# Oscillation numbers for continuous Lagrangian paths
and Maslov index

Julia Elyseeva  Department of Applied Mathematics, Moscow State University of
Technology, Vadkovskii per. 3a, 101472, Moscow, Russia, E-mail address:
<EMAIL_ADDRESS>,  Peter Šepitka  Department of Mathematics and
Statistics, Faculty of Science, Masaryk University, Kotlářská 2, CZ-61137
Brno, Czech Republic, E-mail address<EMAIL_ADDRESS>and  Roman Šimon
Hilscher  Department of Mathematics and Statistics, Faculty of Science,
Masaryk University, Kotlářská 2, CZ-61137 Brno, Czech Republic, E-mail
address<EMAIL_ADDRESS>
###### Abstract.

In this paper we present the theory of oscillation numbers and dual
oscillation numbers for continuous Lagrangian paths in $\mathbb{R}^{2n}$. Our
main results include a connection of the oscillation numbers of the given
Lagrangian path with the Lidskii angles of a special symplectic orthogonal
matrix. We also present Sturmian type comparison and separation theorems for
the difference of the oscillation numbers of two continuous Lagrangian paths.
These results, as well as the definition of the oscillation number itself, are
based on the comparative index theory (Elyseeva, 2009). The applications of
these results are directed to the theory of Maslov index of two continuous
Lagrangian paths. We derive a formula for the Maslov index via the Lidskii
angles of a special symplectic orthogonal matrix, and hence we express the
Maslov index as the oscillation number of a certain transformed Lagrangian
path. The results and methods are based on a generalization of the recently
introduced oscillation numbers and dual oscillation numbers for conjoined
bases of linear Hamiltonian systems (Elyseeva, 2019 and 2020) and on the
connection between the comparative index and Lidskii angles of symplectic
matrices (Šepitka and Šimon Hilscher, 2020).

###### Key words and phrases:

Oscillation number; Lagrangian path; Lidskii angle; Symplectic matrix;
Comparative index; Maslov index

This research was supported by the Czech Science Foundation under grant
GA19–01246S

Version: . Submitted to Journal of Dynamics and Differential Equations on
18.02.2021.

††footnotetext: 2020 Mathematics Subject Classification. Primary 34C10.
Secondary 53D12.

##  1\. Introduction

Let $n\in{\mathbb{N}}$ be a given dimension and $[a,b]\subseteq{\mathbb{R}}$ a
given interval. In this paper we develop the theory of oscillation numbers for
arbitrary continuous Lagrangian paths on $[a,b]$. A continuous matrix-valued
function $Y:[a,b]\to{\mathbb{R}}^{2n\times n}$ is a Lagrangian path if

$Y^{T}(t)\hskip 1.0pt{\mathcal{J}}\hskip
1.0ptY(t)=0,\quad\operatorname{\text{\rm rank}}Y(t)=n,\quad t\in[a,b],$ (1.1)

where ${\mathcal{J}}\in{\mathbb{R}}^{2n\times 2n}$ is the canonical skew-
symmetric matrix. When a Lagrangian path is constant on $[a,b]$, then we call
it a Lagrangian plane. Lagrangian paths arise, among others, as particular
solutions (called conjoined or isotropic bases) of the linear Hamiltonian
differential system

$y^{\prime}={\mathcal{J}}\hskip 1.0pt{\mathcal{H}}(t)\,y,\quad t\in[a,b],$ (H)

where the coefficient matrix ${\mathcal{H}}:[a,b]\to{\mathbb{R}}^{2n\times
2n}$ is symmetric and piecewise continuous. In this case the function $Y$ is
piecewise continuously differentiable on $[a,b]$ and satisfies (1.1). We
partition the matrices ${\mathcal{J}}$, $Y(t)$, and ${\mathcal{H}}(t)$ into
$n\times n$ blocks as

${\mathcal{J}}=\left(\begin{matrix}0&I\\\ -I&0\end{matrix}\right),\quad
Y(t)=\left(\begin{matrix}X(t)\\\
U(t)\end{matrix}\right),\quad{\mathcal{H}}(t)=\left(\begin{matrix}-C(t)&A^{T}(t)\\\
A(t)&B(t)\end{matrix}\right).$ (1.2)

The motivation for the present study comes from the qualitative theory of
canonical systems of the form (H) in [9, 13, 18, 19, 36, 41, 42].

The oscillation number ${\mathcal{N}}(Y,[a,b])$ and the dual oscillation
number ${\mathcal{N}}^{*}(Y,[a,b])$ of a conjoined basis $Y$ of (H) on $[a,b]$
were defined in [18, 19] and [21] as quantities describing the oscillations of
the first component $X$ of the conjoined basis $Y$. They are based on the
notions of the comparative index and the dual comparative index, see [11, 12]
or [10, Section 3], and on piecewise constant symplectic transformations of
solutions of system (H) from [1, 2]. According to the definitions in (2.1)
below, for two (constant) Lagrangian planes $Y$ and $\hat{Y}$ the comparative
index $\mu(Y,\hat{Y})$ and the dual comparative index $\mu^{*}(Y,\hat{Y})$ are
integers between $0$ and $n$, which are defined in algebraic way from the rank
and the index of certain $n\times n$ matrices constructed from the blocks of
the matrices $Y$ and $\hat{Y}$. The utility of the comparative index for the
oscillation and spectral theory of system (H), as well as for the parallel
theory of symplectic difference systems, is documented e.g. in [13, 14, 15,
17, 36, 38, 39, 40] and [10, 20]. If system (H) satisfies the Legendre
condition

$B(t)\geq 0\quad\text{for all }t\in[a,b],$ (1.3)

then the oscillation number ${\mathcal{N}}(Y,[a,b])$ reduces to the total
number of left proper focal points of the conjoined basis $Y$ in the interval
$(a,b]$, while the dual oscillation number ${\mathcal{N}}^{*}(Y,[a,b])$
reduces to the total number of right proper focal points of $Y$ in the
interval $[a,b)$. These notions were introduced in [43] and [33]. The main
results in [19] and [21] provide, without assuming (1.3) and without any
majorant condition on the involved coefficient matrices, the Sturmian type
comparison and separation theorems for conjoined bases of two possibly
uncontrollable linear Hamiltonian systems of the form (H) in terms of the
oscillation numbers.

The main purpose of this paper is to extend the concepts of an oscillation
number ${\mathcal{N}}(Y,[a,b])$ and a dual oscillation number
${\mathcal{N}}^{*}(Y,[a,b])$, following the definitions presented in [18, 19,
21] via the comparative index, to an arbitrary continuous Lagrangian path $Y$
on $[a,b]$ and to connect these notions with the Maslov index. As the main
tools we employ the traditional theory of Lidskii angles (or arguments) for
symplectic matrices introduced in [35, 44, 45] and a new connection of the
Lidskii angles with the comparative index obtained recently in [42]. More
precisely, with a given continuous Lagrangian path $Y$ on $[a,b]$ we associate
the special continuous symplectic and orthogonal matrix $Z_{Y}(t)$ defined by

$Z_{Y}(t):=\left(\begin{matrix}{\mathcal{J}}\hskip
1.0ptY(t)\,K_{Y}(t)&Y(t)\,K_{Y}(t)\end{matrix}\right),\quad
K_{Y}(t):=[Y^{T}(t)\,Y(t)]^{-1/2},\quad t\in[a,b],$ (1.4)

where $K_{Y}(t)>0$ represents a normalization factor, and consider its
continuous Lidskii angles $\varphi_{j}(t)$ on $[a,b]$ for
$j\in\\{1,\dots,n\\}$. We then express (Theorem 3.8) the oscillation number
${\mathcal{N}}(Y,[a,b])$ in terms of the cumulative changes of specific
integers $q_{j}(t)$ at the endpoints of $[a,b]$. For each
$j\in\\{1,\dots,n\\}$ the integer $q_{j}(t)$ has the property that the Lidskii
angle $\varphi_{j}(t)$ belongs to the half-open interval $[2\pi\hskip
1.0ptq_{j}(t),2\pi\hskip 1.0pt(q_{j}(t)+1))$. In a similar way we express the
dual oscillation number ${\mathcal{N}}^{*}(Y,[a,b])$ in terms of the integer
quantities $q_{j}^{*}(t)$ corresponding to the Lidskii angles
$\varphi_{j}(t)$, which are located in the half-open interval $(2\pi
q_{j}^{*}(t),2\pi\hskip 1.0pt(q_{j}^{*}(t)+1)]$. Combining the latter two
results yields a formula (Theorem 3.13) relating the oscillation number and
the dual oscillation number of one Lagrangian path on $[a,b]$. We also prove a
special invariance property of the oscillation numbers under a symplectic
orthogonal transformation (Theorem 3.12). We also present (under some
monotonicity assumptions) formulas for the oscillation and dual oscillation
numbers of the Lagrangian path $Y$ in terms of the changes in the rank of the
first component $X$ of $Y$ (Theorem 4.6).

Using the Lidskii angles in our analysis of the oscillation numbers leads in a
natural way to their connection with the Maslov index of two continuous
Lagrangian paths $Y$ and $\hat{Y}$ on $[a,b]$, denoted by
$\operatorname{\text{\rm Mas}}\hskip 1.0pt(Y,\hat{Y},[a,b])$. Here we use the
analytic definition of the Maslov index from [5], see also [25, 26, 27]. We
show (Theorem 4.2) that the Maslov index of $Y$ and $\hat{Y}$ on $[a,b]$ can
be calculated from the changes at the endpoints of $[a,b]$ of the Lidskii
angles of the symplectic matrix $Z_{Y}^{-1}(t)\hskip 1.0ptZ_{\hat{Y}}(t)$.
This approach is known in [46, Definition 2.2] and [6, Eqs. (2.4)–(2.5)].
Consequently, the Maslov index of $Y$ and $\hat{Y}$ on $[a,b]$ is equal to the
oscillation number of the transformed Lagrangian path $Z_{Y}^{-1}\hskip
1.0pt\hat{Y}$ on $[a,b]$, i.e.,

$\operatorname{\text{\rm Mas}}\hskip
1.0pt(Y,\hat{Y},[a,b])={\mathcal{N}}(Z_{Y}^{-1}\hskip 1.0pt\hat{Y},[a,b]),$
(1.5)

where the symplectic and orthogonal matrix matrices $Z_{Y}(t)$ and
$Z_{\hat{Y}}(t)$ are defined according to (1.4). Equivalently, the number in
(1.5) is equal to the oscillation number ${\mathcal{N}}(Z^{-1}\hat{Y},[a,b])$,
where $Z(t)$ is any continuous symplectic matrix, whose second block column is
equal to $Y(t)$. In particular, the oscillation number of $Y$ on $[a,b]$ can
be expressed via the Maslov index as

${\mathcal{N}}(Y,[a,b])=\operatorname{\text{\rm Mas}}\hskip 1.0pt(E,Y,[a,b]),$
(1.6)

where the matrix $E=\left(\begin{matrix}0&I\end{matrix}\right)^{T}$ represents
the vertical Lagrangian plane. We also discuss (Remark 4.5) a notion of the
dual Maslov index, which is related in analogous way to the dual oscillation
number and to the dual comparative index, and a new monotonicity result for
calculating the Maslov index (Theorem 4.8). Thus, we contribute in the spirit
of [25, 26, 27] to a detailed investigation of the Maslov index for Lagrangian
paths in ${\mathbb{R}}^{2n}$.

The main properties of the oscillation numbers obtained via the Lidskii angles
allow to derive Sturmian type comparison theorems for the oscillation numbers
(Theorem 5.1) of two continuous Lagrangian paths on $[a,b]$, essentially
extending the results in [19, Theorem 4.4] and [21, Theorem 4.3] to the
context of continuous Lagrangian paths. These formulas involve the comparative
index or the dual comparative index of $Y(t)$ and $\hat{Y}(t)$ evaluated at
the endpoints of the interval $[a,b]$. In addition, we derive Sturmian type
separation theorems (Theorems 5.6 and 5.8) for specific continuous Lagrangian
paths, which belong to the set of paths determined by a given continuous
symplectic matrix $\Phi(t)$ on $[a,b]$. In this way we generalize the result
in [40, Theorem 1.1] to the context of continuous Lagrangian paths. The above
mentioned comparison theorem for the oscillation numbers also yields
(Corollary 5.4) the expression of the Maslov index $\operatorname{\text{\rm
Mas}}\hskip 1.0pt(Y,\hat{Y},[a,b])$ in terms of the two reference Maslov
indices $\operatorname{\text{\rm Mas}}\hskip 1.0pt(E,\hat{Y},[a,b])$ and
$\operatorname{\text{\rm Mas}}\hskip 1.0pt(E,Y,[a,b])$ and in terms of the
comparative index of $\hat{Y}$ and $Y$ evaluated at the endpoints of the
interval $[a,b]$.

We are convinced that the results in this paper contribute to the
understanding of the role of the comparative index in the oscillation theory
of continuous Lagrangian paths. We believe that further investigations in this
direction will lead to substantial advancements in several areas of
theoretical mathematics, such as in the spectral theory of linear Hamiltonian
systems, where the monotonicity assumption on the spectral parameter is
dropped, in the oscillation theory on discrete time domains, or in the theory
of continuous symplectic matrices in general. In particular, the presented
methods are instrumental for future development of the oscillation numbers and
the Maslov index on unbounded intervals (including the properties of the
rotation number).

The paper is organized as follows. In Section 2 we recall the definitions of
the comparative index, the dual comparative index, and the Lidskii angles of a
symplectic matrix. We also present their relationship based on special
argument functions derived in [42]. In Section 3 we define the oscillation
number and the dual oscillation number for a continuous Lagrangian path on
$[a,b]$ and study their relationship with the Lidskii angles. In Section 4 we
investigate connections of the oscillation number and the dual oscillation
number with the Maslov index. In Section 5 we present comparison and
separation theorems for the oscillation numbers and the dual oscillation
numbers of two Lagrangian paths. Finally, in Section 6 we comment about the
results of this paper and their future development.

##  2\. Comparative index and Lidskii angles

In this section we recall the definitions of the comparative index, the dual
comparative index, and the Lidskii angles for a symplectic matrix. We also
present formulas for the calculation of the comparative index and the dual
comparative index by means of the Lidskii angles of suitable symplectic
matrices. Following [11, 12] or [10, Section 3], for two constant matrices
$Y,\hat{Y}\in{\mathbb{R}}^{2n\times n}$ satisfying the properties in (1.1) we
define the comparative index $\mu(Y,\hat{Y})$ and the dual comparative index
$\mu^{*}(Y,\hat{Y})$ by the formulas

$\mu(Y,\hat{Y}):=\operatorname{\text{\rm
rank}}{\mathcal{M}}+\operatorname{\text{\rm
ind}}{\mathcal{P}},\quad\mu^{*}(Y,\hat{Y}):=\operatorname{\text{\rm
rank}}{\mathcal{M}}+\operatorname{\text{\rm ind}}\,(-{\mathcal{P}}),$ (2.1)

where the $n\times n$ matrices ${\mathcal{M}}$ and ${\mathcal{P}}$ are given
by

${\mathcal{M}}:=(I-X^{\dagger}X)\,W(Y,\hat{Y}),\quad{\mathcal{P}}:=V\hskip
1.0pt[W(Y,\hat{Y})]^{T}X^{\dagger}{\hat{X}}V,\quad
V:=I-{\mathcal{M}}^{\dagger}{\mathcal{M}},$ (2.2)

and where $W(Y,\hat{Y}):=Y^{T}\\!{\mathcal{J}}\hskip 1.0pt\hat{Y}$ is the
Wronskian of $Y=(X^{T},U^{T})^{T}$ and
$\hat{Y}=({\hat{X}}^{T},{\hat{U}}^{T})^{T}$, following the notation in (1.2).
The dagger in (2.2) denotes the Moore–Penrose pseudoinverse, see e.g. [4, 7].
The notation $\operatorname{\text{\rm ind}}{\mathcal{P}}$ means the number of
negative eigenvalues of the symmetric matrix ${\mathcal{P}}$. Note that
$\operatorname{\text{\rm ind}}{\mathcal{P}}+\operatorname{\text{\rm
ind}}\,(-{\mathcal{P}})=\operatorname{\text{\rm rank}}{\mathcal{P}}$. For
convenience we also define the constant $2n\times n$ matrix $E$ representing
the vertical Lagrangian plane, i.e.,

$E:=\left(\begin{matrix}0&I\end{matrix}\right)^{T}.$ (2.3)

We will use the following basic invariant properties of the comparative
indices defined in (2.1), see [12, Properties 1–2, pg. 448] or [10, Theorem
3.5(i)–(iii)],

