TITLE: Map induced between Pontryagin duals
QUESTION [3 upvotes]: Let $f\colon A\to B$ be a group homomorphism between finite abelian groups.
For abelian group $G$, let $G^\wedge=\operatorname{Hom}_\mathbb{Z}(G,\mathbb{Q}/\mathbb{Z})$ be its Pontryagin dual.
Since $A,B$ are finite abelian groups, we have $A^\wedge=A, B^\wedge=B$.
Now, my question is :
How can we describe $f^\wedge\colon B\to A$ in terms of $f$?
Added: Can we relate $\operatorname{Ker} f^\wedge$, $\operatorname{Coker} f^\wedge$ with $\operatorname{Ker} f $, $\operatorname{Coker} f$?

REPLY [3 votes]: It is important to note that while a finite group is isomorphic to its Pontryagin dual, the isomorphism is non-canonical. To get a map $f^{\wedge} : B \to A$ using $f$ you need to fix an isomorphism $\phi$ from $A$ to $A^{\wedge}$, and $\psi$ from $B$ to $B^{\wedge}$. Then we have
$$f^{\wedge} = \phi^{-1} \circ f^{*} \circ \psi $$
where $f^{*}$ is the canonical map $B^{\wedge} \to A^{\wedge}$.
Added:
As for kernels and cokernels, note that once we have fixed the above isomorphisms, we have a homomorphism $\text{Ker }f \to \text{Coker } f^{\wedge}$ sending $k$ to $k + \text{Im } f^{\wedge}$. A back-of-envelope diagram chase suggests that this map is an isomorphism, and the same works for the other two groups.


text:
\begin{document}
\dedicatory{To my parents and my sister}
\address[Garrisi Daniele]{
Pohang Mathematics Institute,
Department of Mathematics, Postech
San 31, Hyojadong, Namgu, Pohang, Gyungbuk 790-784
Republic of Korea}
\tableofcontents
\title[Spectral flow in Banach spaces]
{On the spectral flow for paths of essentially hyperbolic bounded
operators on Banach spaces}

\begin{abstract}
We give a definition of the spectral flow for paths of
bounded essentially hyperbolic operators on a Banach space. The spectral flow
induces a group homomorphism on the fundamental group of every connected
component of the space of essentially hyperbolic operators. We prove that this
homomorphism completes the exact homotopy sequence of a Serre fibration.
This allows us to characterize its kernel and image and produce examples of
spaces where it is not injective or not surjective, unlike
what happens for Hilbert spaces. For a large class of paths, namely
the essentially splitting, the spectral flow of $ A $ coincides with
$ -\ind(F_A) $, the Fredholm index of the differential operator
$ F_A (u) = u' - A u $.
\end{abstract}
\author[Garrisi,~D.]{Garrisi Daniele}
\thanks{This work was supported by Priority
Research Centers Program through the National Research Foundation of Korea
(NRF) funded by the Ministry of Education, Science and Technology
(Grant 2009-0094068) and from the Scuola Normale Superiore of Pisa}
\keywords{Spectral flow, projectors, hyperplanes}
\subjclass{46B20,58B05,58E05,58J30}
\maketitle
\section*{Introduction}
The spectral flow appeared first in \cite{APS75} for a family
of elliptic and self-adjoint operators $ A_t $, ascribed to a
joint work of M.~Atiyah and G.~Lusztig. We outline their effective description
as ``net number of eigenvalues that change sign (from $ - $ to $ + $)
while the parameter family is completing a period'' in the definition
given by J.~Robbin and D.~Salamon in \cite[Theorem~4.21]{RS95}: In
a neighbourhood $ [t_-,t_+] $ of the real line of a point $ t\in\R $
(called ``crossing'') such that $ 0\in\sigma(A_t) $, $ \sigma(A_s) $ can be
described as a finite family of continuously differentiable curves
$ \lambda_i\colon [t_-,t_+]\ra\R $ such that $ \lambda_i ' (t)\neq 0 $.
The contribution to the spectral flow of a crossing is given by
\[
\sum_{\lambda_i} \sign(\lambda_i (t_+)) - \sign(\lambda_i (t_-))
\]
and the spectral flow is the sum of these contributions over all the
crossings. The spectral flow was used to define a Morse index for
1-periodic hamiltonian orbits and then define the Floer homology.
Such construction requires some differentiability
hypotheses and transversality condition, even though such requirements
are dense in continuous families. P.~Rabier extended their work
for unbounded families of operators in Banach spaces in \cite{Rab04}.
In \cite{Phi96},
J.~Phillips semplified their definition as follows:
$ A_t \in\mathcal{F}\sp{sa} (H) $
was assumed to be a path of Fredholm and self-adjoint operators,
on $ [0,1] $. If $ U $ is a neighbourhood of the origin such that
\begin{itemize}
\item[1)] $ \sigma(A_s)\cap \p U = \emptyset $ for every $ s\in [t_-,t_+] $;
\item[2)] $ \sigma(A_s)\cap U $ is a finite set of eigenvalues,
\end{itemize}
then the contribution to the spectral flow on the interval $ [t_-,t_+] $
is defined as
\[
\dim(P(A(t_+);U)) - \dim(P(A(t_-);U)).
\]
The spectral flow is the sum over a partition of the unit inverval,
$ J_i $, where a neighbourhood $ U_i $ as in 1,2) exists. It is
invariant for fixed-endpoints homotopies and defines a group homomorphism on
each connected component of $ \mathcal{F}\sp {sa} (H) $. If $ H $ is
separable, there are only three, corresponding to $ I $ and $ -I $,
contractible to a point, and $ 2P - I $, where $ P $ is a projector
with infinite-dimensional kernel and image. On the third one the
spectral flow
\[
\Sf\colon\pi_1 (\mathcal{F}\sp{sa} _*)\ra\mb{Z}
\]
is a group isomorphism. J.~Phillips extended afterwards his definition
in \cite{BBLP05} to families of unbounded, Fredholm and self-adjoint operators.
C.~Zhu and Y.~Long in \cite{ZL99} extended such definition to bounded,
admissible operators on Banach spaces. Given a partition $ J_i $
of $ [0,1] $ and spectral sections, that is
\begin{gather*}
Q_i\colon J_i\ra\mathcal{P}(\mathcal{L}(E))\\
Q_i (t) - P\sp + (A(t)) \text{ is compact for } t\in J_i,
\end{gather*}
the contribuition of $ J_i $ to the spectral flow is
\[
[Q_i (t\sp i _-) - P\sp + (A(t\sp i _-))] -
[Q_i (t\sp i _+) - P\sp + (A(t\sp i _+))],
\]
where $ [Q - P] $ denotes the Fredholm index of
$ P\colon\ran(Q)\ra\ran(P) $. The purposes of this work are essentially
three:\par
I) Semplifying further the definition of spectral flow. Given a path
$ A_t \in e\mathcal{H}(E) $ of essentially hyperbolic operators (which
are compact perturbation of hyperbolic operators and correspond to
the admissible ones used in \cite{ZL99}), by the homotopy lifting
property of the Serre fibration
\[
\prc\colon \mathcal{P}(\mathcal{L}(E))\ra\mathcal{P}(\mathcal{C}(E)).
\]
we prove that there exists a continuous path of projectors on $ [0,1] $
such that $ P(t) - P\sp + (A(t)) $ is compact. Thus, we define
\[
\Sf(A) = [P(0) - P\sp + (A(0))] - [P(1) - P\sp + (A(1))].
\]
Therefore, we do not need to partitionate the unit interval as long as
we do not require $ P(t) $ to be a spectral projector of $ A(t) $.
Such definition coincides with the one in \cite{ZL99}.
In \S\ref{Three}, we show that is invariant for fixed-endpoints homotopies
and that $ \Sf(A*B) = \Sf(A) + \Sf(B) $.\par
II) Studying the homomorphism properties of the spectral flow.
We prove that $ e\mathcal{H}(E) $ is homotopically equivalent to
$ \mathcal{P}(\mathcal{C}(E)) $, the space of projectors of the Calkin
algebra and that the spectral flow completes the exact homotopy sequence
of the fibration above. That allows us to characterize the kernel and the
image of the spectral flow $ \Sf_P $ defined on the fundamental group of the
connected component of $ 2P - I\in e\mathcal{H}(E) $. Precisely,
\begin{itemize}
\item[h1)] an integer $ m $ belongs to the image of $ \Sf_P $ if and only
if there exists a projector $ Q $ connected to $ P $ by a path in the space
of projectors $ \mathcal{P}(\mathcal{L}(E)) $, such that $ Q - P $ is compact
and $ [P - Q] = m $;
\item[h2)] $ \text{im}(\prc_*)\cong\ker(\Sf_P) $.
\end{itemize}
The existence of a projector satisfying
properties h1,2) highly depends on the structure of the Banach space $ E $.
When $ E $ is Hilbert, J.~Phillips proved in \cite{Phi96} that for every
projector with infinite-dimensional range and kernel, h1) with
$ m = 1 $ and h2) hold.
We show that this happens for the spaces $ \ell\sp p $ and
$ \ell\sp\infty $. For the spaces satisfying the hypotheses of
Proposition~\ref{prop:surjective} and Proposition~\ref{prop:injective},
such a projector exists. The question is strongly related with
the existence of complemented subspaces isomorphic to closed subspaces
of co-dimension $ m $. In direct sums of spaces isomorphic to hyperplanes
$ \Sf_P $, where $ P $ is a projector over each of the summands,
is surjective; in spaces isomorphic to hypersquares, but not hyperplanes, as
the ones constructed by W.~T.~Gowers and B.~Maurey in \cite{GM97},
the image is $ 2\mb{Z} $ and so on. In a Douady space (cf. \cite{Dou65}),
we prove that $ \Sf_P $ can be not injective or trivial, even in
projectors of infinite-dimensional range and kernel.\par
III) Comparing the spectral flow with the Fredholm index of the
differential operator
\[
F_A\colon W\sp{1,p} (\R)\ra L\sp p (\R),\ \ u\mapsto
\left(\frac{d}{dt} - A(t)\right)u,
\]
We prove in Theorem~5.6 that for a large class of path, which are essentially
hyperbolic and \textsl{essentially splitting} (cf. \cite{AM03}), the equality
\[
\ind(F_A) = - \Sf(A)
\]
holds. The equality above applies, for instance, in the special case where
$ A $ is a continuous, compact perturbation of a path of hyperbolic operators
with some boundary conditions (check \cite[Theorem~E]{AM03}).
That confirms the guess of A.~Abbondandolo and P.~Majer in \S7
of \cite{AM03} that for these paths the equality above hold.
\vskip .4em
We remark that our work deals with \textsl{bounded} operators. The
differential operator $ F_A $ arises naturally from the linearization of
a vector field $ \xi\in C\sp 1 (E,E) $ on a solution
$ v'(t) = \xi(v(t)) $, such that the endpoints are zeroes of
$ \xi $ and $ A(t) = D\xi(v(t)) $. If $ F_A $ is Fredholm and surjective
(and the zeroes are hyperbolic), then the set
\[
W_\xi (p,q) = \{v\colon\R\ra E:v'(t) = \xi(v(t)),
v(-\infty) = p,v(+\infty) = q\}
\]
is a sub-manifold of dimension $ \ind(F_A) $. That constitutes a
landmark for the study of the Morse theory on Banach
manifolds, as in \cite{AM07}. A proof of this can be found in
\cite[\S8]{AM03}. If $ A $ fulfills the hypotheses of Theorem~\ref{thm:final},
$ \Sf(A) $ determines $ \ind(F_A) $.\vskip .4em
The spectral flow also provides naturally an index for 1-periodic solutions
of $ u' (t) = X(u(t)) $. If $ DX(u_t) $ is essentially hyperbolic,
then $ \mathrm{index}_X (u) = \Sf(DX (u_t)) $. Thus, in order
to define a Morse complex, the question whether there are loops with
non-trivial spectral flow becames relevant.

