\addtocontents{toc}{\medskip\noi} 


\chapter*{Introduction}


\subsection*{Upcoming and continuing interest in the Maslov index} Since the legendary work of
\auindex{Maslov,\ V.P.}V.P. Maslov \cite{Maslov} in the mid 1960s and the supplementary
explanations by \auindex{Arnol'd,\ V.I}V. Arnol'd \cite{Arnold:1967}, there has been a continuing
interest in the \subindex{Maslov index}Maslov index for \subindex{Continuously varying!Fredholm
pairs of Lagrangian subspaces}curves of pairs of Lagrangians in symplectic space. As explained by Maslov and
Arnol'd, the interest arises from the study of \subindex{Dynamical systems}dynamical systems in
classical mechanics and related problems in \subindex{Morse theory}Morse theory. This same index
occurs as well in certain asymptotic formulae for solutions of the Schr{\"o}dinger equations. For a
systematic review of the basic vector analysis and geometry and for the physics background, we
refer to \auindex{Arnol'd,\ V.I}Arnol'd \cite{Arnold:1989} and \auindex{de Gosson,\ M.}M. de Gosson
\cite{Go01}.


The \subindex{Morse theory!Morse index theorem}Morse index theorem expresses the Morse index of a
geodesic by the Maslov index. Later, \auindex{Yoshida,\ T.}T. Yoshida \cite{Yo91} and
\auindex{Nicolaescu,\ L.}L. Nicolaescu \cite{Ni95,Ni97} expanded the view by embracing also
spectral problems for \subindex{Elliptic differential operators!Dirac type operators}Dirac type
operators on \subindex{Partitioned manifold}partitioned manifolds and thereby stimulating some
quite new research in that direction. For a short review, we refer to our Section \ref{s:history}
below.

\subsection*{Weak symplectic forms on Banach manifolds}
Early in the 1970s, \auindex{Chernoff,\ P.R.}\auindex{Marsden,\ J.E.}P.~Chernoff, J. Marsden
\cite{CheMa74} and \auindex{Weinstein,\ A.}A. Weinstein \cite{Weinstein:1971} called attention to
the practical and theoretical importance of \subindex{Banach manifold!symplectic forms on
it}symplectic forms on Banach manifolds. See \auindex{Swanson,\ R.C.}R.C. Swanson
\cite{Swanson1,Swanson2,Swanson:1980} for an elaboration of the achievements of that period
regarding \subindex{Banach space!linear symplectic structures on it}linear symplectic structures on
Banach spaces. It seems, however, that rigorous and operational definitions of the Maslov index of
\subindex{Continuously varying!Fredholm pairs of Lagrangian subspaces}curves of
\subindex{Lagrangian subspaces!in spaces of infinite dimension}Lagrangian subspaces in spaces of
infinite dimension was not obtained until 25 years later. Our \auindex{Boo{\ss}--Bavnbek,\
B.}\auindex{Zhu,\ C.}\cite[Section 3.2]{BooZhu:2013} gives an account and compares the various
definitions.

\begin{figure}
\includegraphics[scale=0.95]{Obstructions_crop.pdf}
\caption{Why going weak and what obstructions to circumvent?}\auindex{Swanson,\
R.C.}\auindex{Schmid,\ R.}\symindex{J@$J$  operator associated to symplectic
form}\subindex{Operator!associated to symplectic form}\subindex{Symplectic form!associated almost
complex generating operator}\subindex{Obstructions to straight forward generalization!generator $J$
non-invertible} \subindex{Obstructions to straight forward generalization!no symplectic splitting}
\subindex{Obstructions to straight forward generalization!double annihilator not the identity on
closed subspaces} \subindex{Obstructions to straight forward generalization!non-vanishing index of
Fredholm pair of Lagrangian subspaces} \subindex{Obstructions to straight forward
generalization!$\Ll$ not contractible}\subindex{Obstructions to straight forward
generalization!fundamental group of Fredholm Lagrangian Grassmannian unknown}\subindex{Obstructions
to straight forward generalization!no meaningful Maslov cycle}\subindex{Fredholm Lagrangian
Grassmannian} \label{f:obstructions}
\end{figure}


