#  DyREx: Dynamic Query Representation for Extractive Question Answering

Urchade Zaratiana1,2 , Niama El Khbir2, Dennis Núñez2, Pierre Holat1,2,
Nadi Tomeh2, Thierry Charnois2
1FI Group, 2LIPN, Université Sorbonne Paris Nord - CNRS UMR 7030
Correspondence to<EMAIL_ADDRESS>
###### Abstract

Extractive question answering (ExQA) is an essential task for Natural Language
Processing. The dominant approach to ExQA is one that represents the input
sequence tokens (question and passage) with a pre-trained transformer, then
uses two learned query vectors to compute distributions over the start and end
answer span positions. These query vectors lack the context of the inputs,
which can be a bottleneck for the model performance. To address this problem,
we propose DyREx, a generalization of the vanilla approach where we
dynamically compute query vectors given the input, using an attention
mechanism through transformer layers. Empirical observations demonstrate that
our approach consistently improves the performance over the standard one. The
code and accompanying files for running the experiments are available at
https://github.com/urchade/DyReX.

##  1 Introduction

Extractive question answering is a challenging task where the goal is to
extract the answer span given a question and a passage as inputs (Rajpurkar et
al., 2016; Kwiatkowski et al., 2019). The prevailing approach achieves
Extractive question answerin (ExQA) by firstly producing a contextualized
representation of the input, which is a concatenation of the question and the
passage, using a pre-trained transformer model. Two learned query vectors are
then used to compute a probability distribution over this input sequence
representation to produce the start and end positions of the answer span. This
approach has demonstrated very strong and hard-to-beat results, which makes it
the de facto approach to extractive QA (Devlin et al., 2019; Liu et al., 2019;
Joshi et al., 2020).

However, despite their high performance, we argue that these methods remain
suboptimal since the query vectors used to compute the start and end
distributions are static, i.e., they are independent of the input sequence,
which can be a bottleneck for improving the performance of the model. Hence,
we propose to extend this by allowing the queries to dynamically aggregate
information from the input sequence to better answer the question. Our method,
DyREx, iteratively refines the initial query representations, allowing them to
aggregate information from the source sequence through attention mechanism
(Bahdanau et al., 2015; Vaswani et al., 2017). More specifically, we make use
of an L-layers transformer decoder architecture, which allows (1) interaction
between the queries through self-attention to model the interdependence
between the start and end of the answer span, and allows (2) interaction
between queries and the input sequence through cross-attention, which
specializes the queries to a specific input question and passage, giving more
flexibility than a static representation.

We conduct extensive experiments on several extractive Question Answering
benchmarks, including SQuad (Rajpurkar et al., 2016) and MRQA datasets (Fisch
et al., 2019). Experimental results demonstrate that our approach consistently
improves the performance over the standard approach.

##  2 Model

###  2.1 Background: Vanilla QA model

We describe here the mainstream approach to extractive Question Answering
tasks. In all the following, we call it the ExQA vanilla approach. It is
typically performed by feeding the input text sequence $\\{x_{i}\\}_{i=1}^{N}$
(the concatenation of the question $Q$ and the passage $D$ containing the
answer) into a pre-trained language model such as BERT (Devlin et al., 2019),
producing contextualized token representations
$\\{\textbf{h}_{i}\\}_{i=1}^{N}\in\mathbb{R}^{d}$, $d$ being the embedding
dimension of the model. Then, to compute the probability of the start and end
positions of the answer span, the following estimators are used:

$\displaystyle
p(\textit{start}=i|Q,D)=\frac{\exp(\textbf{q}_{s}^{T}\textbf{h}_{i})}{\sum_{i^{\prime}=1}^{N}\exp(\textbf{q}_{s}^{T}\textbf{h}_{i^{\prime}})}$
$\displaystyle
p(\textit{end}=j|Q,D)=\frac{\exp(\textbf{q}_{e}^{T}\textbf{h}_{j})}{\sum_{j^{\prime}=1}^{N}\exp(\textbf{q}_{e}^{T}\textbf{h}_{j^{\prime}})}$
(1)

Where $\textbf{q}_{s}$ and $\textbf{q}_{e}\in\mathbb{R}^{d}$ are respectively
the start and end queries, randomly initialized and updated during model
learning. The training objective is to minimize the sum of the negative log-
likelihood of the correct start and end positions $(\hat{i},\hat{j})$:

$\displaystyle\mathcal{L}=-\log p(\textit{start}=\hat{i}|Q,D)-\log
p(\textit{end}=\hat{j}|Q,D)$ (2)

This approach was first proposed by Devlin et al. (2019), and is now used by
most of the work on transformer-based extractive question answering (Liu et
al., 2019; Joshi et al., 2020; Shi et al., 2022).

###  2.2 Our model: DyREx

The learned query vectors $\textbf{q}_{s}$ and $\textbf{q}_{e}$ in the vanilla
approach are shared among all sentences and are context insensitive. We
presume that using such static queries is a constraining factor for
performance improvement, so we propose to extend this approach by allowing the
queries to dynamically aggregate information from the input sequence to allow
the model to better adapt to the context.

In our model, the initial start and end query representations
$\textbf{q}_{s}^{0}$ and $\textbf{q}_{e}^{0}$ are concatenated and fed to an
L-layers transformer decoder (Vaswani et al., 2017) to obtain dynamic
representations $\textbf{q}_{s}^{L}$ and $\textbf{q}_{e}^{L}$:

$\textbf{Q}^{L}=\texttt{Trans\\_Dec}_{L}(\textbf{Q}^{0},\textbf{H})$ (3)

with $\textbf{Q}^{i}=[\textbf{q}_{e}^{i},\textbf{q}_{s}^{i}]$ the
concactenated queries at layer $i$ and
$\textbf{H}=[\textbf{h}_{0},\textbf{h}_{1},...,\textbf{h}_{N}]$ the
concatenated token representations, and $\texttt{Trans\\_Dec}_{L}$ being an
L-layers transformer decoder.

More specifically, the i-th transformer layer consists of a bi-directional
self-attention module $\texttt{self-att}_{i}$ applied between the queries to
model the interdependence between the start and the end positions of the
answer, a cross-attention $\texttt{cross-att}_{i}$ which updates the query
representations by aggregating information from the input sequence embeddings,
and finally a two-layer point-wise feedforward network $\texttt{FFN}_{i}$ with
GeLU activation (Hendrycks and Gimpel, 2016):

$\begin{split}\tilde{\textbf{Q}}^{i}=\texttt{self-
att}_{i}(\texttt{Q}=\textbf{Q}^{i},\texttt{K}=\textbf{Q}^{i},\texttt{V}=\textbf{Q}^{i})\\\
\widehat{\textbf{Q}}^{i}=\texttt{cross-
att}_{i}(\texttt{Q}=\tilde{\textbf{Q}}^{i},\texttt{K}=\textbf{H},\texttt{V}=\textbf{H})\\\
\textbf{Q}^{i+1}=\texttt{FNN}_{i}(\widehat{\textbf{Q}}^{i})\end{split}$ (4)

Furthermore, an Add-Norm (skip connection (He et al., 2016) \+ layer
normalization (Ba et al., 2016)) is inserted after each of the components as
in Vaswani et al. (2017), but we do not show it here for better readability.
Moreover, both self-att and cross-att are multi-head scaled dot-product
attention from Vaswani et al. (2017), and the embedding dimension and the
number of attention heads of the decoder layers are the same as for the token
representation layer.

Finally, to compute the start and the end answer position probabilities, we
use the same estimator as the vanilla model in equation 1, substituting
$\textbf{q}_{s}$ and $\textbf{q}_{e}$ by $\textbf{q}_{s}^{L}$ and
$\textbf{q}_{e}^{L}$ respectively. Note that the vanilla model is a particular
case of our model with a number of decoder layers $L=0$.

##  3 Experimental setup

#### Datasets

We evaluate our model on English Machine Reading Comprehension datasets
including SQuAD (Rajpurkar et al., 2016), HotpotQA (Yang et al., 2018),
TriviaQA (Joshi et al., 2017), NewsQA (Trischler et al., 2016), and Natural
Questions (Kwiatkowski et al., 2019). We preprocess the datasets using
standard approaches, then we evaluate our model using the F1 and Exact Match
(EM) metrics implemented on the MRQA Github repository.

#### Hyperparameters

We adopt a similar experimental setup of Ram et al. (2021) and Joshi et al.
(2020). For all experiments, we use SpanBERT for token representations. We
fine-tune our model using hyperparameters from the default configuration of
the HuggingFace Transformers library (Wolf et al., 2020). We use Adam
optimizer with a learning rate of $3.10^{-5}$, where a warm-up stage is set
for the first $10\%$ of the steps, then decay the learning rate linearly for
the rest of the training steps. We employed a batch size of 12, and we trained
for a maximum of 5 epochs for full-sized datasets, and for few-shot models, we
train the models up to either 2500 steps or 10 epochs. For DyREx, unless
specified, we employed a 3-layers transformer decoder for the query
representations, where each attention layer has 8 heads and the same embedding
dimension as the contextualized token embeddings. To run the vanilla ExQA
model, we borrowed the code from (Ram et al., 2021) GitHub repository. We
trained all the models in a server with V100 GPUs.

##  4 Results

Table 1 presents the obtained results for the different datasets, employing
SpanBERT for token representations. Our approach is very effective in few-shot
settings compared to the vanilla approach. For instance, our model exceeds the
vanilla approach by 7.47% F1 score on SQuAD (512 samples), 5.41% on HotpotQA
(1024 samples), 12.76% on TriviaQA (256 samples), and 7.97% on NewsQA (256
samples). These empirical results show the effectiveness of our approach and
demonstrate that contextual queries are important when only a few data are
available.

Train size | Models | Datasets
---|---|---
SQuAD | HotpotQA | TriviaQA | NewsQA | NaturalQs
256 | Vanilla | 65.74 | 53.23 | 28.49 | 35.80 | 41.87
DyREx | 70.75 | 57.08 | 41.66 | 43.77 | 45.57
512 | Vanilla | 69.72 | 58.70 | 45.39 | 43.24 | 48.36
DyREx | 77.19 | 61.15 | 52.57 | 49.20 | 55.37
1024 | Vanilla | 74.01 | 62.54 | 51.87 | 50.61 | 53.42
DyREx | 79.42 | 67.95 | 57.59 | 54.26 | 61.67
Full | Vanilla | 90.64 | 79.95 | 76.31 | 68.02 | 77.79
DyREx | 91.01 | 80.55 | 77.37 | 68.53 | 78.58
Table 1: Main results. We reported experimental results for different sizes of
training datasets using SpanBERT (Joshi et al., 2020) for token
representation.