I wish to thank my advisors, Prof. A.~Abbondandolo and P.~Majer the aid
and helpful suggestions they gave to me.
\section{Preliminaries}

We recall basics definitions, results on spectral theory and
Fredholm operators. Our main references are \cite[Ch.~X]{Rud91} and
\cite[IV.4,5]{Kat95}.
\subsection{Spectral theory}
A \textsl{Banach algebra} is the given of an algebra
$ \mathcal{A} $ with unit, 1, over the real or complex field and a norm
$ \no{\cdot} $ such that
\begin{itemize}
\item[-] $ (\mathcal{A},\no{\cdot}) $ is a Banach space;
\item[-] $ \no{xy}\leq\no{x}\cdot\no{y} $ for every $ x,y\in\mathcal{A} $.
\end{itemize}
we denote with $ G(\mathcal{A}) $ the set of invertible elements of the
algebra; it is an open subset of $ \mathcal{A} $.
Given $ x\in\mathcal{A} $, the subset of the field
\[
\sigma(x) = \{\lambda\in\mb{F}:x - \lambda\cdot 1\not\in G(\mathcal{A})\}
\]
is called \textsl{spectrum} of $ x $. We recall some properties
of the spectrum.
\begin{prop}
\label{prop:spectrum-basics}
For every $ x\in\mathcal{A} $, $ t\in\mb{F} $ and $ \Omega\subset\mb{F} $
open subset,
\begin{enumerate}
\item if $ \sigma(x) $ is non-empty, is closed and bounded;
\item $ \sigma(x + t) = \sigma(x) + t $, $ \sigma(tx) = t\sigma(x) $;
\item there exists $ \delta > 0 $ such that $ \sigma(y)\subset\Omega $
for every $ y\in B(x,\delta) $;
\item if $ \mathcal{A} $ is complex, then $ \sigma(x) $ is non-empty;
\item if $ f\colon\mathcal{A}\ra\mathcal{B} $ is an algebras homomorphism
such that $ f(1) = 1 $, then $ \sigma(f(x))\subseteq\sigma(x) $.
\end{enumerate}
\end{prop}
Given a real algebra, we can consider the complex algebra associated to
it $ \mathcal{A}_{\C} = \mathcal{A}\otimes_{\R} \C $. We have an inclusion
of algebras
\[
\mathcal{A}\hookrightarrow \mathcal{A}_{\C},\ \ x\mapsto x\otimes 1
\]
and $ \sigma(x)\subseteq\sigma (x\otimes 1) $. Hereafter, we will
take $ \sigma(x\otimes 1) $ as spectrum of $ x $.
\begin{defn}
A finite family of closed curves in the complex plane,
$ \Gamma = \{c_i:1\leq i\leq n\} $ is said \textsl{simple} if, for every
$ z\not\in\bigcup_{i = 1}\sp n \text{im}\hskip .1em c_i $,
\[
\ind_{\Gamma} (z) =
\frac{1}{2\pi\imath}\int_{\Gamma} \frac{d\zeta}{z - \zeta}\in\{0,1\}.
\]
we denote with $ \Omega_0 $ and $ \Omega_1 $ the subsets of the complex space
such that $ \ind_{\Gamma} (z) $ is zero or $ 1 $, respectively.
\end{defn}
\begin{defn}
An element $ p\in\mathcal{A} $ is said \textsl{projector} if $ p\sp 2 = p $.
We can associate to it the sub-algebra
$ \mathcal{A}(p) = \{pxp:x\in\mathcal{A}\} $, with the unit $ p $.
We denote with $ \sigma_p $ the spectrum of the elements of $ \mathcal{A}(p) $.
\end{defn}
If $ xp = px $, we have the following property
\begin{equation}
\label{eq:sp}
\sigma(x) = \sigma_p (pxp)\cup\sigma_{1 - p} ((1 - p)x(1 - p)).
\end{equation}
\begin{theorem}
\label{thm:spectrum-projector}
Let $ x\in\mathcal{A} $, $ \Sigma\subset\sigma(x) $ open and closed in
$ \sigma(x) $. Then, there exists a projector, called
\textsl{spectral projector}, that we denote with $ P(x;\Sigma) $ such that
\begin{enumerate}
\item $ \sigma_P (PxP) = \Sigma $;
\item $ Px = xP $;
\item $ I - P = P(x;\Sigma\sp c) $;
\item given an algebras homomorphism, $ f\colon\mathcal{A}\ra\mathcal{B} $,
$ f(P(x;\Sigma)) = P(f(x);\Sigma) $.
\end{enumerate}
Given $ \Gamma $ as above such that
$ \Sigma\subset\Omega_1 $, then
\begin{equation*}
P(x;\Sigma) = P_{\Gamma} (x)
:= \frac{1}{2\pi\imath}\int_{\Gamma} (x - z)\sp {-1} dz.
\end{equation*}
On the open subset $ \mathcal{A}(\Gamma) = \{x\in\mathcal{A}:
\sigma(x)\cap\Gamma = \emptyset\} $, $ P_{\Gamma} (x) $ is a continuous map.
\end{theorem}
For more details check \cite[Theorem~10.27]{Rud91}
or \cite[Theorem~6.17]{Kat95}.
\subsection{Spaces of projectors}
\label{ssect:spaces-of-projectors}
Given a Banach space $ E $, we denote with $ \mathcal{L}(E) $ the space of
bounded operators on $ E $ and with $ \mathcal{L}_c (E) $ the closed linear
subspace of compact operators. We denote the quotient space with
$ \mathcal{C}(E) $. It inherits a structure of Banach algebra and it is called
\textsl{Calkin algebra}. The quotient projection
\[
\prc\colon \mathcal{L}(E)\ra \mathcal{C}(E),\ \ A\mapsto A + \mathcal{L}_c (E)
\]
is an algebras homomorphism.
\begin{defn}
Given a Banach algebra $ \mathcal{A} $, we define the following subsets
\begin{enumerate}
\item $ \mathcal{P}(\mathcal{A}) = \{p\in\mathcal{A}:p\sp 2 = p\} $,
\textsl{projectors};
\item $ \mathcal{Q}(\mathcal{A}) = \{q\in\mathcal{A}:q\sp 2 = 1\} $,
\textsl{square roots of the unit};
\item $ \mathcal{H}(\mathcal{A}) = \{x\in\mathcal{A}:\sigma(x)\cap
\imath\R = \emptyset\} $, \textsl{hyperbolic} elements.
\end{enumerate}
\end{defn}
\subsubsection*{Properties and remarks}
$ \mathcal{P} $ and $ \mathcal{Q} $ are closed subsets, locally
arc-wise connected and analytic sub-manifold. A proof of this can be
found in \cite{AM09} after \textsc{Remark} 1.4.
They are diffeomorphic to each other through the diffeomorphism
$ p\mapsto 2p - 1 $. From (iii) of Proposition
\ref{prop:spectrum-basics}, $ \mathcal{H}(\mathcal{A}) $ is
open.
\begin{theorem}
\label{thm:pair-of-projectors}
Given two projectors $ p,q $ such that $ \no{p - q} < 1 $ or
are in the same connected component of $ \mathcal{P}(\mathcal{A}) $,
there exists $ u\in G_1 (\mathcal{A}) $ such that $ up = qu $.
\end{theorem}
By $ G_1 (\mathcal{A}) $ we denote the connected component of
$ G(\mathcal{A}) $ of the unit. For the proof and details we refer to
\cite[Proposition~1.2]{AM09}. The theorem above has two
consequences
\begin{itemize}
\item[c1)] $ \mathcal{P}(\mathcal{A}) $ is locally arc-wise connected;
\item[c2)] when $ \mathcal{A} = \mathcal{L}(E) $, two projectors in the
same connected component have isomorphic ranges and kernels.
\end{itemize}
The quotient projection $ \prc $
restricts to the subset of projectors and roots of the unit
\begin{gather*}
\mathcal{P}(\prc)\colon\mathcal{P}(\mathcal{L}(E))\ra
\mathcal{P}(\mathcal{C}(E)),\ \
P\mapsto P + \mathcal{L}_c (E)\\
\mathcal{Q}(\prc)\colon\mathcal{Q}(\mathcal{L}(E))\ra
\mathcal{Q}(\mathcal{C}(E)),\ \
Q\mapsto Q + \mathcal{L}_c (E).
\end{gather*}
\begin{defn}
A continuous map $ p\colon E\ra B $ has the
\textsl{homotopy lifting property} w.r.t. a topological space $ X $
if, given continuous maps
\[
h\colon X\times [0,1]\rightarrow B, \ \ \
f\colon X\times\{0\}\rightarrow E,
\]
there exists $ H\colon X\times [0,1]\rightarrow E $ such that
$ H(x,0) = f(x,0) $ and $ p\circ H = h $. If the homotopy
lifting property holds w.r.t. $ [0,1]^n $ for every $ n\geq 0 $, then
$ \prc $, is called \textsl{Serre fibration}.
\end{defn}
\begin{prop}
\label{prop:loc-sect-Calkin}
The maps $ \mathcal{P}(\prc) $ and $ \mathcal{Q}(\prc) $ are surjective
Serre fibrations.
\end{prop}
In general, every surjective algebra homomorphism induces
a Serre fibration, for a proof check \cite[Theorem~2.4]{PR90}.
The surjectivity of the restriction follows, for instance,
\cite[Proposition~4.1]{AM09}. In fact, $ \mathcal{P}(\prc) $ and
$ \mathcal{Q}(\prc) $ are locally trivial fiber bundles, refer for instance
\cite[Proposition~1.3]{AM09} or \cite[Theorem~4.2]{Gar08}.
\subsection{Fredholm operators and relative dimension}
\label{sect:fred}
Let $ T\in\mathcal{L}(E,F) $ be a bounded operator. If the image of
$ T $ is a closed subspace, we have two Banach spaces associated to it,
namely $ \ker(T) $ and $ E/\ran(T) = \coker(T) $.
\begin{defn}
An operator as above is said \textsl{semi-Fredholm} if either $ \ker(T) $
or $ \coker(T) $ are finite-dimensional spaces. If both have finite
dimension, $ T $ is called \textsl{Fredholm} and the integer
\[
\ind(T) = \dim\ker(T) - \dim\coker(T)
\]
is the \textsl{Fredholm index}. If one between the two spaces have infinite
dimension, the index is defined to be $ +\infty $ or $ -\infty $ whether
$ \ker(T) $ or $ \coker(T) $ have infinite dimension.
\end{defn}
we denote with $ \Fred(E,F) $ and $ \Fred(E) $ the set of Fredholm operator
