{2}$ by the Sarason transform.

######  Lemma 3.1.

$(H^{*})^{j}\ 1=(-1)^{j}\frac{(\ln x)^{j}}{j!}$ (3.2)

###### Proof.

We proceed by induction. Clearly, (3.2) holds when $j=0$. Assume $j\geq 0$ and
(3.2) holds. Then

$\displaystyle(H^{*})^{j+1}\ 1$ $\displaystyle=H^{*}((H^{*})^{j}\ 1)$
$\displaystyle=\frac{(-1)^{j}}{j!}H^{*}(\ln x)^{j}$
$\displaystyle=\frac{(-1)^{j}}{j!}\int_{x}^{1}\frac{(\ln t)^{j}}{t}dt$
$\displaystyle=\frac{(-1)^{j}}{j!}\int_{\ln x}^{0}u^{j}du$
$\displaystyle=(-1)^{j+1}\frac{(\ln x)^{j+1}}{(j+1)!}$

∎

The following result is proved in [22]; we include a proof for expository
reasons.

######  Lemma 3.3.

$z^{n}\sim\sum_{j=0}^{n}\binom{n}{j}\frac{(\ln x)^{j}}{j!}$

###### Proof.

We have $S=1-\widehat{H}^{*}$. Therefore,

$z^{n}=S^{n}\ 1\sim(1-H^{*})^{n}\ 1.$

But using Lemma 3.1,

$\displaystyle(1-H^{*})^{n}\ 1$
$\displaystyle=\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}\ (H^{*})^{j}\ 1$
$\displaystyle=\sum_{j=0}^{n}(-1)^{j}\binom{n}{j}\big{(}(-1)^{j}\frac{(\ln
x)^{j}}{j!}\big{)}$ $\displaystyle=\sum_{j=0}^{n}\binom{n}{j}\frac{(\ln
x)^{j}}{j!}.$

∎

Define

$e_{n}(x)\ =\ \sum_{j=0}^{n}\binom{n}{j}\frac{(\ln x)^{j}}{j!}.$

We just proved that $e_{n}=(1-H^{*})^{n}1$. The fact that the functions
$e_{n}$ are an orthonormal basis is already well-known. Indeed, the Laguerre
polynomials

$p_{n}(t)\ =\ \sum_{j=0}^{n}\binom{n}{j}\frac{(-t)^{j}}{j!}$

are orthogonal polynomials of norm $1$ in $L^{2}([0,\infty))$ with weight
$e^{-t}$. By the change of variables $x=\ln\frac{1}{t}$, we get immediately
that $e_{n}(x)$ is an orthonormal basis for ${\rm L}^{2}([0,1])$.

######  Lemma 3.4.

If $f\in{\rm H}^{2}$ extends to be analytic on a neighborhood of 1, then

$f(z)\sim\sum_{j=0}^{\infty}f^{(j)}(1)\frac{(\ln x)^{j}}{(j!)^{2}}$

###### Proof.

If

$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$

is the power series representation of $f$, then by Lemma 3.3

$\displaystyle f(z)$
$\displaystyle\sim\sum_{n=0}^{\infty}a_{n}\big{(}\sum_{j=0}^{n}\binom{n}{j}\frac{(\ln
x)^{j}}{j!}\big{)}$ $\displaystyle=\sum_{j=0}^{\infty}\frac{(\ln
x)^{j}}{(j!)^{2}}\big{(}\sum_{n=j}^{\infty}\frac{n!}{(n-j)!}a_{n}\big{)}$
$\displaystyle=\sum_{j=0}^{\infty}f^{(j)}(1)\frac{(\ln x)^{j}}{(j!)^{2}}.$

∎

As a reality check let us let us verify the formula in Lemma 2.3 using Lemma
3.4. Note that

$k_{\alpha}^{(j)}(1)=\frac{j!\ \bar{\alpha}^{j}}{(1-\bar{\alpha})^{j+1}}$

Therefore, Lemma 3.4 implies that

$\displaystyle k_{\alpha}(z)$ $\displaystyle\sim\sum_{j=0}^{\infty}\frac{j!\
\bar{\alpha}^{j}}{(1-\bar{\alpha})^{j+1}}\frac{(\ln x)^{j}}{(j!)^{2}}$
$\displaystyle=\frac{1}{1-\bar{\alpha}}\ \sum_{j=0}^{\infty}\ \frac{1}{j!}\
\big{(}\frac{\bar{\alpha}}{1-\bar{\alpha}}\ln x\big{)}^{j}$
$\displaystyle=\frac{1}{1-\bar{\alpha}}\
e^{\frac{\bar{\alpha}}{1-\bar{\alpha}}\ln x}$
$\displaystyle=\frac{1}{1-\bar{\alpha}}\
x^{\frac{\bar{\alpha}}{1-\bar{\alpha}}}.$

The formula in Lemma 3.4 reminds one of Bessel functions. Indeed,

$\displaystyle e^{z-1}$ $\displaystyle\sim\sum_{j=0}^{\infty}\frac{(\ln
x)^{j}}{(j!)^{2}}$
$\displaystyle=\sum_{j=0}^{\infty}\frac{(-1)^{j}}{(j!)^{2}}(-\ln x)^{j}$
$\displaystyle=\sum_{j=0}^{\infty}\frac{(-1)^{j}}{(j!)^{2}}\big{(}\frac{\sqrt{\ln
x^{-4}}}{2}\big{)}^{2j}$ $\displaystyle=J_{0}\big{(}{\sqrt{\ln
x^{-4}}}\big{)}$

######  Proposition 3.5.

If $f\in{\rm H}^{2}$ extends to be analytic on a neighborhood of 1, then
$\Phi:(0,1]\to\mathbb{C}$ defined by

$\Phi(x)=U^{*}f(x),\qquad x\in(0,1],$

extends holomorphically to $\mathbb{C}\setminus(-\infty,0]$. Furthermore, if
$f_{-1}$ denotes the function defined by $f_{-1}(z)=f(-z)$, then

$U^{*}f_{-1}(x)=\Phi(\frac{1}{x}),\qquad x\in(0,1].$

###### Proof.

Observe that as $f$ is assumed to be analytic on a neighborhood of 1, the
Cauchy-Hadamard radius of convergence formula implies that $F$, defined by the
formula,

$F(w)=\sum_{k=0}^{\infty}f^{(k)}(1)\frac{w^{k}}{(k!)^{2}},$

is an entire function. Consequently, as Lemma 3.4 implies that

$\Phi(x)=U^{*}f(x)=F(\ln x)$

and $\ln x$ extends holomorphically to $\mathbb{C}\setminus(-\infty,0]$, so
also, $\Phi$ extends holomorphically to $\mathbb{C}\setminus(-\infty,0]$.