##  5 Component analysis for DyREx

We perform a more in-depth analysis to inspect the contribution of the
separate components of our DyREx architecture. We study the influence of the
number of layers of the decoder. We also examine various masking strategies of
self-attention of DyREx decoder: Bidirectional which allows full attention
between queries, Causal with causal masking (the start query cannot attend the
end query), and Independent which fully masks the attention between the
queries. For all the studies, we employed the SQuAD dataset, a SpanBERT
representation, and averaged results across three different seeds.

Figure 1: Queries and tokens interactions. This figure shows the different
interactions (or attention) between the queries ($q_{s}$ and $q_{e}$) and the
input tokens ($\\{\textbf{h}_{i}\\}_{i=1}^{N}$). a) Vanilla: No interaction,
either token-query or query-query (i.e., static queries). b) Bidirectional:
dynamic queries with bidirectional query interaction. c) Causal: dynamic
queries with causal relation ($q_{s}$ influence $q_{e}$). d) Independent:
queries are independent of each other but remain dynamic.

# Layers | F1 | EM
---|---|---
0 | $90.64\pm 0.10$ | $83.08\pm 0.09$
1 | $91.08\pm 0.05$ | $83.56\pm 0.31$
2 | $90.92\pm 0.12$ | $83.50\pm 0.17$
3 | $91.01\pm 0.03$ | $83.35\pm 0.07$
4 | $\textbf{91.17}\pm\textbf{0.05}$ | $83.57\pm 0.14$
5 | $91.08\pm 0.07$ | $\textbf{83.64}\pm\textbf{0.17}$
Table 2: Results for different number of transformer decoder layers.

| F1 | EM
---|---|---
Static |  |
a) Vanilla | $90.64\pm 0.10$ | $83.08\pm 0.09$
Dynamic |  |
b) Bidirectional | $\textbf{91.01}\pm\textbf{0.03}$ | $83.35\pm 0.07$
c) Causal | $91.00\pm 0.05$ | $\textbf{83.47}\pm\textbf{0.17}$
d) Independent | $90.87\pm 0.09$ | $83.14\pm 0.44$
Table 3: Results for different interactions/attentions between the queries.

#### Influence of # of layers

Table 3 shows that performance is generally better with more layers. The best
performance is attained when using 4 or 5 layers, but a lower number of layers
can also obtain competitive results. However, the results are much weaker
without a decoder layer, i.e., without a static query representation.

#### Type of self-attention

Table 3 shows the obtained results using the various attention or masking
strategies for the decoder. We see that fully masking (d) attention between
queries provides the weakest result, while Causal and Bidirectional attention
perform similarly. This shows that the interaction between queries is
important.

##  6 Related Work

Early ExQA models based on deep learning, such as BidAF (Seo et al., 2017),
Match-LSTM (Hu et al., 2017), QaNet (Yu et al., 2018) and others, were highly
specialized and heavily engineered. The arrival of BERT and pre-trained
language models completely transformed the domain by introducing a simple yet
effective approach: the Vanilla ExQA (section 2.1) (Devlin et al., 2019).
Since its introduction, this has been the dominant approach to extractive QA
(Liu et al., 2019; Joshi et al., 2020). For instance, (Fajcik et al., 2021)
proposes to model the joint probability of the spans, instead of modeling the
probability of a span’s start and end positions independently, but it does not
outperform the Vanilla approach. In this work, we extend the vanilla model not
by modifying the objective function but by learning a richer representation of
the query parameters.

##  7 Conclusion

In this paper, we propose DyREx, a method to dynamically compute query
representations to calculate the start and end positions of answer spans in
extractive question answering. Our approach consistently outperforms the
dominant approach on a wide range of QA datasets, and the gain is even more
significant in a few-shot scenario. In future work, it would be interesting to
adapt DyREx for multi-span extraction tasks such as Named Entity Recognition
and Keyphrase Extraction.

## Acknowledgments

This work is partially supported by a public grant overseen by the French
National Research Agency (ANR) as part of the program Investissements d’Avenir
(ANR-10-LABX-0083). This work was granted access to the HPC/AI resources of
[CINES/IDRIS/TGCC] under the allocation 20XX-AD011013682 made by GENCI.

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#  Spectrum of Weighted Composition Operators
Part VIII
Lower semi-Fredholm spectrum of weighted composition operators on $C(K)$. The
case of non-invertible surjections.

Arkady Kitover  Community College of Philadelphia, 1700 Spring Garden St.,
Philadelphia, PA, USA<EMAIL_ADDRESS>and  Mehmet Orhon  University of New
Hampshire, 105 Main Street Durham, NH 03824<EMAIL_ADDRESS>
###### Key words and phrases:

Weighted composition operators, spectrum, Fredholm spectrum, essential spectra

###### 2010 Mathematics Subject Classification:

Primary 47B33; Secondary 47B48, 46B60

##  1\. Introduction

In this paper we continue the study of essential spectra of weighted
composition operators. In [5] the first named author obtained a description of
essential spectra of weighted composition operators on $C(K)$ in the case when
the corresponding map is a homeomorphism of the compact space $K$ onto itself.
That allowed to provide a similar description of essential spectra of
operators of the form $wU$ acting on Banach lattices in the case when $w$ is a
central operator, $U$ is a $d$-isomorphism, and the spectrum of $U$ lies on
the unit circle.

In [7] the authors described essential spectra of weighted composition
operators on $C(K)$ in the case when the corresponding map is a non-invertible
homeomorphism of $K$ into itself.

In [6] we obtained some general results concerning essential spectra of
$d$-homomorphisms of Banach $C(K)$-modules. From this results follows as a
very special case that the upper semi-Fredholm spectrum of a weighted
composition operator on $C(K)$ is rotation invariant provided that the set of
eventually periodic points of the corresponding map is of first category in
$K$.

Despite these partial results, the problem of describing essential spectra of
general weighted composition operators on $C(K)$ and other Banach lattices
remains, to the best of our knowledge, unsolved. The current paper represents
another step toward the solution of this problem: namely, we describe the
lower semi-Fredholm and the Fredholm spectrum of a weighted composition
operator on $C(K)$ in the case when the corresponding map is a surjection.

The paper is organized as follows.

In section 2 we introduce the notations we use throughout the paper and state
some known results needed in the sequel.

In section 3 we obtain a criterion for the operator $\lambda I-T$ to be
Fredholm or lower semi-Fredholm, providing that $|\lambda|<\rho_{min}(T)$. We
refer the reader to formula (2) below for the definition of $\rho_{min}(T)$.

In section 4 we obtain a criterion for the operator $\lambda I-T$ to be
Fredholm or lower semi-Fredholm, providing that
$\rho_{min}(T)<|\lambda|<\rho(T)$. To obtain the corresponding results, and at
the same time to avoid making our statements too cumbersome, we had to impose
the following additional conditions.

  1. (1)

The surjection $\varphi$ is open.

  2. (2)

The weight $w$ is an invertible element of the algebra $C(K)$.

  3. (3)

The set of all eventually $\varphi$-periodic points is of first category in
$K$.

It might be worth noticing that the criterions we obtained involve the notion
of almost homeomorphisms of compact Hausdorff spaces that was introduced and
studied by Louis Friedler and the first named author in [3]. In particular,
Theorem 4.14 states that if the compact Hausdorff space belongs to the class
AH (see Definition 4.13) then the spectrum $\sigma(T)$ coincides with the
Fredholm spectrum $\sigma_{f}(T)$.

##  2\. Notations and preliminaries

Throughout the paper the following notations are used.

$K$ is a compact Hausdorff space.

$C(K)$ is the Banach algebra of all continuous complex-valued functions on $K$
endowed with the supremum norm.

$\varphi$ is a continuous map of $K$ onto itself.

$w\in C(K)$ is the weight.

$\mathds{N}$ is the semigroup of all positive integers.

$\mathds{Z}$ is the group of all integers.

$\mathds{R}$ is the field of all real numbers.

$\mathds{C}$ is the field of all complex numbers.

$\mathds{T}$ is the unit circle.

$\mathds{D}$ is the open unit disk.

$\mathds{U}$ is the closed unit disk.

For any $n\in\mathds{N}$ we denote by $\varphi^{n}$ the $n^{th}$ iteration of
the map $\varphi$.

$\varphi^{0}$ is the identical map: $\varphi^{0}(k)=k,k\in K$.

If the map $\varphi$ is invertible and $n\in\mathds{N}$ then $\varphi^{-n}$ is
the $n^{th}$ iteration of the inverse map $\varphi^{-1}$.

For any subset $F$ of $K$ and for any $n\in\mathds{N}$ we denote by
$\varphi^{(-n)}(F)$ the full $n^{th}$ preimage of $F$, i.e.

$\varphi^{(-n)}(F)=\\{k\in K:\;\varphi^{n}(k)\in F\\}$

For any $n\geq 1$ we denote by $w_{n}$ the function
$w(w\circ\varphi)\ldots(w\circ\varphi^{n-1})$. Thus, $w_{1}=w$.

A point $k\in K$ is called eventually $\varphi$-periodic if for some $n\geq 0$
the point $\varphi^{n}(k)$ is $\varphi$-periodic. When it does not cause any
ambiguity we will write eventually periodic or periodic instead of eventually
$\varphi$-periodic or $\varphi$-periodic, respectively.

For a given $w\in C(K)$ and a continuous surjection $\varphi$ we define the
weighted composition operator $T=wT_{\varphi}$ as

$(Tf)(k)=w(k)f(\varphi(k)),\;f\in C(K),\;k\in K.$

Let $X$ be a Banach space. We denote its Banach conjugate by $X^{\prime}$.

We denote the algebra of all bounded linear operators on $X$ by $L(X)$.

Let $T\in L(X)$. We denote by $\sigma(T,X)$ and $\rho(T,X)$ the spectrum and
the spectral radius of $T$, respectively.

We will use the notations $\sigma(T)$ and $\rho(T)$, as far as it does not
cause any ambiguity.

In the case when $T=wT_{\varphi}$ is a weighted composition operator on $C(K)$
the spectral radius $\rho(T)$ can be computed as (see [4])

$\rho(T)=\max\limits_{\mu\in M_{\varphi}}\exp\int\ln|w|d\mu,$ (1)

where $M_{\varphi}$ is the set of all regular Borel $\varphi$-invariant
probability measures on $K$.