in $ \mathcal{L}(E,F) $ and $ \mathcal{L}(E) $, respectively;
$ \Fred_k (E,F) $ is the set of Fredholm operators of index $ k $. Let
$ T\in\Fred(E,F) $, $ S\in\mathcal{L}(F,G) $ and
$ K\in\mathcal{L}_c (E,F) $. We have
\begin{itemize}
\item[a)] $ \Fred_k (E,F)\subseteq\mathcal{L}(E,F) $ is an open subset;
\item[b)] $ T + K\in\Fred(E,F) $ and $ \ind(T + K) = \ind(T) $;
\item[c)] $ S\circ T \in\Fred(E,G) $ and
$ \ind(S\circ T) = \ind(S) + \ind(T) $;
\item[d)] given $ B\in\mathcal{L}(E,F) $, there exists $ \var > 0 $ such
that the maps $ \dim\ker(T + \lambda B) $ and $ \dim\coker(T + \lambda B) $
are constant on $ B(0,\var)\setminus\{0\} $;
\item[e)] $ T\in\Fred(E,F) $ if and only if there exists
$ S\in\mathcal{L}(F,E) $ such that $ T\circ S - I\in\mathcal{L}_c (F) $
and $ S\circ T - I\in\mathcal{L}_c (E) $.
\end{itemize}
These results hold true for unbounded operators too. Check
\cite[IV.4,5]{Kat95}.
\begin{defn}
A pair of closed subspaces $ (X,Y) $ is \textsl{semi-Fredholm} if and only if
their sum is closed and either $ X\cap Y $ or $ E/(X + Y) $  has a
finite dimension. If both have finite dimension,
then it is defined the \textsl{Fredholm index} of the pair
$ (X,Y) $ and
\[
\ind(X,Y) = \dim X\cap Y -
\codim X + Y.
\]
Otherwise, the index is $ +\infty $ or $ -\infty $.
\end{defn}
Two projectors $ P,Q $ are compact perturbation of each other
if $ P - Q\in\mathcal{L}_c (E) $. In this case, the restriction of
$ Q $ to $ \ran(P) $ is in $ \Fred(\ran(P),\ran(Q)) $. The
\textsl{relative dimension}, between $ P $ and $ Q $ is defined as
\[
[P - Q] := \ind (Q\colon\ran(P)\ra\ran(Q)).
\]
Such definition is meant to generalize the dimension for finite-dimensional
spaces to Banach spaces. The notation above is used by
C.~Zhu and Y.~Long in \cite{ZL99}. Corresponding definitions are known in
Hilbert setting, considered by A.~Abbondandolo and P.~Majer in \cite{AM01}
(see also \cite[Remark~4.9]{BDF73}). A definition for
pairs of closed subspaces $ (X,Y) $, not necessarily complemented,
can be found in \cite[Definition~5.8]{Gar08}.
\begin{theorem}
\label{thm:relative-dimension}
Given pairs of projectors $ (P,Q) $ and $ (Q,R) $ with compact difference,
we have
\begin{enumerate}
\item if $ \ran(P),\ran(Q) $ have finite dimension, then\\
$ [Q - P] = \dim\ran(P) - \dim\ran(Q) $;
\item $ [P - R] = [P - Q] + [Q - R] $;
\item on the subset $ \{(P,Q)\in \mathcal{P}(\mathcal{L}(E)\times
\mathcal{P}(\mathcal{L}(E)):P - Q\in\mathcal{L}_c (E)\} $,
the map $ [P - Q] $ is continuous;
\item $ [P - Q] = [(I - Q) - (I - P)] $;
\item $ (\ran(P),\ker(Q)) $ is a Fredholm pair
and $ \ind(\ran(P),\ker(Q)) = [Q - P] $.
\end{enumerate}
\end{theorem}
Property (iii) follows from stability results for the index of semi-Fredholm
pairs, check \cite[Theorem~4.30]{Kat95} and \cite[Theorem~3.3]{Gar08}.
For a proof of (iv) and (v) check \cite[Lemma~2.3]{ZL99} and
\cite[Proposition~5.3]{Gar08}, respectively;
(i) follows from \cite[Theorem~6.3]{Gar08} and the special case
when $ PQ = Q $ or $ QP = P $.
\section{Essentially hyperbolic operators}
We recall that a bounded operator $ A\in\mathcal{L}(E) $ -
or more generally an element of a Banach algebra $ \mathcal{A} $ - is called
\textsl{hyperbolic} if its spectrum does not meet the imaginary axis.
We denote with $ GL(E) $ the group of invertible operators on $ E $
and with $ GL_I (E) $ the connected component of the identity operator.
\begin{defn}
An operator $ A $ is called \textsl{essentially hyperbolic} if
$ A + \mathcal{L}_c (E) $ is a hyperbolic element in $ \mathcal{C}(E) $.
\end{defn}
The spectrum of $ A + \mathcal{L}_c (E) $ is the \textsl{essential} spectrum,
denoted with $ \sigma_e (A) = \set{\lambda}
{A - \lambda\not\in\Fred(E)} $.
Thus, an operator $ A \in\mathcal{L}(E) $ is essentially
hyperbolic if and only if its essential spectrum does not meet the imaginary
axis. A consequence of this property is the following:
\begin{lemma}
\label{lem:structure}
Let $ D $ be the set of isolated points of
$ \sigma(A) $ and let $ \partial\sigma(A) $ be the set of
boundary points of $ \sigma(A) $. Then
$ \partial \sigma(A)\setminus D $ is a subset of $ \sigma_e(A) $.
\end{lemma}
\begin{proof}
Let $ \lambda\in\p\sigma(A)\setminus D $ and suppose that
$ \lambda\not\in\sigma_e (A) $, thus $ A - \lambda\in\Fred(E) $. Let
$ \var > 0 $ as in d) of \S~\ref{sect:fred}, with $ B = I $. Since
$ \lambda\in\p\sigma(A) $, there exists
$ w\in B(\lambda,\var)\setminus\{\lambda\} $ such that $ A - w $ is invertible.
Thus, for every $ z\in B(\lambda,\var)\setminus\{\lambda\} $, $ A - z $ is
invertible, hence $ \lambda $ is isolated in $ \sigma(A) $.
\end{proof}
We need a well-known fact about the topology of the complex plane:
\begin{prop}
A closed proper subset of the complex plane with a discrete boundary is
discrete.
\end{prop}
By Lemma \ref{lem:structure}, $ \sigma(A)\cap\J $ is discrete and,
by the proposition above, is discrete. Thus, it is a finite set, by
(i) of Proposition \ref{prop:spectrum-basics}.
\begin{prop}
Each of the points of $ \sigma(A)\cap\J $ is an eigenvalue of finite
algebraic multiplicity.
\end{prop}
\begin{proof}
Let $ \lambda\in\sigma(A)\cap\J $.
We infer that $ A - \lambda\in\Fred_0 (E) $. In fact, by a,d) of
\S~\ref{sect:fred}, there exists a neighbourhood
$ \lambda\in V $ such that
\begin{gather*}
A - z\in\Fred_k (E),\ \ z\in V\\
\dim\ker(A - z)\equiv n,\ \ \dim\coker(A - z)\equiv m,\ \ z\in
V\setminus\{\lambda\},
\end{gather*}
where $ k = \ind(A - \lambda) $. Since $ \lambda $ is isolated, there
exists $ z'\in V\setminus\{\lambda\} $
such that $ A - z' $ is invertible, hence $ m = n = 0 $ and $ k = 0 $.
Thus, $ \lambda $ is an eigenvalue and isolated. These two conditions,
by \cite[Theorem~5.10,5.28]{Kat95}, imply that
$ \lambda $ has finite \textsl{multiplicity}, that is the range of
$ P(A;\lambda) $ is a finite-dimensional space.
\end{proof}
Theorem \ref{thm:spectrum-projector} provides us with projectors
$ P_i = P(A;\lambda_i) $ for every $ \lambda_i\in\sigma(A)\cap\J $,
and $ P = P(A;\sigma(A)\cap\{\re z\neq 0\}) $.
We can write
\begin{equation}
\label{eq:structure}
A = \left(A P + \sum_{i = 1} \sp n P_i\right)  + (A - I)\sum_{i = 1}^n P_i.
\end{equation}
According to (i,ii) of Theorem \ref{thm:spectrum-projector},
the term in the brackets is hyperbolic and the last term has finite rank.
Thus, we have proved that an essentially hyperbolic operator is a compact
perturbation of a hyperbolic one. Conversely, a compact perturbation of a
hyperbolic operator is essentially hyperbolic. In fact, let
$ H,K $ be a hyperbolic and a compact operator, respectively.
By b) of \S~\ref{sect:fred},
$ \sigma_e (H + K) = \sigma_e (H)\subseteq\sigma(H) $. Since $ H $ is
hyperbolic, the latter does not meet the imaginary axis, so does
$ \sigma_e (H) $. Thus, we have proved the following
\begin{theorem}
An operator is essentially hyperbolic if and only if it is a compact
perturbation of a hyperbolic one.
\end{theorem}
We denote with $ e\mathcal{H}(E) $ the set of essentially hyperbolic
operators endowed with the topology induced by the operator norm.
By applying (iii) of Proposition \ref{prop:spectrum-basics},
to the algebra $ \mathcal{C}(E) $, it can be checked that
$ e\mathcal{H}(E)\subset\mathcal{L}(E) $ is an open subset, thus
locally arc-wise connected.
\vskip .2em
Let $ \prc $ be the quotient projection
$ \mathcal{L}(E)\rightarrow\mathcal{C}(E) $, and let $ s $ be a continuous
global section of $ \prc $, i.e. $ \prc\circ s $ is the identity map.
Since $ \mathcal{L}_c (E) $ is contractible, classical results
on continuous selections ensure the existence of such a map, refer
\cite[Proposition~A.1]{AM09} for a proof
and references.
\begin{prop}
The space of essentially hyperbolic operators is homeomorphic to the
product $ \mathcal{H}(\mathcal{C}(E))\times\mathcal{L}_c (E) $.
\end{prop}
\begin{proof}
Since $ e\mathcal{H}(E) $ is defined as
$ \prc\sp{-1} (\mathcal{H}(\mathcal{C})) $,
$ s(\mathcal{H}(\mathcal{C}))\subset e\mathcal{H}(E) $. We define the
continuous maps