To see the second assertion of the lemma, note using Lemma 3.4 (with $f$
replaced with $f_{-1}$),

$\displaystyle U^{*}f_{-1}(x)$
$\displaystyle=\sum_{k=0}^{\infty}f_{-1}^{(k)}(1)\frac{(\ln x)^{k}}{(k!)^{2}}$
$\displaystyle=\sum_{k=0}^{\infty}f^{(k)}(1)\frac{(\ln x)^{k}}{(k!)^{2}}$

∎

##  4 Proofs and auxiliary results

###  4.1 Convergence of Subspaces

######  Proposition 4.1.

Let $\mathcal{M}_{n}$ be a sequence of closed subspaces of a Hilbert space
$\mathcal{H}$. The following are equivalent.

(i) $\mathcal{M}_{n}\to\mathcal{M}$

(ii) $P_{\mathcal{M}_{n}}P_{\mathcal{M}}\to P_{\mathcal{M}}$ in the strong
operator topology on $\mathcal{B}(\mathcal{H})$, and there is no larger space
$\mathcal{N}\supsetneq\mathcal{M}$ such that
$P_{\mathcal{M}_{n}}P_{\mathcal{N}}\to P_{\mathcal{N}}$.

(iii) $P_{\mathcal{M}_{n}}P_{\mathcal{M}}\to P_{\mathcal{M}}$ in the weak
operator topology on $\mathcal{B}(\mathcal{H})$, and larger space
$\mathcal{N}\supsetneq\mathcal{M}$ such that
$P_{\mathcal{M}_{n}}P_{\mathcal{N}}\to P_{\mathcal{N}}$.

###### Proof.

(i) $\Rightarrow$ (ii). Let $f\in\mathcal{H}$. Let $P_{\mathcal{M}}f=g$. By
(i), there exist $g_{n}\in\mathcal{M}_{n}$ so that $\|g_{n}-g\|\to 0$.
Therefore

$\|P_{\mathcal{M}_{n}}g-P_{\mathcal{M}}g\|\ \leq\ \|g_{n}-g\|\to 0,$

so $P_{\mathcal{M}_{n}}P_{\mathcal{M}}\to P_{\mathcal{M}}$ SOT.

If a space $\mathcal{N}\supseteq\mathcal{M}$ existed for which
$P_{\mathcal{M}_{n}}P_{\mathcal{N}}\to P_{\mathcal{N}}$, let
$h\in\mathcal{N}\ominus\mathcal{M}$. Let $h_{n}=P_{\mathcal{M}_{n}}h$. Then
$h_{n}\in\mathcal{M}_{n}$ and $h_{n}\to h$. By (i), this means
$h\in\mathcal{M}$, so $h=0$ and $\mathcal{N}=\mathcal{M}$.

(ii) $\Rightarrow$ (i). If $f\in\mathcal{M}$, then $P_{\mathcal{M}_{n}}f\to
f$, so $f\in\lim\mathcal{M}_{n}$. If there were some $h=\lim g_{n}$ for a
sequence $g_{n}\in\mathcal{M}_{n}$, then $\mathcal{N}=\mathcal{M}+\mathbb{C}h$
would satisfy $P_{\mathcal{M}_{n}}P_{\mathcal{N}}\to P_{\mathcal{N}}$ SOT. So
by (ii), this means $h\in\mathcal{M}$, so $\mathcal{M}=\lim\mathcal{M}_{n}$.

(ii) $\Leftrightarrow$ (iii): This is because a sequence of projections in a
Hilbert space converge WOT if and only if they converge SOT. Indeed, suppose
$Q_{n}\to Q$ WOT, and $Q$ and each $Q_{n}$ is a projection. Then

$\|(Q-Q_{n})f\|^{2}\ =\ \langle Qf,f\rangle-2{\mathrm{R}e\,}\langle
Q_{n}f,Qf\rangle+\langle Q_{n}f,f\rangle\ \to 0.$

∎

###  4.2  Proof of Theorem 1.19

###### Proof.

We have already shown that $e_{n}$ are an orthonormal basis. Clearly $J$ is
unitary, so must be given by Theorem 1.17 for some $c(s)$ and $\tau(s)$. As
$J1=1$, we have $\tau(0)=0$ and $c(0)=1$. As $J^{2}=1$, we have
$\tau(\tau(s))=s$ and $c(\tau(s))c(s)=1$.

So $\tau$ is an automorphism of $\mathbb{S}$ that fixes $0$ and is period $2$.
Once we know $\tau^{\prime}(0)$, this will uniquely determine $\tau$. To
calculate $\tau^{\prime}(0)$, note that

$e_{1}(x)\ =\ 1+\frac{\partial}{\partial s}x^{s}|_{s=0}.$

Therefore

$\displaystyle Je_{1}(x)$ $\displaystyle\ =\ $
$\displaystyle-1+\frac{\partial}{\partial
s}\left[c(s)x^{\tau(s)}\right]_{s=0}$ $\displaystyle=$ $\displaystyle
1+c^{\prime}(0)+c(0)\tau^{\prime}(0)\ln x.$

This yields $\tau^{\prime}(0)=-1$, so (1.21) hold. ∎

###  4.3 Proof of Corollary 1.14

######  Lemma 4.2.

Suppose $T$ is a bounded monomial operator given by

$Tx^{s}\ =\ c(s)x^{\tau(s)}.$ (4.3)

Then $\tau$ is a holomorphic function from $\mathbb{S}$ to $\mathbb{S}$, and
$c$ is a holomorphic function on $\mathbb{S}$.