In the case when $|w|>0$ on $K$ we also introduce

$\rho_{min}(T)=\min\limits_{\mu\in M_{\varphi}}\exp\int\ln|w|d\mu.$ (2)

Notice that in the case when the operator $T=wT_{\varphi}$ is invertible on
$C(K)$ we have $\rho_{min}(T)=[\rho(T^{-1})]^{-1}$.

Recall that $T\in L(X)$ is called lower semi-Fredholm if its defect is finite,
i.e. $codim(TX)<\infty$, it is called upper semi-Fredholm if its range is
closed in $X$ and $dim\ker{T}<\infty$, and it is called Fredholm if it is both
lower semi-Fredholm and upper semi-Fredholm

We use the standard notations $\Phi^{-}(X)$, $\Phi^{+}(X)$, and $\Phi(X)$ for
the sets of all lower semi-Fredholm operators, upper semi=Fredholm operators,
and Fredholm operators in $L(X)$, respectively. We will write
$\Phi^{-},\Phi^{+}$, and $\Phi$ instead of $\Phi^{-}(X),\Phi^{+}(X)$, and
$\Phi(X)$ when it will not cause any ambiguity.

We consider the following subsets of $\sigma(T)$.

$\sigma_{a.p.}(T)=\\{\lambda\in\mathds{C}:\exists x_{n}\in
X,\|x_{n}\|=1,Tx_{n}-\lambda x_{n}\rightarrow 0$}. Thus, $\sigma_{a.p.}(T)$ is
the union of the point spectrum and the approximate point spectrum of $T$.

$\sigma_{r}(T)=\sigma(T)\setminus\sigma_{a.p.}(T)$; i.e.,
$\lambda\in\sigma_{r}(T)$ if and only if the operator $\lambda I-T$ is not
invertible but bounded from below.

$\sigma_{usf}(T)$ is the upper semi-Fredholm spectrum of an operator $T\in
L(X)$. It is defined as

$\sigma_{usf}(T)=\\{\lambda\in\mathds{C}:\;\lambda I-T\not\in\Phi^{+}\\}.$

$\sigma_{lsf}(T)$ is the lower semi-Fredholm spectrum of an operator $T\in
L(X)$. It is defined as

$\sigma_{lsf}(T)=\\{\lambda\in\mathds{C}:\;\lambda I-T\not\in\Phi^{-}\\}.$

$\sigma_{f}(T)$ is the Fredholm spectrum of an operator $T\in L(X)$. It is
defined as

$\sigma_{f}(T)=\\{\lambda\in\mathds{C}:\;\lambda I-T\not\in\Phi\\}.$

We will need the following well known characterizations of $\sigma_{usf}(T)$
and $\sigma_{lsf}(T)$ (see e.g. [2]).

######  Proposition 2.1.

Let $X$ be a Banach space and $T\in L(X)$. Let $\lambda\in\mathds{C}$. The
following conditions are equivalent.

  1. (1)

$\lambda\in\sigma_{usf}(T)$

  2. (2)

There is a sequence $x_{n}$, $x_{n}\in X$, $n\in\mathds{N}$, such that
$\|x_{n}\|=1$, $Tx_{n}-\lambda x_{n}\rightarrow 0$, and the sequence $x_{n}$
is singular, i.e. it does not contain any norm convergent subsequence.

######  Proposition 2.2.

Let $X$ be a Banach space and $T\in L(X)$. Let $\lambda\in\mathds{C}$. The
following conditions are equivalent.

  1. (1)

$\lambda\in\sigma_{lsf}(T)$

  2. (2)

There is a sequence $x_{n}^{\prime}$, $x_{n}^{\prime}\in X^{\prime}$,
$n\in\mathds{N}$, such that $\|x_{n}^{\prime}\|=1$,
$T^{\prime}x_{n}^{\prime}-\lambda x_{n}^{\prime}\rightarrow 0$, and the
sequence $x_{n}^{\prime}$ is singular, i.e. it does not contain any norm
convergent subsequence.

######  Definition 2.3.

A subset $\\{k_{n},n\in\mathds{Z}\\}$ of $K$ is called a $\varphi$-string if
$\varphi(k_{n})=k_{n+1},n\in\mathds{Z}$.

Consider the space $\prod\limits_{n=-\infty}^{\infty}K_{n}$ endowed with the
Tikhonov topology, where each $K_{n}$ is a copy of $K$. The set of all
$\varphi$-strings is a closed subset of
$\prod\limits_{n=-\infty}^{\infty}K_{n}$. We will denote the compact space of
all $\varphi$-strings by $\mathbf{K}$.

We define the homeomorphism $\Phi$ of $\mathbf{K}$ onto itself as follows. If
$\mathbf{k}\in\mathbf{K}$ and $\mathbf{k}=(k_{n})_{n=-\infty}^{\infty}$ then
$\Phi(\mathbf{k})=(k_{n+1})_{n=-\infty}^{\infty}$.

We define $\mathbf{w}\in C(\mathbf{K})$ by the equality

$\mathbf{w}(\mathbf{k})=w(k_{0}).$

######  Definition 2.4.

Let $K$ be a compact Hausdorff space, $\varphi:K\rightarrow K$ be a continuous
surjection, and $w\in C(K)$. Let $\mathbf{K}$, $\Phi$, and $\mathbf{w}$ be the
objects introduced above. We will say that $\Phi$ is the homeomorphism
associated with $\varphi$ and that $\mathbf{T}=\mathbf{w}T_{\Phi}$ is the
operator associated with $T=wT_{\varphi}$.

We will need the following two facts about the lower semi-Fredholm spectrum of
weighted automorphisms of $C(K)$ (see [5]).

######  Theorem 2.5.

Let $\varphi$ be a homeomorphism of $K$ onto itself, $w\in C(K)$, and
$T=wT_{\varphi}$. Assume that the set of all $\varphi$-periodic points is of
first category in $K$. Let $\lambda\in\mathds{C}\setminus\\{0\\}$. The
following conditions are equivalent

  1. (1)

$def(\lambda I-T)\neq 0$.

  2. (2)

$\exists k\in K$ such that

$|w_{n}(k)|\leq|\lambda|^{n}\;\text{and}\;|w_{n}(\varphi^{-n}(k)|\geq|\lambda|^{n},\;n\in\mathds{N}.$
(3)

######  Theorem 2.6.

Let $\varphi$ be a homeomorphism of $K$ onto itself, $w\in C(K)$, and
$T=wT_{\varphi}$. Assume that the set of all $\varphi$-periodic points is of
first category in $K$. Let $\lambda\in\sigma(T)\setminus\\{0\\}$. The
following conditions are equivalent

  1. (1)

$(\lambda I-T)C(K)=C(K)$.

  2. (2)

$K=K_{1}\cup K_{2}\cup O$, where $K_{1}$, $K_{2}$, and $O$ are pairwise
disjoint nonempty subsets of $K$ such that

    1. (a)

$K_{1}$ and $K_{2}$ are closed $\varphi$-invariant subsets of $K$.

    2. (b)

$\rho(T,C(K_{1}))<|\lambda|$.

    3. (c)

The operator $T$ is invertible on $C(K_{2})$ and
$\rho(T^{-1},C(K_{2}))<|\lambda|^{-1}$.

    4. (d)

For any closed subset $F$ of $O$ we have

$\bigcap\limits_{n=1}^{\infty}cl\bigcup\limits_{j=n}^{\infty}\varphi^{j}(F)\subseteq
K_{2}$ (4)

and

$\bigcap\limits_{n=1}^{\infty}cl\bigcup\limits_{j=n}^{\infty}\varphi^{-j}(F)\subseteq
K_{1}.$ (5)

We will end this section with the following definition.

######  Definition 2.7.

Let $K$ be a compact Hausdorff space and $\varphi:K\rightarrow K$ be a
surjection. We will say that the point $k$ is perfectly periodic if $k$ is
periodic and
$\varphi^{(-1)}(\\{k,\varphi(k),\ldots\varphi^{p-1}(k)\\})=\\{k,\varphi(k),\ldots\varphi^{p-1}(k)\\}$,
where $p$ is the period of $k$.

##  3\. Conditions for $\lambda I-T$ to be lower semi-Fredholm. The case
$\lambda<\rho_{min}(T)$.

In this section we will obtain a criterion for the operator $T=wT_{\varphi}$
to be lower semi-Fredholm, providing that $\varphi:K\rightarrow K$ is a non-
invertible surjection. The answer involves the notion of almost homeomorphism
of a compact Hausdorff space introduced in [3]

######  Definition 3.1.

Let $K$ be a compact Hausdorff space and $\varphi:K\rightarrow K$ be a
continuous surjection. The map $\varphi$ is called an almost homeomorphism of
$K$ if there is a finite subset $S$ of $K$ such that the map
$\varphi:(K\setminus S)\rightarrow(K\setminus\varphi(S))$ is a homeomorphism.

######  Definition 3.2.

We say that a compact Hausdorff space $K$ belongs to the class $\mathcal{AH}$
if every almost homeomorphism of $K$ is a homeomorphism.

######  Theorem 3.3.

Let $\varphi$ be a continuous non-invertible 111Because the case when
$\varphi$ is a homeomorphism of $K$ onto itself was considered in [5], in all
the statements in the current paper we assume that the surjection $\varphi$ is
not invertible. map of a compact Hausdorff space $K$ onto itself and let $w\in
C(K)$. Assume that $|w|>0$ on $K$. Let $T=wT_{\varphi}$ be the corresponding
weighted composition operator on $C(K)$. The following conditions are
equivalent

  1. (1)

The operator $\lambda I-T$ is lower semi-Fredholm for every
$\lambda\in\rho_{min}(T)\mathds{D}$.

  2. (2)

The operator $\lambda I-T$ is Fredholm for every
$\lambda\in\rho_{min}(T)\mathds{D}$.

  3. (3)

The map $\varphi$ is an almost homeomorphism of $K$.

Moreover, for every $\lambda\in\rho_{min}(T)\mathds{D}$

$ind(\lambda I-T)=def(\lambda I-T)=\\\ =card\\{(p,q):\;p,q\in K,p\neq
q,\varphi(p)=\varphi(q)\\}.$ (6)

###### Proof.

$(2)\Rightarrow(1)$. This implication is trivial.