\begin{align*}
e\mathcal{H}(E)\rightarrow\mathcal{H}(\mathcal{C}(E))\times\mathcal{L}_c (E),
\ \ \
&A\mapsto (\prc(A), A - s(\prc(A))), \\
\mathcal{H}(\mathcal{C}(E))\times\mathcal{L}_c (E)\rightarrow
e\mathcal{H}(E), \ \ \
&(x,K)\mapsto s(x) + K.
\end{align*}
It is easy to check that their compositions in both orders are the
identity.
\end{proof}
\begin{defn}
Given $ x\in\mathcal{A} $ such that
$ \sigma\sp + (x) = \sigma(x)\cap\{\mathrm{re}\hskip .1em z > 0\} $
disconnects $ \sigma(x) $, we denote with $ p^+ (x) $ the projector
$ P(x;\{\re z > 0\}) $. Similarly, we define $ p\sp - (x) $.
\end{defn}
\begin{prop}
\label{prop:H-P}
The map
$ p\sp + \colon\mathcal{H}(\mathcal{A})\rightarrow\mathcal{P}(\mathcal{A})$
defines a homotopy equivalence, a homotopy inverse being the map
$ j\colon\mathcal{P}(A)\rightarrow\mathcal{H}(A) $, $ j(p) = 2p - 1 $.
\end{prop}
\begin{proof}
Given $ x\in\mathcal{H}(A) $, $ \sigma\sp + (x) $ is a spectral subset.
There is a continuous, positively oriented and simple curve $ c $, such
that $ \text{im}(c) $ is the boundary of a rectangle
$ Q = (0,a)\times (-b,b) $ and $ \sigma\sp + (x)\subset Q $. Thus,
\[
\sigma(x)\subset\big(\R\times [-b,b]\big)\sp c
\cap\text{im}(c)\sp c := \Omega.
\]
By (iii) of Proposition \ref{prop:spectrum-basics}, there exists
$ \delta > 0 $ such that, if $ d(y,x) < \delta $, then
$ y\in\mathcal{H}(\mathcal{A}) $ and $ \sigma(y)\subset\Omega $. Thus
$ \Omega\cap\sigma(y) = \sigma\sp + (y) $ and $ p\sp + (y) = P_c (y) $
which is continuous by Theorem \ref{thm:spectrum-projector}. Thus,
$ p\sp + $ is continuous in a neighbourhood of $ x $, namely $ B(x,\delta) $,
hence it is continuous. Given a projector $ p $, $ j(p) $ is a square root of
the unit, thus $ \sigma(j(p))\subseteq\{-1,1\} $, hence is hyperbolic.
Given $ \zeta\in\mathbb{C}\setminus\{-1,1\} $, we have
\[
(j(p) - \zeta)^{-1} = \frac{\zeta}{1 - \zeta^2} + \frac{j(p)}{1 - \zeta^2}
= \frac{1}{2}\left(-\frac{1}{\zeta + 1} + \frac{1}{1 - \zeta}\right) -
\frac{1}{2}\left(-\frac{1}{\zeta + 1} - \frac{1}{1 - \zeta}\right) j(p).
\]
If we integrate both sides over a curve surrounding the compact set
$ \{1\} $ in $ \mathbb{C}\setminus\{-1\} $, we obtain
\[
2\pi i\cdot p^+ (j(p)) =
2\pi i\left(\frac{1}{2} + \frac{1}{2} (2p - 1)\right) = 2\pi i\cdot p.
\]
Thus, $ p^+ \circ j $ is the identity map on $ \mathcal{H}(\mathcal{A}) $.
In order to prove that $ j\circ p^+ $ is homotopically equivalent to the
identity on $ \mathcal{P}(\mathcal{A}) $, we define
\[
H(t,x) = \big((1 - t) x + t\big) p^+ (x) + \big((1 - t) x - t\big) p^- (x).
\]
By (\ref{eq:sp}), we have
\begin{equation}
\label{spectrum}
\begin{split}
\sigma(H(t,x)) &=
\sigma_{p\sp + (x)} \big((1 - t) xp^+ (x) + tp^+ (x)\big)
\cup\sigma_{p\sp - (x)}\big((1 - t) xp^- (x) - tp^- (x) \big) \\
&= \{(1 - t)\sigma^+ (x) + t\}
\cup\{ (1 - t)\sigma^- (x) - t\}.
\end{split}
\end{equation}
Since the subsets of the complex plane $ \{\mathrm{re}\hskip .1em z > 0\} $
and $ \{\mathrm{re}\hskip .1em z < 0\} $ are convex, the sets in the
second line of (\ref{spectrum}) do not meet the imaginary axis, thus
$ H(t,x) $ is hyperbolic. Moreover, $ H(0,\cdot) $ is the
identity map and $ H(1,x) = j\circ p^+ (x) $.
\end{proof}
Since $ \mathcal{L}_c (E) $ is a vector space, thus it is
contractible to a point, the projection onto the first factor in
$ \mathcal{H}(\mathcal{C}(E))\times\mathcal{L}_c (E) $ is
a homotopy equivalence. Together with the last two propositions we have
proved the following
\begin{cor}
\label{cor:homotopy-equivalence}
The map $ \Psi\colon e\mathcal{H}(E)\rightarrow\mathcal{P}(\mathcal{C}(E)) $,
$ A\mapsto p^+ (A + \mathcal{L}_c (E)) $ is a homotopy equivalence.
\end{cor}
\begin{prop}
\label{prop:componentschar}
Given a connected component $ \ccH\subset e\mathcal{H}(E) $, there exists
$ P\in\mathcal{P}(\mathcal{L}(E)) $ such that $ 2P - I\in\ccH $. Moreover,
$ A,B\in\ccH $ if and only if there exists $ T\in GL_I (E) $
such that
\[
T P\sp + (A) T^{-1} - P\sp + (B)\in\mathcal{L}_c (E).
\]
\end{prop}
\begin{proof}
Given $ A\in\ccH $, by (\ref{eq:structure}) there exists a hyperbolic
operator $ H $ such that $ A - H\in\mathcal{L}_c (E) $. Using the convex
combination $ t\mapsto H + t(A - H) $, we obtain $ H\in\ccH $. By
Proposition \ref{prop:H-P}, $ 2P\sp + (H) - I\in\ccH $. Conversely,
if $ A,B\in\ccH $, there exists a path
$ \alpha\colon [0,1]\ra\mathcal{P}(\mathcal{C}(E)) $
such that $ \alpha(0) = \Psi(A) $ and $ \alpha(1) = \Psi(B) $. By
Proposition \ref{prop:loc-sect-Calkin}, there exists a path of projectors
$ P $ such that $ P(0) = P\sp + (A) $ and $ p(P(t)) = \alpha(t) $.
By Theorem \ref{thm:pair-of-projectors}, there exists $ T\in GL_I (E) $
such that
\[
TP\sp + (A) T\sp {-1} = P(1).
\]
By the lifting properties of $ P $, $ p(P(1)) = \Psi(B) =
p\sp + (B + \mathcal{L}_c (E)) $; the second equality follows from
the definition $ \Psi $. By (iv) of Proposition \ref{thm:spectrum-projector},
$ p\sp + (B + \mathcal{L}_c (E)) = P\sp + (B) + \mathcal{L}_c (E) $,
hence $ P(1) - P\sp + (B) \in\mathcal{L}_c (E) $.
\end{proof}
\section{The spectral flow}
\label{Three}
Let $ A\colon [0,1]\ra e\mathcal{H}(E) $ be a continuous path. By
Proposition \ref{prop:loc-sect-Calkin}, there exists a continuous path
of projectors $ P $, such that $ \prc(P(t)) = P\sp + (A(t)) $.
We have an integer associated to it
\begin{equation*}
\Sf(A;P) := \big[P(0) - P\sp + (A(0))\big] - \big[P(1) - P\sp + (A(1))\big].
\end{equation*}
Moreover, given another path $ Q $ such that $ \prc(Q(t)) = \prc(P(t)) $,
by (iii) of Theorem \ref{thm:relative-dimension}, we have
\[
\begin{split}
\Sf(A;Q) &= \big[Q(0) - P\sp + (A(0))\big] - \big[Q(1) - P\sp + (A(1))\big]
\\
&= [Q(0) - P(0)] + \big[P(0) - P\sp + (A(0))\big] \\
&- [Q(1) - P(1)] - \big[P(1) - P\sp + (A(1))\big] = \Sf(A;P).
\end{split}
\]
\begin{defn}
Given $ A $ as above, we define the \textsl{spectral flow} as
the integer $ \Sf(A;P) $ where $ P $ is any of the paths of projectors
such that $ \prc(P(t)) = P\sp + (A(t)) $. We denote it by $ \Sf(A) $.
\end{defn}
\begin{prop}
\label{prop:sf-properties}
The spectral flow satisfies the following properties:
\begin{enumerate}
\item It is well behaved with respect to the catenation of paths;
\item the spectral flow of a constant path or a path in
$ \mathcal{H}(\mathcal{L}(E)) $
is zero;
\item it is invariant for free-endpoints homotopies in
$ \mathcal{H}(\mathcal{L}(E)) $ and for fixed-endpoints homotopies in
$ e\mathcal{H}(E) $;
\item if $ A_i \in C([0,1],e\mathcal{H}(E_i)) $ for $ 1\leq i\leq n $,
then $ \Sf(\oplus_{i = 1} \sp n A_i) = \sum_{i = 1} \sp n \Sf(A_i) $;
\item if $ E $ is an $ n $-dimensional linear space then, for
every integer $ - n\leq k\leq n $, there is a path such that $ \Sf(A) = k $;
\item if $ E $ has infinite dimension, then, for every $ k $ there is
$ A $ such that $ \Sf(A) = k $.
\end{enumerate}
\end{prop}
\begin{proof}
(i). Let $ A,B $ be two paths such that $ A(1) = B(0) $. We can choose
paths of projectors $ P $ and $ Q $ such that $ \prc(P) = \Psi(A) $ and
$ \prc(Q) = \Psi(B) $, with $ Q(0) = P(1) $. Denote by $ C $ and $ R $ the
catenation of $ A,B $ and $ P,Q $ respectively. Then,
\[
\begin{split}
\Sf(A*B) &= [R(0) - P\sp + (C(0))]
- [R(1) - P\sp + (C(1))]\\
&= [P(0) - P\sp + (A(0))] -
[Q(1) - P\sp + (B(1))] = [P(0) - P\sp + (A(0))] \\
&- [P(1) - P\sp + (A(1))]
+ [Q(0) - P\sp + (B(0))] - [Q(1) - P\sp + (B(1))] \\
&= \Sf(A) + \Sf(B).
\end{split}
\]\par
\noindent (ii). If $ A $ is constant, the path $ P $ can be chosen to
be constant. Hence, the spectral flow is zero. If $ A $ is hyperbolic,
$ P\sp + (A(t)) $ is continuous and can be chosen as lifting path of
$ \Psi(A) $. Hence,
\[
\Sf(A) = [P\sp + (A(0)) - P\sp + (A(0))] - [P\sp + (A(1)) - P\sp + (A(1))] = 0.
\]\par
\noindent (iii). Let $ H\colon I\times I \ra e\mathcal{H}(E) $ be a continuous
map. There exists $ P\colon I\times I\ra \mathcal{P}(E) $ such that
\[
P(t,s) - P\sp + (H(t,s))\in\mathcal{L}_c (E),\text{ for every } t,s.
\]
Let $ H(\cdot,0) = A $ and $ H(\cdot,1) = B $. We have