###### Proof.

The map $s\mapsto x^{s}$ is a holomorphic map from $\mathbb{S}$ to ${\rm
L}^{2}([0,1])$. Therefore, for each $t\in\mathbb{S}$, the map

$s\ \mapsto\ \langle x^{s},T^{*}x^{t}\rangle\ =\
\frac{c(s)}{1+\tau(s)+\bar{t}}$

is holomorphic. Letting $t=0$ and $1$ and taking the quotient, we get

$\frac{2+\tau(s)}{1+\tau(s)}\ =\ 1+\frac{1}{1+\tau(s)}$

is a meromorphic function of $s$. Hence $\tau$ is meromorphic in $\mathbb{S}$.
Moreover, $\tau$ cannot have a pole, since otherwise in a neighborhood of this
pole it would take on all values in a neighborhood of $\infty$, including ones
not in $\mathbb{S}$. Therefore $\tau$ is holomorphic, and consequently so is
$c(s)$ since we have

$c(s)\ =\ (1+\tau(s))\langle x^{s},T^{*}1\rangle.$

∎

###### Proof.

(Of Corollary 1.14). If $T$ maps ${\rm Lat(H)}$ to ${\rm Lat}(H)$, it must be
a monomial operator, since each monomial functions spans a one-dimensional
$H$-invariant subspace.

Conversely, suppose $T$ is a monomial operator given by (4.3), and
$\mathcal{M}\in{\rm Lat}(H)$. By Lemma 4.2, the function $\tau$ is a
holomorphic map from $\mathbb{S}$ to $\mathbb{S}$. Define the function
$\phi\in H^{\infty}({\mathbb{D}}(1,1))$ by

$\phi(z)\ =\ \frac{1}{1+\tau(\frac{1-z}{z})}.$

Then for every $s\in\mathbb{S}$ we have

$\phi(\frac{1}{1+s})\ =\ \frac{1}{1+\tau(s)}.$

Therefore $HT=T\phi(H)$, since they agree on all monomials, so
$HT\mathcal{M}=T\phi(H)\mathcal{M}\subseteq T\mathcal{M}$, as required. ∎

######  Remark 4.4.

We used the fact that if $\phi\in H^{\infty}({\mathbb{D}}(1,1))$ then
$\phi(H)$ is a bounded operator. We define $\phi(H)$ to be the monomial
operator

$\phi(H):x^{s}\ \mapsto\ \phi(\frac{1}{1+s})x^{s}.$

This will be bounded by $M$ if and only if $M^{2}-\phi(H)^{*}\phi(H)\geq 0$,
which is equivalent to

$\frac{M^{2}-\phi(\frac{1}{1+s})\overline{\phi(\frac{1}{1+t})}}{1+s+\bar{t}}\
\geq\ 0.$ (4.5)

The fact that (4.5) is equivalent to the assertion that $\phi$ has norm at
most $M$ in $H^{\infty}({\mathbb{D}}(1,1))$ is, after a change of variables,
the content of Pick’s theorem [25].

###  4.4 Proof of Theorem 1.22

###### Proof.

Let $T$ be a unitary monomial operator, given by

$Tx^{s}\ =\ c(s)x^{\tau(s)}.$ (4.6)

By Theorem 1.17, we know that $\tau$ is a holomorphic automorphism of
$\mathbb{S}$. It is well-known that holomorphic automorphisms of the upper
half plane are given by linear fractional transformations with coefficients
from $SL(2,\mathbb{R})$. So any holomorphic automorphism of $\mathbb{S}$ is of
the form

$\tau(s)\ =\ \frac{A(s+\frac{1}{2})-iB}{iC(s+\frac{1}{2})+D}-\frac{1}{2},$
(4.7)

where $\begin{pmatrix}A&B\\\ C&D\end{pmatrix}$ is in $SL(2,\mathbb{R})$. Let
$\sigma(s)=\frac{-s}{1+2s}$.

Case (i): $\tau(\infty)=\infty$. Then $\tau$ is of the form $\tau(s)=\alpha
s+\beta+i\gamma$, where $\alpha>0$, $\beta,\gamma\in\mathbb{R}$, and
$\beta=\frac{\alpha-1}{2}$. As $T$ is given by (4.6) and

$c(s)\ =\ c_{0}\frac{1+\bar{\beta}+\alpha s+\beta}{1+s}\ =\ c_{0}\alpha,$

we have

$T:x^{s}\ \mapsto\ c_{0}\alpha x^{\beta}x^{\alpha s}.$

Therefore

$T:f(x)\ \mapsto\ c_{0}\alpha x^{\beta}f(x^{\alpha}).$

Therefore

$T\ L^{2}([\rho,1])\ =\ L^{2}[\rho^{\frac{1}{\alpha}},1]\ =\
{\mathcal{A}}_{1,\frac{1}{2\alpha}\log\frac{1}{\rho}}.$

Case (ii): $\tau(\infty)=-\frac{1}{2}$. Then $\sigma\circ\tau(\infty)=\infty$.
So by Case (i), we have

$JT\ L^{2}([\rho,1])\ =\ A_{1,w}$

for some $w$. Therefore

$T\ L^{2}([\rho,1])\ =\ A_{-1,w}.$

Case (iii) $\tau(\infty)=-\frac{1}{2}+i\delta$. Then

$M_{x^{-i\delta}}T:x^{s}\mapsto\tilde{c}(s)x^{\tilde{\tau}(s)},$

where $\tilde{\tau}(\infty)=-\frac{1}{2}$. By Case (ii), we have

$T\ L^{2}([\rho,1])\ =\ M_{x^{i\delta}}{\mathcal{A}}_{-1,w^{\prime}}\ =\
{\mathcal{A}}_{\tau,w},$

for $\tau=\frac{2i\delta+1}{2i\delta-1}$ and
$w=\frac{w^{\prime}}{1+4\delta^{2}}$.

Uniqueness of representation: We need to show that if
${\mathcal{A}}_{\tau,w}={\mathcal{A}}_{\tau^{\prime},w^{\prime}}$, then
$\tau^{\prime}=\tau$ and $w^{\prime}=w$. Observe that if $U$ is a unitary, and
$P_{\mathcal{M}}f=g$, then $P_{U\mathcal{M}}Uf=Ug$.