$(1)\Rightarrow(2)$. It is sufficient to prove that $\ker{(\lambda I-T)}=0$ if
$|\lambda|<\rho_{min}(T)$. Moreover, we will prove that
$\rho_{min}(T)\mathds{D}\subseteq\sigma_{r}(T)$. Indeed, if
$\lambda\in\sigma_{a.p.}(T)$ then (see [4]) there is $k\in K$ such that for
any $u\in\varphi^{(-n)}(k)$ and for any $n\in\mathds{N}$ we have

$|w_{n}(u)|\leq|\lambda|^{n}.$ (7)

It is immediate to see from (7) and (2) that $|\lambda|\geq\rho_{min}(T)$.

$(2)\Rightarrow(3)$. This implication is also trivial.

$(3)\Rightarrow(2)$. First notice that $T_{\varphi}C(K)$ is a closed unital
$C^{\star}$ subalgebra of $C(K)$ and therefore $(3)$ implies that the operator
$T_{\varphi}$ is Fredholm and that

$ind(T_{\varphi})=card\\{(p,q):\;p,q\in K,p\neq q,\varphi(p)=\varphi(q)\\}.$

Because the operator of multiplication by $w$ is invertible in $C(K)$ we see
that $T$ is Fredholm and $ind(T)=ind(T_{\varphi})$.

To prove the last statement of the theorem recall that the index of a semi-
Fredholm operator is stable under small norm perturbations. Therefore the set
$A=\\{\lambda\in\rho_{min}(T)\mathds{D}:ind(\lambda I-T)=ind(T)\\}$ is open in
$\mathds{C}$. Assume, contrary to our claim, that there is
$\alpha\in\rho_{min}(T)\mathds{D}\cap\partial A$. The operator $\alpha I-T$ is
upper semi-Fredholm (because $\alpha\in\sigma_{r}(T)$) and $ind(\alpha
I-T)\neq ind(T)$. But it contradicts the stability of index under small norm
perturbations. ∎

If we do not assume that the weight $w$ is invertible in $C(K)$ the
corresponding statement becomes more complicated. Let us denote by $Z(w)$ the
set of all zeros of $w$ in $K$, by $P$ the subset of $Z(w)$ that consists of
$\varphi$-periodic points, and by $S$ the smallest $\varphi^{(-1)}$-invariant
subset of $K$ that contains $P$. Let $K_{1}=K\setminus S$.

######  Theorem 3.4.

Let $\varphi$ be a continuous non-invertible map of a compact Hausdorff space
$K$ onto itself and let $w\in C(K)$. Let $T=wT_{\varphi}$ be the corresponding
weighted composition operator on $C(K)$. The following conditions are
equivalent

  1. (1)

$T$ is lower semi-Fredholm.

  2. (2)

$T$ is Fredholm.

  3. (3)

$\varphi$ is an almost homeomorphism of $K$ and every point of the set $Z(w)$
is isolated in $K$.

Moreover, if $T$ is Fredholm then $\lambda I-T$ is Fredholm for any
$\lambda\in\rho_{min}(T,C(K_{1}))\mathds{D}$ 222It follows from (3) that
$K_{1}$ is a compact subspace of $K$. and

$ind(\lambda I-T)=def(\lambda I-T)=\\\ =card\\{(p,q):\;p,q\in K,p\neq
q,\varphi(p)=\varphi(q)\\}.$ (8)

###### Proof.

$(3)\Rightarrow(2)$. Let $f_{n}\in C(K)$ be such that $\|f_{n}\|=1$ and
$Tf_{n}\rightarrow 0$. Condition (3) guarantees that $|w|>c>0$ on $K\setminus
Z(w)$. Therefore $f_{n}\rightarrow 0$ uniformly on the set $\varphi(K\setminus
Z(w))$. Indeed, $|f_{n}(\varphi(k))|\leq c^{-1}\|Tf_{n}\|,k\in K\setminus
Z(w)$. It follows from the equality $\varphi(K\setminus
Z(w))\cup\varphi(Z(w))=K$ that
$K\setminus\varphi(Z(w))\subseteq\varphi(K\setminus Z(w))$.

Let $B$ be the subset of $\varphi(z(w))$ that consists of points isolated in
$K$. Notice that the set $B$ is either finite or empty. Let
$A=\varphi(z(w)\setminus B$. We claim that $|f_{n}(t)|\leq
c^{-1}\|Tf_{n}\|,t\in A$. Indeed, let us fix $t\in A$. Because the point $t$
is not isolated in $K$ and the set $\varphi(z(w))$ is finite there is a net
$t_{\alpha}$ in $K$ that converges to $t$ and such that
$t_{\alpha}\not\in\varphi(z(w))$. Thus, $|f_{n}(t_{\alpha})|\leq
c^{-1}\|Tf_{n}\|$ and therefore, $|f_{n}(t)|\leq c^{-1}\|Tf_{n}\|,t\in A$. The
sequence $f_{n}$ converges uniformly to zero on the set $K\setminus B$.
Because the set $B$ is either empty or finite and consists of points isolated
in $K$, there is a subsequence of the sequence $f_{n}$ that converges in
$C(K)$. By Proposition 2.1 $T$ is upper semi-Fredholm.

Assume now that there is a sequence $\mu_{n}$ in $C(K)^{\prime}$ such that
$\|\mu_{n}\|=1$ and $\mu_{n}\rightarrow 0$. Let $\nu_{n}=\mu_{n}|Z(w)$ and
$\tau_{n}=\mu_{n}-\nu_{n}$. Then $T^{\prime}\nu_{n}=0$, and therefore
$T^{\prime}\tau_{n}\rightarrow o$. Let

$w_{1}(k)=\left\\{\begin{array}[]{ll}w(k),&\hbox{if $k\in K\setminus
Z(w)$;}\\\ 1,&\hbox{if $k\in Z(w)$,}\end{array}\right.$

and let $T_{1}=w_{1}T_{\varphi}$. Then
$T_{1}^{\prime}\tau_{n}=T^{\prime}\tau_{n}\rightarrow 0$ and by Theorem 3.3
the sequence $\tau_{n}$ contains a convergent subsequence. Because the
sequence $\nu_{n}$ obviously has a convergent subsequence it follows that
$\mu_{n}$ contains a convergent subsequence. By Proposition 2.2 $T$ is lower
semi-Fredholm and therefore, Fredholm.

$(2)\Rightarrow(1)$. This implication is trivial.

$(1)\rightarrow(3)$. Assume $(1)$. If $k\in Z(w)$ and $k$ is not isolated in
$K$ we can find pairwise distinct points $k_{n}\in K$ such that
$|w(k_{n})|\leq\frac{1}{n},n\in\mathds{N}$. The sequence $\delta_{k_{n}}\in
C(K)^{\prime}$ is singular and $T^{\prime}\delta_{k_{n}}\rightarrow 0$, a
contradiction. Let us fix an $\varepsilon>0$ and let

$w_{\varepsilon}(k)=\left\\{\begin{array}[]{ll}w(k),&\hbox{if $k\in K\setminus
Z(w)$;}\\\ \varepsilon,&\hbox{if $k\in Z(w)$,}\end{array}\right.$ (9)

For any sufficiently small $\varepsilon$ the operator
$T_{\varepsilon}=w_{\varepsilon}T_{\varphi}$ is lower semi-Fredholm, and by
Theorem 3.3 $\varphi$ is an almost homeomorphism of $K$.

Having proved the equivalence of (1), (2), and (3) we will prove now the last
statement of the theorem.

Assume one of equivalent conditions (1) - (3). Let
$S=\bigcup\limits_{n=1}^{\infty}\varphi^{(-n)}(P)$. Because $\varphi$ is an
almost homeomorphism and the points of $P$ are isolated in $K$, the set $S$ is
an at most countable open subset of $K$. Hence, the set $K_{1}=K\setminus S$
is a compact subset of $K$ and $\varphi(K_{1})=\varphi^{(-1)}(K_{1})=K_{1}$.

Let $\varepsilon>0$ and let $T_{\varepsilon}$ be as above. By Theorem 3.3 for
every $\lambda\in\rho_{min}(T_{\varepsilon})\mathds{D}$ the operator $\lambda
I-T_{\varepsilon}$ is Fredholm. But
$\rho_{min}(T,C(K_{1}))\leq\rho_{min}(T_{\varepsilon})$ and the operator
$T-T_{\varepsilon}$ is finite dimensional. Hence the last statement of the
theorem follows. ∎

##  4\.  Conditions for $\lambda I-T$ to be lower semi-Fredholm. The case
$|\lambda|>\rho_{mim}(T)$.

We start with the following lemma.

######  Lemma 4.1.

Let $K$ be a compact Hausdorff space and $\varphi:K\rightarrow K$ be a
continuous non-invertible surjection. Let $\Phi$ be the homeomorphism of
$\mathbf{K}$ associated with $\varphi$ (see Definition 2.4). The following
conditions are equivalent

  1. (1)

The set of perfectly $\varphi$-periodic points (see Definition 2.7) is of
first category in K.

  2. (2)

The set of $\Phi$-periodic points is of first category in $\mathbf{K}$.

###### Proof.

Denote by $P_{n}$ (respectively, $\mathbf{P}_{n}$) the set of all
$\varphi$-periodic (respectively, $\Phi$-periodic) points of period $n$.

$(1)\Rightarrow(2)$. Assume (1) and assume to the contrary that for some
$n\in\mathds{N}$ we have $Int\;\mathbf{P}_{n}\neq\emptyset$. Considering, if
necessary, the map $\Phi^{n}$ instead of $\Phi$ we can assume without loss of
generality that $n=1$. The set $E=\\{\mathbf{k}_{0}:\mathbf{k}\in
Int\;\mathbf{P}_{1}\\}$ is an open subset of $P_{1}=\\{s\in K:\varphi(s)=s\\}$
and $\varphi(E)=E$. Therefore, by (1) there is an $s_{1}\in K$ such that
$s_{1}$ is not a $\varphi$-periodic point and $\varphi(s_{1})=k\in E$. Let
$\mathbf{k}\in\mathbf{K}$ be such that $(\mathbf{k}_{n})=k,n\in\mathds{Z}$.
Let $s_{n}:n\geq 2$ be a sequence of points in $K$ such that
$\varphi(s_{n+1})=s_{n},n\in\mathds{N}$. For any $m\in\mathds{N}$ consider the
point $\mathbf{k}^{m}\in\mathbf{K}$ such that

$\mathbf{k}^{m}_{n}=\left\\{\begin{array}[]{ll}k,&\hbox{if $n\geq-m$};\\\
s_{-(n+m)},&\hbox{if $n<-m$.}\end{array}\right.$

The points $\mathbf{k}^{m},m\in\mathds{N}$, are not $\Phi$-periodic and they
converge to $\mathbf{k}$ in $\mathbf{K}$, a contradiction.