$\displaystyle\mu(YC_{1},\hat{Y}C_{2})=\mu(Y,\hat{Y}),\quad\mu^{*}(YC_{1},\hat{Y}C_{2})=\mu^{*}(Y,\hat{Y}),\quad\det
C_{1}\neq 0,\ \det C_{2}\neq 0,$ (2.4)
$\displaystyle\mu(LY,L\hat{Y})=\mu(Y,\hat{Y}),\quad\mu^{*}(LY,L\hat{Y})=\mu^{*}(Y,\hat{Y}),\quad
L=\left(\begin{matrix}P&0\\\ K&P^{T-1}\end{matrix}\right),\
L^{T}\\!{\mathcal{J}}L={\mathcal{J}},$ (2.5)
$\displaystyle\mu(Y,\hat{Y})=\mu^{*}(Z^{-1}E,Z^{-1}\hat{Y}),\quad\mu^{*}(Y,\hat{Y})=\mu(Z^{-1}E,Z^{-1}\hat{Y}),\quad
Z^{T}\\!{\mathcal{J}}Z={\mathcal{J}},\ ZE=Y.$ (2.6)

Next we recall the definition of Lidskii angles. Consider a real symplectic
matrix

$S=\left(\begin{matrix}S_{11}&S_{12}\\\ S_{21}&S_{22}\end{matrix}\right),\quad
S_{ij}\in{\mathbb{R}}^{n\times n},\quad S^{T}\\!{\mathcal{J}}S={\mathcal{J}}.$
(2.7)

We define, according to [35, 45, 3] or [28, Section 3.1], the complex $n\times
n$ matrix

$W_{S}:=(S_{11}-iS_{12})^{-1}\hskip 1.0pt(S_{11}+iS_{12}),$ (2.8)

where $i$ is the imaginary unit ($i^{\hskip 1.0pt2}=-1$). The matrix $W_{S}$
is well-defined, symmetric, and unitary, and in particular its eigenvalues
$w_{j}$ lie on the unit circle ${\mathbb{U}}$ in the complex plane. Hence,
$w_{j}=\exp\hskip 1.0pt(i\varphi_{j})$ with the real arguments $\varphi_{j}$.
The numbers $\varphi_{j}$ for $j\in\\{1,\dots,n\\}$ are called the Lidskii
angles corresponding to the symplectic matrix $S$. The angles $\varphi_{j}$
are thus defined uniquely up to an additive term $2\pi m_{j}$ for
$m_{j}\in{\mathbb{Z}}$ and

$\sum_{j=1}^{n}\varphi_{j}=\arg\det W_{S}=\big{(}2\hskip 1.0pt\arg\det\hskip
1.0pt(S_{11}+iS_{12})\big{)}\\!\\!\\!\\!\mod 2\pi.$ (2.9)

In view of (2.9) the Lidskii angles $\varphi_{j}$ satisfy

$\frac{1}{2}\,\sum_{j=1}^{n}\varphi_{j}=\operatorname{\text{\rm
Arg}}_{3}(S)\\!\\!\\!\\!\mod\pi,$ (2.10)

where $\operatorname{\text{\rm Arg}}_{3}(S)$ is one of the arguments of the
symplectic matrix $S$ considered by Yakubovich in [44, pg. 263], see also [29,
Lemma 5.6], namely

$\operatorname{\text{\rm Arg}}_{3}(S):=\arg\det\hskip
1.0pt(S_{11}+iS_{12})=\arg\det\\!\bigg{(}\\!\\!\left(\begin{matrix}I&0\end{matrix}\right)S\left(\begin{matrix}I\\\
iI\end{matrix}\right)\\!\\!\bigg{)}.$ (2.11)

In [42] we introduced the following argument function, which is motivated by
(2.10). Given a real interval ${\mathcal{I}}$ and a real symplectic matrix
$S$, we consider the Lidskii angles $\varphi_{j}$ of $S$, which belong to the
interval ${\mathcal{I}}$ for all $j\in\\{1,\dots,n\\}$. Then we set

$\operatorname{\text{\rm Arg}}_{\hskip 1.0pt{\mathcal{I}}}(S\hskip
1.0pt):=\frac{1}{2}\,\sum_{j=1}^{n}\varphi_{j}\qquad\text{with
$\varphi_{j}\in{\mathcal{I}}$ for all $j\in\\{1,\dots,n\\}$}.$ (2.12)

We will use the above argument function $\operatorname{\text{\rm Arg}}_{\hskip
1.0pt{\mathcal{I}}}$ with the particular choice of the half-open intervals
${\mathcal{I}}=[2\pi q,2\pi\hskip 1.0pt(q+1))$ and ${\mathcal{I}}=(2\pi
q,2\pi\hskip 1.0pt(q+1)]$ for $q\in{\mathbb{Z}}$.

######  Remark 2.1.

It is known that the number of the Lidskii angles $\varphi_{j}$ of $S$, which
are integer multiples of $2\pi$, is equal to the defect of the block $S_{12}$
in (2.7), see e.g. [22, Proposition 2.5] or [41, Eq. (4.10)].

The main result in [42] provides a connection between the comparative index
(or the dual comparative index) and the Lidskii angles. The connection is
based on the symplectic and orthogonal matrices of the form (1.4). More
precisely, for a real constant $2n\times n$ matrix $Y$ satisfying (1.1) we
define the constant $2n\times 2n$ matrix

$Z_{Y}:=\left(\begin{matrix}{\mathcal{J}}\hskip
1.0ptYK_{Y}&YK_{Y}\end{matrix}\right),\quad K_{Y}:=(Y^{T}Y)^{-1/2},$ (2.13)

where the symmetric matrix $K_{Y}$ obeys the condition $K_{Y}>0$. Then the
matrix $Z_{Y}$ is symplectic and orthogonal, i.e.,
$Z_{Y}^{T}{\mathcal{J}}Z_{Y}={\mathcal{J}}$ and $Z_{Y}^{T}Z_{Y}=I$. This
implies that the matrices $Z_{Y}$ and ${\mathcal{J}}$ commute. Note also that
$Z_{E}=I$.

######  Remark 2.2.

Observe that for an arbitrary symplectic matrix $S$ we have

$S=Z_{SE}\hskip 1.0ptL,$ (2.14)

where $L=Z_{SE}^{-1}\hskip 1.0ptS=Z_{SE}^{T}\hskip 1.0ptS$ is a symplectic
lower block triangular matrix in the form given by (2.5). This fact follows
from equation (2.13) with $Y:=SE$. Indeed, the right upper block of the matrix
$L$ is equal to $(I\ 0)\hskip 1.0ptLE=(I\ 0)\hskip
1.0ptZ_{SE}^{T}SE=-K_{Y}(Y^{T}\\!{\mathcal{J}}Y)=0$. Moreover, if $S$ is a
symplectic and orthogonal matrix, then in (2.14) we have $L=I$ and hence by
(2.13) we get $Z_{SY}=SZ_{Y}$ and $Z_{SE}=S$.

The following result is derived in [42, Theorem 1.1].

######  Proposition 2.3 (Comparative index and Lidskii angles).

Let $Y$ and $\hat{Y}$ be real constant $2n\times n$ matrices satisfying (1.1)
and define the symplectic matrices $Z_{Y}$ and $Z_{\hat{Y}}$ by (2.13). Then
the comparative index and the dual comparative index defined in (2.1) with
(2.2) satisfy

$\displaystyle\mu(Y,\hat{Y})$
$\displaystyle=\frac{1}{\pi}\,\Big{\\{}\operatorname{\text{\rm Arg}}_{\hskip
1.0pt[0,2\pi)}(Z_{\hat{Y}})-\operatorname{\text{\rm Arg}}_{\hskip
1.0pt[0,2\pi)}(Z_{Y})+\operatorname{\text{\rm Arg}}_{\hskip
1.0pt[0,2\pi)}(Z_{\hat{Y}}^{-1}Z_{Y})\Big{\\}},$ (2.15)
$\displaystyle\mu^{*}(Y,\hat{Y})$
$\displaystyle=n-\frac{1}{\pi}\,\Big{\\{}\operatorname{\text{\rm Arg}}_{\hskip
1.0pt(0,2\pi]}(Z_{\hat{Y}})-\operatorname{\text{\rm Arg}}_{\hskip
1.0pt(0,2\pi]}(Z_{Y})+\operatorname{\text{\rm Arg}}_{\hskip
1.0pt(0,2\pi]}(Z_{\hat{Y}}^{-1}Z_{Y})\Big{\\}},$ (2.16)

where $\operatorname{\text{\rm Arg}}_{\hskip 1.0pt{\mathcal{I}}}$ is the
special argument function defined in (2.12).

In the last comment in this section we discuss the continuity property of the
Lidskii angles.

######  Remark 2.4.

Consider a continuous Lagrangian path $Y$ on the interval $[a,b]$ and define
the symplectic and orthogonal matrix $Z_{Y}(t)$ on $[a,b]$ by (1.4). Note that
this matrix corresponds to the matrix $Z_{Y(t)}$ in (2.13). Then the matrix
$Z_{Y}(t)$ is continuous on $[a,b]$, which implies that the corresponding
matrix $W_{Z_{Y}(t)}$ defined through (2.8) is also continuous on $[a,b]$.
Therefore, the Lidskii angles $\varphi_{j}(t)$ of the symplectic matrix
$Z_{Y}(t)$ can be chosen to be continuous functions on $[a,b]$, see also the
proof of [28, Theorem 3.6]. This fact is utilized in Definition 3.4 below. In
the next sections we will always work with such continuous Lidskii angles
$\varphi_{j}(t)$ on $[a,b]$ for all $j\in\\{1,\dots,n\\}$.

##  3\. Oscillation numbers for Lagrangian paths

In this section we generalize the notion of the oscillation number and the
dual oscillation number from [17, pp. 17–18], [19, pp. 311–312], and [21,
Definition 3.1], which are defined through the comparative index and the dual
comparative index, in the context of continuous Lagrangian paths on $[a,b]$
and study their properties in terms of the Lidskii angles. For a given
continuous Lagrangian path $Y$ on $[a,b]$, let
$D:=\\{a=t_{0}<t_{1}<\dots<t_{p-1}<t_{p}=b\\}$ be a finite partition of the
interval $[a,b]$ and let $\\{R_{k}(t)\\}_{k=0}^{p-1}$ be a finite system of
real symplectic matrices on $[a,b]$ such that for every
$k\in\\{0,\dots,p-1\\}$ the matrix $R_{k}(t)$ is continuous on the interval
$[t_{k},t_{k+1}]$ and the transformed Lagrangian path
$\tilde{Y}_{k}=({\tilde{X}}_{k}^{T},{\tilde{U}}_{k}^{T})^{T}:=R_{k}^{-1}Y$
satisfies

$\left.\begin{array}[]{c}\operatorname{\text{\rm
rank}}{\tilde{X}}_{k}(t)=\operatorname{\text{\rm
rank}}W(R_{k}(t)E,Y(t))=\operatorname{\text{\rm
const}},\quad\operatorname{\text{\rm rank}}\\!\big{(}(I\ 0)\hskip
1.0ptR_{k}(t)E\big{)}=\operatorname{\text{\rm const}},\\\\[2.84526pt]
t\in[t_{k},t_{k+1}],\quad k\in\\{0,1,\dots,p-1\\}.\end{array}\\!\right\\}$
(3.1)

For a continuous Lagrangian path $Y$ on $[a,b]$ and a partition
$D=\\{t_{k}\\}_{k=0}^{p}$ of $[a,b]$ with the system of symplectic matrices
$R:=\\{R_{k}(t)\\}_{k=0}^{p-1}$ satisfying (3.1) we define the following
quantities, compare with [17, Eq. (2.6)], [19, Eq. (3.4)], and [21, Eq.
(3.1)],

$\displaystyle{\mathcal{N}}(Y,D,R,[a,b])$
$\displaystyle:=\sum_{k=0}^{p-1}\mu(Y(t),R_{k}(t)E)\hskip
1.0pt\big{|}_{t_{k}}^{t_{k+1}},$ (3.2)
$\displaystyle{\mathcal{N}}^{*}(Y,D,R,[a,b])$
$\displaystyle:=\sum_{k=0}^{p-1}\mu^{*}(Y(t),R_{k}(t)E)\hskip
1.0pt\big{|}_{t_{k+1}}^{t_{k}},$ (3.3)

where we use the comparative index in (3.2) and the dual comparative index in
(3.3). Here for a function $f:[a,b]\to{\mathbb{R}}$ we use the standard
notation

$f(t)\big{|}_{\tau_{1}}^{\tau_{2}}:=f(\tau_{2})-f(\tau_{1}),\quad\tau_{1},\tau_{2}\in[a,b].$

Based on (3.1) and property (2.6) of the comparative index we derive the
connections

$\left.\begin{array}[]{rl}{\mathcal{N}}(Y,D,R,[a,b])&\\!\\!\\!\\!=-{\mathcal{N}}^{*}(Z^{-1}E,D,{\tilde{R}},[a,b]),\\\\[2.84526pt]
{\mathcal{N}}^{*}(Y,D,R,[a,b])&\\!\\!\\!\\!=-{\mathcal{N}}(Z^{-1}E,D,{\tilde{R}},[a,b]).\end{array}\\!\right\\}$
(3.4)

Here $Z(t)$ is any continuous symplectic matrix with $Z(t)\hskip 1.0ptE=Y(t)$
on $[a,b]$ and ${\tilde{R}}=\\{{\tilde{R}}_{k}(t)\\}_{k=0}^{p-1}$ is the
transformed system of symplectic matrix-valued functions
${\tilde{R}}_{k}(t):=Z^{-1}(t)\hskip 1.0ptR_{k}(t)$ obtained from the original
system $R=\\{R_{k}(t)\\}_{k=0}^{p-1}$, compare with [21, Proposition 3.3(vi)].
Indeed, by applying (2.6) we rewrite equations (3.2) and (3.3) as

$\displaystyle{\mathcal{N}}(Y,D,R,[a,b])$
$\displaystyle=\sum_{k=0}^{p-1}\mu^{*}(Z^{-1}(t)\hskip 1.0ptE,Z^{-1}(t)\hskip
1.0ptR_{k}(t)E)\hskip 1.0pt\big{|}_{t_{k}}^{t_{k+1}},$ (3.5)
$\displaystyle{\mathcal{N}}^{*}(Y,D,R,[a,b])$
$\displaystyle=\sum_{k=0}^{p-1}\mu(Z^{-1}(t)\hskip 1.0ptE,Z^{-1}(t)\hskip
1.0ptR_{k}(t)E)\hskip 1.0pt\big{|}_{t_{k+1}}^{t_{k}},$ (3.6)

where the pair of matrices $Z^{-1}(t)\hskip 1.0ptE$ and $\tilde{R}_{k}(t)$
satisfies the properties in assumption (3.1), replacing respectively the
matrices $Y(t)$ and $R_{k}(t)$ therein. Incorporating the order of the
substitutions in (3.2) and (3.3), we derive the result in (3.4) from equations
(3.5) and (3.6).

######  Remark 3.1.

(i) In [17] we considered a partition $D$ and a system
$R=\\{R_{k}\\}_{k=0}^{p-1}$ of constant symplectic matrices such that instead
of (3.1) we have that

$\text{${\tilde{X}}_{k}(t)$ is invertible on the interval $[t_{k},t_{k+1}]$}.$
(3.7)

For the special case of

$R_{k}:=R_{\alpha_{k}}=\left(\begin{matrix}(\cos\alpha_{k})\hskip
1.0ptI&(\sin\alpha_{k})\hskip 1.0ptI\\\ -(\sin\alpha_{k})\hskip
1.0ptI&(\cos\alpha_{k})\hskip 1.0ptI\end{matrix}\right)$

a direct proof of the existence of a partition $D$ with property (3.7) can be
found in [2, pg.41] and [19, Lemma 3.5]. Using the representation
${\tilde{X}}_{k}(t)=-W(R_{k}E,Y(t))$, condition (3.7) can be rewritten in the
form

$\operatorname{\text{\rm rank}}W(R_{k}E,Y(t))=n,\,\quad t\in[t_{k},t_{k+1}].$
(3.8)

Condition (3.8) then means that the Lagrangian path $Y$ is transversal to the
Lagrangian plane $R_{k}E$ on the interval $[t_{k},t_{k+1}]$.

(ii) In [19, Eq. (3.13)] we introduced an equivalent definition of the
oscillation numbers from [17] for conjoined bases $Y$ of system (H). Namely,
consider a partition $a=s_{0}<s_{1}<\dots<s_{r-1}<s_{r}=b$ of $[a,b]$ such
that for any $\ell\in\\{0,\dots,r-1\\}$ there exists a conjoined basis
$Y_{\ell}$ of system (H) with the nonsingular upper block $X_{\ell}(t)$ on
$[s_{\ell},s_{\ell+1}]$, i.e.,

$\det X_{\ell}(t)\neq 0,\quad
t\in[s_{\ell},s_{\ell+1}],\quad\ell\in\\{0,\dots,r-1\\}.$ (3.9)

The oscillation numbers in [19, Eq. (3.13)] are defined by (3.2) with
$R_{\ell}(t):=Z_{\ell}(t)$, where $Z_{\ell}(t)$ is a symplectic fundamental
matrix of system (H) such that $Z_{\ell}(t)\hskip 1.0ptE=Y_{\ell}(t)$ on
$[s_{\ell},s_{\ell+1}]$. Note that for this choice of $R_{\ell}(t)$ all
conditions in (3.1) are satisfied, in particular $Y$ and $R_{\ell}\hskip
1.0ptE=Y_{\ell}$ are conjoined bases of (H) and then the first condition in
(3.1) automatically holds. Hence, we can see that condition (3.1) is a proper
generalization of conditions (3.7) and (3.9).