Let us calculate $P_{{\mathcal{A}}_{\tau,w}}x^{s}$. If $\tau=1$, then

$P_{{\mathcal{A}}_{1,w}}x^{s}\ =\ \chi_{[e^{-2w},1]}x^{s},$

and

$\|P_{{\mathcal{A}}_{1,w}}x^{s}\|^{2}\ =\
\frac{1}{1+2{\mathrm{R}e\,}s}\left[1-e^{-2w(1+2{\mathrm{R}e\,}s)}\right].$
(4.8)

Otherwise, $\tau=\frac{2i\delta+1}{2i\delta-1}$ for some
$\delta\in\mathbb{R}$. Then

$P_{{\mathcal{A}}_{\tau,w}}x^{s}\ =\
M_{x^{i\delta}}JP_{{\mathcal{A}}_{1,w^{\prime}}}Jx^{s-i\delta}.$

From (1.21),

$Jx^{s-i\delta}\ =\
\frac{1}{1+2s-2i\delta}x^{\frac{-s+i\delta}{1+2s-2i\delta}}.$

Therefore

$\displaystyle\|P_{{\mathcal{A}}_{\tau,w}}x^{s}\|^{2}$ $\displaystyle\ =\ $
$\displaystyle\|P_{{\mathcal{A}}_{1,w^{\prime}}}Jx^{s-i\delta}\|^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{|1+2s-2i\delta|^{2}}\int_{e^{-2w^{\prime}}}^{1}x^{2{\mathrm{R}e\,}\frac{-s+i\delta}{1+2s-2i\delta}}\
dx$

When $s=u+iv$ , this gives

$\|P_{{\mathcal{A}}_{\tau,w}}x^{s}\|^{2}\ =\
\frac{1}{1+2u}\left[1-e^{-2w^{\prime}\frac{1+2u}{(1+2u)^{2}+4(\delta-v)^{2}}}\right].$
(4.9)

Comparing (4.8) and (4.9), we see that $\tau$ and $w$ are completely
determined by $\|P_{{\mathcal{A}}_{\tau,w}}x^{s}\|^{2}$. ∎

##  5 Proofs using Hardy space theory

Although Theorems 1.16 and 1.23 are stated without using the language of Hardy
spaces, the authors do not know how to prove them directly.

###  5.1 Proof of 1.16

###### Proof.

Let $\mathcal{M}$ be in ${\rm Lat}(H)$. Define a sequence $S\subset\mathbb{S}$
by $S=\\{s:\langle f,x^{s}\rangle=0\ \forall\ f\in\mathcal{M}\\}$. The number
$s$ will occur in $S$ with multiplicity $m$ where $m$ is the largest number so
that $\mathcal{M}\perp\\{x^{s},(\ln x)x^{s},\dots,(\ln x)^{m-1}x^{s}\\}$. Let
$\mathcal{M}_{0}=\mathcal{M}(S)$.

To see that $\mathcal{M}=\overline{\mathcal{M}_{0}+\mathcal{M}_{1}}$ for some
singular space $\mathcal{M}_{1}$, we use the Sarason transform to move to
$H^{2}$. Then $\mathcal{M}$ becomes $(BSH^{2})^{\perp}$, where $B$ is a
Blaschke product and $S$ is a singular inner function. As $BSH^{2}=BH^{2}\cap
SH^{2}$, we have

$(BSH^{2})^{\perp}\ =\ \overline{(BH^{2})^{\perp}+(SH^{2})^{\perp}},$

and $\mathcal{M}_{1}$ is the inverse Sarason transform of $(SH^{2})^{\perp}$.
∎

###  5.2 Proof of 1.23

Let $S_{\tau,w}$ denote the singular inner function

$S_{\tau,w}(z)\ =\ \exp\left(-w\frac{\tau+z}{\tau-z}\right).$

Let $U:L^{2}\to H^{2}$ be the Sarason transform. By Lemma 2.11 we have that

$U{\mathcal{A}}_{1,w}U^{*}=\left(S_{1,w}H^{2}\right)^{\perp}.$

We wish to extend this to other values of $\tau$.

######  Lemma 5.1.

$UM_{x^{ic}}U^{*}(S_{-1,w^{\prime}})\ =\ FS_{\tau,w},$ (5.2)

where $\tau=\frac{2ic+1}{2ic-1}$, $w=\frac{1}{1+4c^{2}}w^{\prime}$ and
$F(z)=\exp(-2icw)\frac{1}{1+ic-icz}$.

###### Proof.

Observe first that $\widehat{M_{x^{-ic}}}$ is a unitary operator that takes
$k_{\alpha}$ to $\overline{\phi(\alpha)}k_{\psi(\alpha)}$, where

$\displaystyle\phi(z)$ $\displaystyle\ =\ $ $\displaystyle\frac{1}{1+ic-icz}$
$\displaystyle\psi(z)$ $\displaystyle\ =\ $
$\displaystyle\frac{(1-ic)z+ic}{1+ic-icz}.$

Therefore

$\widehat{M_{x^{-ic}}}\ =\ C_{\psi}^{*}M_{\phi}^{*},$

and so

$\widehat{M_{x^{ic}}}\ =\ M_{\phi}C_{\psi}.$ (5.3)

We have

$C_{\psi}S_{-1,w^{\prime}}(z)\ =\ \exp\left(w^{\prime}\
\frac{-1+\psi(z)}{-1-\psi(z)}\right).$

A calculation shows that

$\frac{-1+\psi(z)}{-1-\psi(z)}\ =\ \frac{1}{1+4c^{2}}\
\frac{\tau+z}{\tau-z}-\frac{2ic}{1+4c^{2}}.$

Therefore $C_{\psi}S_{-1,w^{\prime}}(z)$ is a unimodular constant times
$S_{\tau,w}$, and (5.2) holds. ∎

######  Lemma 5.4.

$U{\mathcal{A}}_{\tau,w}U^{*}=\left(S_{\tau,w}H^{2}\right)^{\perp}.$ (5.5)