$(2)\Rightarrow(1)$. Assume (2) and assume to the contrary that for some
$n\in\mathds{N}$ there is an open in $K$ nonempty subset $E$ of $P_{n}$ such
that $E=\varphi^{(-1)}(E)$. Let
$\mathbf{E}=\\{\mathbf{k}\in\mathbf{K}:\mathbf{k}_{0}\in E\\}$. Then
$\mathbf{E}$ is an open nonempty subset of $\mathbf{K}$ and
$\mathbf{E}\subseteq\mathbf{P}_{n}$, a contradiction. ∎

######  Corollary 4.2.

Let $K$ be a compact Hausdorff space and $\varphi:K\rightarrow K$ be a
continuous non-invertible surjection. Let $\Phi$ be the homeomorphism of
$\mathbf{K}$ associated with $\varphi$ (see Definition 2.4).

If the set of all eventually $\varphi$-periodic points is of first category in
$K$, then the set of $\Phi$-periodic points is of first category in
$\mathbf{K}$.

We will proceed with proving a series of lemmas needed for our main result in
this section, Theorem 4.7.

######  Lemma 4.3.

. Let $K$ be a compact Hausdorff space, $\varphi:K\rightarrow K$ be a
continuous non-invertible surjection, $w\in C(K)$, and $T=wT_{\varphi}$.
Assume that

  1. (1)

$\lambda\in\sigma(T)\setminus\sigma(\mathbf{T})$ and
$|\lambda|>\rho_{min}(T)$.

  2. (2)

The operator $\lambda I-T$ is lower semi-Fredholm.

  3. (3)

The set of eventually $\varphi$-periodic points is of first category in $K$.
Then $K=K_{1}\cup K_{2}$ where $K_{1}$ and $K_{2}$ are nonempty closed
disjoint subsets of $K$ such that

  * •

$\varphi(K_{i})=K_{i},i=1,2$.

  * •

$K_{2}\neq\emptyset$ and $\rho_{min}(T,C(K_{2}))>|\lambda|$.

  * •

The restriction of $\varphi$ on $K_{2}$ is an almost homeomorphism, but not a
homeomorphism of $K_{2}$.

  * •

$\rho(T,C(K_{1}))<|\lambda|$ .

###### Proof.

By (3), Lemma 4.3, and Theorem 3.7 in [4] the spectrum $\sigma(\mathbf{T})$ is
rotation invariant and therefore
$\lambda\mathds{T}\cap\sigma(\mathbf{T})=\emptyset$. By Theorem 3.10 in [4] we
have $\mathbf{K}=\mathbf{K}_{1}\cup\mathbf{K}_{2}$ where $\mathbf{K}_{1}$ and
$\mathbf{K}_{2}$ are disjoint $\Phi$-invariant closed subsets of $\mathbf{K}$,
$\rho(\mathbf{T},C(\mathbf{K}_{1}))<|\lambda|$, the operator $\mathbf{T}$ is
invertible on $C(\mathbf{K}_{2})$ and
$\rho(\mathbf{T}^{-1},C(\mathbf{K}_{2}))<|\lambda|$. Notice that the sets
$\mathbf{K}_{1}$ and $\mathbf{K}_{2}$ cannot be empty because on the one hand,
$\rho_{min}(\mathbf{T})=\rho_{min}(T)>|\lambda|$, while on the other hand,
$\rho(\mathbf{T})=\rho(T)>|\lambda|$. Let $p:\mathbf{K}\rightarrow K$ be the
map defined as $p(\mathbf{k})=\mathbf{k}_{0}$ and let
$K_{1}=p(\mathbf{K_{1}})$ and $K_{2}=p(\mathbf{K}_{2})$. It is immediate to
see that $K_{1}$ and $K_{2}$ are disjoint $\varphi$-invariant closed subsets
of $K$ and that $\rho(T,C(K_{1}))<|\lambda|$ while
$\rho_{min}(T,C(K_{2}))>|\lambda|$. The map $\varphi$ cannot be a
homeomorphism of $K_{2}$ onto itself because by theorem 3.10 in [4] it would
imply that $\lambda\not\in\sigma(T)$, and by Theorem 3.3 it must be an almost
homeomorphism of $K_{2}$. ∎

######  Lemma 4.4.

Let $K$ be a compact Hausdorff space, $\varphi:K\rightarrow K$ be a continuous
non-invertible surjection, $w\in C(K)$, and $T=wT_{\varphi}$. Assume that

  1. (1)

$w$ is invertible in $C(K)$.

  2. (2)

The map $\varphi$ is open.

  3. (3)

$\lambda\in\sigma(T)\cap\sigma(\mathbf{T})$ and $|\lambda|>\rho_{min}(T)$.

  4. (4)

The operator $\lambda I-T$ is lower semi-Fredholm.

  5. (5)

$(\lambda I-\mathbf{T})C(\mathbf{K})=C(\mathbf{K})$.

  6. (6)

The set of eventually $\varphi$-periodic points is of first category in $K$.

Then there are subsets $K_{1}$, $K_{2}$, and $Q$ of $K$ such that

  1. (a)

The set $K_{1}$ is closed in $K$ and
$\varphi(K_{1})=\varphi^{(-1)}(K_{1})=K_{1}$.

  2. (b)

$\rho(T,C(K_{1}))<|\lambda|$.

  3. (c)

The set $K_{2}$ is closed in $K$, $\varphi(K_{2})=K_{2}$,
$\rho_{min}(T,C(K_{2}))>|\lambda|$ and $\varphi$ is an almost homeomorphism of
$K_{2}$.

  4. (d)

If

$K=K_{1}\cup\bigcup\limits_{n=0}^{\infty}\varphi^{(-n)}(K_{2})$

then the set $K_{2}$ is open in $K$.

  5. (e)

The set $Q$,

$Q=K\setminus\Big{(}K_{1}\cup\bigcup\limits_{n=0}^{\infty}\varphi^{(-n)}(K_{2})\Big{)}$

is open in $K$ and $\varphi(Q)=\varphi^{(-1)}(Q)=Q$.

  6. (f)

The sets $K_{1},K_{2}$, and $Q$ are pairwise disjoint and

$K=K_{1}\cup\bigcup\limits_{n=0}^{\infty}\varphi^{(-n)}(K_{2})\cup Q.$

  7. (g)

Assume that $Q\neq\emptyset$ and let $E$ be a closed in $K$ subset of $Q$ and
$V_{1},V_{2}$ be open neighborhoods of $K_{1}$ and $K_{2}$, respectively. Then
there is an $m\in\mathds{N}$ such that for any $n\geq m$ we have
$\varphi^{n}(E)\subset V_{2}$ and $\varphi^{(-n)}(E)\subset V_{1}$.

  8. (h)

Assume that $Q\neq\emptyset$. Then there is an open neighborhood $V$ of
$K_{2}$ such that $\varphi(V\cap Q)\subset V\cap Q$ and the restriction of
$\varphi$ on $V\cap Q$ is one-to-one.

###### Proof.

Conditions (3) and (5) combined with Lemma 4.1 and Theorem 2.6 provide that
$\mathbf{K}$ is the union of three disjoint nonempty $\Phi$ and
$\Phi^{-1}$-invariant subsets $\mathbf{K}_{1},\mathbf{K}_{2}$, and
$\mathbf{O}$ with the properties

  1. (I)

The sets $\mathbf{K}_{1}$ and $\mathbf{K}_{2}$ are closed in $\mathbf{K}$.

  2. (II)

$\mathbf{T}$ is invertible on $C(\mathbf{K}_{2})$ and
$\rho(\mathbf{T}^{-1},C(\mathbf{K}_{2}))<1$.

  3. (III)

$\rho(\mathbf{T},C(\mathbf{K}_{1}))<1$.

  4. (IV)

If $\mathbf{V}_{1}$ and $\mathbf{V}_{2}$ are open neighborhoods of
$\mathbf{K}_{1}$ and $\mathbf{K}_{2}$, respectively, and $\mathbf{E}$ is a
closed subset of $\mathbf{O}$ than there is an $m\in\mathds{N}$ such that for
any $n\geq m$ we have

$\Phi^{n}(\mathbf{E})\subseteq\mathbf{V}_{2}\;\text{and}\;\Phi^{-n}(\mathbf{E})\subseteq\mathbf{V}_{1}.$

Let $p:\mathbf{K}\rightarrow K$ be the map $p(\mathbf{k})=k_{0}$ introduced in
the proof of Lemma 4.3 and let $K_{1}=p(\mathbf{K_{1}})$,
$K_{2}=p(\mathbf{K}_{2})$, and $O=p(\mathbf{O})$. We proceed with proving the
statements (a) - (i) of the lemma.

(a). Obviously the set $K_{1}$ is closed. It follows from the equalities
$\Phi(\mathbf{K}_{1})=\mathbf{K}_{1}$, $\Phi(\mathbf{O})=\mathbf{O}$, and
$\Phi(\mathbf{K}_{2})=\mathbf{K}_{2}$ and the definition of the map $p$ that
$\varphi(K_{1})=K_{1}$, $\varphi(O)=O$, and $\varphi(K_{2})=K_{2}$. Moreover,
it follows from properties (II) - (IV) that $K_{1}$ does not intersect with
$K_{2}\cup O$ and therefore, $\varphi(K_{1})=K_{1}=\varphi^{-1}(K_{1})$.

(b). Follows from (III) and the definition of $p$.

(c). The inequality $\rho(\mathbf{T}^{-1},C(\mathbf{K}_{2}))<1$ follows from
(II). It follows from the fact that the operator $\lambda I-T$ is lower semi-
Fredholm and Theorem 3.3 that $\varphi$ is an almost homeomorphism of $K_{2}$.