At the same place we emphasized a couple of rather serious obstructions (see Figure
\ref{f:obstructions}) to applying these concepts to arbitrary systems of \subindex{Elliptic
differential operators!of non-Dirac type}elliptic differential equations of non-Dirac type:
Firstly, some of the key section spaces for studying \subindex{Boundary value problems}boundary
value problems (the \subindex{Sobolev spaces!with weak symplectic forms}Sobolev space
\symindex{H@$H^{1/2}(\Si;E\mid_{\Sigma})$ weak symplectic Sobolev space}$H^{1/2}(\Si;E|_{\Si})$
containing the traces over the boundary $\Si=\partial M$ of sections over the whole manifold
\symindex{M@$M, M(s)$ smooth compact manifold!with boundary $\Si,\Si(s)$}$M$) are not carrying a
\subindex{Symplectic form!strong}strong symplectic structure, but are naturally equipped with a
\subindex{Symplectic form!weak}weak structure not admitting the rule \symindex{J@$J$  operator
associated to symplectic form}\subindex{Operator!associated to symplectic form}\subindex{Symplectic
form!associated almost complex generating operator} $J^2=-I$. Secondly, in
\auindex{Boo{\ss}--Bavnbek,\ B.}\auindex{Zhu,\ C.}\cite{BooZhu:2013} our definition of the Maslov
index in weak symplectic spaces requires a \subindex{Symplectic splitting}symplectic splitting
which does not always exist, is not canonical, and therefore, in general, not obtainable in a
continuous way for continuously varying symplectic structures. Recall that a \textit{symplectic
splitting} of a symplectic Banach space $(X,\w)$ is a decomposition $X=X^-\oplus X^+$ with $-i\w$
negative, respectively, positive definite on $X^{\mp}$ and vanishing on $X^-\times X^+$. Thirdly, a
priori, a \subindex{Symplectic reduction!finite-dimensional}symplectic reduction to finite
dimensions is not obtainable for weak symplectic structures in the setting of \cite{BooZhu:2013}.

An additional incitement to investigate weak symplectic structures comes from a stunning
observation of \auindex{Witten,\ E.}E. Witten (explained by \auindex{Atiyah,\ M.F.}M.F. Atiyah in
\cite{Atiyah:1985} in a heuristic way). He considered a weak presymplectic form on the loop space
\symindex{M@$\operatorname{Map}(S^1,M)$ loop space}\subindex{Geodesic!loop space of a
manifold}$\operatorname{Map}(S^1,M)$ of a finite-dimensional closed orientable Riemannian manifold
$M$ and noticed that a (future) thorough understanding of the infinite-dimensional symplectic
geometry of that loop space ``should lead rather directly to the \subindex{Index!Atiyah-Singer
Index Theorem!via infinite-dimensional symplectic geometry}index theorem for Dirac operators"
(l.c., p. 43). Of course, restricting ourselves to the linear case, i.e., to the geometry of
Lagrangian subspaces instead of \subindex{Lagrangian manifold}Lagrangian manifolds, we can only
marginally contribute to that program in this Memoir.

\subsection*{Symplectic reduction}\subindex{Symplectic reduction!history, origin and meaning}
In their influential paper \cite[p. 121]{Marsden-Weinstein}, \auindex{Marsden,\
J.E.}\auindex{Weinstein,\ A.}J. Marsden and A. Weinstein describe
the purpose of symplectic reduction in the following way:
{\small
\begin{quotation}
``... when we have a \subindex{Symplectic geometry!symplectic manifold}symplectic manifold on which
a group acts symplectically, we can reduce this \subindex{Symplectic reduction!reduced phase
space}phase space to another symplectic manifold in which, roughly speaking, the symmetries are
divided
out."
\end{quotation}}
and {\small
\begin{quotation}
``When one has a \subindex{Hamiltonian system}Hamiltonian system on the phase space which is
invariant under the group, there is a Hamiltonian system canonically induced on the reduced phase
space."
\end{quotation}}
The basic ideas go back to the work of \auindex{Hamel,\ G.}G. Hamel \cite{Hamel:1904,Hamel:1978}
and \auindex{Carath\'eodory,\ C.}C. Carath{\'e}odory \cite{Caratheodory} in \subindex{Dynamical
systems}dynamical systems at the beginning of the last century, see also \auindex{Souriau,\
J.-M.}J.-M. Souriau \cite{Souriau}. For \subindex{Symplectic reduction!in low-dimensional
geometry}symplectic reduction in low-dimensional geometry see the monographs by
\auindex{Donaldson,\ S.K.}\auindex{Kronheimer,\ P.B.}S.K. Donaldson and P.B. Kronheimer, and by
\auindex{McDuff,\ D.}\auindex{Salamon,\ D.}D. McDuff and D. Salamon \cite{Donaldson-Kronheimer,
McDuff-Salamon}.