###### Proof.

We have already proved the case $\tau=1$, so assume $\tau\neq 1$. Consider
next $\tau=-1$. Then

$\displaystyle U{\mathcal{A}}_{-1,w}U^{*}$ $\displaystyle\ =\ $ $\displaystyle
UJ{\mathcal{A}}_{1,w}U^{*}$ $\displaystyle=$ $\displaystyle
UJU^{*}U{\mathcal{A}}_{1,w}U^{*}$ $\displaystyle=$ $\displaystyle
UJU^{*}\left(S_{1,w}H^{2}\right)^{\perp}.$

As $UJU^{*}f(z)=f(-z)$, we have

$UJU^{*}\left(S_{1,w}H^{2}\right)\ =\ \left(S_{-1,w}H^{2}\right),$

so

$UJU^{*}\left(S_{1,w}H^{2}\right)^{\perp}\ =\
\left(S_{-1,w}H^{2}\right)^{\perp}.$

For $\tau\neq\pm 1$, we have, with $\phi$ and $\psi$ as in (5.3) and $F$ as in
Lemma 5.1,

$\displaystyle U{\mathcal{A}}_{\tau,w}^{\perp}U^{*}$ $\displaystyle\ =\ $
$\displaystyle UM_{x^{ic}}J{\mathcal{A}}_{1,w^{\prime}}^{\perp}U^{*}$
$\displaystyle=$ $\displaystyle
UM_{x^{ic}}U^{*}\left(S_{-1,w^{\prime}}H^{2}\right)$ $\displaystyle\ =\ $
$\displaystyle\\{\phi(z)F(z)S_{\tau,w}(z)h(\psi(z)):h\in H^{2}\\}.$

As $F$ and $\phi$ are outer and $\psi$ is an automorphism of ${\mathbb{D}}$,
this proves that

$U{\mathcal{A}}_{\tau,w}^{\perp}U^{*}\ =\ S_{\tau,w}H^{2},$

and hence (5.5). ∎

###### Proof.

(Of Theorem 1.23.) From Lemma 5.4, we have, for distinct points $\tau_{k}$,

$\displaystyle U\ \mathcal{M}(\sum_{k=1}^{n}w_{k}\delta_{\tau_{k}})\ U^{*}$
$\displaystyle\ =\ $
$\displaystyle\sum_{k=1}^{n}\left(S_{\tau_{k},w_{k}}H^{2}\right)^{\perp}$
$\displaystyle=$ $\displaystyle\left((\prod_{k=1}^{n}S_{\tau_{k},w_{k}})\
H^{2}\right)^{\perp}.$

Suppose that $\mu_{n}\to\mu$ weak-*, where $\mu$ and each $\mu_{n}$ are
singular. Define singular inner functions by

$\displaystyle\varphi_{n}(z)$ $\displaystyle\ =\ $
$\displaystyle\exp\left[-\int\frac{e^{i\theta}+z}{e^{i\theta}-z}d\mu_{n}(\theta)\right]$
$\displaystyle\varphi(z)$ $\displaystyle\ =\ $
$\displaystyle\exp\left[-\int\frac{e^{i\theta}+z}{e^{i\theta}-z}d\mu(\theta)\right].$

Then $\|\varphi_{n}-\varphi\|_{H^{2}}\to 0$. Indeed, $\varphi_{n}$ tends to
$\varphi$ weakly in $H^{2}$, since the functions all have norm $1$ and
converge pointwise on ${\mathbb{D}}$. Therefore

$\displaystyle\|\varphi_{n}-\varphi\|^{2}$ $\displaystyle\ =\ $ $\displaystyle
2-2{\mathrm{R}e\,}\langle\varphi_{n},\varphi\rangle$ $\displaystyle\to$
$\displaystyle 0.$

This means that not only do the Toeplitz operators
$T_{\overline{\varphi}_{n}}$ converge to $T_{\overline{\varphi}}$ in the
strong operator topology, but $T_{{\varphi_{n}}}T_{\overline{\varphi}_{n}}$
converges to $T_{\varphi}T_{\bar{\varphi}}$ SOT. This is proved in [24, p.
34]; for the convenience of the reader, we include the proof. Let $f\in
H^{2}$. Then

$\displaystyle\|T_{{\varphi_{n}}}T_{\bar{\varphi}_{n}}f-T_{{\varphi}}T_{\overline{\varphi}}f\|$
$\displaystyle\ \leq\ $
$\displaystyle\|T_{{\varphi_{n}}}(T_{\overline{\varphi}_{n}}-T_{\overline{\varphi}})f\|+\|(T_{{\varphi_{n}}}-T_{\varphi})T_{\overline{\varphi}})f\|$
$\displaystyle\leq$
$\displaystyle\sup_{n}\|\varphi_{n}\|_{H^{\infty}}\|(T_{\overline{\varphi}_{n}}-T_{\overline{\varphi}})f\|+\left(\int|\varphi_{n}-\varphi|^{2}|T_{\bar{\varphi}}f|^{2}\right)^{\frac{1}{2}}.$

The first term tends to zero because $T_{\overline{\varphi}_{n}}$ tends to
$T_{\bar{\varphi}}$ in the SOT, and the second term tends to $0$ because
$\varphi_{n}$ tends to $\varphi$ in measure and $|\varphi_{n}-\varphi|\leq 2$.
As $T_{{\varphi_{n}}}T_{\overline{\varphi}_{n}}$ is the projection onto
$(\varphi_{n}H^{2})^{\perp}$, this means by Proposition 4.1 that the spaces
$(\varphi_{n}H^{2})^{\perp}$ converge to $(\varphi H^{2})^{\perp}$. Applying
the inverse Sarason transform, we conclude that $\mathcal{M}(\mu_{n})$
converges to $\mathcal{M}(\mu)$. ∎

## References

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# Variable range hopping in thin film with large dielectric constant

B. I. Shklovskii  Theoretical Physics Institute, University of Minnesota,
Minneapolis, Minnesota 55455

###### Abstract

In a film with large dielectric constant $\kappa$ the electric field of an
electron spreads inside the film before exiting the film at large distances of
order of $\kappa d$ ($d$ is the film width). This leads to the logarithmic
Coulomb repulsion between electrons and modifies the shape of the Coulomb gap
in the density of localized states in a doped film. As a result the variable
range hopping conductivity in such a film has a peculiar temperature
dependence, where the domain of the $\ln\sigma(T)\propto(T_{0}/T)^{p}$
dependence, with the index $p\simeq 0.7$, is sandwiched between the two
domains with $p=1/2$.