(d). It follows from Baire Category Theorem and the fact that $K_{1}\cap
F=\emptyset$ that there is an $n\in\mathds{N}$ such that
$Int\,\varphi^{(-n)}(K_{2})\neq\emptyset$. Then, because $\varphi$ is open,
$Int\,K_{2}\neq\emptyset$. Assume, contrary to our claim, that
$Int\,K_{2}\varsubsetneqq K_{2}$. Let
$\tilde{K}=K\setminus\bigcup\limits_{n=0}^{\infty}\varphi^{(-n)}(Int\,K_{2})$.
Notice that
$\tilde{K}=K_{1}\cup\bigcup\limits_{n=0}^{\infty}\varphi^{(-n)}(E)$ where
$E=K_{2}\setminus Int\,K_{2}$. Applying again Baire category theorem we see
that there is an $n\in\mathds{N}$ such that
$Int_{\tilde{K}}\varphi^{(-n)}(E)\neq\emptyset$. Therefore, there is an open
subset $V$ of $K$ such that $V\cap\varphi^{(-n)}\subset\varphi^{(-n)}$. Then
$\varphi^{n}(V)\cap E\subset E\neq\emptyset$. The set
$Int\,K_{2}\cup\varphi^{n}(V)\cap E$ is open in $K$ and $Int\,K_{2}\subsetneqq
Int\,K_{2}\cup(\varphi^{n}(V)\cap E)$, a contradiction.

(e) - (h). For brevity let
$F=\bigcup\limits_{n=0}^{\infty}\varphi^{(-n)}(K_{2})$. We assume that
$Q\neq\emptyset$, otherwise the statement is trivial. Let $k\in Q$ and let
$E=p^{-1}(\\{k\\})$. $E$ is a compact subset of $\mathbf{O}$ and therefore by
Theorem 2.6 for arbitrary open neighborhoods $\mathbf{V}_{1}$ and
$\mathbf{V}_{2}$ of, respectively, $\mathbf{K}_{1}$ and $\mathbf{K}_{2}$ there
is an $N\in\mathds{N}$ such that $\Phi^{n}(E)\subseteq\mathbf{V}_{2}$ and
$\Phi^{-n}(E)\subseteq\mathbf{V}_{1}$ for any $n\geq N$. Define a function $f$
on $K$ as follows,

$f(x)=\begin{cases}1,&\text{if $x=k$,}\\\ \lambda^{i}/w_{i}(k),&\text{if
$x=\varphi^{(i)}(k),i\in\mathds{N}$,}\\\ \lambda^{-i}w_{i}(x),&\text{if
$x\in\varphi^{(-i)}(\\{k\\}),i\in\mathds{N}$,}\\\ 0,&\text{
otherwise,}\end{cases}$

It is immediate to see that $f$ is an element of $C(K)^{\prime\prime}$ and
$T^{\prime\prime}f=\lambda f$. Because we assumed that $\lambda I-T$ is a
semi-Fredholm operator it follows that (see e.g. [1, Theorem 4.42])
$\ker(\lambda I-T)\neq\mathbf{0}$. Let $f\in C(K)$, $f\neq 0$, and $Tf=\lambda
f$. Let $U=\\{k\in K:\;f(k)\neq 0\\}$. It follows from $Tf=\lambda f$,
$\lambda\neq 0$, and $w\in C(K)^{-1}$ that

$\varphi(U)=U=\varphi^{(-1)}(U)\;\text{and}\;U\cap(K_{1}\cup F)=\emptyset.$
(10)

By Zorn’s lemma there is a maximal by inclusion open subset $V$ of $K$ with
properties (10). We claim that

$V\cup K_{1}\cup F=K.$ (11)

To prove it consider the compact space $G=K\setminus V$. It follows from (10)
that $\varphi(G)=G$ and therefore the operator $T$ is defined on $C(G)$.
Moreover, the operator $\lambda I-T$ is lower semi-Fredholm. Assume that
$K_{1}\cup F\varsubsetneqq G$. Then our previous reasoning shows that there is
$g\in C(G)$ such that $g\neq 0$ and $Tg=\lambda g$. Let $H=\\{k\in G:g(k)\neq
0\\}$. Then the set $V\cup H$ is open in $K$ and satisfies (10) in
contradiction with the maximality of $V$.

We claim that for any compact subset $W$ of $V$ and for any open neighborhoods
$U_{1}$ and $U_{2}$ of $K_{1}$ and $K_{2}$, respectively, there is an
$N\in\mathds{N}$ such that for any $n>N$ we have

$\varphi^{n}(W)\subset U_{2}\;\text{and}\;\varphi^{(-n)}(W)\subset U_{1}.$
(12)

Indeed, (12) follows from the fact that $p^{(-1)}(W)$ is a compact subset of
$\mathbf{O}$.

Notice that it follows from (12) that $clV\cap K_{2}\neq\emptyset$.

Next we claim that there is an open neighborhood $Q$ of $K_{2}$ in $K$ such
that $\varphi(Q\cap V)\subseteq Q\cap V$ and the restriction of $\varphi$ on
$Q\cap V$ is a homeomorphism of $Q\cap V$ onto its image. First notice that
$\ker{(\lambda I-T^{\prime})}=\ker{((\lambda I-T^{\prime}),C(K_{2}))}$.
Indeed, let $\mu\in C(K)^{\prime}$ be such that $T"\mu=\mu$. Let $U$ be an
open neighborhood of $K_{2}$ and let $f\in C(K)$ be such that
$supp\,f\subseteq K\setminus U$. Then $\int fd\mu=\lambda^{-n}\int
T^{n}fd\mu\mathop{\rightarrow}\limits_{n\to\infty}0$, because
$supp\,T^{n}f\subseteq K\setminus\varphi^{(-n)}(U)$,
$\bigcap\limits_{n=1}^{\infty}(K\setminus\varphi^{(-n)}(U))=K_{1}$, and
$\rho(T,C(K_{1}))<|\lambda|$. It follows that $supp\,\mu\subseteq K_{2}$.

Assume, contrary to our claim, that for any open neighborhood $Q$ of $K_{2}$
there are $p,q\in Q\cap V$ such that $\varphi(p)=\varphi(q)$. The proof of
Lemma 5.9 in [5] shows that there is a sequence $\mu_{n}\in(C(K)^{\prime}$
such that $\|\mu_{n}\|=1$, $T^{\prime}\mu_{n}-\lambda\mu_{n}\rightarrow 0$,
and for each $n$ the measure $\mu_{n}$ is a finite linear combination of point
measures $\delta_{s_{i}}$ where $S_{i}\in V$. Because the operator $\lambda
I-T^{\prime}$ is lower semi-Fredholm the sequence $\mu_{n}$ must contain a
norm convergent subsequence. Let the limit of such a subsequence be $\nu$ then
$T^{\prime}\nu=\nu$ and by our previous step $supp\,\nu\subseteq K_{2}$. That
means that for every $n$ we have $|\nu|\wedge|\mu_{n}|=0$, a contradiction.

Thus, there is an open neighborhood $Q$ of $K_{2}$ such that the restriction
of $\varphi$ on $Q\cap V$ is one-to-one. The proofs of Lemma 5.10 and
Corollary 5.11 in [5] show that we can choose $Q$ in such a way that
$\varphi(V\cap Q)\subset V\cap Q$. ∎

We will now consider what happens when the operator $\lambda I-T$ is lower
semi-Fredholm and $\lambda\in\sigma_{a.p.}(\mathbf{T}^{\prime})$. In this case
by Theorem 2.5 there is $\mathbf{k}\in\mathbf{K}$ such that

$|\mathbf{w}_{n}(\mathbf{k})|\leq|\lambda|^{n},\;|\mathbf{w}_{n}(\Phi^{-n}(\mathbf{k}))|\geq|\lambda|^{n},n\in\mathds{N}.$
(13)

######  Lemma 4.5.

Assume conditions of Lemma 4.4. Let the operator $\lambda I-T$ be lower semi-
Fredholm and $\lambda\in\sigma_{a.p.}(\mathbf{T}^{\prime})$. Let $\mathbf{S}$
be the set of $\Phi$-strings defined as

$\mathbf{S}=\\{\mathbf{s}:\mathbf{s}=\\{\Phi^{n}(\mathbf{k}):n\in\mathds{Z}\\}\\\
|\mathbf{w}_{n}(\mathbf{k})|\leq|\lambda|^{n},\;|\mathbf{w}_{n}(\Phi^{-n}(\mathbf{k}))|\geq|\lambda|^{n},n\in\mathds{N}\\}.$

Then the set $\mathbf{S}$ is finite and for every
$\mathbf{s}=\\{\Phi^{n}(\mathbf{k}):n\in\mathds{Z}\\}\in\mathbf{S}$ the point
$\mathbf{k}$ is isolated in $\mathbf{K}$.

###### Proof.

Let $\mathbf{k}$ be a point in $\mathbf{K}$ such that inequalities (13) are
satisfied. Let $\\{k_{n}:n\in\mathds{Z}\\}$ be the corresponding
$\varphi$-string. We have to consider several cases.

Case 1. The set $\\{k_{n}:n\in\mathds{N}\\}$ is infinite, i.e. the point
$k_{0}$ is not eventually periodic. We claim that

$\sum\limits_{n=1}^{\infty}\frac{|\lambda|^{n}}{|w_{n}(k_{-n})|}+\sum\limits_{n=0}^{\infty}\frac{|w_{n}(k_{0})|}{|\lambda|^{n}}<\infty.$
(14)

To prove (14) assume first to the contrary that

$\sum\limits_{n=1}^{\infty}\frac{|\lambda|^{n}}{|w_{n}(k_{-n})|}=\infty.$ (15)

For every $m\in\mathds{N}$ consider the discrete measure
$\mu_{m}=\sum\limits_{n=-m}^{0}\frac{\lambda^{n}}{w_{n}(k_{n})}\delta_{k_{n}}$.
Then

$\|\mu_{m}\|=\sum\limits_{n=0}^{m}\frac{|\lambda|^{n}}{|w_{n}(k_{-n})|}\mathop{\rightarrow}\limits_{m\rightarrow\infty}\infty$
(16)

and

$\|T^{\prime}\mu_{m}-\lambda\mu_{m}\|=\|\frac{\lambda^{m}}{w_{m}(k_{-m})}\delta_{k_{-m}}+w(k_{0})\delta_{k_{1}}\|\leq
1+\|w\|_{C(K)}.$ (17)

Let $\nu_{m}=\mu_{m}/\|\mu_{m}\|$ It follows from (16) and (17) that we can
find positive integers $m_{l}$ such that
$\|T^{\prime}\nu_{m_{l}}-\lambda\nu_{m_{l}}\|\mathop{\rightarrow}\limits_{l\rightarrow\infty}0$
and the sequence $\nu_{m_{l}}$ is singular, in contradiction with our
assumption that the operator $\lambda I-T$ is lower semi-Fredholm.