(iii) For an arbitrary continuous Lagrangian path $Y$ on $[a,b]$ one can
present an analog of the definition from part (ii) by putting
$R_{k}(t):=Z_{Y(t)}C_{k}$, where $Z_{Y(t)}$ are symplectic and orthogonal
matrices given by (1.4) and $C_{k}$ for $k\in\\{0,1,\dots,p-1\\}$ are constant
symplectic matrices such that the upper block of $R_{k}(t)E$ is nonsingular
for all $t\in[t_{k},t_{k+1}]$ or, more generally, it has constant rank on
$[t_{k},t_{k+1}]$.

(iv) For the subsequent proofs it is important that condition (3.1) remains
valid for the matrices $Z_{R_{k}E}E=R_{k}EK_{R_{k}E}$ and
$Z_{Y(t)}E=Y(t)\hskip 1.0ptK_{Y(t)}$, which are associated with $R_{k}E$ and
$Y(t)$ via (2.13). Indeed, according to the definition in (1.4) the matrices
$K_{R_{k}E}$ and $K_{Y(t)}$ are nonsingular and the values
$\operatorname{\text{\rm rank}}W(Z_{R_{k}E}\hskip
1.0ptE,Z_{Y(t)}E)=\operatorname{\text{\rm rank}}W(R_{k}E,Y(t))$ and
$\operatorname{\text{\rm rank}}((I\,\ 0)\hskip 1.0ptR_{k}(t)\hskip
1.0ptEK_{R_{k}E})$ are then constant on $[t_{k},t_{k+1}]$.

For the cases described in Remark 3.1(i), (ii) the numbers defined in (3.2)
and (3.3) are invariant with respect to the partition $D$ of $[a,b]$ and with
respect to the special choice of the system $R$ of symplectic matrices
satisfying condition (3.7) or (3.9), see [19, Lemma 2.2] and [21, Proposition
3.3(i)]. One can prove the same invariant properties for the general case
(3.1), based only on the properties of the comparative index and the dual
comparative index. Note that the same will follow from our Theorem 3.8 below
(see Remark 3.9). The invariant properties of the numbers in (3.2) and (3.3)
then justify the following definition.

######  Definition 3.2 (Oscillation numbers).

Let $Y:[a,b]\to{\mathbb{R}}^{2n\times n}$ be a continuous Lagrangian path. The
quantity defined in (3.2) is called the oscillation number of $Y$ on $[a,b]$
and it is denoted by ${\mathcal{N}}(Y,[a,b])$. The quantity defined in (3.3)
is called the dual oscillation number of $Y$ on $[a,b]$ and it is denoted by
${\mathcal{N}}^{*}(Y,[a,b])$.

######  Remark 3.3.

(i) Since by (2.4) the comparative index and the dual comparative index are
invariant under the multiplication of its arguments by invertible $n\times n$
matrices from the right, it follows that the same property holds for the
oscillation number and the dual oscillation number. More precisely, if $Y$ is
a continuous Lagrangian path on $[a,b]$ and $C:[a,b]\to{\mathbb{R}}^{n\times
n}$ is a continuous invertible matrix function on $[a,b]$, then $YC$ is also a
continuous Lagrangian path on $[a,b]$ with

${\mathcal{N}}(YC,[a,b])={\mathcal{N}}(Y,[a,b]),\quad{\mathcal{N}}^{*}(YC,[a,b])={\mathcal{N}}^{*}(Y,[a,b]).$
(3.10)

(ii) The definition of the oscillation number and the dual oscillation number
through (3.2) and (3.3) yields their additivity with respect to the base
interval, see also [19, Remark 3.7]. Namely, for any point $c\in(a,b)$ we have

${\mathcal{N}}(Y,[a,b])={\mathcal{N}}(Y,[a,c])+{\mathcal{N}}(Y,[c,b]),\quad{\mathcal{N}}^{*}(Y,[a,b])={\mathcal{N}}^{*}(Y,[a,c])+{\mathcal{N}}^{*}(Y,[c,b]).$
(3.11)

We will now present the main results of this section, which use the following
terminology based on Remark 2.4.

######  Definition 3.4 (Lidskii angles of Lagrangian path).

Let $Y$ be a continuous Lagrangian path on $[a,b]$ and let $Z_{Y}(t)$ be the
associated symplectic (and orthogonal) matrix defined in (1.4). Then the
continuous Lidskii angles $\varphi_{j}(t)$ for $j\in\\{1,\dots,n\\}$ and
$t\in[a,b]$ of $Z_{Y}(t)$ are called the Lidskii angles of the Lagrangian path
$Y$.

With the above terminology, for fixed continuous branches of the Lidskii
angles $\varphi_{j}(t)$ we consider the uniquely defined integers $q_{j}(t)$
and $q_{j}^{*}(t)$, which satisfy the properties

$\displaystyle\varphi_{j}(t)\in[2\pi\hskip 1.0ptq_{j}(t),2\pi\hskip
1.0pt(q_{j}(t)+1)),\quad t\in[a,b],\quad j\in\\{1,\dots,n\\},$ (3.12)
$\displaystyle\varphi_{j}(t)\in(2\pi\hskip 1.0ptq_{j}^{*}(t),2\pi\hskip
1.0pt(q_{j}^{*}(t)+1)],\quad t\in[a,b],\quad j\in\\{1,\dots,n\\},$ (3.13)

that is, the Lidskii angle $\varphi_{j}(t)$ of $Y$ belongs to the indicated
half-open interval of the length $2\pi$. This means that the integers
$q_{j}(t)$ and $q_{j}^{*}(t)$ are given by

$q_{j}(t)=\Big{\lfloor}\frac{\varphi_{j}(t)}{2\pi}\Big{\rfloor},\quad
q_{j}^{*}(t)=\Big{\lceil}\frac{\varphi_{j}(t)}{2\pi}\Big{\rceil}-1,\quad
t\in[a,b],$ (3.14)

where for $x\in{\mathbb{R}}$ the notation $\lfloor x\rfloor$ and $\lceil
x\rceil$ stand for the greatest integer which is smaller or equal to $x$ (the
floor function) and for the smallest integer which is greater or equal to $x$
(the ceiling function). It follows from this definition that
$q_{j}(t)=q_{j}^{*}(t)$ if and only if the angle $\varphi_{j}(t)$ belongs to
the interior of the intervals in (3.12) or (3.13), while
$q_{j}(t)=q_{j}^{*}(t)+1$ holds when the angle $\varphi_{j}(t)$ is an integer
multiple of $2\pi$. This means in view of Remark 2.1 that

$\sum_{j=1}^{n}q_{j}(t)=\operatorname{\text{\rm def}\hskip
1.0pt}X(t)+\sum_{j=1}^{n}q_{j}^{*}(t)\quad\text{for all }t\in[a,b].$ (3.15)

Observe that for the argument function in (2.10) we then have

$\displaystyle\operatorname{\text{\rm
Arg}}_{3}(Z_{Y}(t))\big{|}_{\tau_{1}}^{\tau_{2}}=\frac{1}{2}\,\sum_{j=1}^{n}\varphi_{j}(t)\big{|}_{\tau_{1}}^{\tau_{2}}$
$\displaystyle=\operatorname{\text{\rm Arg}}_{\hskip
1.0pt[0,2\pi)}(Z_{Y}(t))\big{|}_{\tau_{1}}^{\tau_{2}}+\pi\sum_{j=1}^{n}q_{j}(t)\big{|}_{\tau_{1}}^{\tau_{2}},$
(3.16) $\displaystyle\operatorname{\text{\rm
Arg}}_{3}(Z_{Y}(t))\big{|}_{\tau_{1}}^{\tau_{2}}=\frac{1}{2}\,\sum_{j=1}^{n}\varphi_{j}(t)\big{|}_{\tau_{1}}^{\tau_{2}}$
$\displaystyle=\operatorname{\text{\rm Arg}}_{\hskip
1.0pt(0,2\pi]}(Z_{Y}(t))\big{|}_{\tau_{1}}^{\tau_{2}}+\pi\sum_{j=1}^{n}q_{j}^{*}(t)\big{|}_{\tau_{1}}^{\tau_{2}}$
(3.17)

for all $\tau_{1},\tau_{2}\in[a,b]$ with $\tau_{1}<\tau_{2}$. We have the
following important property of the integers $q_{j}(t)$ and $q_{j}^{*}(t)$ in
(3.12) and (3.13).

######  Lemma 3.5.

Let $Y$ be a continuous Lagrangian path on $[a,b]$ with the associated Lidskii
angles $\varphi_{j}(t)$ on $[a,b]$ for $j\in\\{1,\dots,n\\}$ according to
Definition 3.4. If the upper block $X(t)$ of $Y(t)$ has constant rank on
$[\tau_{1},\tau_{2}]\subseteq[a,b]$, then

$\text{$q_{j}(t)\equiv q_{j}$ and $q_{j}^{\hskip 1.0pt*}(t)\equiv
q_{j}^{\hskip 1.0pt*}$ are constant on $[\tau_{1},\tau_{2}]$ for all
$j\in\\{1,\dots,n\\}$},$ (3.18)

where $q_{j}(t)$ and $q_{j}^{\hskip 1.0pt*}(t)$ are given by (3.12) and
(3.13).

###### Proof.

According to Remark 2.1 and using the continuity of $\varphi_{j}(t)$ on
$[a,b]$ it follows that each angle $\varphi_{j}(t)$ either remains in the open
interval $(2\pi\hskip 1.0ptq_{j},2\pi\hskip 1.0pt(q_{j}+1))=(2\pi\hskip
1.0ptq_{j}^{\hskip 1.0pt*},2\pi\hskip 1.0pt(q_{j}^{\hskip 1.0pt*}+1))$ on
$[a,b]$, or it is constant on the interval $[\tau_{1},\tau_{2}]$ with the
value $\varphi_{j}(t)\equiv 2\pi\hskip 1.0ptq_{j}=2\pi\hskip
1.0pt(q_{j}^{\hskip 1.0pt*}+1)$. Hence, the property in (3.18) holds. ∎

Combining Proposition 2.3 with equations (3.16) and (3.17) we derive the
following main property of the comparative index for two continuous Lagrangian
paths.

######  Proposition 3.6 (Comparative index for continuous Lagrangian paths).

Let $Y$ and $\hat{Y}$ be continuous Lagrangian paths on $[a,b]$ with the
associated Lidskii angles $\varphi_{j}(t)$ and $\hat{\varphi}_{j}(t)$ on
$[a,b]$ for $j\in\\{1,\dots,n\\}$ according to Definition 3.4. Consider the
continuous symplectic matrix

$\tilde{S}(t):=Z_{\hat{Y}}^{-1}(t)\,Z_{Y}(t)=Z_{\hat{Y}}^{T}(t)\,Z_{Y}(t),\quad
t\in[a,b],$ (3.19)

and its continuous Lidskii angles $\tilde{\varphi}_{j}(t)$, where $Z_{Y}(t)$
and $Z_{\hat{Y}}(t)$ are the symplectic and orthogonal matrices associated
with $Y(t)$ and $\hat{Y}(t)$ through (1.4). Given the integers $q_{j}(t)$,
$\hat{q}_{j}(t)$, $\tilde{q}_{j}(t)$ and $q_{j}^{*}(t)$,
${\hat{q}}_{j}^{*}(t)$, ${\tilde{q}}_{j}^{*}(t)$ associated through (3.12) and
(3.13) with the angles $\varphi_{j}(t)$, $\hat{\varphi}_{j}(t)$,
$\tilde{\varphi}_{j}(t)$, then for any two points $\tau_{1},\tau_{2}\in[a,b]$
the comparative index and the dual comparative index defined in (2.1) with
(2.2) satisfy

$\displaystyle\mu(Y(t),\hat{Y}(t))\big{|}_{\tau_{1}}^{\tau_{2}}$
$\displaystyle=\sum_{j=1}^{n}\big{(}q_{j}(t)-{\hat{q}}_{j}(t)-\tilde{q}_{j}(t)\big{)}\big{|}_{\tau_{1}}^{\tau_{2}},$
(3.20) $\displaystyle\mu^{*}(Y(t),\hat{Y}(t))\big{|}_{\tau_{1}}^{\tau_{2}}$
$\displaystyle=-\sum_{j=1}^{n}\big{(}q_{j}^{*}(t)-\hat{q}_{j}^{*}(t)-\tilde{q}_{j}^{*}(t)\big{)}\big{|}_{\tau_{1}}^{\tau_{2}}.$
(3.21)

###### Proof.

According to [24, pg. 163], for any symplectic and orthogonal matrices $Z(t)$
and $\hat{Z}(t)$ we have the multiplicative property

$\left(\begin{matrix}I&0\end{matrix}\right)\hat{Z}^{-1}(t)\hskip
1.0ptZ(t)\left(\begin{matrix}I\\\
iI\end{matrix}\right)=\left(\begin{matrix}I&0\end{matrix}\right)\hat{Z}^{-1}(t)\left(\begin{matrix}I\\\
iI\end{matrix}\right)\\!\left(\begin{matrix}I&0\end{matrix}\right)\\!Z(t)\left(\begin{matrix}I\\\
iI\end{matrix}\right).$

Then the argument function $\operatorname{\text{\rm Arg}}_{3}$ defined by
(2.11) for the matrix $\hat{Z}^{-1}(t)Z(t)$ is the sum of the argument
functions of $\hat{Z}^{-1}(t)$ and $Z(t)$, and hence

$\operatorname{\text{\rm
Arg}}_{3}(\hat{Z}(t))\big{|}_{\tau_{1}}^{\tau_{2}}-\operatorname{\text{\rm
Arg}}_{3}(Z(t))\big{|}_{\tau_{1}}^{\tau_{2}}+\operatorname{\text{\rm
Arg}}_{3}(\hat{Z}^{-1}(t)\hskip 1.0ptZ(t))\big{|}_{\tau_{1}}^{\tau_{2}}=0.$
(3.22)

For any continuous Lagrangian paths $Y$ and $\hat{Y}$ on $[a,b]$ we then have
by (2.15) and (3.16) that

$\displaystyle\pi\,\mu(Y(t),\hat{Y}(t))\big{|}_{\tau_{1}}^{\tau_{2}}$
$\displaystyle\overset{\eqref{E:mu.Lidskii}}{=}\operatorname{\text{\rm
Arg}}_{\hskip
1.0pt[0,2\pi)}(Z_{\hat{Y}}(t))\big{|}_{\tau_{1}}^{\tau_{2}}-\operatorname{\text{\rm
Arg}}_{\hskip
1.0pt[0,2\pi)}(Z_{Y}(t))\big{|}_{\tau_{1}}^{\tau_{2}}+\operatorname{\text{\rm
Arg}}_{\hskip 1.0pt[0,2\pi)}(Z_{\hat{Y}}^{-1}(t)\hskip
1.0ptZ_{Y}(t))\big{|}_{\tau_{1}}^{\tau_{2}}$
$\displaystyle\overset{\eqref{EJ:Arg3.ArgL}}{=}\operatorname{\text{\rm
Arg}}_{3}(Z_{\hat{Y}}(t))\big{|}_{\tau_{1}}^{\tau_{2}}-\operatorname{\text{\rm
Arg}}_{3}(Z_{Y}(t))\big{|}_{\tau_{1}}^{\tau_{2}}+\operatorname{\text{\rm
Arg}}_{3}(Z_{\hat{Y}}^{-1}(t)\hskip
1.0ptZ_{Y}(t))\big{|}_{\tau_{1}}^{\tau_{2}}$ $\displaystyle\hskip
28.45274pt+\pi\sum_{j=1}^{n}\big{(}q_{j}(t)-{\hat{q}}_{j}(t)-{\tilde{q}}_{j}(t)\big{)}\big{|}_{\tau_{1}}^{\tau_{2}}.$

Then by (3.22) we derive the equality in (3.20). The proof of equality (3.21)
is similar and it follows from (2.16), (3.17), and (3.22). ∎

######  Corollary 3.7.

Under the assumptions and the notation of Proposition 3.6 we have the
following implications. If $\operatorname{\text{\rm rank}}{\hat{X}}(t)$ is
constant on $[\tau_{1},\tau_{2}]$, then

$\mu(Y(t),\hat{Y}(t))\big{|}_{\tau_{1}}^{\tau_{2}}=\sum_{j=1}^{n}\big{(}q_{j}(t)-\tilde{q}_{j}(t)\big{)}\big{|}_{\tau_{1}}^{\tau_{2}},\quad\mu^{*}(Y(t),\hat{Y}(t))\big{|}_{\tau_{1}}^{\tau_{2}}=\sum_{j=1}^{n}\big{(}q_{j}^{*}(t)-\tilde{q}_{j}^{*}(t)\big{)}\big{|}_{\tau_{2}}^{\tau_{1}},$
(3.23)

while if $\operatorname{\text{\rm rank}}W(\hat{Y}(t),Y(t))$ is constant on
$[\tau_{1},\tau_{2}]$, then

$\mu(Y(t),\hat{Y}(t))\big{|}_{\tau_{1}}^{\tau_{2}}=\sum_{j=1}^{n}\big{(}q_{j}(t)-{\hat{q}}_{j}(t)\big{)}\big{|}_{\tau_{1}}^{\tau_{2}},\quad\mu^{*}(Y(t),\hat{Y}(t))\big{|}_{\tau_{1}}^{\tau_{2}}=\sum_{j=1}^{n}\big{(}q_{j}^{*}(t)-{\hat{q}}_{j}^{*}(t)\big{)}\big{|}_{\tau_{2}}^{\tau_{1}}.$
(3.24)

In particular, if both $\operatorname{\text{\rm rank}}{\hat{X}}(t)$ and
$\operatorname{\text{\rm rank}}W(\hat{Y}(t),Y(t))$ are constant on
$[\tau_{1},\tau_{2}]$, then

$\mu(Y(t),\hat{Y}(t))\big{|}_{\tau_{1}}^{\tau_{2}}=\sum_{j=1}^{n}q_{j}(t)\big{|}_{\tau_{1}}^{\tau_{2}},\quad\mu^{*}(Y(t),\hat{Y}(t))\big{|}_{\tau_{1}}^{\tau_{2}}=\sum_{j=1}^{n}q_{j}^{*}(t)\big{|}_{\tau_{2}}^{\tau_{1}}.$
(3.25)

###### Proof.

Under the stated assumptions, the results in (3.23), (3.24), and (3.25) follow
from Proposition 3.6 and Lemma 3.5. ∎

Our main result presented below shows that the oscillation number and the dual
oscillation number of $Y$ on $[a,b]$ count the cumulative change in the
differences of the integers $q_{j}(t)$ and of the integers $q_{j}^{*}(t)$ at
the endpoints of the interval $[a,b]$.