Variable range hopping (VRH) is the generic mechanism of the low temperature
transport in systems with localized electron states. When electrons repel each
other via the Coulomb potential energy

$V(r)=\frac{e^{2}}{\kappa r},$ (1)

where $\kappa$ is the dielectric constant of the solid the density of
localized states $g(E)$ has the soft Coulomb gap. As a result the VRH
conductivity $\sigma$ obeys the Efros-Shklovskii (ES) law ES ; SE84

$\sigma(T)=\sigma_{0}\exp\left[-\left(\frac{T_{0}}{T}\right)^{1/2}\right]$ (2)

both in two-dimensional (2D) and three-dimensional (3D) cases. Here the
characteristic temperature

$T_{0}=\frac{Ce^{2}}{\kappa a},$ (3)

(we use the energy units for the temperature $T$), $a$ is the localization
length of electrons and $C$ is the numerical coefficient SE84 .

This paper deals with the situation when the Coulomb interaction between
electrons has a more complicated form. We consider a thin film with the
thickness $d\gg a$ and the large dielectric constant $\kappa\gg 1$. The film
is surrounded by the media with much smaller dielectric constant, for example
just by the air with $\kappa_{ext}=1$.

The energy of the Coulomb repulsion of two electrons in the film was
calculated exactly in Ref. Keldysh . Here we present only asymptotic results
with their physics interpretation. Let us assume that the film is defined by
the surfaces $z=\pm d/2$ and one electron is at $z=x=y=0$. Then at distances
$r\ll d$ its electric field (induction) spreads isotropically. At larger
distances the electric field lines are forced by the large $\kappa$ to stay
inside the film, so that the field spreads along the radius
$\rho=\sqrt{x^{2}+y^{2}}$ of the cylindrical coordinate system with the same
$z$-axis. At $\rho\sim\kappa d$ electric field lines exit from the film and
eventually spread uniformly over the whole $4\pi$ body angle again.

Let us discuss the potential energy of repulsion of two electrons in the film.
At the distance $\rho\gg\kappa d$ two electrons interact via the ”external”
Coulomb interaction

$V(\rho)=\frac{e^{2}}{\rho}.$ (4)

On the other hand, in intermediate range of distances $d\ll\rho\ll\kappa d$

$V(\rho)=2\epsilon_{d}\ln\left(\frac{\kappa d}{\rho}\right),$ (5)

where $\epsilon_{d}=e^{2}/\kappa d$. Finally at even smaller distances
$\rho\ll d$ we arrive at the the ”internal” Coulomb interaction

$V(r)=\frac{e^{2}}{\kappa r}+2\epsilon_{d}\ln\kappa,$ (6)

which differs from Eq. (1) by the logarithmically large energy accumulated
when the second electron is moved from infinity to $\rho=d$. The resulting
potential energy $V(\rho)$ is plotted as a function of $\rho$ in Fig. 1 in all
three ranges.

Figure 1: A schematic plot of the potential energy of repulsion of two charges
located inside the film with a large dielectric constant as a function of the
distance $\rho$ between electrons. Equation numbers describing different
segments of this plot are shown next to each segment.

Let us assume that the film is uniformly doped by a concentration of donors
$N_{D}$ and compensated by smaller concentration of acceptors so that low
temperature transport is due to VRH on donors. How does the two-dimensional
electrostatics affect ES law?

This question was first addressed in Ref. Larkin  for another object, a weakly
disordered two-dimensional gas, for example, in a silicon inversion layer. In
this case, according to Ref. Abrahams  localized states have an exponentially
large localization length. Large localized states play the dual role in this
theory. First, VRH conductivity is related to long distance hopping between
those of them which are close to the Fermi level. Second, all other large
localized states according to Ref. Larkin  contribute to the large effective
dielectric constant which keeps the electric lines of a charge in the plane of
the two-dimensional gas. In order to calculate VRH conductivity the authors of
Ref. Larkin  used an intuitive shortcut avoiding discussion of the Coulomb
gap. The authors found two ranges of the temperature dependence of the VRH
conductivity $\sigma(T)$. At very low temperatures where the characteristic
length of the hop $r_{h}$ is much larger than $\kappa d$ they arrived at ES
law with $\kappa=1$ and and at higher temperature range where $d\ll
r_{h}\ll\kappa d$ they obtained activated conductivity.

In this paper, we calculate the VRH conductivity in the framework of our
simpler model, where large $\kappa$ is of the lattice origin or is a result of
close three-dimensional metal-insulator transition. In this way we avoid the
controversial subject of the 2D metal-insulator transition Kravchenko . We
follow the ”orthodox” ES logic ES ; SE84  deriving first the Coulomb gap and
then the conductivity. Let us first formulate our results moving from high to
lower temperatures.

At high temperatures $T\gg\epsilon_{d}a/d$ the VRH hopping conductivity is
determined by the ”internal” ES law

$\sigma(T)=\sigma_{0}\exp\left[-\left(\frac{T_{0}}{T}\right)^{1/2}-\frac{\epsilon_{d}\ln\kappa}{T}\right],$
(7)

which at high enough temperatures coincides with Eq. (2).

It is followed by the range of intermediate temperatures

$\frac{\epsilon_{d}a}{\kappa d}\ll T\ll\frac{\epsilon_{d}a}{d}$ (8)

with ”activated” VRH conductivity

$\sigma(T)=\sigma_{0}\exp\left[-\frac{T_{1}(T)}{T}\right],$ (9)

where the ”activation energy”

$T_{1}(T)=\epsilon_{d}\ln\left(\frac{T}{\epsilon_{d}}\frac{\kappa
d}{a}\right).$ (10)

of the intermediate regime is weakly temperature dependent. If one
approximates Eq. (9) as

$\sigma(T)=\sigma_{0}\exp\left[-\left(\frac{T_{p}}{T}\right)^{p}\right],$ (11)

the power

$p=1-\frac{1}{\ln(T\kappa d/\epsilon_{d}a)}.$ (12)

Close to $T=\epsilon_{d}a/d$ we get $p=1-1/\ln\kappa$. For example, at
$\kappa=40$ one gets $p=0.7$.