Our aim is less intricate, but not at all trivial: Following \auindex{Nicolaescu,\ L.}L. Nicolaescu
\cite{Ni97} and \auindex{Boo{\ss}--Bavnbek,\ B.}\auindex{Furutani,\ K.}K. Furutani \cite{BoFu99}
(joint work with the first author) we are interested in the finite-dimensional reduction of
\subindex{Fredholm pair!of Lagrangian subspaces}Fredholm pairs of Lagrangian \textit{linear}
subspaces in infinite-dimensional Banach space. The \subindex{Symplectic reduction!general
procedure}general procedure is well understood, see also \auindex{Kirk,\ P.}\auindex{Lesch,\ M.}P.
Kirk and M. Lesch in \cite[Section 6.3]{KiLe00}: let $W\< X$ be a closed \subindex{Co-isotropic
subspaces}co-isotropic subspace of a symplectic Banach space \symindex{X@$(X,\w)$ symplectic vector
or Banach space}$(X,\w)$. Then \symindex{W@$W/W^{\w}$ reduced symplectic space}$W/W^{\w}$ inherits
a symplectic form from $\w$ such that
\[\symindex{R@$R_W(\la)$ symplectic reduction of isotropic $\la$ along co-isotropic $W$}
 R_W(\la)\ :=\  \frac{(\la +W^{\w})\cap W}{W^{\w}}\
\<\ \frac W{W^{\w}} \text{ isotropic for $\la$ isotropic}.
\]
Here $W^\w$ denotes the annihilator of $W$ with respect to the symplectic form $\w$ (see Definition
\ref{d:symplectic-space}c).

In general, however, the reduced space $R_W(\la)$ does not need to be Lagrangian in $W/W^{\w}$ even
for Lagrangian $\la$ unless we have $W^\w\<\la\< W$ (see Proposition \ref{p:red-fredholm}). In
\auindex{Nicolaescu,\ L.}\auindex{Boo{\ss}--Bavnbek,\ B.}\auindex{Furutani,\ K.}\cite{Ni97,BoFu99}
a closer analysis of the \subindex{Symplectic reduction!reduction map}reduction map $R_W$ is given
within the setting of strong symplectic structures; with emphasis on the topology of the space of
Fredholm pairs of Lagrangians; and for fixed $W$. Now we drop the restriction to strong symplectic
forms; our goal is to define the Maslov index for \subindex{Continuously varying!Fredholm pairs of
Lagrangian subspaces}continuous \subindex{Fredholm pair!of Lagrangian subspaces!curve}curves $s\to
(\la(s),\mu(s))$ of Fredholm pairs of Lagrangians with respect to continuously varying symplectic
forms $\w(s)$; and, at least locally (for $s\in (t-\e,t+\e)$ around $t\in [0,1]$), we let the pair
$(\la(t),\mu(t))$ induce  the reference space $W(t)$ for the \subindex{Symplectic
reduction}symplectic reduction and the pair $(\la(s),\mu(s))$ induce the reduction map
$R^{(s)}_{W(t)}$ in a natural way. The key to finding the reference spaces $W(t)$ and defining a
suitable reduction map $R_{W(t)}$ is our Proposition \ref{p:reduction-prop}. It is on
decompositions of symplectic Banach spaces, naturally induced by a given Fredholm pair of
Lagrangians of vanishing index. It might be, as well, of independent interest. The assumption of
vanishing index is always satisfied for Fredholm pairs of Lagrangian subspaces in strong symplectic
Hilbert spaces, and by additional global analysis arguments in our applications as well.

Thus for each path $\{(\la(s),\mu(s))\}_{s\in[0,1]}$ of Fredholm pairs of Lagrangian subspaces of
vanishing index, we receive a finite-dimensional symplectic reduction \textit{intrinsically}, i.e.,
without any other assumption. The reduction transforms the given path into a path of pairs of
Lagrangians in finite-dimensional symplectic space. The main part of the Memoir is then to prove
the invariance under symplectic reduction and the independence of choices made. That permits us a
conservative view in this Memoir. Instead of defining the Maslov index in infinite dimensions via
spectral theory of unitary generators of the Lagrangians as we did in \auindex{Boo{\ss}--Bavnbek,\
B.}\auindex{Long,\ Y.}\auindex{Zhu,\ C.}\cite{BooZhu:2013}, we elaborate the concept of the
\subindex{Maslov index!in finite dimensions}Maslov index in finite dimensions and reduce the
infinite-dimensional case to the finite-dimensional case, i.e., we take the symplectic reduction as
our beginning for re-defining the Maslov index instead of deploring its missing.

\begin{figure}
\includegraphics[scale=0.95]{NaturalLevels.pdf}
\caption{Natural levels of treatment}\subindex{Levels of treatment}\label{f:natural-levels}
\end{figure}