######  Theorem 3.8.

Let $Y$ be a continuous Lagrangian path on $[a,b]$ with the associated Lidskii
angles $\varphi_{j}(t)$ for $j\in\\{1,\dots,n\\}$ on $[a,b]$ according to
Definition 3.4. Given the integers $q_{j}(t)$ and $q_{j}^{*}(t)$ satisfying
conditions (3.12) and (3.13), then the oscillation number of $Y$ on $[a,b]$
and the dual oscillation number of $Y$ on $[a,b]$ satisfy

$\displaystyle{\mathcal{N}}(Y,[a,b])$
$\displaystyle=\sum_{j=1}^{n}\big{(}q_{j}(b)-q_{j}(a)\big{)},$ (3.26)
$\displaystyle{\mathcal{N}}^{*}(Y,[a,b])$
$\displaystyle=\sum_{j=1}^{n}\big{(}q_{j}^{*}(b)-q_{j}^{*}(a)\big{)}.$ (3.27)

###### Proof.

According to Definition 3.2 we choose a partition $D=\\{t_{k}\\}_{k=0}^{p}$ of
the interval $[a,b]$ and a system $R=\\{R_{k}(t)\\}_{k=0}^{p-1}$ of symplectic
matrix-valued functions such that (3.1) holds. By (3.25) in Corollary 3.7
applied to $\hat{Y}(t):=R_{k}(t)E$ and $\tau_{1}:=t_{k}$, $\tau_{2}:=t_{k+1}$
for $k\in\\{0,1,\dots,p-1\\}$ we have for the variations of the comparative
indices in (3.2) and (3.3) that

$\mu(Y(t),R_{k}(t)E)\big{|}_{t_{k}}^{t_{k+1}}=\sum_{j=1}^{n}q_{j}(t)\big{|}_{t_{k}}^{t_{k+1}},\quad\mu^{*}(Y(t),R_{k}(t)E)\big{|}_{t_{k+1}}^{t_{k}}=\sum_{j=1}^{n}q_{j}^{*}(t)\big{|}_{t_{k}}^{t_{k+1}},$
(3.28)

where we used the facts that the rank of the upper block of $R_{k}(t)E$ and
rank of the Wronskian $W(R_{k}(t)E,Y(t))$ are constant for
$t\in[t_{k},t_{k+1}]$ according to (3.1). By the first equation in (3.28) we
obtain, with the telescope summation (and using $t_{0}=a$ and $t_{p}=b$), that

${\mathcal{N}}(Y,[a,b])\overset{\eqref{E:Nab.D.R.def}}{=}\sum_{k=0}^{p-1}\mu(Y(t),R_{k}(t)E)\big{|}_{t_{k}}^{t_{k+1}}\overset{\eqref{E:osc.number.q.hlp0}}{=}\sum_{j=1}^{n}\sum_{k=1}^{p-1}q_{j}(t)\big{|}_{t_{k}}^{t_{k+1}}=\sum_{j=1}^{n}\big{(}q_{j}(b)-q_{j}(a)\big{)},$

which completes the proof of equation (3.26). Analogously, by the second
equality in (3.28) and by (3.3) we obtain with the telescope summation that

${\mathcal{N}}^{*}(Y,[a,b])\overset{\eqref{E:Nab*.D.R.def}}{=}\sum_{k=0}^{p-1}\mu^{*}(Y(t),R_{k}(t)E)\big{|}_{t_{k+1}}^{t_{k}}\overset{\eqref{E:osc.number.q.hlp0}}{=}\sum_{j=1}^{n}\sum_{k=1}^{p-1}q_{j}^{*}(t)\big{|}_{t_{k}}^{t_{k+1}}=\sum_{j=1}^{n}\big{(}q_{j}^{*}(b)-q_{j}^{*}(a)\big{)},$

which completes the proof of equation (3.27). ∎

######  Remark 3.9.

The results in Theorem 3.8 confirm that the values in (3.2) and (3.3) indeed
do not depend on the chosen partition $D=\\{t_{k}\\}_{k=0}^{p}$ of $[a,b]$ as
well as on the chosen system $R=\\{R_{k}(t)\\}_{k=0}^{p-1}$ of symplectic
matrix-valued functions satisfying (3.7).

As a consequence of Theorem 3.8 we obtain the value of the oscillation number
and the dual oscillation number of $Y$, when the upper block $X(t)$ of $Y(t)$
has constant rank on $[a,b]$, see also [19, Proposition 3.2(ii)] and [21,
Proposition 3.3(iii)].

######  Corollary 3.10.

Assume that $Y$ is a continuous Lagrangian path on $[a,b]$ such that its upper
block $X(t)$, according to the notation in (1.2), has constant rank on
$[a,b]$. Then we have

${\mathcal{N}}(Y,[a,b])=0,\quad{\mathcal{N}}^{*}(Y,[a,b])=0.$ (3.29)

###### Proof.

The result follows from Theorem 3.8 and from Lemma 3.5 (with $\tau_{1}:=a$ and
$\tau_{2}:=b$). ∎

The following result presents the invariance of the oscillation number and the
dual oscillation number under a special continuous symplectic transformation.

######  Corollary 3.11.

Let $Y$ be a continuous Lagrangian path on $[a,b]$ and let
$L:[a,b]\to{\mathbb{R}}^{2n\times 2n}$ be a continuous symplectic lower block
triangular matrix-valued function, i.e., the matrix $L(t)$ has the form as in
(2.5). Then we have

$\displaystyle{\mathcal{N}}(Y,[a,b])$ $\displaystyle={\mathcal{N}}(L\hskip
1.0ptY,[a,b]),$ (3.30) $\displaystyle{\mathcal{N}}^{*}(Y,[a,b])$
$\displaystyle={\mathcal{N}}^{*}(L\hskip 1.0ptY,[a,b]).$ (3.31)

###### Proof.

The main role in the proof is played by the invariant property of the
comparative index with respect to block lower triangular symplectic
transformations, see (2.5). Obviously, $L\hskip 1.0ptY$ is a continuous
Lagrangian path on $[a,b]$. Under the assumptions and the notation of Theorem
3.8, we show that equations (3.26) and (3.27) are also valid for
${\mathcal{N}}(L\hskip 1.0ptY,[a,b])$ and ${\mathcal{N}}^{*}(L\hskip
1.0ptY,[a,b])$. Applying Definition 3.2 to the path $L\hskip 1.0ptY$ we choose
a partition $D=\\{t_{k}\\}_{k=0}^{p}$ of $[a,b]$ and a system
${\tilde{R}}:=\\{{\tilde{R}}_{k}(t)\\}_{k=0}^{p-1}$ of symplectic matrix-
valued functions $\tilde{R}_{k}(t)$ such that

$\operatorname{\text{\rm rank}}W(\tilde{R}_{k}(t)\hskip 1.0ptE,L(t)\hskip
1.0ptY(t))\text{ and }\operatorname{\text{\rm rank}}((I\ 0)\hskip
1.0pt{\tilde{R}}_{k}(t)\hskip 1.0ptE)\text{ are constant on }[t_{k},t_{k+1}]$
(3.32)

for every $k\in\\{0,\dots,p-1\\}$. By using the assumption that the matrix
$L(t)$ is symplectic block lower triangular one can rewrite (3.32) in the
equivalent form

$W(L^{-1}(t)\hskip 1.0pt\tilde{R}_{k}(t)\hskip 1.0ptE,Y(t))\text{ and
}\operatorname{\text{\rm rank}}((I\ 0)\hskip 1.0ptL^{-1}(t)\hskip
1.0pt\tilde{R}_{k}(t)\hskip 1.0ptE)\text{ are constant on }[t_{k},t_{k+1}],$
(3.33)

where we used that ${\mathcal{J}}L(t)=L^{T-1}(t){\mathcal{J}}$. It follows
from (3.33) that one can use the same partition for $L(t)\hskip 1.0ptY(t)$ and
$Y(t)$ with the corresponding transformation matrices $\tilde{R}_{k}(t)$ and
$L^{-1}(t)\hskip 1.0pt\tilde{R}_{k}(t)$. Finally, by (2.5) we have, instead of
(3.28), the equalities

$\displaystyle\mu(L(t)\hskip 1.0ptY(t),\tilde{R}_{k}(t)\hskip
1.0ptE)\big{|}_{t_{k}}^{t_{k+1}}$
$\displaystyle\overset{\eqref{prop2}}{=}\mu(Y(t),L^{-1}(t)\hskip
1.0pt\tilde{R}_{k}(t)\hskip
1.0ptE)\big{|}_{t_{k}}^{t_{k+1}}=\sum_{j=1}^{n}q_{j}(t)\big{|}_{t_{k}}^{t_{k+1}},$
$\displaystyle\mu^{*}(L(t)\hskip 1.0ptY(t),\tilde{R}_{k}(t)\hskip
1.0ptE)\big{|}_{t_{k+1}}^{t_{k}}$
$\displaystyle\overset{\eqref{prop2}}{=}\mu^{*}(Y(t),L^{-1}(t)\hskip
1.0pt\tilde{R}_{k}(t)\hskip
1.0ptE)\big{|}_{t_{k+1}}^{t_{k}}=\sum_{j=1}^{n}q_{j}^{*}(t)\big{|}_{t_{k}}^{t_{k+1}},$

where by analogy with the proof of Theorem 3.8 we applied Corollary 3.7 with
the matrix $\hat{Y}(t):=L^{-1}(t)\hskip 1.0pt\tilde{R}_{k}E$. Summing the
equalities derived above for all $k\in\\{0,\dots,p-1\\}$ we complete the proof
of (3.30) and (3.31). ∎

Based on Corollary 3.11 we are able to prove the invariance of the oscillation
number and the dual oscillation number in the sense that every continuous
symplectic transformation of a continuous Lagrangian path can be realized with
a special continuous symplectic and orthogonal transformation.

######  Theorem 3.12.

Let $Y$ be a continuous Lagrangian path on $[a,b]$ and let $S(t)$ be a
continuous symplectic matrix on $[a,b]$. Then we have the equalities

$\displaystyle{\mathcal{N}}(S^{-1}\hskip 1.0ptY,[a,b])$
$\displaystyle={\mathcal{N}}(Z_{SE}^{-1}\hskip 1.0pt\hskip 1.0ptY,[a,b]),$
(3.34) $\displaystyle{\mathcal{N}}^{*}(S^{-1}\hskip 1.0ptY,[a,b])$
$\displaystyle={\mathcal{N}}^{*}(Z_{SE}^{-1}\hskip 1.0pt\hskip 1.0ptY,[a,b]),$
(3.35)

where the symplectic and orthogonal matrix $Z_{SE}(t)=Z_{S(t)E}$ is defined
according to (1.4).

###### Proof.

By property (2.14) in Remark 2.2 we have $S(t)=Z_{S(t)E}\hskip 1.0ptL(t)$,
where $L(t)$ is a continuous symplectic block lower triangular matrix. Then in
view of Corollary 3.11 we have

$\displaystyle{\mathcal{N}}(S^{-1}\hskip 1.0ptY,[a,b])$
$\displaystyle={\mathcal{N}}(L^{-1}Z_{SE}^{-1}\hskip 1.0pt\hskip
1.0ptY,[a,b])\overset{\eqref{E:osc.number.L}}{=}{\mathcal{N}}(Z_{SE}^{-1}\hskip
1.0pt\hskip 1.0ptY,[a,b]),$ $\displaystyle{\mathcal{N}}^{*}(S^{-1}\hskip
1.0ptY,[a,b])$ $\displaystyle={\mathcal{N}}^{*}(L^{-1}Z_{SE}^{-1}\hskip
1.0pt\hskip
1.0ptY,[a,b])\overset{\eqref{E:osc.number.L*}}{=}{\mathcal{N}}^{*}(Z_{SE}^{-1}\hskip
1.0pt\hskip 1.0ptY,[a,b]),$

which completes the proof. ∎

Next we present a formula relating the oscillation number
${\mathcal{N}}(Y,[a,b])$ and the dual oscillation number
${\mathcal{N}}^{*}(Y,[a,b])$, see also [21, Proposition 3.3(iv)] and [36,
Theorem 5.1].

######  Theorem 3.13.

Let $Y$ be a continuous Lagrangian path on $[a,b]$. Then the oscillation
number and the dual oscillation number of $Y$ on $[a,b]$ are related by the
formula

${\mathcal{N}}(Y,[a,b])+\operatorname{\text{\rm
rank}}X(b)={\mathcal{N}}^{*}(Y,[a,b])+\operatorname{\text{\rm rank}}X(a),$
(3.36)

where $X(t)$ is the upper block of $Y(t)$ as in (1.2).

###### Proof.

The result follows from Theorem 3.8 and from the relationship between the
integers $q_{j}(t)$ and $q_{j}^{*}(t)$ in (3.15). Namely, we have

$\displaystyle{\mathcal{N}}(Y,[a,b])+\operatorname{\text{\rm rank}}X(b)$
$\displaystyle\overset{\eqref{E:osc.number.q}}{=}\operatorname{\text{\rm
rank}}X(b)+\sum_{j=1}^{n}\big{(}q_{j}(b)-q_{j}(a)\big{)}$
$\displaystyle\overset{\eqref{E:qj.qj*}}{=}\operatorname{\text{\rm
rank}}X(b)+\operatorname{\text{\rm def}\hskip
1.0pt}X(b)-\operatorname{\text{\rm def}\hskip
1.0pt}X(a)+\sum_{j=1}^{n}\big{(}q_{j}^{*}(b)-q_{j}^{*}(a)\big{)}$
$\displaystyle\overset{\eqref{E:osc.number.q*}}{=}{\mathcal{N}}^{*}(Y,[a,b])+\operatorname{\text{\rm
rank}}X(a),$

which proves the result in (3.36). ∎

######  Corollary 3.14.

Let $S(t)$ be a continuous symplectic matrix on $[a,b]$, which is partitioned
into $n\times n$ blocks as in (2.7). Then we have the equalities

$\displaystyle{\mathcal{N}}(SE,[a,b])+{\mathcal{N}}(S^{-1}E,[a,b])$
$\displaystyle=\operatorname{\text{\rm rank}}S_{12}(a)-\operatorname{\text{\rm
rank}}S_{12}(b),$ (3.37)
$\displaystyle{\mathcal{N}}^{*}(SE,[a,b])+{\mathcal{N}}^{*}(S^{-1}E,[a,b])$
$\displaystyle=\operatorname{\text{\rm rank}}S_{12}(b)-\operatorname{\text{\rm
rank}}S_{12}(a).$ (3.38)

###### Proof.

According to (3.4), for the continuous Lagrangian path $Y(t):=S(t)E$ on
$[a,b]$ we have

${\mathcal{N}}(SE,[a,b])=-{\mathcal{N}}^{*}(S^{-1}E,[a,b]),\quad{\mathcal{N}}^{*}(SE,[a,b])=-{\mathcal{N}}(S^{-1}E,[a,b]).$
(3.39)

Equations (3.37) and (3.38) now follow directly from formula (3.36) (with
$Y(t):=S(t)\hskip 1.0ptE$ and $X(t):=S_{12}(t)$) and from the relations in
(3.39). The proof is complete. ∎

In the final part of this section we will discuss some additional properties
of the oscillation number and the dual oscillation number, which are based on
Theorem 3.8. At the first place we obtain the following representations of the
changes in the corresponding special arguments $\operatorname{\text{\rm
Arg}}_{\hskip 1.0pt[0,2\pi)}$ and $\operatorname{\text{\rm Arg}}_{\hskip
1.0pt(0,2\pi]}$ of the matrix $Z_{Y}(t)$.