At even smaller temperatures $T\ll\epsilon_{d}a/\kappa d$ we arrive at the
”external” ES law:

$\sigma(T)=\sigma_{0}\exp\left[-\left(\frac{T_{0}^{ext}}{T}\right)^{1/2}\right],$
(13)

where $T_{0}^{ext}=Ce^{2}/a$. Thus, the ”activated” VRH conductivity Eq. (9)
is sandwiched between the two different ES regimes of $\sigma(T)$, the
”internal” one, Eq. (7), at the high temperature side, and the ”external” one,
Eq. (13), on the low temperature side. The two low temperature regimes Eq. (9)
and Eq. (13) are in agreement with Ref. Larkin . The new high temperature
”internal” ES regime exists only if $d\gg a$.

The experimental literature on the VRH conductivity in thin films is
controversial (see Refs. Goldman ; Baturina  and references therein.) How
large is the film dielectric constant and how important is contribution of
large localized states Larkin  in most of cases is not clear. On the
insulating side of the superconductor-insulator transition in ultrathin
quench-condensed Ag, Bi, Pb and Pd films Goldman  VRH data agree with Eq. (11)
with $p=2/3$. On the other hand, for relatively thick TiN films Baturina  the
crossover from $p\simeq 1/2$ to $p\simeq 1$ is observed with the decreasing
temperature. Finding an explanation for this crossover is challenging Baturina
because one would normally expect that the ES law emerges at the low
temperature limit. Our theory shows that in relatively thick film with the
decreasing temperature one can see the crossover from the ”internal” ES law to
the activated transport. This may explain results of Ref. Baturina .

In order to simulate a film with a large dielectric constant one can make a
two-dimensional array of isolated metallic islands overhanging each other
Delsing ; Fistul . Although these arrays were originally designed as arrays of
Josephson junctions they perfectly simulate a large dielectric constant in the
normal state. Indeed, such array keeps electric lines in its plane if the
capacitance between two islands is larger than the capacitance of each island
to the ground. As a result such a normal array in presence of some disorder
should show activated VRH discussed above.

Before switching to the derivation of our results we would like to dwell on
the related theoretical paper Fisher  which deals with the VRH transport of
point like vortexes responsible for the low temperature resistance of a
superconductor film in the external magnetic film. Two vortexes interact via
the logarithmic potential at small distances, while again at large distances
their interaction follows the ”external” Coulomb potential Pearl , so that one
could expect to see the two low temperature ranges discussed above, the
”activated” regime and the ”external” ES law. However, Ref. Fisher  argues
that ES approach is not valid in this case because ”logarithmic interaction
grows without bound with particle separation” and, therefore, ”single-particle
energies can not be defined” so that ”multi-vortex hopping dominates the above
single-particle effects”. The multi-particle estimate Fisher  leads to Eq.
(11) with $2/3<p<4/5$. This is close to what we got for $p$ at $\kappa=40$.

In contrary to the above statements of Ref. Fisher  Fig. 1 clearly shows that
the repulsion energy vanishes at infinity similarly to the standard Coulomb
potential Eq. (4). Thus, there is no problem to introduce a single particle
energy for a system of localized electrons interacting with the pairwise
potential $V(\rho)$. We can proceed with the ES argument and study the new
shape of the Coulomb gap in the density of states (DOS) of single particle
excitations and eventually the VRH conductivity. We return to the discussion
of the role of multi-particle processes in the end of this paper.

Below we calculate the zero temperature DOS $g(\epsilon)$ following Refs. ES ;
SE84 ; ES85 . In the ground state of the system we define the single electron
energy $\epsilon_{i}$ of an occupied donor $i$ as the energy necessary to
extract an electron from this state to the state with the energy right at the
Fermi level at infinity. The single electron energy $\epsilon_{j}$ of an empty
in the ground state donor $j$ is defined as the energy necessary to bring to
it an electron from a state at infinity with the energy right at the Fermi
level $\epsilon_{F}$. By the definition of the ground state
$\epsilon_{j}>\epsilon_{F}>\epsilon_{i}$. Another stronger stability condition
can be formulated for each pair of occupied and empty states as follows

$\epsilon_{j}-\epsilon_{i}-V(r_{ij})>0.$ (14)

Here $V(r_{ij})$ is the repulsion energy of two electrons on sites $i$ and $j$
and $-V(r_{ij})$ is the Coulomb energy of attraction of the electron which
moved to the site $j$ by the hole it has left at the site $i$. In other words,
this term describes the exciton effect. Eq. (14) requires that any two states
close in energy to $\epsilon_{F}$ should be far enough in the space. This
limits the density of states (DOS) close to the Fermi level. For the Coulomb
potential the result for the DOS is known and we do not repeat derivations
from ES ; SE84 ; ES85 , but list the results.

The ”external” Coulomb potential Eq. (4) limits two-dimensional density of
states at the level

$g(\epsilon)=\frac{2}{\pi}\frac{\epsilon}{e^{4}},$ (15)

For the ”internal” Coulomb potential Eq. (6) we get

$g(\epsilon)=\frac{3}{\pi}\frac{\kappa^{3}\epsilon^{2}d}{e^{6}},$ (16)

where the factor $d$ converts the three-dimensional DOS to the two-dimensional
one. Apparently the asymptote Eq. (15) is valid at large distances, i. e. at
small energies $\epsilon$ and the asymptote Eq. (15) describes small lengths
or large energies. Let us now derive the DOS for the intermediate range of
distances and energies. To this end we have to calculate the number of states
in the band of the width $2\epsilon$ around $\epsilon_{F}$. Using Eq. (14) and
Eq. (5) we get $\epsilon_{d}\ln(\kappa d/\rho)<\epsilon$. This means that in
this energy band there is no more than one state in the disc with radius