\[
\Sf(A) = [P(0,0) - P\sp + (H(0,0))] - [P(1,0) - P\sp + (H(1,0))].
\]
For $ i=0,1 $ and every $ s $, the operator $ P(i,s) - P\sp + (H(i,s)) $
is compact. The right summand is constant or continuous,
whether the homotopy has fixed endpoints in $ e\mathcal{H}(E) $ or laying in
$ \mathcal{H}(\mathcal{L}(E)) $. In both cases
\[
[P(i,s) - P\sp + (H(i,s))] = k_i
\]
for every $ 0\leq s\leq 1 $ and $ i=0,1 $. Thus,
$ \Sf(A) = k_0 - k_1 = \Sf(B) $.\par
\noindent (iv). Let $ P_i $ be continuous paths of projectors such that
$ P_i (t) - P\sp + (A_i (t))\in\mathcal{L}_c (E_i) $.
\[
\begin{split}
\Sf(\oplus_{i = 1}\sp n A_i) =& \left[\oplus_{i = 1} \sp n P_i (0) -
\oplus_{i = 1} \sp n P\sp + (A_i (0))\right] -
\left[\oplus_{i = 1} \sp n P_i (1) -
\oplus_{i = 1} \sp n P\sp + (A_i (1))\right] \\
=& \sum_{i = 1}\sp n [P_i (0) - P\sp + (A_i (0))] -
[P_i (1) - P\sp + (A_i (1))] = \sum_{i = 1} \sp n \Sf(A_i).
\end{split}
\]
\par
\noindent (v). Given $ 0\leq k\leq n $, the spectral flow of
\[
A(t) = (2t - 1) I_k \oplus I_{n - k}
\]
can be computed using $ P(t)\equiv I $. Since $ P\sp + (A(1)) = I $
and $ P\sp + (A(0)) = 0\oplus I_{n - k} $, we have $ \Sf(A) = k $ and
$ \Sf(A(-\cdot)) = -k $.\par
\noindent (vi). Given $ k\in\mb{Z} $, let $ E = X\sp k \oplus R_k $
where $ X\sp k $ is a closed subspace and $ \dim(R_k) = k $. Thus, the
spectral flow of $ A(t) = I_{X\sp k} \oplus (2t - 1) I_{R_k} $ can be computed
with $ P(t)\equiv I $ and $ \Sf(A) = k $ and $ \Sf(A(-\cdot)) = -k $.
\end{proof}
\subsection{spectral sections}
The definition of spectral flow we used corresponds to the one given
by C.~Zhu and Y.~Long in \cite{ZL99} for paths of \textsl{admissible}
operators, refer \cite[Definition~2.3]{ZL99},
which are essentially hyperbolic. We briefly recall their definition
\begin{defn}
\label{defn:section}
A \textsl{s-section}, for a path of projectors $ Q $ on
$ J\subset [0,1] $ is a continuous path $ P $ such that
$ P(t) - Q(t)\in\mathcal{L}_c (E) $.
\end{defn}
The authors show in \textsc{Corollary}~2.1 that
there exists a partition of the unit interval $ (J_k)_{k = 1} \sp n $
and a continuous path $ Q_k (\cdot) $ on $ J_k $ such that $ Q_k (t) $
is a spectral projector of $ A(t) $ and is an s-section for
$ P\sp + (A(\cdot)) $ on $ J_k $.
Then, they define
\[
\Sf(A) = \sum_{k = 1} \sp n \Sf(A_k;Q_k)
\]
where $ A_k $ is the restriction of $ A $ to $ J_k $.
Using Proposition \ref{prop:loc-sect-Calkin} we showed that an s-section
of the path $ P\sp + (A(\cdot)) $ on $ [0,1] $ exists,
even though the projectors need not be spectral.
That allows us to simplify the definition of spectral flow, provide
simpler proofs of well known properties, such as
the homotopy invariance, from the original proof in
\cite[Proposition~2.2]{ZL99} or in \cite{Phi96}.
It also makes easy to produce examples, as we will show in the next section.
\section{Spectral flow as group homomorphism}
\label{Four}
By (ii) and (i) of Proposition \ref{prop:sf-properties}, the spectral
flow determines a $ \mb{Z} $-valued group homomorphism on the fundamental
group of each connected component of $ e\mathcal{H}(E) $. Given
a projector $ P $, we denote with $ \Sf_P $ the spectral flow on
fundamental group of the connected component of $ 2P - I $.\vskip .4em
The fiber of $ \prc\colon \mathcal{P}(\mathcal{L}(E))\rightarrow
\mathcal{P}(\mathcal{C}(E)) $ over a point of the base space,
$ P + \mathcal{L}_c (E) $ is the set
\begin{gather*}
\mathcal{P}_c (E;P) = \{Q\in\mathcal{P}(\mathcal{L}(E)):
Q - P\in\mathcal{L}_c (E)\}\\
i\colon\mathcal{P}_c (E;P)\hookrightarrow \mathcal{P}(\mathcal{L}(E)).
\end{gather*}
\begin{prop}
\label{prop:PcE}
For every projector $ P $, the connected components of $ \mathcal{P}_c (E;P) $
correspond to $ \mb{Z} $ through the bijection $ Q\mapsto [P - Q] $.
Moreover, if both the range and the kernel have infinite dimension,
$ \pi_1 (\mathcal{P}_c (E;P),P)\cong\mb{Z}_2 $.
\end{prop}
It follows from \cite[Theorem~6.3]{Gar08} or
\cite{AM09} and from \cite[Theorem~7.3]{Gar08}.
Since $ \prc $ is a Serre fibration, the sequence of homomorphisms
\begin{equation}
\label{eq:exact-1}
\xymatrix{
\pi_1 (\mathcal{P}_c (E;P),P) \ar[r]^-{i_*} &
\pi_1 (\mathcal{P}(\mathcal{L}(E)),P) \ar[r]^-{\prc_*} &
\pi_1 (\mathcal{P}(\mathcal{C}(E)),P + \mathcal{L}_c (E))
}
\end{equation}
is exact. The homotopy equivalence $ \Psi $ defined in Corollary
\ref{cor:homotopy-equivalence} determines a group isomorphism
\[
\Psi_* \colon\pi_1 (e\mathcal{H}(E),2P - I)\ra
\pi_1 (\mathcal{P}(\mathcal{C}(E)),P + \mathcal{L}_c (E)).
\]
We set $ \vfi_P = \Sf_P\circ (\Psi_*) \sp{-1} $.
\begin{theorem}
The sequence of homomorphisms
\begin{equation}
\label{eq:sf=index}
\xymatrix@C+0.4cm{
\pi_1 (\mathcal{P}(\mathcal{L}(E)),P) \ar[r]^-{\prc_*} &
\pi_1 (\mathcal{P}(\mathcal{C}(E)),P + \mathcal{L}_c (E))
\ar[r]^-{\vfi_P} & \mb{Z},
}
\end{equation}
is exact.
\end{theorem}
\begin{proof}
We prove that $ \ker(\vfi_P)\subseteq\text{im}(\prc_*) $.
Let $ a $ be a loop in $ \mathcal{P}(\mathcal{C}(E)) $ at the base
point $ P + \mathcal{L}_c (E) $ and $ A_t \in e\mathcal{H}(E) $
such that $ \Psi(A(t)) = a(t) $. Thus, there exists a path of projectors
$ P_t $ such that $ P(0) = P $ and $ \prc(P(t)) = P\sp + (A(t)) $.
Hence,
\[
\vfi_P (a) = \Sf_P (A) = [P(0) - P\sp + (A(0))] - [P(1) - P\sp + (A(1))] =
[P(0) - P(1)] = 0;
\]
the third inequality follows from (ii) of Theorem
\ref{thm:relative-dimension}. By Proposition
\ref{prop:PcE}, there exists a continuous path $ Q $ such that
\[
Q(0) = P(1),\ \ Q(1) = P,\ \ Q(t) - P(t)\in\mathcal{L}_c (E).
\]
Thus, $ \prc_* (P * Q) = a $. We prove that
$ \text{im}(\prc_*)\subseteq\ker(\vfi_P) $. Given a loop
$ P_t \in\mathcal{P}_c (E;P) $,
\[
\vfi_P (i_* (P)) = [P(0) - P(1)] = 0
\]
because $ P(0) = P(1) $.
\end{proof}
\begin{cor}
Given $ P\in\mathcal{P}(\mathcal{L}(E)) $, we have the following properties of
the kernel and the image of the spectral flow $ \Sf_P $:
\begin{itemize}
\item[h1)] $ m\in\text{im}(\Sf_P) $ if and only
if there exists a projector $ Q $ in the same connected component of
$ P $ such that $ Q - P $ is compact and $ [Q - P] = m $;
\item[h2)] $ \text{im}(\prc_*)\cong\ker(\Sf_P) $.
\end{itemize}
\end{cor}
The isomorphism classes of the kernel and the image of $ \Sf_P $ depend
only on the conjugacy class of $ P + \mathcal{L}_c (E) $ in
$ \mathcal{P}(\mathcal{C}(E)) $. We show that in many cases
we can find a projector $ P $ such that $ \Sf_P $ is an isomorphism.
\begin{lemma}
\label{lem:X+Y}
Let $ E $ be a Banach space, and $ X,Y\subset E $ closed subspaces such
that $ X\cong Y $ and $ X\oplus Y = E $. Two projectors
$ P_X,P_Y $ with ranges $ X $ and $ Y $ respectively, are connected by a
continuous path on $ \mathcal{P}(\mathcal{L}(E)) $.
\end{lemma}
A proof of this can be found in \cite[\S9]{PR87} or in \cite{Mit70}.
\begin{prop}
\label{prop:surjective}
Let $ X,Y\subset E $ be as above. Suppose that
$ X $ is isomorphic to its closed subspaces of co-dimension $ m $.
Let $ P $ be the projector onto $ X $ with kernel $ Y $. Then $ P $
satisfies the condition {\rm h1)} with the integer $ m $.
\end{prop}
\begin{proof}
Let $ X\sp m,R_m\subset X $ be closed subspaces such that $ \dim(R_m) = m $
and $ X\sp m\cong X $. We have the decomposition and isomorphism
\[
E = R_m \oplus X\sp m\oplus Y,\ X\sp m\cong Y.
\]
By applying Lemma \ref{lem:X+Y} with $ E = X\sp m \oplus Y $, we obtain that
$ P_{X\sp m} $ is connected to $ P_Y $. By applying it a second time to $ E $
and subspaces $ X = R_m\oplus X\sp m $ and $ Y $, we obtain that $ P_X $ is
connected to $ P_Y $. Hence, $ P_X $ is connected to
$ P_{X\sp m} $.
\end{proof}
We required $ E $ to be isomorphic to a cartesian product of a
space $ X $ with itself, but it suffices that $ E $ has a complemented
subspace $ F $ fulfilling the requirements of
Proposition \ref{prop:surjective}. In fact, if $ A_t \in e\mathcal{H}(F) $
is such that $ \Sf(A_t) = m $, then $ \Sf(I\oplus A_t) = m $, by (iv) of
Proposition \ref{prop:sf-properties}.