######  Corollary 3.15.

Let $Y$ be a continuous Lagrangian path on $[a,b]$ and let $Z_{Y}(t)$ be the
continuous symplectic and orthogonal matrix defined in (1.4). Then we have the
representations

$\displaystyle\operatorname{\text{\rm Arg}}_{\hskip
1.0pt[0,2\pi)}(Z_{Y}(b))-\operatorname{\text{\rm Arg}}_{\hskip
1.0pt[0,2\pi)}(Z_{Y}(a))$ $\displaystyle=\operatorname{\text{\rm
Arg}}_{3}(Z_{Y}(b))-\operatorname{\text{\rm
Arg}}_{3}(Z_{Y}(a))-\pi\,{\mathcal{N}}(Y,[a,b]),$ (3.40)
$\displaystyle\operatorname{\text{\rm Arg}}_{\hskip
1.0pt(0,2\pi]}(Z_{Y}(b))-\operatorname{\text{\rm Arg}}_{\hskip
1.0pt(0,2\pi]}(Z_{Y}(a))$ $\displaystyle=\operatorname{\text{\rm
Arg}}_{3}(Z_{Y}(b))-\operatorname{\text{\rm
Arg}}_{3}(Z_{Y}(a))-\pi\,{\mathcal{N}}^{*}(Y,[a,b]).$ (3.41)

###### Proof.

The result in (3.40) follows by combining formula (3.16) at $t=a$ and $t=b$
with (3.26) in Theorem 3.8, while the result in (3.41) follows by combining
formula (3.17) at $t=a$ and $t=b$ with (3.27) in Theorem 3.8. ∎

The representations of the oscillation number and the dual oscillation number
in Theorem 3.8 yield their additive property with respect to a block diagonal
structure of the components $X$ and $U$ of the Lagrangian path $Y$. More
precisely, consider the dimensions $n_{1},n_{2}\in{\mathbb{N}}$ and the
permutation matrix

$\Pi:=\left(\begin{matrix}I_{n_{1}}&0&0&0\\\ 0&0&I_{n_{2}}&0\\\
0&I_{n_{1}}&0&0\\\ 0&0&0&I_{n_{2}}\end{matrix}\right)$ (3.42)

of dimension $2(n_{1}+n_{2})\times 2(n_{1}+n_{2})$. Here $I_{k}$ denotes the
$k\times k$ identity matrix. The following result will be useful in particular
for the construction of higher dimensional examples. We recall the convention
that $\operatorname{\text{\rm diag}}\\{A,B\\}$ denotes the block diagonal
matrix with the matrices $A$ and $B$ on the diagonal.

######  Theorem 3.16.

Assume that $Y_{1}=(X_{1}^{T},U_{1}^{T})^{T}$ and
$Y_{2}=(X_{2}^{T},U_{2}^{T})^{T}$ are continuous Lagrangian paths on $[a,b]$
with values in ${\mathbb{R}}^{2n_{1}\times n_{1}}$ and
${\mathbb{R}}^{2n_{2}\times n_{2}}$, respectively. Consider the continuous
Lagrangian path on $[a,b]$ defined by

$Y:=\Pi\,\operatorname{\text{\rm diag}}\\{Y_{1},\hskip
1.0ptY_{2}\\}=\left(\begin{matrix}\operatorname{\text{\rm
diag}}\\{X_{1},X_{2}\\}\\\\[1.42262pt] \operatorname{\text{\rm
diag}}\\{U_{1},U_{2}\\}\end{matrix}\right)$ (3.43)

with values in ${\mathbb{R}}^{2n\times n}$, where $n:=n_{1}+n_{2}$ and where
the matrix $\Pi$ is given by (3.42). Then

$\displaystyle{\mathcal{N}}(Y,[a,b])$
$\displaystyle={\mathcal{N}}(Y_{1},[a,b])+{\mathcal{N}}(Y_{2},[a,b]),$ (3.44)
$\displaystyle{\mathcal{N}}^{*}(Y,[a,b])$
$\displaystyle={\mathcal{N}}^{*}(Y_{1},[a,b])+{\mathcal{N}}^{*}(Y_{2},[a,b]).$
(3.45)

###### Proof.

According to (1.4) applied to the Lagrangian paths $Y_{1}$, $Y_{2}$, and to
the Lagrangian path $Y$ defined in (3.43), for the corresponding symplectic
and orthogonal matrices $S_{1}(t):=Z_{Y_{1}}(t)$, $S_{2}(t):=Z_{Y_{2}}(t)$,
and $S(t):=Z_{Y}(t)$ we have

$S(t)=\Pi\,\operatorname{\text{\rm diag}}\\{S_{1}(t),\,S_{2}(t)\\}\,\Pi,\quad
t\in[a,b].$

This implies that the matrices $W_{S_{1}}(t)$, $W_{S_{2}}(t)$, and $W_{S}(t)$
defined through equation (2.8) satisfy

$W_{S}(t)=\operatorname{\text{\rm diag}}\\{W_{S_{1}}(t),\hskip
1.0ptW_{S_{2}}(t)\\},\quad t\in[a,b].$ (3.46)

Formula (3.46) shows that the Lidskii angles $\varphi_{j}(t)$ for
$j\in\\{1,\dots,n\\}$ of the Lagrangian path $Y$ according to Definition 3.4
consists exactly of the Lidskii angles $\varphi_{j}^{[1]}(t)$ for
$j\in\\{1,\dots,n_{1}\\}$ and $\varphi_{j}^{[2]}(t)$ for
$j\in\\{1,\dots,n_{2}\\}$ of the Lagrangian paths $Y_{1}$ and $Y_{2}$.
Therefore, equalities (3.44) and (3.45) follow respectively from formula
(3.26) for the oscillation numbers ${\mathcal{N}}(Y,[a,b])$,
${\mathcal{N}}(Y_{1},[a,b])$, ${\mathcal{N}}(Y_{2},[a,b])$ and from formula
(3.27) for the dual oscillation numbers ${\mathcal{N}}^{*}(Y,[a,b])$,
${\mathcal{N}}^{*}(Y_{1},[a,b])$, ${\mathcal{N}}^{*}(Y_{2},[a,b])$. ∎

##  4\. Oscillation numbers and Maslov index

In this section we make a connection of the oscillation number and the dual
oscillation number of two continuous Lagrangian paths $Y$ and $\hat{Y}$ on
$[a,b]$ with the Maslov index. Here we use the definition of the Maslov index
$\operatorname{\text{\rm Mas}}\hskip 1.0pt(Y,\hat{Y},[a,b])$ from [5,
Definition 1.5], which we recall.

Let $Y$ and $\hat{Y}$ be continuous Lagrangian paths on $[a,b]$ with their
partitions into $n\times n$ blocks as in (1.2). According to the latter
reference, see also [26, Section 1], we consider the complex $n\times n$
matrix

$\Gamma(t):=-[X(t)+i\hskip 1.0ptU(t)]\,[X(t)-i\hskip
1.0ptU(t)]^{-1}\,[{\hat{X}}(t)-i\hskip
1.0pt{\hat{U}}(t)]\,[{\hat{X}}(t)+i\hskip 1.0pt{\hat{U}}(t)]^{-1},\quad
t\in[a,b].$ (4.1)

Then the matrix $\Gamma(t)$ is well-defined, continuous, and unitary on
$[a,b]$. The last property also follows from the proof of Lemma 4.1 below, see
equation (4.7). Let $\gamma_{j}(t)$ for $j\in\\{1,\dots,n\\}$ be the
eigenvalues of the matrix $\Gamma(t)$, which are continuous on $[a,b]$ and lie
on the unit circle ${\mathbb{U}}$ in the complex plane. Let us fix a point
$\tau_{0}\in[a,b]$. Since the matrix $\Gamma(\tau_{0})$ has $n$ (i.e.,
finitely many) eigenvalues $\gamma_{j}(\tau_{0})$, there exists a neighborhood
${\mathcal{O}}(\tau_{0})$ of $\tau_{0}$ and a number $\varepsilon\in(0,\pi)$
such that the matrix $\Gamma(t)-\exp\hskip 1.0pt\\{i\hskip
1.0pt(\pi\pm\varepsilon)\\}\,I$ is invertible for all
$t\in{\mathcal{O}}(\tau_{0})$. This implies by the compactness of $[a,b]$ that
there exists a finite partition $D=\\{a=t_{0}<t_{1}\dots<t_{p}=b\\}$ of
$[a,b]$ along with numbers $\varepsilon_{k}\in(0,\pi)$ such that for every
$k\in\\{1,\dots,p\\}$ the matrix $\Gamma(t)-\exp\hskip 1.0pt\\{i\hskip
1.0pt(\pi\pm\varepsilon_{k})\\}\,I$ is invertible for all
$t\in(t_{k-1},t_{k})$, i.e., $\exp\hskip 1.0pt\\{i\hskip
1.0pt(\pi\pm\varepsilon_{k})\\}$ is not an eigenvalue of $\Gamma(t)$ for
$t\in(t_{k-1},t_{k})$. Moreover, for each $t\in[t_{k-1},t_{k}]$ there are at
most $n$ angles $\theta\in[0,\varepsilon_{k}]\subseteq[0,\pi)$ such that
$\exp\hskip 1.0pt\\{i\hskip 1.0pt(\pi+\theta)\\}$ is an eigenvalue of
$\Gamma(t)$. This allows to define the number

$\ell(t,\varepsilon_{k}):=\sum_{\theta\in[0,\varepsilon_{k})}\\!\\!\operatorname{\text{\rm
def}\hskip 1.0pt}\Big{(}\Gamma(t)-\exp\hskip 1.0pt\\{i\hskip
1.0pt(\pi+\theta)\\}\,I\Big{)},\quad t\in[t_{k-1},t_{k}].$ (4.2)

Hence, $\ell(t,\varepsilon_{k})$ is equal to the number of the eigenvalues of
$\Gamma(t)$, which lie on the arc

$A_{k}:=\\{\exp\hskip 1.0pt(i\hskip 1.0pt\theta),\
\theta\in[\pi,\pi+\varepsilon_{k})\\}$

of the unit circle ${\mathbb{U}}$. By [5, Definition 1.5] or [26, Definition
1.4] the Maslov index of the continuous Lagrangian paths $Y$ and $\hat{Y}$ is
defined as the integer number

$\operatorname{\text{\rm Mas}}\hskip
1.0pt(Y,\hat{Y},[a,b]):=\sum_{k=1}^{p}\Big{(}\ell(t_{k},\varepsilon_{k})-\ell(t_{k-1},\varepsilon_{k})\Big{)}.$
(4.3)

Note that this definition does not depend on the choice of the partition
$D=\\{t_{k}\\}_{k=0}^{p}$ of $[a,b]$ and on the choice of the numbers
$\varepsilon_{k}$, as long as they satisfy the above properties, see e.g. [5,
pg. 10]. This property will also follow from our main result below (see
Theorem 4.2 and Remark 4.3).

######  Lemma 4.1.

Let $Y$ and $\hat{Y}$ be continuous Lagrangian paths on $[a,b]$ and define the
matrix $\Gamma(t)$ by (4.1). Consider the symplectic matrix

$S(t):=Z_{Y}^{-1}(t)\,Z_{\hat{Y}}(t)=Z_{Y}^{T}(t)\,Z_{\hat{Y}}(t),\quad
t\in[a,b],$ (4.4)

where $Z_{Y}(t)$ and $Z_{\hat{Y}}(t)$ are the symplectic and orthogonal
matrices associated with $Y(t)$ and $\hat{Y}(t)$ through (1.4). Moreover,
consider the unitary and symmetric matrix $W_{S}(t):=W_{S(t)}$ on $[a,b]$
defined in (2.8) which is associated with the above matrix $S(t)$, i.e.,

$W_{S}(t)=K_{\hat{Y}}^{-1}(t)\,[\hskip 1.0ptY^{T}(t)\hskip
1.0pt\hat{Y}(t)+i\hskip 1.0ptW(Y(t),\hat{Y}(t))]^{-1}\,[\hskip
1.0ptY^{T}(t)\hskip 1.0pt\hat{Y}(t)-i\hskip
1.0ptW(Y(t),\hat{Y}(t))]\,K_{\hat{Y}}(t).$ (4.5)

Then for each $t\in[a,b]$ the matrices $W_{S}(t)$ and $-\Gamma(t)$ are
similar.

###### Proof.

Let us fix any $t\in[a,b]$. For brevity we will suppress the argument $t$ in
the following calculations. Consider the auxiliary Lagrangian paths

$Y_{*}:=YK_{Y}=\left(\begin{matrix}X_{*}\\\
U_{*}\end{matrix}\right),\quad\hat{Y}_{*}:=\hat{Y}K_{\hat{Y}}=\left(\begin{matrix}\hat{X}_{*}\\\
\hat{U}_{*}\end{matrix}\right),$ (4.6)

which form the second block columns of the matrices $Z_{Y}$ and $Z_{\hat{Y}}$,
i.e., $Y_{*}=Z_{Y}E$ and $\hat{Y}_{*}=Z_{\hat{Y}}E$. Since $X_{*}=XK_{Y}$,
$U_{*}=UK_{Y}$ and $\hat{X}_{*}={\hat{X}}K_{\hat{Y}}$,
$\hat{U}_{*}={\hat{U}}K_{\hat{Y}}$, where the matrices $K_{Y}$ and
$K_{\hat{Y}}$ are invertible, it follows from the definition of $\Gamma$ in
(4.1) that

$\Gamma=-(X_{*}+i\hskip 1.0ptU_{*})\,(X_{*}-i\hskip
1.0ptU_{*})^{-1}\,(\hat{X}_{*}-i\hskip 1.0pt\hat{U}_{*})\,(\hat{X}_{*}+i\hskip
1.0pt\hat{U}_{*})^{-1}.$ (4.7)

Note that the orthogonality of the matrices $Z_{Y}$ and $Z_{\hat{Y}}$ yields
that

$(X_{*}\pm i\hskip 1.0ptU_{*})^{-1}=X_{*}^{T}\mp i\hskip
1.0ptU_{*}^{T},\quad(\hat{X}_{*}\pm i\hskip
1.0pt\hat{U}_{*})^{-1}=\hat{X}_{*}^{T}\mp i\hskip 1.0pt\hat{U}_{*}^{T}.$ (4.8)

We express the matrix $S$ in the form

$S\overset{\eqref{E:St.ZY.ZhY.def}}{=}\left(\begin{matrix}K_{Y}\hskip
1.0ptY^{T}\hat{Y}K_{\hat{Y}}&-K_{Y}\hskip 1.0ptW(Y,\hat{Y})\hskip
1.0ptK_{\hat{Y}}\\\\[2.84526pt] K_{Y}\hskip 1.0ptW(Y,\hat{Y})\hskip
1.0ptK_{\hat{Y}}&K_{Y}\hskip
1.0ptY^{T}\hat{Y}K_{\hat{Y}}\end{matrix}\right)\overset{\eqref{E:GGamma.WS.hlp1}}{=}\left(\begin{matrix}Y_{*}^{T}\hat{Y}_{*}&-W(Y_{*},\hat{Y}_{*})\\\\[2.84526pt]
W(Y_{*},\hat{Y}_{*})&Y_{*}^{T}\hat{Y}_{*}\end{matrix}\right).$ (4.9)

Then according to the form of the matrix $W_{S}$ in (4.5) we have

$W_{S}\overset{\eqref{E:GGamma.WS.hlp4}}{=}[\hskip
1.0ptY_{*}^{T}\hat{Y}_{*}+i\hskip 1.0ptW(Y_{*},\hat{Y}_{*})]^{-1}\,[\hskip
1.0ptY_{*}^{T}\hat{Y}_{*}-i\hskip
1.0ptW(Y_{*},\hat{Y}_{*})]=L^{T-1}\bar{L}^{T},$ (4.10)

where the $n\times n$ matrix $L$ is defined by

$L:=\hat{Y}_{*}^{T}Y_{*}-i\hskip 1.0ptW(\hat{Y}_{*},Y_{*})=[\hskip
1.0ptY_{*}^{T}\hat{Y}_{*}+i\hskip 1.0ptW(Y_{*},\hat{Y}_{*})]^{T},\quad
L^{-1}=\bar{L}^{T}.$ (4.11)

This means that $L$ is a unitary matrix. Then we obtain that

$\displaystyle-\Gamma$
$\displaystyle\overset{\eqref{E:GGamma.WS.hlp2}}{=}(X_{*}+i\hskip
1.0ptU_{*})\,(X_{*}-i\hskip 1.0ptU_{*})^{-1}\,(\hat{X}_{*}-i\hskip
1.0pt\hat{U}_{*})\,(\hat{X}_{*}+i\hskip 1.0pt\hat{U}_{*})^{-1}$
$\displaystyle\overset{\eqref{E:GGamma.WS.hlp3}}{=}(X_{*}+i\hskip
1.0ptU_{*})\,\underbrace{(X_{*}^{T}+i\hskip
1.0ptU_{*}^{T})\,(\hat{X}_{*}-i\hskip
1.0pt\hat{U}_{*})}\,\underbrace{(\hat{X}_{*}^{T}-i\hskip
1.0pt\hat{U}_{*}^{T})\,(X_{*}+i\hskip 1.0ptU_{*})}\,(X_{*}+i\hskip
1.0ptU_{*})^{-1}$ $\displaystyle\hskip 3.69885pt=(X_{*}+i\hskip
1.0ptU_{*})\,[\hskip 1.0ptY_{*}^{T}\hat{Y}_{*}-i\hskip
1.0ptW(Y_{*},\hat{Y}_{*})]\,[\hskip 1.0pt\hat{Y}_{*}^{T}Y_{*}+i\hskip
1.0ptW(\hat{Y}_{*},Y_{*})](X_{*}+i\hskip 1.0ptU_{*})^{-1}$
$\displaystyle\overset{\eqref{E:GGamma.WS.hlp6}}{=}(X_{*}+i\hskip
1.0ptU_{*})\,L^{-1}\bar{L}\,(X_{*}+i\hskip
1.0ptU_{*})^{-1}\overset{\eqref{E:GGamma.WS.hlp5}}{=}(X_{*}+i\hskip
1.0ptU_{*})\,L^{-1}\,W_{S}\,L\,(X_{*}+i\hskip 1.0ptU_{*})^{-1},$

where in the last step we used the symmetry of the matrix $W_{S}$. Therefore,
we showed that the matrices $-\Gamma$ and $W_{S}$ are similar, which completes
the proof of this lemma. ∎

Based on the above preliminary considerations we can now prove the main result
of this section, which connects the Maslov index with the Lidskii angles and
hence with the oscillation number through equation (1.5). More precisely, the
Maslov index of $Y$ and $\hat{Y}$ over the interval $[a,b]$ can be calculated
as the total change in the interval $[a,b]$ of the integers $q_{j}(t)$, which
are associated through (3.12) or (3.14) with the continuous Lidskii angles
$\varphi_{j}(t)$ of the symplectic matrix $S(t)$ defined in (4.4), compare
with [46, Definition 2.2] and [6, Eqs. (2.4)–(2.5)]. Consequently, the Maslov
index is equal to the oscillation number of the transformed Lagrangian path
$Z_{Y}^{-1}\hskip 1.0pt\hat{Y}$ on $[a,b]$.