$\rho(\epsilon)=\kappa d\exp\left[-\frac{\epsilon}{\epsilon_{d}}\right].$ (17)

This leads to the following estimate of the DOS for
$\epsilon_{d}\ll\epsilon\ll\epsilon_{d}\ln\kappa$:

$g(\epsilon)\sim\frac{1}{\rho(\epsilon)^{2}\epsilon}=\frac{1}{(d\kappa)^{2}\epsilon}\exp\left[\frac{2\epsilon}{\epsilon_{d}}\right].$
(18)

DOS Eq. (18) matches DOS Eq. (15) at $\epsilon=\epsilon_{d}$ and Eq. (16) at
$\epsilon=\epsilon_{d}\ln\kappa$ (see Fig. 2)). At large energies the
parabolic range of the Coulomb gap, Eq. (16), is limited by the total width of
the impurity band
$\epsilon_{max}=e^{2}N_{D}^{1/3}/\kappa+\epsilon_{d}\ln\kappa$. At
$\epsilon\gg\epsilon_{max}$ the DOS decreases (see the dashed line in Fig. 2)
similarly to the three-dimensional DOS of the classical impurity band (see
Ch.14 of Ref. SE84 ).

Figure 2: Schematic plot of the DOS of localized electrons limited by Coulomb
interaction of electrons as a function of energy. Equation numbers describing
different segments of this plot are shown next to each segment. Dashed line
shows beginning of the DOS decline at large energies

Now we can apply the calculated above DOS in order to estimate the exponential
term of the VRH conductivity of the film at different temperatures. This
estimate closely follows the original Mott’s approach Mott . We define an
energy band around the Fermi level and estimate contribution of this band to
the VRH conductivity. Then we can optimize result with respect of the band
width. This calculations are quite straightforward and we will not go through
them here. Results are already formulated in the beginning of our paper.

Let us return to the multi-electron effects on ES law. For the standard
Coulomb potential Eq. (1) they were studied in Refs. Efros ; Baran ; SE84 ;
ES85 . It was shown that they may change only the coefficient $C$ in Eq. (2).
Here we only briefly remind what was done. In 3D case a small energy single-
electron excitation strongly interacts with dipole moments of surrounding
compact electron-hole excitations forming together with them a composite
charged multi-electron excitation, the electronic polaron. Polarons being
charged particles obey stability criterion Eq. (14), have the Coulomb gap and
lead to ES law. The only difference is that coefficient $C$ can be somewhat
larger because every hop of a single-electron excitation in 3D is accompanied
by the tunnelling depolarization of many polaron pairs. A simple estimate
showed that the total length of these small hops is of the order of the length
of the main long ES VRH hop. This gives no more than the factor 2 in the
expression for $C$. In 2D number of dipole excitations in the polaron
atmosphere is of order one and polaron effects provide only small corrections
to $C$.

We estimated similar effects in the framework of the complex potential of this
paper (Eqs. (4), (5), (6)). It turns out that in the low temperature range
polaron effects provide only small corrections to $C$ in Eq. (13). On the
other hand, for the two higher temperature ranges (Eqs. (9) and (2) the
situation is similar to the 3D case studied in Refs. Baran ; SE84 ; ES85 .
Namely, at these temperatures multi-electron effects may add a numerical
coefficient in the exponential. For Eq. (9) this means that $T_{1}$ may become
twice larger. Thus, multi-electron effects do not change the power $p$ in Eq.
(11). This conclusion is in disagreement with Ref. Fisher  which assumes that
even in 2D the total length of small hops is much larger than the ES hop and,
therefore, overestimates importance of the multi-electron effects.

Above we discussed only the ohmic transport in a weak electric field. If the
electric field is so strong that $eEa/2\gg T$ one can replace $T$ by the
effective VRH temperature $eEa/2$ in Eqs. (2), (9), (11) and (13) to obtain
non-ohmic current-voltage characteristics Shklovskii  (see also Larkin ;
Marianer ). For the intermediate activated regime we then arrive at the
current-voltage characteristics $J\propto J_{0}\exp(-E_{0}/E)^{p}$, where $p$
is close to unity. This result is in agreement with the earlier theory Nelson
of the VRH transport of pinned vortexes in superconductors under the influence
of a strong current.

Until now we dealt with a film. Now let us briefly discuss the similar physics
in a long cylindrical nano-wire or nano-rod with radius $d$ made from a
semiconductor with a large dielectric constant $\kappa\gg 1$, for simplicity,
in the air environment. Here again the electric field of an electron at
distances $r\ll d$ spreads isotropically, then stays inside nano-rod for a
distance $x<\xi=d\kappa^{1/2}$ along the cylinder axis, and then leaks from
the cylinder and eventually spreads isotropically in the air at large enough
$r$. Thus, the potential energy of repulsion of two electrons again changes
from the very short range ”internal” Coulomb interaction Eq. (4) to the very
long range ”external” Coulomb interaction Eq. (1) with the large intermediate
range of $x$, where interaction has the one-dimensional character. The
interaction in this range was studied Kamenev ; Zhang  for an ion channel in a
lipid membrane, where the cylindrical pore with radius $d$ is filled by water
with $\kappa=81$ and is surrounded by lipids with $\kappa=2$. Translated to
our problem the potential energy of two electrons located at the nano-rod axis
at the intermediate distance $x$ from each other is well approximated by

$V(x)=eE_{0}\xi[\exp(-x/\xi)-1],$ (19)

where $E_{0}=2e/\kappa d^{2}$. (see Sec. VIII of Ref. Zhang ). Using this
potential for repulsion of two electrons in the nano-rod and following the
Coulomb gap based derivation similar to one used above for a film (or the
shortcut approach of Ref. Larkin ) one can calculate the temperature
dependence of the VRH conductivity of the nano-rod in the intermediate
temperature range. The result is the strict activation regime with the
temperature independent activation energy $T_{a}=eE_{0}\xi$. It is, of course,
sandwiched between the two ES laws, the ”internal” one on the high temperature
side and the ”external” one Eq. (13) on the low temperature side.

I am grateful to T. Baturina, A. M. Goldman, A. Kamenev, D. E. Khmelnitskii
and A. I. Larkin for useful discussions.

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