\subsection*{Structure of presentation}
This Memoir is divided into four chapters and one appendix. The first three chapters present a
rigorous definition of the Maslov index in Banach bundles by symplectic reduction. In Chapter
\ref{s:banach}, we fix the notation and establish our key technical device, namely the mentioned
\subindex{Banach space!natural decomposition}natural decomposition of a symplectic Banach space
into two symplectic spaces, induced by a pair of co-isotropic subspaces with finite codimension of
their sum and finite dimension of the intersection of their annihilators. We introduce the
\subindex{Symplectic reduction}symplectic reduction of arbitrary linear subspaces via a fixed
co-isotropic subspace $W$ and prove the transitivity of the symplectic reduction when replacing $W$
by a larger co-isotropic subspace $W'$. For Fredholm pairs of Lagrangian subspaces of vanishing
\subindex{Fredholm pair!index}index, that yields an identification of the two naturally defined
symplectic reductions. In Chapter \ref{s:maslov-hilbert}, we recall and elaborate the
\subindex{Maslov index!in finite dimensions}\subindex{Maslov index!in strong symplectic Hilbert
space}Maslov index in strong symplectic Hilbert space, particularly in finite dimensions, to prove
the invariance of our definition of the Maslov index under different symplectic reductions. In
Chapter \ref{s:maslov-general}, we investigate the symplectic reduction to finite dimensions for a
given path of Fredholm pairs of Lagrangian subspaces in fixed \subindex{Banach space!with varying
weak symplectic structure}Banach space with varying symplectic structures and define the Maslov
index in the general case via \subindex{Symplectic reduction!finite-dimensional}finite-dimensional
symplectic reduction. In Section \ref{ss:sympl-invariance}, we show that the Maslov index is
invariant under symplectic reduction in the general case. For a first review of the entangled
levels of treatment see Figure \ref{f:natural-levels}.

Chapter \ref{s:gsff} is devoted to an application in \subindex{Global analysis}global analysis.
We summarize the predecessor formulae, we prove a
wide generalization of the \subindex{Yoshida-Nicolaescu's Theorem}Yoshida-Nicolaescu spectral flow
formula, namely the identity \subindex{Spectral flow}\subindex{Maslov index!spectral flow
formula}Maslov index=spectral flow, both in general terms of \subindex{Banach bundle}Banach bundles
and for \subindex{Elliptic differential operators!on smooth manifolds with boundary}elliptic
differential operators of arbitrary positive order on smooth manifolds with boundary. That involves
weak symplectic Hilbert spaces like the Sobolev space $H^{1/2}$ over the boundary. Applying
substantially more advanced results we derive a corresponding spectral flow formula in all
\subindex{Sobolev spaces}Sobolev spaces $H^{\sigma}$ for $\sigma\ge 0$, so in particular in the
familiar strong symplectic $L^2$.


In the Appendix \ref{s:closed-subspaces} on \subindex{Banach space!closed subspace}closed subspaces
in Banach spaces, we address the \subindex{Banach space!continuity of operations}continuity of
operations of linear subspaces. In \subindex{Gap topology}gap topology, we prove some sharp
estimates which might be of independent interest. E.g., they yield the following basic convergence
result for sums and intersections of permutations in the space $\Ss(X)$ of closed linear subspaces
in a Banach space $X$ in Proposition \ref{p:close-to} (\auindex{Neubauer,\ G.}\cite[Lemma 1.5 (1),
(2)]{Ne68}): Let $(M'_j)_{j=1,2.\dots}$ be a sequence in $\Ss(X)$ converging to $M\in\Ss(X)$ in the
gap topology, shortly \symindex{zz@$\to$ converging to}$M'\to M$, let similarly $N'\to N$ and $M+N$
be closed. Then $M'\cap N'\to M\cap N$ iff $M'+N'\to M+N$. For each of the three technical main
results of the Appendix, some applications are given to the \subindex{Global analysis!of elliptic
problems on manifolds with boundary}global analysis of elliptic problems on manifolds with
boundary.


\subsection*{Relation to our previous results}
With this Memoir we conclude a series of our mutually related previous approaches to
\subindex{Symplectic geometry}symplectic geometry, \subindex{Dynamical systems}dynamical systems,
and \subindex{Global analysis}global analysis; in chronological order
\auindex{Boo{\ss}--Bavnbek,\ B.}\auindex{Chen,\ G.}
\auindex{Furutani,\ K.}\auindex{Otsuki,\ N.}\auindex{Lesch,\ M.}
\auindex{Phillips,\ J.}\auindex{Long,\ Y.}\auindex{Zhu,\ C.}\cite{BoFu98, BoFu99, ZhLo99, Zh00,
Zh01, BoFuOt01, BoLePh01, Zh05, BoZh05, BoLeZh08, BCLZ, BooZhu:2013}.