######  Theorem 4.2.

Let $Y$ and $\hat{Y}$ be continuous Lagrangian paths on $[a,b]$ and let the
symplectic and orthogonal matrices $Z_{Y}(t)$, $Z_{\hat{Y}}(t)$ together with
the invertible matrices $K_{Y}(t)$, $K_{\hat{Y}}(t)$ be defined according to
(1.4) on $[a,b]$. Then

$\operatorname{\text{\rm Mas}}\hskip
1.0pt(Y,\hat{Y},[a,b])=\sum_{j=1}^{n}\big{(}\hskip
1.0ptq_{j}(b)-q_{j}(a)\big{)},$ (4.12)

where $q_{j}(t)$ are the integers associated through (3.12) with the
continuous Lidskii angles $\varphi_{j}(t)$ of the symplectic matrix $S(t)$
defined in (4.4). Consequently, for any continuous symplectic matrix $Z(t)$
such that $Y(t)=Z(t)\hskip 1.0ptE$ on $[a,b]$ we have

$\operatorname{\text{\rm Mas}}\hskip
1.0pt(Y,\hat{Y},[a,b])={\mathcal{N}}(Z^{-1}\hat{Y},[a,b])={\mathcal{N}}(\tilde{Y},[a,b]),\quad\tilde{Y}:=Z_{Y}^{-1}\hskip
1.0pt\hat{Y}=\left(\begin{matrix}-K_{Y}\hskip 1.0ptW(Y,\hat{Y})\\\ K_{Y}\hskip
1.0ptY^{T}\hat{Y}\end{matrix}\right).$ (4.13)

###### Proof.

Let the matrices $Z_{Y}(t)$, $Z_{\hat{Y}}(t)$ and $K_{Y}(t)$, $K_{\hat{Y}}(t)$
be as in the theorem. Since the matrix $Z_{Y}(t)$ is orthogonal, the form of
the transformed Lagrangian path $\tilde{Y}$ defined in (4.13) follows from the
same calculation as in (4.9). Consider the matrix $\Gamma(t)$ defined in (4.1)
on $[a,b]$ and a partition $D=\\{t_{k}\\}_{k=0}^{p}$ of $[a,b]$ with the
numbers $\varepsilon_{k}\in(0,\pi)$, which are used in (4.2) in order to
define the numbers $\ell(t,\varepsilon_{k})$. With the symplectic matrix
$S(t)$ in (4.4) we consider the corresponding matrix $W_{S}(t)$ in (4.5) on
$[a,b]$. Denote by $\varphi_{j}(t)$ for $j\in\\{1,\dots,n\\}$ the continuous
Lidskii angles of the symplectic matrix $S(t)$ on $[a,b]$ with the
corresponding integers $q_{j}(t)$ satisfying (3.12). Since by Lemma 4.1 the
matrices $\Gamma(t)$ and $-W_{S}(t)$ are similar for all $t\in[a,b]$, then
these matrices have the same eigenvalues. And since the arguments of the
eigenvalues of the matrices $-W_{S}(t)$ and $W_{S}(t)$ differ by $\pi$, we
obtain from (4.2) that

$\ell(t,\varepsilon_{k})=\sum_{\theta\in[0,\varepsilon_{k})}\\!\\!\operatorname{\text{\rm
def}\hskip 1.0pt}\Big{(}W_{S}(t)-\exp\hskip 1.0pt(i\hskip
1.0pt\theta)\,I\Big{)},\quad t\in[t_{k-1},t_{k}].$ (4.14)

Equation (4.14) shows that $\ell(t,\varepsilon_{k})$ is equal to the number of
the eigenvalues of the matrix $W_{S}(t)$, which lie on the arc
$B_{k}:=\\{\exp\hskip 1.0pt(i\hskip 1.0pt\theta),\
\theta\in[0,\varepsilon_{k})\\}$ of the unit circle ${\mathbb{U}}$. Hence, the
changes (i.e., incrementing or decrementing) of the integers
$\ell(t,\varepsilon_{k})$ when the eigenvalues of $\Gamma(t)$ pass through
$-1$, as it is commented in [26, pp. 796–797], can be calculated by the same
changes when the eigenvalues of $W_{S}(t)$ pass through $1$. More precisely,
the number $\ell(t,\varepsilon_{k})$ increases by one if and only if there is
a corresponding Lidskii angle $\varphi_{j}(t)$ of the matrix $S(t)$, which
arrives from below to an integer multiple of $2\pi$. And the number
$\ell(t,\varepsilon_{k})$ decreases by one if and only if there is a
corresponding Lidskii angle $\varphi_{j}(t)$ of $S(t)$, which leaves an
integer multiple of $2\pi$ in the downward direction. Consequently, the sum of
the numbers $q_{j}(t)$ increases or decreases in the interval
$[t_{k-1},t_{k}]$ by the same amount as $\ell(t,\varepsilon_{k})$, and thus

$\ell(t_{k},\varepsilon_{k})-\ell(t_{k-1},\varepsilon_{k})=\sum_{j=1}^{n}\big{(}\hskip
1.0ptq_{j}(t_{k})-q_{j}(t_{k-1})\big{)}.$ (4.15)

By the definition of the Maslov index of $Y$ and $\hat{Y}$ on $[a,b]$ in (4.3)
we then obtain

$\operatorname{\text{\rm Mas}}\hskip
1.0pt(Y,\hat{Y},[a,b])\overset{\eqref{E:osc.Maslov.hlp4}}{=}\sum_{k=1}^{p}\sum_{j=1}^{n}\big{(}\hskip
1.0ptq_{j}(t_{k})-q_{j}(t_{k-1})\big{)}=\sum_{j=1}^{n}\big{(}\hskip
1.0ptq_{j}(b)-q_{j}(a)\big{)},$ (4.16)

which shows the result in (4.12). By combining equality (4.16) and (3.26) in
Theorem 3.8 (with the continuous Lagrangian path $Y:=SE=\tilde{Y}K_{\hat{Y}}$)
we conclude that

$\operatorname{\text{\rm Mas}}\hskip
1.0pt(Y,\hat{Y},[a,b])\overset{\eqref{E:osc.Maslov.hlp5}}{=}\sum_{j=1}^{n}\big{(}\hskip
1.0ptq_{j}(b)-q_{j}(a)\big{)}\overset{\eqref{E:osc.number.q}}{=}{\mathcal{N}}(\tilde{Y}K_{\hat{Y}},[a,b])\overset{\eqref{E:osc.number.multiple}}{=}{\mathcal{N}}(\tilde{Y},[a,b]),$

where in the last step we used the invariance of the oscillation number with
respect to the multiplication of $\tilde{Y}$ by a continuous invertible
$n\times n$ matrix function from the right (see Remark 3.3(i)). Therefore, the
second equality in (4.13) is proved. Next we apply Theorem 3.12 with
$S(t):=Z(t)$ to prove the first equality in (4.13) for an arbitrary continuous
symplectic matrix $Z(t)$ such that $Z(t)\hskip 1.0ptE=Y(t)$. The proof is
complete. ∎

######  Remark 4.3.

(i) Equation (4.12) also proves the independence of the value
$\operatorname{\text{\rm Mas}}\hskip 1.0pt(Y,\hat{Y},[a,b])$ defined in (4.3)
on the choice of the therein used partition $D=\\{t_{k}\\}_{k=0}^{p}$ and
numbers $\varepsilon_{k}$.

(ii) According to the definition in (4.1) the Maslov index is invariant under
the multiplication of its arguments $Y$ and $\hat{Y}$ by invertible $n\times
n$ matrices from the right, i.e.,

$\operatorname{\text{\rm Mas}}\hskip
1.0pt(YC,\hat{Y}\hat{C},[a,b])=\operatorname{\text{\rm Mas}}\hskip
1.0pt(Y,\hat{Y},[a,b])$

for arbitrary continuous nonsingular matrices $C(t)$ and $\hat{C}(t)$ on
$[a,b]$. This fact also follows from formula (4.13) and from the invariant
properties of the oscillation numbers according to Remark 3.3(i) and Corollary
3.11. In particular, the symplectic matrix $Z(t)$ in Theorem 4.2 can be chosen
to be in a more general form $Z(t)\hskip 1.0ptE=Y(t)\hskip 1.0ptC(t)$ on
$[a,b]$.

The result in Theorem 4.2 leads to the evaluation of the oscillation number in
terms of the Maslov index, as we announced in equation (1.6).

######  Corollary 4.4.

Let $Y$ be a continuous Lagrangian path on $[a,b]$. Then

${\mathcal{N}}(Y,[a,b])=\operatorname{\text{\rm Mas}}\hskip 1.0pt(E,Y,[a,b]),$
(4.17)

where the matrix $E$ defined in (2.3) represents the constant vertical
Lagrangian path.

###### Proof.

Formula (4.17) follows from equation (4.13) in Theorem 4.2 (with $Y:=E$ and
$\hat{Y}:=Y$), since in this case the matrix $Z_{Y}(t)=Z_{E}(t)\equiv I$ on
$[a,b]$. ∎

Next we discuss a dual notion to the Maslov index $\operatorname{\text{\rm
Mas}}\hskip 1.0pt(Y,\hat{Y},[a,b])$, which reflects the results regarding the
dual oscillation numbers obtained in Sections 3 and 5.

######  Remark 4.5.

Let $Y$ and $\hat{Y}$ be two given continuous Lagrangian paths on $[a,b]$.
Assume that $D=\\{a=t_{0}<t_{1}\dots<t_{p}=b\\}$ is a partition of $[a,b]$,
which is used along with numbers $\varepsilon_{k}\in(0,\pi)$ in the definition
of the Maslov index $\operatorname{\text{\rm Mas}}\hskip
1.0pt(Y,\hat{Y},[a,b])$ in (4.3). Then instead of the numbers
$\ell(t,\varepsilon_{k})$ defined in (4.2) we consider the numbers

$\ell^{\hskip
1.0pt*}(t,\varepsilon_{k}):=\sum_{\theta\in[0,\varepsilon_{k})}\\!\\!\operatorname{\text{\rm
def}\hskip 1.0pt}\Big{(}\Gamma(t)-\exp\hskip 1.0pt\\{i\hskip
1.0pt(\pi-\theta)\\}\,I\Big{)},\quad t\in[t_{k-1},t_{k}].$ (4.18)

This means that $\ell^{\hskip 1.0pt*}(t,\varepsilon_{k})$ is equal to the
number of the eigenvalues of $\Gamma(t)$, which lie on the arc
$A_{k}^{*}:=\\{\exp\hskip 1.0pt(i\hskip 1.0pt\theta),\
\theta\in(\pi-\varepsilon_{k},\pi]\\}$ of the unit circle ${\mathbb{U}}$. By
analogy with (4.3) we define the dual Maslov index of the continuous
Lagrangian paths $Y$ and $\hat{Y}$ as the integer

$\operatorname{\text{\rm Mas}}\hskip
1.0pt^{\\!*}(Y,\hat{Y},[a,b]):=\sum_{k=1}^{p}\Big{(}\ell^{\hskip
1.0pt*}(t_{k-1},\varepsilon_{k})-\ell^{\hskip
1.0pt*}(t_{k},\varepsilon_{k})\Big{)}.$ (4.19)

Then similarly as in Theorem 4.2 we obtain that for an arbitrary continuous
symplectic matrix $Z(t)$ associated with $Y(t)$ via the condition
$Y(t)=Z(t)\hskip 1.0ptE$

$\operatorname{\text{\rm Mas}}\hskip
1.0pt^{\\!*}(Y,\hat{Y},[a,b])={\mathcal{N}}^{*}(Z^{-1}\hat{Y},[a,b])={\mathcal{N}}^{*}(Z_{Y}^{-1}\hat{Y},[a,b]).$
(4.20)

In view of (3.4), the representation formulas (4.13) and (4.20) for the Maslov
index and the dual Maslov index imply that

$\operatorname{\text{\rm Mas}}\hskip
1.0pt^{\\!*}(Y,\hat{Y},[a,b])={\mathcal{N}}^{*}(Z^{-1}\hat{Y},[a,b])=-{\mathcal{N}}(\hat{Z}^{-1}Y,[a,b])=-\operatorname{\text{\rm
Mas}}\hskip 1.0pt(\hat{Y},Y,[a,b]),$ (4.21)

where the continuous symplectic matrix $\hat{Z}(t)$ is such that
$\hat{Y}(t)=\hat{Z}(t)\hskip 1.0ptE$. This yields, in view of Theorem 3.8, the
geometric interpretation of the dual Maslov index $\operatorname{\text{\rm
Mas}}\hskip 1.0pt^{\\!*}(Y,\hat{Y},[a,b])$ as the total change in the interval
$[a,b]$ of the integers $q_{j}^{*}(t)$, which are associated through (3.13) or
(3.14) with the continuous Lidskii angles $\varphi_{j}(t)$ of the symplectic
matrix $S(t)$ defined in (4.4). From (4.20) we then obtain the dual version of
Corollary 4.4 in the form

${\mathcal{N}}^{*}(Y,[a,b])=\operatorname{\text{\rm Mas}}\hskip
1.0pt^{\\!*}(E,Y,[a,b]).$ (4.22)

Finally, from Theorem 3.13 we obtain the equality

$\operatorname{\text{\rm Mas}}\hskip
1.0pt^{\\!*}(Y,\hat{Y},[a,b])=\operatorname{\text{\rm Mas}}\hskip
1.0pt(Y,\hat{Y},[a,b])+\operatorname{\text{\rm
rank}}W(Y(b),\hat{Y}(b))-\operatorname{\text{\rm rank}}W(Y(a),\hat{Y}(a)).$
(4.23)

In view of formulas (4.17) and (4.22), the results in (3.29) and (4.21),
(4.23) correspond to the vanishing property and to the flipping property (or
the symmetry) of the Maslov index and the dual Maslov index, see [6,
Proposition 2.3.1 (e), (f)] and [8, Property XI, pg. 130].

In [17, Theorem 2.3], [19, Propositions 3.2 and 3.4], and [21, Proposition
3.3(v)] we proved that the oscillation number and the dual oscillation number
count the left and right proper focal points of a conjoined basis $Y$ of
system (H) satisfying the Legendre condition (1.3). Below we extend these
properties to the case when $Y$ is an arbitrary piecewise continuously
differentiable Lagrangian path on $[a,b]$. For this purpose we recall that if
$A(t)$ is a given matrix defined on the interval $[a,b]$, then
$\operatorname{\text{\rm rank}}A(t_{0}^{\pm})$ denote the left-hand and the
right-hand limits of the quantity $\operatorname{\text{\rm rank}}A(t)$ at the
point $t_{0}\in[a,b]$.