Similarly we can prove that

$\sum\limits_{n=0}^{\infty}\frac{|w_{n}(k_{0})|}{|\lambda|^{n}}<\infty.$

Let

$\mu=\sum\limits_{n=1}^{\infty}\frac{\lambda^{n}}{w_{n}(k_{-n})}\delta_{k_{-n}}+\sum\limits_{n=0}^{\infty}\frac{w_{n}(k_{0})}{\lambda^{n}}\delta_{k_{n}}.$

Then $T^{\prime}\mu=\lambda\mu$. We claim that the point $k_{0}$ is isolated
in $K$. Indeed, otherwise using (14) and the fact that the map $\varphi$ is
open we can find points $k_{i,m},m\in\mathds{N},-m\leq i\leq m$ such that

  1. (a)

$\varphi(k_{i,m}=k_{i+1,m},m\in\mathds{N},-m\leq i\leq m-1$.

  2. (b)

The points $k_{i,m}$ are pairwise distinct.

  3. (c)

$\mu_{m}-\mu\mathop{\rightarrow}\limits_{m\rightarrow\infty}0$, where

$\mu_{m}=\sum\limits_{i=-m}^{-1}\frac{\lambda^{|i|}}{w_{i}(k_{m,i})}\delta_{k_{m,i}}+\sum\limits_{i=0}^{m}\frac{w_{i}(k_{m,i})}{\lambda^{i}}\delta_{k_{m,i}}.$

Let $\nu_{m}=\mu_{m}/\|\mu_{m}\|$. It follows from (a) - (c) that
$T^{\prime}\nu_{m}-\lambda\nu_{m}\rightarrow 0$ and the sequence $\nu_{m}$ is
singular in contradiction with $\lambda I-T$ assumed to be lower semi-
Fredholm.

Case 2. The set $\\{k_{n}:n\in\mathds{Z}\\}$ is infinite but the set
$\\{k_{n}:n\in\mathds{N}\\}$ is finite. In other words, there is
$s\in\mathds{Z}$ such that the point $k_{s}$ is $\varphi$-periodic, but the
point $k_{s-1}$ is not. Let $p$ be the period of the point $k_{s}$. There are
two possibilities.

(1) $w_{p}(k_{s})=\lambda^{p}$. Let

$\mu=\sum\limits_{i=0}^{p-1}\lambda^{p-i-1}(T^{\prime})^{i}\delta_{s}.$

Then $T^{\prime}\mu=\lambda\mu$.

(2) $w_{p}(k_{s})\neq\lambda^{p}$. Like in case 1 we can prove that

$\sum\limits_{n=1}^{\infty}\frac{|\lambda|^{n}}{|w_{n}(k_{s-n})|}<\infty.$

Let

$\mu=(\lambda^{p}-w_{p}(k_{s})\sum\limits_{n=1}^{\infty}\frac{\lambda^{n}}{|w_{n}(k_{s-n})|}\delta_{k_{s-n}}+\sum\limits_{i=0}^{p-1}\lambda^{p-i-1}(T^{\prime})^{i}\delta_{s}.$

Then $T^{\prime}\mu=\lambda\mu$.

In both cases we can prove that the point $k_{s}$ is isolated in $K$ using the
same reasoning as in Case 1.

Case 3. The set $\\{k_{n}:n\in\mathds{Z}\\}`$ is finite. Assume that the point
$k_{0}$ is not isolated in $K$. Then there are the following possibilities.

1\. For every open neighborhood $V$ of $k_{0}$ and for every $n\in\mathds{N}$
the neighborhood $V$ contains either a point that is not $\varphi$-periodic or
a $\varphi$-periodic point of period at least $n$. In this case we can
construct a singular sequence $\mu_{n}$ such that
$T^{\prime}\mu_{n}-\lambda\mu_{n}\rightarrow 0$.

Indeed we can find points $t_{n,i}\in K,n\in\mathds{N}-n\leq i\leq n$ such
that

  1. (I)

$\varphi(t_{n,i})=t_{n,i+1},-n\leq i<n$,

  2. (II)

The points $t_{n,i},n\in\mathds{N},-n\leq i\leq n$ are pairwise distinct,

  3. (III)

$|w_{j}(t_{n,0})|\leq 2|\lambda|^{n}$ and $|w_{j}(t_{n,-j})|\geq
1/2|\lambda|^{n},,j=1,\ldots,n$.

Let

$\mu_{n}=\sum\limits_{j=0}^{n-1}\big{(}1-\frac{1}{\sqrt{n}}\big{)}^{j}\lambda^{-j}w_{j}(t_{j,0})\delta_{t_{j,n}}+\sum\limits_{j=1}^{n-1}\big{(}1-\frac{\lambda^{j}}{\sqrt{n}}\big{)}^{j}\frac{1}{w_{j}(t_{n,-j})}\delta_{t_{n,-j}}.$
(18)

It is not difficult to see from (III) and (18) that

$\|T^{\prime}\mu_{n}-\lambda\mu_{n}\|=o(\|\mu_{n}\|),n\rightarrow\infty.$ (19)

But the measures $\mu_{n}$ are pairwise disjoint in virtue of II and therefore
the sequence $\nu_{n}=\frac{\mu_{n}}{\|\mu_{n}\|}$ is singular, a
contradiction.

2\. There are an open neighborhood $V$ of $k_{0}$ and an $n\in\mathds{N}$ such
that every point in $V$ is $\varphi$-periodic and has period less or equal to
$n$. That obviously contradicts our assumption that the set of eventually
$\varphi$-periodic points is of first category in $K$.

Thus it follows from our assumption that the set of all eventually
$\varphi$-periodic points is of first category that only the first case is
possible. Moreover, because in this case to each $\mathbf{s\in\mathbf{S}}$
there is a unique up to a constant factor discrete measure $\mu$ on the set
$p(\mathbf{s})$ such that $T^{\prime}\mu=\lambda\mu$, we see that the set
$\mathbf{S}$ is finite. ∎

######  Corollary 4.6.

Assume conditions of Lemma 4.4. Let the operator $\lambda I-T$ be lower semi-
Fredholm and let the set $\mathbf{S}$ be not empty. Then there is a countable
open subset $S$ of $K$ such that $S$ is the union of finite number of strings,
$\varphi(K\setminus S)=K\setminus S$ and the operator $\lambda I-T$ is lower
semi-Fredholm on $C(K\setminus S)$.

###### Proof.

We take $S=p(\mathbf{S})$ and apply Lemma 4.5. ∎

Finally we can state our main result.

######  Theorem 4.7.

Let $K$ be a compact Hausdorff space and $\varphi$ be an open continuous non-
invertible map of $K$ onto itself. Let $w$ be an invertible element of the
algebra $C(K)$. Assume that the set of all eventually $\varphi$-periodic
points is of first category in $K.$ Let

$(Tf)(k)=w(k)f(\varphi(k)),f\in C(K),k\in K.$

Let $\lambda\in\mathds{C}$ be such that $\lambda\in\sigma(T)$ and
$|\lambda|>\rho_{min}(T)$.

The following conditions are equivalent.

(1) The operator $\lambda I-T$ is lower semi-Fredholm.

(2) One of the following conditions is satisfied.

  1. (I)

The compact space $K$ is the union of two disjoint $\varphi$-invariant clopen
subsets $K_{1}$ and $K_{2}$ such that $\rho(T,C(K_{1}))<|\lambda|$,
$\rho_{min}(T,C(K_{2}))>|\lambda|$, and $\varphi$ is an almost homeomorphism
but not a homeomorphism of $K_{2}$ onto itself.

  2. (II)

There are closed subsets $K_{1}$ and $K_{2}$ of $K$ such that
$\varphi^{(-1)}(K_{1})=\varphi(K_{1})=K_{1}$, $\rho(T,C(K_{1})<|\lambda|$,
$\varphi(K_{2})=K_{2}$, $\rho_{min}(T,C(K_{2}))>|\lambda|$, and $\varphi$ is
an almost homeomorphism but not a homeomorphism of $K_{2}$ onto itself.

Moreover, the set $K_{2}$ is a clopen subset of $K$,
$K_{2}\subsetneqq\varphi^{(-1)}(K_{2})$, and

$K=K_{1}\cup\bigcup\limits_{n=0}^{\infty}\varphi^{(-n)}(K_{2}).$

  3. (III)

$K=K_{1}\cup\bigcup\limits_{n=0}^{\infty}\varphi^{(-n)}(K_{2})\cup Q,$

where the sets $K_{1}$, $K_{2}$, and $Q$ have properties described in the
statement of Lemma 4.4.

  4. (IV)

$K=K_{1}\cup\bigcup\limits_{n=0}^{\infty}\varphi^{(-n)}(K_{2})\cup Q\cup S,$

where the sets $K_{1}$, $K_{2}$, and $Q$ have properties described in the
statement of Lemma 4.4 and $S$ is an open countable subset of $K$ disjoint
with the set $K_{1}\cup\bigcup\limits_{n=0}^{\infty}\varphi^{(-n)}(K_{2})\cup
Q$. Moreover, the set $S$ is the union of finite number of $\varphi$-strings
and for every $s\in S$ we have

$|w_{n}(s_{0})|\leq|\lambda|^{n},\;|w_{n}(s_{-n})|\geq|\lambda|^{n},n\in\mathds{N}$

###### Proof.

The implication $(1)\Rightarrow(2)$ follows from Lemmas 4.3 \- 4.5.

To prove the implication $(2)\Rightarrow(1)$ we will prove that every of
conditions (I) - (IV) implies (2).

$(I)\Rightarrow(2)$. This implication follows from Theorem 3.3.

$(II)\Rightarrow(2)$. First we notice that (II) implies that
$\lambda\in\sigma_{r}(T)$, and therefore the image $(\lambda I-T)C(K)$ is
closed in $C(K)$. Indeed, assume to the contrary that
$\lambda\in\sigma_{a.p.}(T)$. Then (see [4]) there is $k\in K$ such that

$|w_{n}(k)|\geq|\lambda|^{n}\;\text{and}\;\forall
t\in\varphi^{(-n)}(k),|w_{n}(t)|\leq|\lambda|^{n},n\in\mathds{N}.$ (20)

But it is immediate to see that the existence of a point $k$ satisfying (20)
contradicts condition (II).