The model for our various approaches was developed in joint work with K. Furutani and N. Otsuki in
\cite{BoFu98, BoFu99, BoFuOt01}. Roughly speaking, there we deal with a \textit{strong} symplectic
Hilbert space $(X,\lla\cdot,\cdot\rra,\w)$, so that $\w(x,y)=\lla Jx,y\rra$ with $J^*=-J\tand
J^2=-I$, possibly after continuous deformation of the inner product $\lla \cdot,\cdot\rra$. Then
the space \subindex{Lagrangian Grassmannian}\symindex{L@$\Ll(X,\w)$ Lagrangian
Grassmannian}$\Ll(X,\w)$ of all Lagrangian subspaces is contractible and, for fixed
$\la\in\Ll(X,\w)$, the fundamental group of the \subindex{Fredholm Lagrangian Grassmannian}
\symindex{FL@$\Ff\Ll(X,\w)$ Fredholm Lagrangian Grassmannian}Fredholm Lagrangian Grassmannian
$\Ff\Ll(X,\w,\la)$ of all Fredholm pairs $(\la,\mu)$ with $\mu\in\Ll(X,\w)$ is cyclic, see
\auindex{Boo{\ss}--Bavnbek,\ B.}\auindex{Furutani,\ K.}\cite[Section 4]{BoFu99} for an elementary
proof. By the induced \subindex{Symplectic splitting}{\em symplectic splitting} $X=X^+\oplus X^-$
with $X^{\pm}:= \ker(J\mp iI)$ we obtain
\begin{enumerate}\symindex{g@$\Graph(A)$ graph of operator $A$}
\item  $\forall \la\in\Ll(X,\w) \ \exists U\colon X^+\to X^-$ unitary with $\la=\Graph(U)$;

\item $(\la,\mu)\in\Ff\Ll(X,\w)\iff UV\ii -I_{X^-}\in\Ff(X)$; and
\item $\Mas(\la(s),\mu(s))_{s\in [0,1]}:=\SF_{(0,\infty)}\bigl(U_sV_s\ii\bigr)_{s\in [0,1]}$ well
defined.
\end{enumerate}\symindex{Mas@$\Mas(\la(s),\mu(s))_{s\in [0,1]}$ Maslov index}\symindex{sfl@$\SF_{\ell}$
spectral flow through gauge curve $\ell$}
Here \symindex{F@$\Ff(X)$ space of bounded Fredholm operators on $X$}$\Ff(X)$ denotes the space of
bounded Fredholm operators on $X$ and \symindex{FL@$\Ff\Ll(X,\w)$ Fredholm Lagrangian
Grassmannian}$\Ff\Ll(X,\w)$ the set of Fredholm pairs of Lagrangian subspaces of $(X,\w)$ (see
Definition \ref{d:fredholm-lag-pair}).

This setting is suitable for the following application in operator theory: Let $\mathcal{H}$ be a
complex separable \subindex{Hilbert space}Hilbert space and $A$ a \subindex{Closed symmetric
operator}closed symmetric operator. We extend slightly the frame of the \auindex{Birman,\
M.S.}\auindex{Krein,\ M.G.}\auindex{Vishik,\ M.I.}\subindex{Birman-Kre\u\i n-Vishik
theory}{B}irman-{K}re\u\i n-{V}ishik theory of \subindex{Extension!self-adjoint}self-adjoint
extensions of semi-bounded operators (see the review \cite{Alonso-Simon} by \auindex{Alonso,\
A.}\auindex{Simon,\ B.}A. Alonso and B. Simon). Consider the space \symindex{\beta@$\beta(A)$ space
of abstract boundary values}$\beta(A):=\dom(A^*)/\dom(A)$ of \subindex{Abstract boundary
values}abstract boundary values. It becomes a \subindex{Hilbert space!strong symplectic}strong
symplectic Hilbert space with
\[
 \omega(\gamma(x),\gamma(y))\ :=\ \lla A^*x,y\rra-\lla x,A^*y\rra,
\]
and the projection \symindex{\gamma@$\gamma$ abstract trace map}$\gamma\colon
\dom(A^*)\to\beta(A)$, $x\mapsto [x]:=x+\dom(A)$. The inner product $\lla \gamma(x),\gamma(y)\rra$
is induced by the graph inner product $\lla x,y\rra_{\Gg} := \lla x,y\rra + \lla A^*x,A^*y\rra$
that makes $\dom(A^*)$ and, consequently, $\beta(A)$ to Hilbert spaces. Introduce the
\subindex{Cauchy data space!abstract (or reduced)}abstract \textit{Cauchy data space}
\symindex{CD@$\operatorname{CD}(A)$ abstract Cauchy data space}$\operatorname{CD}(A):=
\left(\ker(A^*)+\dom(A)\right)/\dom(A) = \{\gamma(x)\mid x\in\ker A^*\}$. From \auindex{Neumann,
von,\ J.}von Neumann's \subindex{von Neumann program}famous \cite{Neu} we obtain
the correspondence
\[\subindex{Lagrangian subspaces!and self-adjoint extensions}\symindex{AD@$A_D$ extension
(realization) of operator $A$ with domain $D$} A_D \text{  self-adjoint extension $\ \iff\ [D]\<
\beta(A)$ Lagrangian},
\]
for $\dom(A) \< D \< \dom(A^*)$. Now let $A_D$ be a \subindex{Extension!self-adjoint
Fredholm}\textit{self-adjoint Fredholm extension}, $\{C(s)\}_{s\in [0,1]}$ a $C^0$
\subindex{Continuously varying!bounded self-adjoint operators}curve in
\symindex{Bs@$\mathcal{B}^{\sa}(\mathcal {H})$ space of bounded self-adjoint
operators}$\mathcal{B}^{\sa}(\mathcal {H})$, the space of bounded self-adjoint operators, and
assume \subindex{Weak inner Unique Continuation Property (wiUCP)}\textit{weak inner Unique
Continuation Property (UCP)}, i.e., $\ker(A^*+C(s)+\e)\cap \dom(A)=\{0\}$ for small positive $\e$.
Then, \auindex{Boo{\ss}--Bavnbek,\ B.}\auindex{Furutani,\ K.}\cite{BoFu98} shows that