######  Theorem 4.6.

Let $Y$ be a piecewise continuously differentiable Lagrangian path $[a,b]$
with the partition as in (1.2) and assume that

$[Y^{\prime}(t)]^{T}\\!{\mathcal{J}}\hskip 1.0ptY(t)=-Y^{T}(t)\hskip
1.0pt{\mathcal{J}}\hskip 1.0ptY^{\prime}(t)\geq 0,\quad t\in[a,b].$ (4.24)

Then the subspace $\operatorname{\text{\rm Im}}X(t)$ is piecewise constant on
$[a,b]$ and

$\displaystyle{\mathcal{N}}(Y,[a,b])$
$\displaystyle=\sum_{t_{0}\in(a,b]}\\!\\!\big{(}\operatorname{\text{\rm
rank}}X(t_{0}^{-})-\operatorname{\text{\rm rank}}X(t_{0})\big{)}\geq 0,$
(4.25) $\displaystyle{\mathcal{N}}^{*}(Y,[a,b])$
$\displaystyle=\sum_{t_{0}\in[a,b)}\\!\\!\big{(}\operatorname{\text{\rm
rank}}X(t_{0}^{+})-\operatorname{\text{\rm rank}}X(t_{0})\big{)}\geq 0,$
(4.26)

where the sums in equations (4.25) and (4.26) are finite.

###### Proof.

The main idea of the proof is to construct a linear Hamiltonian system in form
(H) with the condition ${\mathcal{H}}(t)\geq 0$ on $[a,b]$ for its coefficient
matrix ${\mathcal{H}}(t)$ and such that $YQ$ is a conjoined basis of this
system for a suitably chosen nonsingular matrix-valued function
$Q:[a,b]\to{\mathbb{R}}^{n\times n}$. Then by a classical result for such
systems, see e.g. [32, Theorem 3], [23, Proof of Lemma 3.6(a)], or [16,
Theorem 2.4], [10, Theorems 1.79 and 1.81], the condition
${\mathcal{H}}(t)\geq 0$ will imply that the sets $\operatorname{\text{\rm
Ker}}X(t)\hskip 1.0ptQ(t)$ and $\operatorname{\text{\rm Im}}X(t)$ are
piecewise constant on $[a,b]$. Hence, the quantity $\operatorname{\text{\rm
rank}}X(t)$ is also piecewise constant on $[a,b]$, giving a correct meaning to
the right-hand sides of equations (4.25) and (4.26).

A construction of such Hamiltonian system is given in [16], see also [10,
Section 5.2.1], where the authors considered symplectic spectral problems with
self-adjoint boundary conditions depending on spectral parameter
$\lambda\in{\mathbb{R}}$. In the notation of this paper and according to the
proof of [16, Lemma 4.1], we consider the solution $Q(t)$ of the linear
differential system

$Q^{\prime}=-K_{Y}^{2}(t)\hskip 1.0ptY^{T}(t)\hskip
1.0ptY^{\prime}(t)\,Q,\quad t\in[a,b],\quad Q(a)=I.$ (4.27)

Since the coefficient matrix in (4.27) is piecewise continuous on $[a,b]$, it
follows that the matrix $Q(t)$ is piecewise continuously differentiable and
invertible on $[a,b]$. Consider the invertible matrix $M(t):=K_{Y}(t)\hskip
1.0ptQ^{T-1}(t)$ and modify the symplectic matrix $Z_{Y}(t)$ in (1.4) to
become

$\tilde{Z}_{Y}(t):=Z_{Y}(t)\operatorname{\text{\rm
diag}}\\{M(t),M^{T-1}(t)\\}=\left(\begin{matrix}{\mathcal{J}}\hskip
1.0ptY(t)\hskip 1.0ptK_{Y}^{2}(t)\hskip 1.0ptQ^{T-1}(t)&Y(t)\hskip
1.0ptQ(t)\end{matrix}\right)$ (4.28)

on $[a,b]$. Then the matrix $\tilde{Z}_{Y}(t)$ is piecewise continuously
differentiable on $[a,b]$ and symplectic, as a product of two symplectic
matrices. Therefore, the piecewise continuous matrix

${\mathcal{H}}(t):=-{\mathcal{J}}\hskip 1.0pt[\tilde{Z}_{Y}(t)]\hskip
1.0pt^{\prime}\hskip 1.0pt\tilde{Z}_{Y}^{-1}(t)$ (4.29)

is symmetric and satisfies with the above defined $\tilde{Z}_{Y}(t)$ the
linear Hamiltonian system

$[\tilde{Z}_{Y}(t)]\hskip 1.0pt^{\prime}={\mathcal{J}}\hskip
1.0pt{\mathcal{H}}(t)\,\tilde{Z}_{Y}(t),\quad t\in[a,b].$ (4.30)

Note that ${\mathcal{H}}(t)=\Psi(\tilde{Z}_{Y}(t))$ with the notation from
[16, Propositions 2.2 and 2.3]. Upon calculating the derivative of the matrix
$\tilde{Z}_{Y}(t)$ in (4.28), we get (suppressing the argument $t$)

$(\tilde{Z}_{Y})^{\prime}=\left(\begin{matrix}{\mathcal{J}}\hskip
1.0ptY^{\prime}K_{Y}^{2}\hskip 1.0ptQ^{T-1}+{\mathcal{J}}\hskip
1.0ptY(K_{Y}^{2})^{\prime}\hskip 1.0ptQ^{T-1}+{\mathcal{J}}\hskip
1.0ptYK_{Y}^{2}\hskip 1.0pt(Q^{T-1})^{\prime}&\quad
Y^{\prime}Q+YQ^{\prime}\end{matrix}\right).$ (4.31)

Then by combining equations (4.27)–(4.31) with (1.1) and assumption (4.24) we
conclude that

$\tilde{Z}_{Y}^{T}(t)\hskip 1.0pt{\mathcal{H}}(t)\hskip
1.0pt\tilde{Z}_{Y}(t)=-V^{T}(t)\,\operatorname{\text{\rm
diag}}\\!\big{\\{}Y^{T}(t)\hskip 1.0pt{\mathcal{J}}\hskip
1.0ptY^{\prime}(t),\,Y^{T}(t)\hskip 1.0pt{\mathcal{J}}\hskip
1.0ptY^{\prime}(t)\big{\\}}\,V(t)\overset{\eqref{derWr}}{\geq}0$ (4.32)

on $[a,b]$, where the matrix $V(t):=\operatorname{\text{\rm
diag}}\\{K_{Y}^{2}(t)\hskip 1.0ptQ^{T-1}(t),\,Q(t)\\}$ is invertible. Note
that we also used that the matrix $Q^{T-1}(t)$ solves on $[a,b]$ the adjoint
system to (4.27), which is the linear differential system with the coefficient
matrix equal to $[Y^{\prime}(t)]^{T}\hskip 1.0ptY(t)\hskip 1.0ptK_{Y}^{2}(t)$.
Thus, according to (4.28) and (4.32) the function $YQ$ is a conjoined basis of
the linear Hamiltonian system (4.30) with ${\mathcal{H}}(t)\geq 0$ on $[a,b]$,
so that the Legendre condition (1.3) for this system holds. Applying [17,
Theorem 2.3] we obtain that the oscillation number for the piecewise
continuously differentiable Lagrangian path $YQ$ is equal to the total number
of left proper focal points of $YQ$ in the interval $(a,b]$, i.e.,

$\displaystyle{\mathcal{N}}(YQ,[a,b])$
$\displaystyle=\sum_{t_{0}\in(a,b]}\\!\\!\big{\\{}\\!\operatorname{\text{\rm
rank}}\\!\big{(}X(t_{0}^{-})\hskip
1.0ptQ(t_{0}^{-})\big{)}-\operatorname{\text{\rm
rank}}\\!\big{(}X(t_{0})\hskip 1.0ptQ_{0}(t_{0})\big{)}\big{\\}}$
$\displaystyle=\sum_{t_{0}\in(a,b]}\\!\\!\big{(}\operatorname{\text{\rm
rank}}X(t_{0}^{-})-\operatorname{\text{\rm rank}}X(t_{0})\big{)}\geq 0.$

In a similar way, by applying [21, Proposition 3.3(v)] we obtain that the dual
oscillation number for $YQ$ is equal to the total number of right proper focal
points of $YQ$ in the interval $[a,b)$, i.e.,

$\displaystyle{\mathcal{N}}^{*}(YQ,[a,b])$
$\displaystyle=\sum_{t_{0}\in[a,b)}\\!\\!\big{\\{}\\!\operatorname{\text{\rm
rank}}\\!\big{(}X(t_{0}^{+})\hskip
1.0ptQ(t_{0}^{+})\big{)}-\operatorname{\text{\rm
rank}}\\!\big{(}X(t_{0})\hskip 1.0ptQ_{0}(t_{0})\big{)}\big{\\}}$
$\displaystyle=\sum_{t_{0}\in[a,b)}\\!\\!\big{(}\operatorname{\text{\rm
rank}}X(t_{0}^{+})-\operatorname{\text{\rm rank}}X(t_{0})\big{)}\geq 0.$

However, since the matrix $Q(t)$ is nonsingular on $[a,b]$, it follows from
Remark 3.3(i) that ${\mathcal{N}}(YQ,[a,b])={\mathcal{N}}(Y,[a,b])$ and
${\mathcal{N}}^{*}(YQ,[a,b])={\mathcal{N}}^{*}(Y,[a,b])$, which completes the
proof of this theorem. ∎

######  Remark 4.7.

The proof of Theorem 4.6 shows that the conclusion of the piecewise constant
image of $X(t)$ on $[a,b]$ in the statement of this theorem follows from the
stronger property that the matrix $X(t)\hskip 1.0ptQ(t)$ has piecewise
constant kernel on $[a,b]$, where $Q(t)$ is the solution of (4.27).

By combining Theorems 4.2 and 4.6 we derive the following monotonicity
property of the Maslov index of two continuously differentiable Lagrangian
paths on $[a,b]$. Note that here we do not make any strict monotonicity
assumption.

######  Theorem 4.8.

Let $Y$ and $\hat{Y}$ be piecewise continuously differentiable Lagrangian
paths $[a,b]$. Assume that there exist a piecewise continuously differentiable
symplectic matrix $Z(t)$ and a nonsingular piecewise continuously
differentiable $n\times n$ matrix $P(t)$ such that

$[\hskip 1.0pt\bar{Y}^{\prime}(t)]^{T}\\!{\mathcal{J}}\hskip
1.0pt\bar{Y}(t)\geq 0,\quad\bar{Y}(t):=Z^{-1}(t)\hskip 1.0pt\hat{Y}(t),\quad
Z(t)\hskip 1.0ptE=Y(t)\hskip 1.0ptP(t),\quad t\in[a,b].$ (4.33)

Then the subspace $\operatorname{\text{\rm Im}}W(Y(t),\hat{Y}(t))$ is
piecewise constant on $[a,b]$ and

$\displaystyle\operatorname{\text{\rm Mas}}\hskip 1.0pt(Y,\hat{Y},[a,b])$
$\displaystyle=\sum_{t_{0}\in(a,b]}\\!\big{(}\operatorname{\text{\rm
rank}}W(Y(t_{0}^{-}),\hat{Y}(t_{0}^{-}))-\operatorname{\text{\rm
rank}}W(Y(t_{0}),\hat{Y}(t_{0})\big{)}\geq 0,$ (4.34)
$\displaystyle\operatorname{\text{\rm Mas}}\hskip
1.0pt^{\\!*}(Y,\hat{Y},[a,b])$
$\displaystyle=\sum_{t_{0}\in[a,b)}\\!\big{(}\operatorname{\text{\rm
rank}}W(Y(t_{0}^{+}),\hat{Y}(t_{0}^{+}))-\operatorname{\text{\rm
rank}}W(Y(t_{0}),\hat{Y}(t_{0})\big{)}\geq 0.$ (4.35)

In particular, assumption (4.33) is satisfied under the conditions

$[\hskip 1.0ptY^{\prime}(t)]^{T}\\!{\mathcal{J}}\hskip 1.0ptY(t)\leq
0,\quad[\hskip 1.0pt\hat{Y}^{\prime}(t)]^{T}\\!{\mathcal{J}}\hskip
1.0pt\hat{Y}(t)\geq 0,\quad t\in[a,b].$ (4.36)

###### Proof.

The function $\bar{Y}$ defined in (4.33) is a piecewise continuously
differentiable Lagrangian path on $[a,b]$. Then by applying formula (4.13) in
Theorem 4.2 and Remark 4.3(ii) we get

$\displaystyle\operatorname{\text{\rm Mas}}\hskip 1.0pt(Y,\hat{Y},[a,b])$
$\displaystyle=\operatorname{\text{\rm Mas}}\hskip
1.0pt(YP,\hat{Y},[a,b])={\mathcal{N}}(\bar{Y},[a,b]),$
$\displaystyle\operatorname{\text{\rm Mas}}\hskip
1.0pt^{\\!*}(Y,\hat{Y},[a,b])$ $\displaystyle=\operatorname{\text{\rm
Mas}}\hskip 1.0pt^{\\!*}(YP,\hat{Y},[a,b])={\mathcal{N}}^{*}(\bar{Y},[a,b]).$

For the calculation of these oscillation numbers we apply Theorem 4.6 (with
$Y:=\bar{Y}$). Note that the upper block $\bar{X}(t)$ of $\bar{Y}(t)$ has the
form $\bar{X}(t)=-P^{T}(t)\,W(Y(t),\hat{Y}(t))$. Then, by the nonsingularity
of the matrix $P(t)$, the results in (4.34) and (4.35) follow from equations
(4.25) and (4.26). Finally, we assume that (4.36) holds and consider the
matrices $Q(t)$, $\tilde{Z}_{Y}(t)$, and ${\mathcal{H}}(t)$ given by (4.27),
(4.28), and (4.29). Then we have (repeating the proof of Theorem 4.6) from the
first condition in (4.36) that ${\mathcal{H}}(t)\leq 0$ (pay attention to the
sign change) on $[a,b]$. Next we put $P(t):=Q(t)$ and $Z(t):=\tilde{Z}_{Y}(t)$
in (4.33), so that

$\displaystyle[\hskip 1.0pt\bar{Y}^{\prime}(t)]^{T}\\!{\mathcal{J}}\hskip
1.0pt\bar{Y}(t)$ $\displaystyle=\big{(}\\!-\tilde{Z}_{Y}^{-1}(t)\hskip
1.0pt[\tilde{Z}_{Y}(t)]\hskip 1.0pt^{\prime}\hskip
1.0pt\tilde{Z}_{Y}^{-1}(t)\hskip 1.0pt\hat{Y}(t)+\tilde{Z}_{Y}^{-1}(t)\hskip
1.0pt\hat{Y}^{\prime}(t)\big{)}^{T}\\!{\mathcal{J}}\tilde{Z}_{Y}^{-1}(t)\hskip
1.0pt\hat{Y}(t)$ $\displaystyle=[\hskip
1.0pt\hat{Y}^{\prime}(t)]^{T}\\!{\mathcal{J}}\hskip
1.0pt\hat{Y}(t)-\hat{Y}^{T}(t)\hskip 1.0pt{\mathcal{H}}(t)\hskip
1.0pt\hat{Y}(t)\geq 0$

on $[a,b]$, where we used that the matrix $\tilde{Z}_{Y}^{-1}(t)$ is
symplectic and that the matrix ${\mathcal{H}}(t)$ is symmetric. The last
inequality then follows from ${\mathcal{H}}(t)\leq 0$ and the second condition
in (4.36). The proof is complete. ∎

######  Remark 4.9.

In [26, Section 4] the authors consider monotonicity properties of the
eigenvalues of the matrix $\Gamma(t)$ in (4.1) in the following sense. As the
parameter $t\in[a,b]$ varies in a fixed direction, the eigenvalues of the
matrix $\Gamma(t)$ move monotonically around unit circle ${\mathbb{U}}$. These
results demand strict monotonicity assumptions for the derivatives in (4.33)
and (4.24), see [26, Lemma 4.2] in particular. In this case we would have the
strict inequality ${\mathcal{H}}(t)>0$ on $[a,b]$ for the Hamiltonian of the
differential system (4.30) associated with $Y(t)$ in the proof of Theorem 4.6.
Therefore, in this case the system (4.30) would be completely controllable on
$[a,b]$. The approach without a strict monotonicity assumption and hence
without the complete controllability condition, as presented in (4.36) above,
is known in [41] in the analysis of proper focal points of conjoined bases of
system (H).

##  5\. Comparison theorems for Lagrangian paths

In this section we derive Sturmian type comparison theorems for two continuous
Lagrangian paths $Y$ and $\hat{Y}$ on $[a,b]$. Thus we extend the results in
[19, Theorem 4.4] and [21, Theorem 4.3] to the context of arbitrary continuous
Lagrangian paths on $[a,b]$. At the same time we utilize the more general
definition of the oscillation number and the dual oscillation number as
presented in Section 3. Compared with the results in the latter two
references, which were proven by using the properties of the comparative
index, we employ the results from Section 3 based on the theory of Lidskii
angles.