Assume that $\mu\in C(K)^{\prime}$, $\mu\neq 0$, and
$T^{\prime}\mu=\lambda\mu$. The proof of Theorem 5.14 in [5] shows that
$supp\;\mu\subseteq K_{2}$. Because $\varphi$ is an almost homeomorphism of
$K_{2}$ we see that $def(\lambda I-T)<\infty$.

$(III)\Rightarrow(2)$. Assume (III) and assume to the contrary that there is a
singular sequence $\mu_{n}\in C(K)^{\prime}$ such that
$T^{\prime}\mu_{n}-\lambda\mu_{n}\rightarrow 0$. From the previous step we
conclude that without loss of generality we can assume that
$supp\;\mu_{n}\subset Q$. Assume first that the compact space $K$ is
extremally disconnected. Then the restriction of $\varphi$ on $clV$ is one-to
one. Then we come to contradiction as in the proof of Theorem 5.14 in [5].

If $K$ is not extremally disconnected we can consider the operator
$T^{\prime\prime}$ on the second dual $C(K)^{\prime\prime}\cong C(\mathds{Q})$
where the compact space $\mathds{Q}$ is extremally disconnected. Notice that
$T^{\prime\prime}=w^{\prime\prime}T_{\psi}$ where $\psi$ is a continuous map
of $\mathds{Q}$ onto itself. It is not difficult to see that $\psi$ satisfies
condition (III).

$(IV)\Rightarrow(2)$. Assume (IV) and assume to the contrary that there is a
singular sequence $\mu_{n}\in C(K)^{\prime}$ such that
$T^{\prime}\mu_{n}-\lambda\mu_{n}\rightarrow 0$. Without loss of generality we
can assume that $supp\mu_{n}\subseteq S$ Let $S^{\star\star}$ be the set
$j^{-1}(S)$ where $j:Q\rightarrow K$ is the surjection corresponding to the
isometric embedding of $C(K)$ into C(Q). The set $S^{\star\star}$ is a finite
union of $\psi$-strings and points of $S^{\star\star}$ are isolated in $Q$.
The map $\psi$ extends to a homeomorphism of $clS^{\star\star}$ onto itself.
It follows from Theorem 2.11 in [5] that the sequence $\mu_{n}$ contains a
convergent subsequence, a contradiction. ∎

######  Corollary 4.8.

Assuming one of conditions (I) - (IV) from Theorem 4.7 is satisfied, the
defect of $\lambda I-T$ can be computed as

$def(\lambda I-T)=card(\\{(p,q):p,q\in K_{2},p\neq
q,\varphi(p)=\varphi(q)\\})+card(\mathbf{S}),$ (21)

where $\mathbf{S}$ is the set introduced in the statement of Lemma 4.5.

In particular, $def(\lambda I-T)=0$, i.e. $(\lambda I-T)C(K)=C(K)$ if the
following two conditions are satisfied

  * •

The set $\mathbf{S}$ is empty,

  * •

The map $\varphi$ is a homeomorphism of $K_{2}$ onto itself. 333See also
Theorem 5.14 in [5]

From Theorem 4.7 easily follows the following criterion for the operator
$\lambda I-T$ (where $\lambda>\rho_{min}(T)$) to be Fredholm.

######  Theorem 4.9.

Let $K$ be a compact Hausdorff space, $\varphi$ be a continuous open non-
invertible map of $K$ onto itself. Let $w\in C(K)^{-1}$ and

$(Tf)(k)=w(k)f(\varphi(k)),\;f\in C(K),k\in K.$

Assume that the set of all eventually $\varphi$-periodic points is of first
category in $K$. Let $\lambda\in\sigma(T)$ and $\lambda>\rho_{min}(T)$. The
following conditions are equivalent.

(1) The operator $\lambda I-T$ is Fredholm.

(2) One of conditions (I) - (IV) is satisfied. Moreover, if the set $Q$ is not
empty, then every point of $Q$ is isolated in $K$ and there is a finite subset
$\\{k_{1},\ldots k_{p}\\}$ of $Q$ such that the sets $A_{1},\ldots,A_{p}$ are
pairwise disjoint and $Q=\bigcup\limits_{j=1}^{p}A_{j}$, where $A_{j}$ is the
smallest $\varphi$ and $\varphi^{(-1)}$ invariant subset of $Q$ that contains
$k_{j}$, i.e.
$A_{j}=\bigcup\limits_{n=0}^{\infty}\bigcup\limits_{m=1}^{\infty}\varphi^{(-m)}(\varphi^{n}(k_{j}))$.

######  Corollary 4.10.

Assume condition (2) of Theorem 4.9. Then

$ind(\lambda I-T)=p-def(\lambda I-T),$ (22)

where $p$ is from the statement of Theorem 4.9 and $def(\lambda I-T)$ is
defined by (21).

######  Corollary 4.11.

Assume either conditions of Theorem 4.7, or of Theorem 4.9. Then the spectrum
$\sigma_{sfl}(T)$ or, respectively, the Fredholm spectrum $\sigma_{f}(T)$ is
rotation invariant

######  Corollary 4.12.

Let $K$ be a compact Hausdorff space, $\varphi$ be a continuous open map of
$K$ onto itself. Let $w\in C(K)^{-1}$ and

$(Tf)(k)=w(k)f(\varphi(k)),\;f\in C(K),k\in K.$

Assume that the set of all eventually $\varphi$-periodic points is of first
category in $K$. Assume that either $|\lambda|=\rho(T)$ or
$|\lambda|=\rho_{min}(T)$. Then the operator $\lambda I-T$ cannot be lower
semi-Fredholm.

###### Proof.

Let $|\lambda|=\rho(T)$. If the operator $\lambda I-T$ is semi-Fredholm then
by the punctured neighborhood theorem (see [9]) $\lambda$ is an isolated point
in $\sigma(T)$. On the other hand, it follows from our assumption that the set
of all eventually $\varphi$-periodic points is of first category in $K$ that
$\sigma(T)$ is rotation invariant (see [4]). As $\rho(T)>0$, we have a
contradiction.

Next, assume that $|\lambda|=\rho_{min}(T)$ and the operator $\lambda I-T$ is
lower semi-Fredholm. Then there are $\lambda_{n}\in\mathds{C},n\in\mathds{N}$
such that $|\lambda_{n}|\downarrow|\lambda|$ and either
$\lambda_{n}\not\in\sigma(T),n\in\mathds{N}$, or $\lambda_{n}\in\sigma(T)$ and
the operator $\lambda_{n}I-T$ is lower semi-Fredholm for any $n\in\mathds{N}$.

Applying in the first case Theorem 3.10 from [4] and in the second case
Theorem 4.7 we can see that there is a closed subset $K_{\infty}$ of $K$ such
that $\varphi(K_{\infty})=K_{\infty}$ and
$\rho(T,C(K_{\infty}))=|\lambda|=\rho_{min}(T)$.

We claim that the operator $\lambda I-T$ considered on the space
$C(K_{\infty})$ is lower semi-Fredholm. Indeed, otherwise there is a singular
sequence $\mu_{n}\in C(K_{\infty})^{\prime}$ such that $\|\mu_{n}\|=1$ and
$T^{\prime}\mu_{n}-\lambda\mu_{n}\rightarrow 0$. Considering the measures
$\mu_{n}$ as elements of $C(K)^{\prime}$ we come to a contradiction.

Applying the punctured neighborhood theorem to the operator $\lambda I-T$ on
$C(K_{\infty})$ we see that the set $\sigma(T,C(K_{\infty}))$ is not rotation
invariant. By Theorem 3.12 from [4] there is a $\varphi$-periodic point $k\in
K_{\infty}$ such that $|w_{p}(k)|<|\lambda|^{p}$, where $p$ is the period of
$k$. But the last inequality contradicts the definition of $\rho_{min}(T)$. ∎

To state our next result we have to recall the following definition introduced
in [3]

######  Definition 4.13.

Let $K$ be a compact Hausdorff space. We say that $K\in AH$ if every almost
homeomorphism of $K$ onto itself is a homeomorphism.

######  Theorem 4.14.

Let $K$ be a compact Hausdorff space, $\varphi$ be a continuous non-invertible
map of $K$ onto itself, and $w\in C(K)^{-1}$. Assume that the set of all
eventually $\varphi$-periodic points is of first category in $K$.

Assume that $K\in AH$. Then,

$\sigma_{f}(T)=\sigma(T).$

###### Proof.

Consider $\lambda\in\sigma(T)$. we have to consider three cases.

  1. (1)

$|\lambda|<\rho_{min}(T)$. The operator $\lambda I-T$ cannot be Fredholm by
Theorem 3.3.

  2. (2)

$|\lambda|=\rho_{min}(T)$ or $|\lambda|=\rho(T)$. The operator $\lambda I-T$
cannot be Fredholm by Corollary 4.12

  3. (3)

$\rho_{m}in(T)<|\lambda|<\rho(T)$. The operator $\lambda I-T$ cannot be
Fredholm by Theorem 4.9.

∎

While a complete characterization of the class AH remains unknown, numerous
sufficient conditions guaranteeing that $K\in AH$ were obtained by Friedler
and Kitover in [3] and by Vermeer in [10]. We list some of this conditions in
Corollary 4.15.

######  Corollary 4.15.

Let $K$ be a compact Hausdorff space, $\varphi$ be a continuous non-invertible
map of $K$ onto itself, and $w\in C(K)^{-1}$. Assume that the set of all
$\varphi$-periodic points is of first category in $K$.

Assume one of the following conditions.

  * •

The compact space $K$ is extremally disconnected and has no isolated points.

  * •

The compact space $K$ is an $F$-space without isolated points that satisfies
the countable chain condition.

  * •

The compact space $K$ is arcwise connected and one of the following conditions
is satisfied

    1. (a)

The fundamental group $\Pi_{1}(X)$ is finite.

    2. (b)

The fundamental group $\Pi_{1}(X)$ is abelian.

    3. (c)

The fundamental group $\Pi_{1}(X)$ is finitely generated.

  * •

$K$ is a convex, compact subset of a linear topological space.

  * •

$K$ is a compact, arcwise connected subset of the plane $\mathds{R}^{2}$ such
that $\mathds{R}^{2}\setminus K$ consists of a finite number of components.

  * •

$K$ is a locally simply connected compact subset of $\mathds{R}^{2}$.

  * •

$K=X\times Y$ where $X$ and $Y$ are locally connected compact spaces that have
no isolated points.

Then $\sigma_{f}(T)=\sigma(T)$.

## References

  * [1] Abramovich Y.A. and Aliprantis C.D., An invitation to operator theory, AMS, 2002.
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