\begin{enumerate}

\item $\{\operatorname{CD}(A+C(s)),\gamma(D)\}_{s\in [0,1]}$ is a continuous curve of Fredholm pairs of Lagrangians
in the gap topology, and

\item $\SF\{(A+C(s))_D\}_{s\in [0,1]} =
\Mas\{\operatorname{CD}(A+C(s)),\gamma(D)\}_{s\in [0,1]}$.
\end{enumerate}

On one side, the approach of \auindex{Boo{\ss}--Bavnbek,\ B.}\auindex{Furutani,\ K.}\cite{BoFu98}
has considerable strength: It is ideally suited both to \subindex{Hamiltonian system}Hamiltonian
systems of ordinary differential equations of first order over an interval $[0,T]$ with varying
lower order coefficients, and to \subindex{Continuously varying!Dirac type operators}curves of
\subindex{Dirac type operators!curves of}Dirac type operators on a Riemannian partitioned manifold
or manifold $M$ with boundary $\Si$ with fixed Clifford multiplication and Clifford module (and so
fixed principal symbol), but symmetric bounded perturbation due to varying affine connection
(background field). Hence it explains \auindex{Nicolaescu,\ L.}\subindex{Yoshida-Nicolaescu's
Theorem}Nicolaescu's Theorem (see below Section \ref{s:history}) in purely functional analysis
terms and elucidates the decisive role of weak inner UCP. For such curves of Dirac type operators,
the $\beta$-space remains fixed and can be described as a subspace of the distribution space
$H^{-1/2}(\Si)$ with ``half" component in $H^{1/2}(\Si)$. As shown in \auindex{Boo{\ss}--Bavnbek,\
B.}\auindex{Furutani,\ K.}\cite{BoFu99}, the Maslov index constructed in this way is invariant
under finite-dimensional symplectic reduction. Moreover, the approach admits varying boundary
conditions and varying symplectic forms, as shown in \auindex{Boo{\ss}--Bavnbek,\
B.}\auindex{Lesch,\ M.}\auindex{Phillips,\ J.}\auindex{Zhu,\ C.}\cite{BoLePh01,BoZh05} and can be
generalized to a \subindex{Spectral flow!spectral flow formula}spectral flow formula in the common
$L^2(\Si)$ as shown in \auindex{Furutani,\ K.}\auindex{Otsuki,\ N.}\cite{BoFuOt01}.

Unfortunately, that approach has severe limitations since it excludes \subindex{Varying maximal
domain}varying maximal domain: there is no $\beta$-space when variation of the highest order
coefficients is admitted for the \subindex{Continuously varying!elliptic differential
operators}curve of elliptic differential operators.

The natural alternative (here for first order operators) is to work with the Hilbert space
\symindex{H@$H^{1/2}(\Si;E\mid_{\Si})$
weak symplectic Sobolev space}
\[
H^{1/2}(\Si;E|_{\Si})\ \cong\ H^1(M;E)/H^1_0(M;E)
\]
which remains fixed as long as we keep our underlying Hermitian vector bundle $E\to M$ fixed. So,
let $A(s)\colon \Ci_0(M;E)\to\Ci_0(M;E),\/s\in [0,1]$ be a \subindex{Continuously varying!elliptic
differential operators}\subindex{Elliptic differential operators!symmetric!curve}curve of symmetric
elliptic first order differential operators. Green's form for $A(s)$ induces on $L^2(\Si;E|_{\Si})$
a \textit{strong} symplectic form $\w(s)_{\operatorname{Green}}(x,y):=-\lla J(s)x,y\rra_{L^2}$. On
\symindex{H@$H^{1/2}(\Si;E\mid_{\Si})$ weak symplectic Sobolev space}$H^{1/2}(\Si;E|_{\Si})$ the
induced symplectic form \symindex{\omega s@$\w(s)_{\operatorname{Green}}$ by Green's form induced
symplectic form}$\w(s)(x,y):=\w(s)_{\operatorname{Green}}(x,y)= -\lla J'(s)x,y\rra_{H^{1/2}}$ is
\textit{weak}. To see that, we choose a formally self-adjoint elliptic operator $B$ of first order
on $\Si$ to generate the metric on \symindex{H@$H^{1/2}(\Si;E\mid_{\Si})$ weak symplectic Sobolev
space}$H^{1/2}$ according to \subindex{G{\aa}rding's Inequality}G{\aa}rding's Theorem. Then we find
$J'(s)=(I+|B|)^{-1/2} J(s)$, which is a compact operator and so not invertible. This we emphasized
already in our \auindex{Boo{\ss}--Bavnbek,\ B.}\auindex{Long,\ Y.}\auindex{Zhu,\ C.}\cite{BoZh04}
where we raised the following questions:

\begin{description}
\item[Q1] How to define $\Mas(\la(s),\mu(s))_{s\in [0,1]}$ for curves of Fredholm pairs of
Lagrangian subspaces?

\item[Q2] How to calculate?

\item[Q3] What for?\qquad

\item[Q4] Dispensable? Non-trivial example?
\end{description}

Questions Q3 and Q4 are addressed below in  Chapter \ref{s:gsff} (see also our
\auindex{Boo{\ss}--Bavnbek,\ B.}\auindex{Zhu,\ C.}\cite{BoZh04}). There we point to the necessity
to work with the weak symplectic Hilbert space $H^{1/2}(\Si)$. Such work is indispensable when we
are looking for \subindex{Spectral flow!spectral flow formula}spectral flow formulae for
\subindex{Partitioned manifold}partitioned manifolds with curves of \subindex{Continuously
varying!elliptic differential operators}\subindex{Elliptic differential operators!of not-Dirac
type}elliptic operators which are \textit{not} of Dirac type.

To answer questions Q1 and Q2, we recall the following list of obstructions and open problems,
partly from \auindex{Boo{\ss}--Bavnbek,\ B.}\auindex{Zhu,\ C.}\cite{BoZh04} (see also Figures
\ref{f:obstructions}, \ref{f:examples}). For simplicity, we specify for Hilbert spaces instead of
Banach spaces:

Let $(X,\w)$ be a fixed complex Hilbert space with weak symplectic form $\w(x,y)=\lla Jx,y\rra$,
and $(X(s),\w(s)),s\in[0,1]$ a \subindex{Continuously varying!symplectic
forms}\subindex{Continuously varying!Hilbert inner products}curve of weak symplectic Hilbert
spaces, parametrized over the interval $[0,1]$ (other parameter spaces could be dealt with). Then
in general we have in difference to strong symplectic forms:


\begin{enumerate}\subindex{Symplectic splitting}\subindex{Annihilator!double}
\subindex{Obstructions to straight forward generalization!generator $J$ non-invertible}


\renewcommand{\labelenumi}{(\Roman{enumi})}
\item $J^2\ne -I$;

\subindex{Obstructions to straight forward generalization!no symplectic splitting}
\item so, in general $X\ne X^-\oplus X^+$ with $X^{\pm}:=\ker(J\mp iI)$; more generally, our
Example \ref{ex:no-splitting} shows that there exist strong symplectic Banach spaces that do not
admit any symplectic splitting;

\subindex{Obstructions to straight forward generalization!symplectic splitting not continuously
varying even when existing}
\item in general, for continuously varying $\w(s)$ it does not hold that $X^{\mp}(s)$ is continuously varying;

\subindex{Obstructions to straight forward generalization!double annihilator not the identity on
closed subspaces}
\item as shown in our Example \ref{ex:double-annihilator}, we have $\la^{\w\w}\supsetneqq \la$ for some closed linear subspaces $\la$;
according to our Lemma \ref{l:double-annihilator}, the double annihilator, however, is the identity
for $\w$-closed subspaces, where the topology is defined by the semi-norms $p_y(x):=|\w(x,y)|$
(based on \auindex{Schmid,\ R.}R. Schmid, \cite{Schmid:1987});

\subindex{Obstructions to straight forward generalization!non-vanishing index of Fredholm pair of
Lagrangian subspaces}
\item by Corollary \ref{c:fred-pair-index-vanishing} we have $\Index(\la,\mu)\le 0$ for $(\la,\mu)\in \Ff\Ll$;
our Example \ref{ex:negative-index} shows that there exist Fredholm pairs of Lagrangian subspaces
with truly negative index; hence, in particular, the concept of the \textit{Maslov cycle}
\symindex{M@$\Mm(X,\w,\la_0)$ Maslov cycle}\subindex{Symplectic geometry!Maslov
cycle}$\Mm(X,\w,\la_0):=\Ff\Ll(\la_0,\cdot)\setminus\Ff\Ll^0(\la_0,\cdot)$ of a fixed Lagrangian
subspace $\la_0$ (comprising all Lagrangians that form a Fredholm pair with $\la_0$ but do not
intersect $\la_0$ transversally) is invalidated: \subindex{Obstructions to straight forward
generalization!no meaningful Maslov cycle}we can no longer conclude complementarity of $\mu$ and
$\la_0$ from $\mu\cap\la_0=\{0\}$;