\begin{prop}
\label{prop:injective}
Given $ P\in\mathcal{L}(\mathcal{P}(E)) $, the map
$ \pi\colon GL(E)\rightarrow \mathcal{P}(\mathcal{L}(E)),
\ \ T\mapsto TP T^{-1} $ defines a principal bundle with fiber
$ GL(X)\times GL(Y) $, where $ X = \ran(P) $ and $ Y = \ker(P) $.
\end{prop}
A proof of this can be found in
\cite{PR90,AM09}.
\begin{cor}
If $ GL(E) $ is simply-connected and $ GL(X),GL(Y) $ are connected, then
$ \Sf_P $ is injective.
\end{cor}
\begin{proof}
Since a locally trivial bundle is a Serre fibration, we have a long exact
sequence of homomorphisms which ends
\begin{equation*}
\xymatrix{
\pi_1 (GL(E),I) \ar[r]^-{\pi_*} &
\pi_1 (\mathcal{P}(\mathcal{L}(E)),P) \ar[r]^-{\Delta} &
\pi_0 (GL(X)\times GL(Y),I).
}
\end{equation*}
Thus, if $ GL(E) $ is simply connected and $ GL(X) $ and $ GL(Y) $ are
connected, the middle group is trivial, hence in (\ref{eq:sf=index})
$ \prc_* $ is the trivial map, thus $ \vfi_P $ is injective and $ \Sf_P $
is injective.
\end{proof}
Then, we have sufficient conditions to a Banach space in order to
have at least one projector $ P $ such that $ \Sf_P $ is an isomorphism.
\begin{theorem}
Let $ E = X\oplus X $ such that $ X $ is isomorphic to its hyperplanes and
$ GL(E) $ is simply-connected and $ GL(X) $ is connected. Then
$ \Sf_{P_X} $ is an isomorphism.
\end{theorem}
\begin{proof}
That $ \Sf_{P_X} $ is surjective follows from Proposition
\ref{prop:surjective}. From the corollary above, $ \Sf_{P_X} $ is also
injective.
\end{proof}
A special case of the theorem above is when $ E\cong E\times E $, it
is isomorphic to a hyperplane and has a \textsl{contractible} general
linear group. This, in fact, is the case of the most common
infinite-dimensional spaces as separable Hilbert spaces,
$ c_0,\ell\sp p $ with $ p\geq 1 $, and $ L\sp p (\Omega,\mu) $ with $ p > 1 $,
and $ C(K,F) $ for large classes of compact spaces $ K $ and Banach spaces
$ F $, and many others. For a richer list, check \textsc{Theorem}~2
of \cite{Sch98} and \cite{Kui65,Art66,Neu67,Mit70}. Sequence spaces
$ \ell\sp p,\ell\sp\infty $ and $ c_0 $ are also \textsl{prime}
(see \cite[Theorem~2.2.4]{AK06} and \cite{Lin67})
that is they are isomorphic to their complemented, infinite-dimensional
subspaces. Thus, for \textsl{every} projector $ P $ such that
$ \ran(P) $ and $ \ker(P) $ have infinite dimension $ \Sf_P $ is
an isomorphism.
\subsubsection*{Trivial spectral flow}
When $ P $ is a projector with a finite-dimensional range or kernel,
$ P + \mathcal{L}_c (E) $ is 0 or 1, thus
its connected component in $ \mathcal{P}(\mathcal{C}(E)) $ is $ \{0\} $
or $ \{1\} $, thus $ \Sf_P = 0 $. This is the case of finite-dimensional
spaces. A spaces is said \textsl{undecomposable} if the only projectors
are as above. In \cite{GM93}, the authors showed the existence of
a infinite-dimensional, undecomposable space.