\subindex{Obstructions to straight forward generalization!$\Ll$ not contractible}

\item in general, the space $\Ll(X,\w)$ is \subindex{Counterexamples!$\Ll$ not contractible}

not contractible and even not connected according to Swanson's arguments for counterexamples
\auindex{Swanson,\ R.C.}\cite[Remarks after Theorem 3.6]{Swanson:1980}, based on \auindex{Douady,\
A.}A. Douady, \cite{Douady:1965};

\subindex{Obstructions to straight forward generalization!fundamental group of Fredholm Lagrangian
Grassmannian unknown}\subindex{Fredholm Lagrangian Grassmannian}
\item \symindex{\pi@$\pi_1(\Ff\Ll(\la,\cdot))$ fundamental group of Fredholm Lagrangian Grassmannian}
$\pi_1(\Ff\Ll_0(X,\la))\fequal{?}\ZZ$ for $\la\in\Ll(X,\w)$; valid for strong symplectic Hilbert
space $(X,\w)$.
\end{enumerate}


\begin{figure}
\includegraphics[scale=0.95]{Examples_crop.pdf}
\caption{Three counterexamples}\label{f:examples}
\end{figure}


\subsection*{Limited value of our previous pilot study}
Anyway, our previous \auindex{Boo{\ss}--Bavnbek,\ B.}\auindex{Zhu,\ C.}\cite{BooZhu:2013} deals
with a continuous family of \subindex{Continuously varying!weak symplectic forms}weak symplectic
forms $\w(s)$ on a curve of \subindex{Continuously varying!Banach spaces}Banach spaces $X(s)$,
$s\in[0,1]$. It gives a definition of the \subindex{Maslov index}Maslov index for a path
$(\lambda(s),\mu(s))_{s\in [0,1]}$ of Fredholm pairs of Lagrangian subspaces of index $0$ under the
assumption of a \subindex{Continuously varying!symplectic splittings}continuously varying
symplectic splitting $X=X^+(s)\oplus X^-(s)$. The definition is inspired by the careful
distinctions of planar intersections in \auindex{Long,\ Y.}\auindex{Zhu,\ C.}\cite{ZhLo99, Zh00,
Zh01, Zh05}. Then it is shown that all nice properties of the Maslov index are preserved for this
general case. However, that approach has four serious \subindex{Obstructions to straight forward
generalization}drawbacks which render this definition incalculable:

\begin{enumerate}\subindex{Symplectic splitting}\subindex{Annihilator!double}
\renewcommand{\labelenumi}{\arabic{enumi}.}
\item In Section \ref{ss:maslov-unitary}, our Example \ref{ex:no-splitting} provides a strong symplectic Banach space that
does not admit a symplectic splitting.

\item Even when a single \subindex{Symplectic splitting!continuously varying}symplectic splitting is
guaranteed, there is no way to establish such splitting for families in a continuous way (see also
our obstruction III above).

\item The Maslov index, as defined in \cite{BooZhu:2013} becomes independent of the choice of the splitting only for strong symplectic
forms.

\item That construction admits finite-dimensional \subindex{Symplectic reduction}symplectic reduction
only for strong symplectic forms.

\end{enumerate}

To us, our \auindex{Boo{\ss}--Bavnbek,\ B.}\auindex{Zhu,\ C.}\cite{BooZhu:2013} is a highly
valuable pilot study, but the preceding limitations explain why in this Memoir we begin again from
scratch. For that purpose, an encouraging result was obtained in \auindex{Boo{\ss}--Bavnbek,\
B.}\auindex{Zhu,\ C.}\cite{BoLeZh08} combined with \auindex{Chen,\ G.}\cite{BCLZ}: the continuous
variation of the \subindex{Calder{\'o}n projection}Calder{\'o}n projection in $L^2(\Si)$ for a
\subindex{Continuously varying!elliptic differential operators}curve of elliptic differential
operators of first order. We shall use this result in our Section \ref{ss:gsff}.


\subsection*{Acknowledgements} {\small We thank Prof. K. Furutani (Tokyo), Prof. M. Lesch
(Bonn), and Prof. R. Nest (Copenhagen) for inspiring discussions about this subject. Last but not least,
we would like to thank the referees of this paper for their critical reading and very helpful comments and suggestions.
The second author was partially supported by NSFC (No.11221091 and No. 11471169), LPMC of MOE of China.}
{"config": "arxiv", "file": "1406.0569/introduction.tex"}