\subsubsection*{Non-trivial and not surjective spectral flow}
In \cite{GM97}, W.~T.~Gowers and B.~Maurey proved the existence of a space
isomorphic to their hypersquares (that is, subspaces of co-dimension 2),
but not their hyperplanes. If we denote with $ X $ a space with such
property and by $ P $ the projector onto the first factor in
$ E = X\oplus X $, then $ 2\in\text{im}(\Sf_P) $ by Proposition
\ref{prop:surjective}. However, if $ 1\in\text{im}(\Sf_P) $, by
condition h1) and c2) in \S\ref{ssect:spaces-of-projectors},
$ X $ would be isomorphic to its hyperplanes.
\subsection{The Douady space}
\label{ssect:douady}
We show the existence of a projector $ P $ such that $ \vfi_P $, and
thus $ \Sf_P $, is not injective.
\begin{prop}
\label{prop:delta}
Let $ E = X\oplus X $. Given $ T\in GL(X) $, there exists
a loop $ x $ in the space of projectors such that
\[
\Delta(x) =
\begin{pmatrix}
T & 0 \\
0 & T^{-1}
\end{pmatrix}.
\]
\end{prop}
\begin{proof}
Let $ M $ be the operator defined in the line above. Let $ U_t \in GL(E) $ be
such that $ U(1) = M $ and $ U(0) = I $. In general,
$ T\oplus T' $ is connected to $ TT' \oplus I $, check \cite{Mit70}.
Since $ T $ commutes with $ P_X $, the path
\[
P(t) = U(t) P_X U(t) \sp{-1}
\]
is a loop in $ \mathcal{P}(\mathcal{L}(E)) $ with base point $ P $.
We denote with $ x $ its homotopy class. The path $ U_t $
is a lifting path for $ P $. Hence $ \Delta(x) = U(1) = M $.
\end{proof}
Let $ F $ and $ G $ be Banach spaces such that
\begin{enumerate}
\item every bounded map $ G\rightarrow F $ is compact;
\item both $ F $ and $ G $ are isomorphic to their hyperplanes.
\end{enumerate}
We have the following:
\begin{lemma}
Let $ X = F\oplus G $, $ F $ and $ G $ as above.
There exists a continuous, surjective map
$ j\colon GL(X)\rightarrow\mb{Z} $.
\end{lemma}
\begin{proof}
The Lemma follows from a more general result of A.~Douady, \cite{Dou65}. We
briefly sketch the proof of B.~S.~Mitjagin in \cite{Mit70}.
Let $ T\in GL(X) $ be an invertible operator and $ S $ be its inverse. We have
\[
\begin{pmatrix}
I_F & 0 \\
0 & I_G
\end{pmatrix}
= TS =
\begin{pmatrix}
T_{11} S_{11} + T_{12} S_{21} & T_{11} S_{12} + T_{12} S_{22} \\
T_{21} S_{11} + T_{22} S_{21} & T_{22} S_{22} + T_{21} S_{12}.
\end{pmatrix}
\]
A similar equality holds for $ ST $. Since
$ S_{21} $ and $ T_{21} $ are compact operators, $ T_{11} $ and $ T_{22} $
satisfy e) of \S\ref{sect:fred} with $ S_{11} $ and $ S_{22} $, respectively.
Thus, they are Fredholm operators. We define
\[
j(T) = \ind(T_{11}).
\]
By a) of \S\ref{sect:fred}, there exists $ \var > 0 $ such that, if
$ \no{T' _{11} - T_{11}} < \var $, then $ T' _{11} $ is a Fredholm
operator and $ \ind(T' _{11}) = \ind(T_{11}) $. This proves the continuity.
Moreover, given two invertible operators $ T $ and $ S $, we have
\[
\begin{split}
j(TS) &= \ind(TS)_{11} = \ind(T_{11} S_{11} + T_{12} S_{21}) \\
&= \ind(T_{11} S_{11}) = \ind(T_{11}) + \ind(S_{11}),
\end{split}
\]
where b,c) of \S\ref{sect:fred} have been used. Thus, $ j $ is a group
homomorphism.
Let $ F\sp 1 $ and $ G\sp 1 $ be
hyperplanes of $ F $ and $ G $, respectively. We define
\begin{gather*}
\sigma\colon F\ra F\sp 1,\ \ \tau\colon G\sp 1 \ra G\\
F = \langle v\rangle\oplus F\sp 1 = F,\ \ G = \langle w\rangle\oplus G\sp 1\\
B\colon G\ra F,\ \ tw + y\mapsto tv.
\end{gather*}
where $ \sigma,\tau $ are isomorphisms, which exist by (i). We define
\[
T(x,y) = (\sigma(x) + B(y),\tau(Py));
\]
$ T $ is invertible and $ \ind(T_{11}) = \ind(\sigma) = 1 $. Since
$ j $ is a homomorphism, it is surjective.
\end{proof}
\begin{prop}
If $ E = X\oplus X $, where $ X $ is a direct sum of two spaces
$ F $ and $ G $ fulfilling the two conditions i) and ii) above,
then $ \Sf_{P_X} $ is surjective, but not injective.
\end{prop}
\begin{proof}
From the lemma above, for every $ k\in\mb{Z} $, there exists $ T_k \in GL(X) $
such that $ j(T_k) = k $. By Proposition \ref{prop:delta} there exists
$ x_k \in\pi_1 (\mathcal{P}_c (E;P),P_X) $ such that $ \Delta(x_k) = T_k $
and $ x_k \neq x_h $ if $ k\neq h $. Then
$ \pi_1 (\mathcal{P}(\mathcal{L}(E)),P_X) $ has infinite elements, while
$ \pi_1 (\mathcal{P}_c (E;P),P_X) $ is a finite group, by Proposition
\ref{prop:PcE}. Hence, in (\ref{eq:exact-1}), $ i_* $ is not surjective, thus
$ x_k\not\in\text{im}(i_*) $ for infinitely many $ k\in\mb{Z} $. Hence,
\[
\prc_* (x_k)\neq 0,\ \ \vfi_P (\prc_* (x_k)) = 0.
\]
because the sequence (\ref{eq:exact-1}) is exact; therefore,
the kernel of $ \vfi_P $ is not trivial.
Since $ E $ and $ X $ fulfill the hypothesis of
Proposition~\ref{prop:surjective}, $ \Sf_{P_X} $ is surjective.
\end{proof}
Pairs of Banach spaces with the properties i) and ii) are
$ (\ell\sp p,\ell\sp 2) $, with $ p > 2 $. We refer to
\cite[Theorem~4.23]{Rya02}.\vskip .4em
We show that even when $ P $ has infinite-dimensional range and kernel,
$ \Sf_P $ can be trivial.
\begin{prop}
Let $ X $ be as above. Then, $ \Sf_{P_F} = 0 $.
\end{prop}
\begin{proof}
Let $ 0 \leq m\in\text{im}(\Sf) $. Then, $ P_F $ is connected to a
projector $ Q\in\mathcal{L}(\mathcal{P}(X)) $ onto a subspace of
$ F\oplus\{0\} $ of co-dimension $ m $, $ F\sp m\oplus\{0\} $.
By Theorem \ref{thm:pair-of-projectors}, there exists a continuous
path $ U_t \in GL(X) $ such that
\[
U(0) = I,\ \ U(1)P_F = Q U(1).
\]
From these relations it follows that $ U(1)_{11} (F) = F\sp m $, hence
$ j(U(1)) = -m $, and $ j(U(0)) = 0 $. By the lemma above, $ j(U(t)) $ is
constant, therefore $ m = 0 $.
\end{proof}
\section{The Fredholm index of $ F_A $ and the spectral flow}
\begin{defn}
A path $ A\colon\R\ra\mathcal{L}(E) $ of bounded operators is said to be
\textsl{asymptotically hyperbolic} if the limits $ A(+\infty) $
and $ A(-\infty) $ exist and are hyperbolic operators.
\end{defn}
If $ A $ is also essentially hyperbolic, we can define the spectral flow as
follows: Since the set of hyperbolic operators is an open subset of
$ \mathcal{L}(E) $, there exists
$ \delta > 0 $ such that $ A(t) $ is hyperbolic for every
$ t\in (-\infty,-\delta]\cup[\delta,+\infty) $. We set
\begin{equation}
\label{sf_hyp}
\Sf(A) = \Sf(A,[-\delta,\delta]).
\end{equation}
The definition does not depend on the choice of $ \delta $ by (i,ii) of
Proposition \ref{prop:sf-properties}.
Let $ P\sp + (A) $ be the spectral projector $ P(A;\{\re z > 0\}) $.
\begin{defn}
An asymptotically hyperbolic path is called \emph{essentially splitting}
if the following properties
\begin{enumerate}
\item $ P^+ (A(+\infty)) - P^+ (A(-\infty)) $ is compact;
\item $ [A(t),P^+ (A(+\infty))] $ is compact for every $ t $,
\end{enumerate}
holds, where $ [A,P] = A P - P A $.
\end{defn}
The definition above corresponds to the one given in
\cite[Theorem~6.3]{AM03} when the property (i) above is added. Thus, we refer
to the paths above as essentially splitting, as well.
\begin{lemma}
\label{essential_splitting}
Let $ A $ be an asymptotically hyperbolic and essentially hyperbolic path.
It is also essentially splitting if and only if the projectors
of the subset $ \{P^+ (A(t)):t\in\mathbb{R}\} $ are compact perturbation
of each other.
\end{lemma}
\begin{proof}
Suppose $ A $ is essentially splitting and consider the restriction on the
positive half-line. We denote with $ E^+ $ and $ E^- $ the ranges
of $ P^+ $ and $ P^- $, respectively. With respect to the decomposition
$ E = E^+ \oplus E^- $ we can write the operators of the path block-wise:
\begin{equation*}
A(t) =
\begin{pmatrix}
A_+ (t) & K_{\pm} (t) \\
K_{\mp} (t) & A_{-} (t)
\end{pmatrix}
\end{equation*}
where $ K_{\pm} (t) $ and $ K_{\mp} (t) $ are compact operators because
$ A $ is essentially splitting. Thus
\begin{equation}
\label{eq:essential-splitting:2}
P^+ (A(t)) - P^+ (A_+ (t)) \oplus P^+ (A_- (t))
\end{equation}
is compact: in fact if two operators have compact difference, then
their spectral projectors are compact perturbation of each other.
Since $ A(+\infty) $ is
hyperbolic, and $ K_{\pm} (+\infty),K_{\mp} (+\infty) $ are null operators,
$ A_+ (+\infty) $ is hyperbolic and there exists  $ t_+ > 0 $ such that for
every $ t\geq t_+ $ the operator $ A_+ (t) $ is hyperbolic on $ E^+ $ and
\[
\| P^+ (A_+ (t)) - P^+ (A_+ (+\infty))\| < 1.
\]
Since $ \sigma(A_+ (+\infty))\subset \{\re z > 0\} $, then
$ P^+ (A_+ (+\infty)) = I_{E\sp +} $.
Thus, $ P\sp + (A_+ (t)) = I_{E\sp +} $. Hence, the path
\[
A_+\colon [-s,t_+]\rightarrow e\mathcal{H}(E^+),\ \forall s
\]
lies in the connected component of the identity in $ e\mathcal{H} (E^+) $.
By Proposition (\ref{prop:componentschar}), all the projectors
$ P^+ (A_+ (t)) $ are
compact perturbation of the identity.
Similarly, the spectrum of
$ A_- (+\infty) $ is contained in the negative
complex half-plane, hence $ P\sp + (A_- (+\infty)) = 0 $ and there exists
$ t_- > 0 $ such that, for every $ t\geq t_- $,
\[
P^+ (A_- (t)) = 0.
\]
The path $ A_- \colon [-s,t_-]\ra e\mathcal{H}(E\sp -) $ lies in the
connected component of $ 0\in e\mathcal{H}(E^-) $ for every $ s > 0 $.
By Proposition (\ref{prop:componentschar}) all the projectors
$ P^+ (A_- (t)) $ have finite rank. To recap, for every $ t\in\R $,
\[
P^+ (A_+ (t)) - I_{E\sp +} \in\mathcal{L}_c (E^+), \ \
P^+ (A_- (t))\in\mathcal{L}_c (E^-).
\]
The direct sum $ I_{E^+} \oplus 0_{E^-} $ is $ P^+ (A(+\infty)) $. Hence,
\[
P^+ (A_+ (t))\oplus P^+ (A_- (t)) - P^+ (A(+\infty))
\]
is compact. Together with (\ref{eq:essential-splitting:2}), we conclude.
Conversely, suppose that each of the projectors of the
set $ \{P^+ (A(t)):t\in\mathbb{R}\} $ is compact perturbation of each other
and let $ a $ be such that $ A(s) $ is hyperbolic for every $ |s| > a $. Thus
$ P\sp + $ is continuous and
\[
P^+ (A(+\infty)) - P^+ (A(-\infty)) =
\lim_{\genfrac{}{}{0pt}{}{s\rightarrow+\infty}{|s| > a}}
\Big(P^+ (A(s)) - P^+ (A(-s))\Big).
\]
because $ A $ is asymptotically hyperbolic. Hence, the left member is the
limit of a sequence of compact operators. Since
$ \mathcal{L}_c (E) $ is a closed subset of $ \mathcal{L}(E) $, we have
proved (i). Property (ii) follows from the equality
\[
[A(t),P^+ (A(s))] = [A(t),P^+ (A(t))] + [A(t),P^+ (A(s)) - P^+ (A(t))]
\]
where the first summand
of the right member is zero and the second one is
compact by hypothesis. In particular, the equality holds for $ s > a $, so
we finish our proof by taking the limit as $ s\rightarrow +\infty $.
\end{proof}
\subsection{The spectral flow and the Fredholm index of $ F_A $}
Given a continuous, bounded path $ A_t \in\mathcal{L}(E) $, we
denote with $ F_A $ the differential operator
\[
F_A \colon W\sp{1,p} (\R,E)\rightarrow L\sp p (\R,E), \ \
F_A (u) = \frac{du}{dt} - A(\cdot)u.
\]
\begin{theorem}
\label{thm:index-FA}
Let $ A $ be an asymptotically hyperbolic and essentially splitting path.
Then $ F_A $ is a Fredholm operator and
\[
\ind(F_A) = [P^- (A(-\infty)) - P^- (A(+\infty))]
\]
\end{theorem}
A.~Abbondandolo and P.~Majer proved it in
\cite[Theorem~6.3]{AM03} when $ E $ is a Hilbert space and $ p = 2 $. However,
the theorem, as much of the content of their work can be generalized
to Banach spaces.
\begin{theorem}
\label{thm:sf-ess-split}
Let $ A $ be an asymptotically hyperbolic, essentially splitting and
essentially hyperbolic path. Then,
\[
\Sf(A) = - [P^- (A(-\infty)) - P^- (A(+\infty))]
\]
\end{theorem}
\begin{proof}
Let $ \delta > 0 $ as in (\ref{sf_hyp}). By Lemma \ref{essential_splitting},
the constant path $ P\equiv P\sp + (A(\delta)) $ is a section for
$ P\sp + (A_t) $ on $ [-\delta,\delta] $ in the sense of Definition
\ref{defn:section}. Hence,
\[
\Sf(A) = [P\sp + (A(\delta)) - P\sp + (A(-\delta))].
\]
Since $ A $ is hyperbolic on $ (-\infty-\delta]\cup [\delta,+\infty) $ the
path $ P^+ (A_t) $ is continuous on this subset.
By (iii) of Theorem \ref{thm:relative-dimension},
\[
[P^- (A(-\infty)) - P^- (A(+\infty))] =
- [P^+ (A(\delta)) - P^+ (A(-\delta))] = -\Sf(A).
\]
\end{proof}
From Theorem \ref{thm:sf-ess-split} and Theorem \ref{thm:index-FA} we
have the final result
\begin{theorem}
\label{thm:final}
If $ A $ is essentially hyperbolic, essentially splitting and
asymptotically hyperbolic, then
\begin{equation}
\label{eq:final}
\ind(F_A) = -\Sf(A).
\end{equation}
\end{theorem}
Let $ A(t) = A_0 (t) + K(t) $, where $ A_0 (t) $ is hyperbolic and
$ A_0,A $ are asymptotically hyperbolic and $ K(t) $ is compact and
\begin{gather*}
A_0 (t) E\sp - \subseteq E\sp -,\ \ A_0 (t) E\sp + \subseteq E\sp +;\\
E\sp - = E\sp - (A_0 (\pm\infty)),\ \ E\sp + = E\sp + (A_0 (\pm\infty)).
\end{gather*}
The second line tells us that
$ P\sp + (A_0 (+\infty)) = P\sp + (A_0 (-\infty)) $, thus
$ P\sp + (A(+\infty)) - P\sp + (A(-\infty)) $ is compact. From the
first line, it follows that $ [A(t),P\sp + (A(+\infty))] $ is compact.
Thus, by Theorem \ref{thm:final}, we confirm the guess of A.~Abbondandolo and
P.~Majer in \cite[\S7]{AM03}, that for paths satisfying the hypotheses
of Theorem~E, corresponding to those listed above, the relation
(\ref{eq:final}) holds.

When $ A $ is not essentially splitting, the authors provided in
\cite[Example~6,7]{AM03} counterexamples to the (\ref{eq:final}).

\bibliographystyle{plain}

config: arxiv


TITLE: functions with floors
QUESTION [4 upvotes]: If
$$z = \frac{ \left\{ \sqrt{3} \right\}^2 - 2 \left\{ \sqrt{2} \right\}^2 }{ \left\{ \sqrt{3} \right\} - 2 \left\{ \sqrt{2} \right\} }$$
find $\lfloor z \rfloor$.
I don't rely know how to do this, but I was thinking about multiplying the denominator by it's conjugate, but idk.

REPLY [3 votes]: Assuming $\{\cdot \}$ is fractional part,
$\{\sqrt{3}\} = \sqrt{3}-1$ and $\{\sqrt{2}\} = \sqrt{2}-1$, so
$$ z = \frac{(\sqrt{3}-1)^2 - 2 (\sqrt{2}-1)^2}{\sqrt{3} + 1 - 2 \sqrt{2}}$$
Expand out the numerator and you should recognize that it is a certain integer times the denominator.


TITLE: Sign in the Free Propagator
QUESTION [2 upvotes]: The free propagator in Zee's book for free scalar field theory is
$$D(x-y)=\int \frac{d^4k}{(2\pi)^4}\frac{\exp[ik(x-y)]}{k^2-m^2+i\epsilon}.$$
He then says that the sign of $k$ in the exponential does not matter here by the symmetry $k \rightarrow -k$. Why is that? Where does this symmetry explicitly come from?

REPLY [2 votes]: $$
\frac{\exp[ik(x-y)]}{k^2-m^2+i\epsilon} = \frac{\cos(k(x-y)) + i\sin(k(x-y))}{k^2-m^2+i\epsilon}
$$
The $k$ odd $sin$ part drops out upon integration, leaving only the $k$ even $cos$ part. Therefore, the sign of $k$ does not matter.


TITLE: Probability - Removing a ball and replacing with the same colour.
QUESTION [0 upvotes]: Question:
A box contains g green balls and v violet balls.
A ball is drawn at random.
It's colour is recorded and is returned to the box with another ball of the same colour.
The process is repeated indefinitely.

What is the probability the second ball taken out the box is violet?
What is the probability the first ball is violet given that the
second ball is violet?

My attempt, though I feel like I am going down the entirely wrong path as . I probably should be working with permutations:
R = V B = G

Would anyone be able to provide a explained solution as I am quite confused by this question.

REPLY [1 votes]: For the first question, there are two alternatives:
First ball is green, second is violet.
First ball is violet, second is violet.
$\frac{g}{v+g} \cdot\frac{v}{g+v+1} + \frac{v}{v+g} \cdot\frac{v+1}{g+v+1}$
For the second question use Bayes' theorem.
So you need basicaly three values. One you have from previous point ( $P(second=v)$).  The other are  $P(first=v)=\frac{v}{g+v}$
$P(second=v|first=v) = \frac{v+1}{v+g+1}$
Now we are done: $P(first=v|second=v) = \frac{P(s=v|f=v)\cdot P(f=v)}{P(s=v)} = \frac{v+1}{g+v+1}$