######  Theorem 5.1 (Comparison theorem for oscillation numbers).

Let $Y$ and $\hat{Y}$ be continuous Lagrangian paths on $[a,b]$. Then

$\displaystyle{\mathcal{N}}(Y,[a,b])-{\mathcal{N}}(\hat{Y},[a,b])$
$\displaystyle=\mu(Y(b),\hat{Y}(b))-\mu(Y(a),\hat{Y}(a))+{\mathcal{N}}(Z_{\hat{Y}}^{-1}\hskip
1.0ptY,[a,b]),$ (5.1)
$\displaystyle{\mathcal{N}}^{*}(Y,[a,b])-{\mathcal{N}}^{*}(\hat{Y},[a,b])$
$\displaystyle=\mu^{*}(Y(a),\hat{Y}(a))-\mu^{*}(Y(b),\hat{Y}(b))+{\mathcal{N}}^{*}(Z_{\hat{Y}}^{-1}\hskip
1.0ptY,[a,b]),$ (5.2)

where $Z_{\hat{Y}}(t)=Z_{\hat{Y}(t)}$ is the symplectic and orthogonal matrix
defined in (1.4), which is associated with the Lagrangian path $\hat{Y}(t)$.
Moreover, the matrix $Z_{\hat{Y}}(t)$ in (5.1) and (5.2) can be replaced by an
arbitrary continuous symplectic matrix $\hat{Z}(t)$ with $\hat{Z}(t)\hskip
1.0ptE=\hat{Y}(t)$ on $[a,b]$.

###### Proof.

Applying Proposition 3.6 (with the special case of $\tau_{1}:=a$ and
$\tau_{2}:=b$) we derive by (3.20) and (3.21) that

$\displaystyle\mu(Y(t),\hat{Y}(t))\big{|}_{a}^{b}$
$\displaystyle=\sum_{j=1}^{n}q_{j}(t)\big{|}_{a}^{b}-\sum_{j=1}^{n}{\hat{q}}_{j}(t)\big{|}_{a}^{b}-\sum_{j=1}^{n}{\tilde{q}}_{j}(t)\big{|}_{a}^{b},$
(5.3) $\displaystyle\mu^{*}(Y(t),\hat{Y}(t))\big{|}_{a}^{b}$
$\displaystyle=-\sum_{j=1}^{n}q_{j}^{*}(t)\big{|}_{a}^{b}+\sum_{j=1}^{n}{\hat{q}}_{j}^{*}(t)\big{|}_{a}^{b}+\sum_{j=1}^{n}{\tilde{q}}_{j}^{*}(t)\big{|}_{a}^{b},$
(5.4)

where according to Theorem 3.8 we have

$\displaystyle{\mathcal{N}}(Y,[a,b])=\sum_{j=1}^{n}q_{j}(t)\big{|}_{a}^{b},\quad{\mathcal{N}}(\hat{Y},[a,b])=\sum_{j=1}^{n}{\hat{q}}_{j}(t)\big{|}_{a}^{b},\quad{\mathcal{N}}(Z_{\hat{Y}}^{-1}\hskip
1.0ptY,[a,b])=\sum_{j=1}^{n}{\tilde{q}}_{j}(t)\big{|}_{a}^{b},$
$\displaystyle{\mathcal{N}}^{*}(Y,[a,b])=\sum_{j=1}^{n}q_{j}^{*}(t)\big{|}_{a}^{b},\quad{\mathcal{N}}^{*}(\hat{Y},[a,b])=\sum_{j=1}^{n}{\hat{q}}_{j}^{*}(t)\big{|}_{a}^{b},\quad{\mathcal{N}}^{*}(Z_{\hat{Y}}^{-1}\hskip
1.0ptY,[a,b])=\sum_{j=1}^{n}{\tilde{q}}_{j}^{*}(t)\big{|}_{a}^{b}.$

By substituting the last representations into formulas (5.3) and (5.4) we
complete the proofs of the results in (5.1) and (5.2). Moreover, by using
(3.34) and (3.35) in Theorem 3.12 (with $S(t):=\hat{Z}(t)$ and $S(t)\hskip
1.0ptE=\hat{Y}(t)$ on $[a,b]$) we see that the matrix $Z_{\hat{Y}}(t)$ in
(5.1) and (5.2) can be replaced by an arbitrary continuous symplectic matrix
$\hat{Z}(t)$ with $\hat{Z}(t)\hskip 1.0ptE=\hat{Y}(t)$ on $[a,b]$. ∎

If the Lagrangian paths $Y$ and $\hat{Y}$ have the same values at the
endpoints of the interval $[a,b]$, then we obtain from Theorem 5.1 the
following.

######  Corollary 5.2.

Let $Y$ and $\hat{Y}$ be continuous Lagrangian paths on $[a,b]$ such that
$Y(a)=\hat{Y}(a)$ and $Y(b)=\hat{Y}(b)$. Then

$\displaystyle{\mathcal{N}}(Y,[a,b])-{\mathcal{N}}(\hat{Y},[a,b])$
$\displaystyle={\mathcal{N}}(Z_{\hat{Y}}^{-1}\hskip 1.0ptY,[a,b]),$ (5.5)
$\displaystyle{\mathcal{N}}^{*}(Y,[a,b])-{\mathcal{N}}^{*}(\hat{Y},[a,b])$
$\displaystyle={\mathcal{N}}^{*}(Z_{\hat{Y}}^{-1}\hskip 1.0ptY,[a,b]),$ (5.6)

where $Z_{\hat{Y}}(t)=Z_{\hat{Y}(t)}$ is the symplectic and orthogonal matrix
defined in (1.4), which is associated with the Lagrangian path $\hat{Y}(t)$.
Moreover, the matrix $Z_{\hat{Y}}(t)$ in (5.5) and (5.6) can be replaced by an
arbitrary continuous symplectic matrix $\hat{Z}(t)$ with $\hat{Z}(t)\hskip
1.0ptE=\hat{Y}(t)$ on $[a,b]$.

###### Proof.

Under the assumptions $Y(a)=\hat{Y}(a)$ and $Y(b)=\hat{Y}(b)$ the comparative
indices and the dual comparative indices appearing in (5.1) and (5.2) are
zero. Hence, the results in (5.5) and (5.6), as well as the last statement of
this corollary, follow from Theorem 5.1. ∎

Equations (5.1) and (5.2) can be simplified for special Lagrangian paths on
$[a,b]$, for which the involved comparative indices vanish. For this purpose
we introduce the notation $Y_{a}$ and $Y_{b}$ for continuous Lagrangian paths,
which satisfy the initial conditions

$Y_{a}(a)=E=Y_{b}(b).$ (5.7)

When $Y_{a}$ and $Y_{b}$ correspond to conjoined bases of system (H), then
they are uniquely determined by (5.7) as solutions of (H). In this case
$Y_{a}$ and $Y_{b}$ are called the principal solutions of system (H) at the
points $a$ and $b$. In the general situation of continuous Lagrangian paths on
$[a,b]$ we can still derive some important properties of the paths $Y_{a}$ and
$Y_{b}$. Then we have the following extension of [21, Corollary 4.6].

######  Corollary 5.3.

Let $Y_{a}$ and $Y_{b}$ be continuous Lagrangian paths on $[a,b]$ satisfying
condition (5.7). Then their oscillation numbers and dual oscillation numbers
satisfy the relations

$\displaystyle{\mathcal{N}}(Y_{b},[a,b])-{\mathcal{N}}^{*}(Y_{a},[a,b])$
$\displaystyle={\mathcal{N}}(Z_{Y_{a}}^{-1}Y_{b},[a,b]),$ (5.8)
$\displaystyle{\mathcal{N}}^{*}(Y_{b},[a,b])-{\mathcal{N}}(Y_{a},[a,b])$
$\displaystyle={\mathcal{N}}^{*}(Z_{Y_{a}}^{-1}Y_{b},[a,b]).$ (5.9)

###### Proof.

We partition the Lagrangian paths $Y_{a}$ and $Y_{b}$ on $[a,b]$ according to
(1.2). We apply equality (5.1) with $Y:=Y_{b}$ and $\hat{Y}:=Y_{a}$. Then
$\mu(Y_{b}(b),Y_{a}(b))=\operatorname{\text{\rm rank}}X_{a}(b)$ and
$\mu(Y_{b}(a),Y_{a}(a))=0$, so that by Theorem 3.13 (with $Y:=Y_{a}$) we
obtain

$\displaystyle{\mathcal{N}}(Y_{b},[a,b])$
$\displaystyle\overset{\eqref{E:compare.osc.number}}{=}{\mathcal{N}}(Y_{a},[a,b])+\operatorname{\text{\rm
rank}}X_{a}(b)+{\mathcal{N}}(Z_{Y_{a}}^{-1}Y_{b},[a,b])$
$\displaystyle\overset{\eqref{E:osc.dual.osc.number}}{=}{\mathcal{N}}^{*}(Y_{a},[a,b])+{\mathcal{N}}(Z_{Y_{a}}^{-1}Y_{b},[a,b]).$

This shows (5.8). Next we apply (5.2) with $Y:=Y_{b}$ and $\hat{Y}:=Y_{a}$.
Then $\mu^{*}(Y_{b}(a),Y_{a}(a))=0$ and
$\mu^{*}(Y_{b}(b),Y_{a}(b))=\operatorname{\text{\rm rank}}X_{a}(b)$, so that
by Theorem 3.13 (with $Y:=Y_{a}$) we get

$\displaystyle{\mathcal{N}}^{*}(Y_{b},[a,b])$
$\displaystyle\overset{\eqref{E:compare.osc.number*}}{=}{\mathcal{N}}^{*}(Y_{a},[a,b])-\operatorname{\text{\rm
rank}}X_{a}(b)+{\mathcal{N}}^{*}(Z_{Y_{a}}^{-1}Y_{b},[a,b])$
$\displaystyle\overset{\eqref{E:osc.dual.osc.number}}{=}{\mathcal{N}}(Y_{a},[a,b])+{\mathcal{N}}^{*}(Z_{Y_{a}}^{-1}Y_{b},[a,b]).$

This shows (5.9) and the proof is complete. ∎

Based on the connections of the oscillation number and the Maslov index
derived in Section 4 we can now reformulate the comparison theorem for the
oscillation numbers (Theorem 5.1) in terms of the Maslov index. More
precisely, we obtain a formula calculating the Maslov index
$\operatorname{\text{\rm Mas}}\hskip 1.0pt(Y,\hat{Y},[a,b])$ in terms of the
two reference Maslov indices $\operatorname{\text{\rm Mas}}\hskip
1.0pt(E,\hat{Y},[a,b])$ and $\operatorname{\text{\rm Mas}}\hskip
1.0pt(E,Y,[a,b])$ and in terms of the comparative index of $\hat{Y}$ and $Y$
evaluated at the endpoints of $[a,b]$.

######  Corollary 5.4.

Let $Y$ and $\hat{Y}$ be continuous Lagrangian paths on $[a,b]$. Then we have

$\left.\begin{array}[]{rl}\operatorname{\text{\rm Mas}}\hskip
1.0pt(Y,\hat{Y},[a,b])&\\!\\!\\!=\operatorname{\text{\rm Mas}}\hskip
1.0pt(E,\hat{Y},[a,b])-\operatorname{\text{\rm Mas}}\hskip
1.0pt(E,Y,[a,b])\\\\[2.84526pt] &\hskip
42.67912pt+\,\mu(\hat{Y}(a),Y(a))-\mu(\hat{Y}(b),Y(b)).\end{array}\\!\right\\}$
(5.10)

If in addition $Y(a)=\hat{Y}(a)$ and $Y(b)=\hat{Y}(b)$ hold, then

$\operatorname{\text{\rm Mas}}\hskip
1.0pt(Y,\hat{Y},[a,b])=\operatorname{\text{\rm Mas}}\hskip
1.0pt(E,\hat{Y},[a,b])-\operatorname{\text{\rm Mas}}\hskip 1.0pt(E,Y,[a,b]).$
(5.11)

###### Proof.

Let $Y$ and $\hat{Y}$ be continuous Lagrangian paths on $[a,b]$. We apply
formula (5.1) in Theorem 5.1, in which we interchange the roles of $Y$ and
$\hat{Y}$. Then we get

${\mathcal{N}}(\hat{Y},[a,b])-{\mathcal{N}}(Y,[a,b])=\mu(\hat{Y}(b),Y(b))-\mu(\hat{Y}(a),Y(a))+{\mathcal{N}}(Z_{Y}^{-1}\hskip
1.0pt\hat{Y},[a,b]).$ (5.12)

If we now replace the oscillation numbers appearing in (5.12) by the
corresponding Maslov indices from (4.17) and (4.13), then we obtain the result
in (5.10). Finally, equation (5.11) follows from identity (5.10), in which the
comparative indices vanish under the assumptions that $Y(a)=\hat{Y}(a)$ and
$Y(b)=\hat{Y}(b)$. ∎

######  Remark 5.5.

Combining equations (4.20) and (4.22) with the comparison theorem for the dual
oscillation numbers in (5.2) we derive a dual version of Corollary 5.4 in the
form

$\left.\begin{array}[]{rl}\operatorname{\text{\rm Mas}}\hskip
1.0pt^{\\!*}(Y,\hat{Y},[a,b])&\\!\\!\\!=\operatorname{\text{\rm Mas}}\hskip
1.0pt^{\\!*}(E,\hat{Y},[a,b])-\operatorname{\text{\rm Mas}}\hskip
1.0pt^{\\!*}(E,Y,[a,b])\\\\[2.84526pt] &\hskip
42.67912pt+\,\mu^{*}(\hat{Y}(b),Y(b))-\mu^{*}(\hat{Y}(a),Y(a)).\end{array}\\!\right\\}$
(5.13)

In addition, if $Y(a)=\hat{Y}(a)$ and $Y(b)=\hat{Y}(b)$ hold, then (5.13)
reduces to

$\operatorname{\text{\rm Mas}}\hskip
1.0pt^{\\!*}(Y,\hat{Y},[a,b])=\operatorname{\text{\rm Mas}}\hskip
1.0pt^{\\!*}(E,\hat{Y},[a,b])-\operatorname{\text{\rm Mas}}\hskip
1.0pt^{\\!*}(E,Y,[a,b]).$ (5.14)

Comparison formulas (5.10) and (5.13) appear to be new in the context of the
Maslov index.

Next we consider the corresponding Sturmian type separation theorems for the
oscillation numbers and the dual oscillation numbers. These results extend the
special situation, when both Lagrangian paths $Y$ and $\hat{Y}$ are conjoined
bases of one linear Hamiltonian system (H). We refer to [13, Theorems 2.2 and
2.3] and [36, Theorem 4.1] for the case when the Legendre condition (1.3)
holds, and to [18, Theorem 4.1] and [21, Theorem 4.4] for the case without
assumption (1.3).

Let us fix a continuous symplectic matrix $\Phi(t)$ on $[a,b]$. Consider the
set

${\mathcal{F}}(\Phi):=\big{\\{}\Phi(\cdot)\hskip 1.0ptC,\text{ where $C$ is
a~{}Lagrangian plane}\big{\\}}.$ (5.15)

Then the elements $Y\in{\mathcal{F}}(\Phi)$ are continuous Lagrangian paths on
$[a,b]$, which are constant multiples of the given symplectic matrix $\Phi(t)$
on $[a,b]$, that is, $Y(t)=\Phi(t)\,C$ on $[a,b]$ for some matrix
$C\in{\mathbb{R}}^{2n\times n}$ with $C^{T}\\!{\mathcal{J}}C=0$ and
$\operatorname{\text{\rm rank}}C=n$. In the context of system (H) the set
${\mathcal{F}}(\Phi)$ corresponds to the set of all conjoined bases of (H).
With the notation in (5.15) we can formulate the following.

######  Theorem 5.6 (Separation theorem).

Let $\Phi(t)$ be a continuous symplectic matrix on $[a,b]$. For any continuous
Lagrangian paths $Y$ and $\hat{Y}$ belonging to the set ${\mathcal{F}}(\Phi)$
defined in (5.15) we have

$\displaystyle{\mathcal{N}}(Y,[a,b])-{\mathcal{N}}(\hat{Y},[a,b])$
$\displaystyle=\mu(Y(b),\hat{Y}(b))-\mu(Y(a),\hat{Y}(a)),$ (5.16)
$\displaystyle{\mathcal{N}}^{*}(Y,[a,b])-{\mathcal{N}}^{*}(\hat{Y},[a,b])$
$\displaystyle=\mu^{*}(Y(a),\hat{Y}(a))-\mu^{*}(Y(b),\hat{Y}(b)).$ (5.17)

###### Proof.

Formulas (5.16) and (5.17) follow from the comparison theorem (Theorem 5.1)
and from Corollary 3.10. Indeed, considering the continuous Lagrangian path
$\tilde{Y}:=\hat{Z}^{-1}Y$ on $[a,b]$, then its upper block ${\tilde{X}}$ has
the form ${\tilde{X}}(t)=-W(\hat{Y}(t),Y(t))$, which is a constant matrix on
$[a,b]$ in the setting of this theorem. Hence, we have
${\mathcal{N}}(\tilde{Y},[a,b])=0$ and ${\mathcal{N}}^{*}(\tilde{Y},[a,b])=0$
by Corollary 3.10 and then equations (5.1) and (5.2) yield the results in
(5.16) and (5.17). ∎

The results in Theorem 5.6 show that for a given continuous Lagrangian path
$Y\in{\mathcal{F}}(\Phi)$ it is possible to calculate the value
${\mathcal{N}}(Y,[a,b])$ from the oscillation number of a suitable reference
Lagrangian path from the set ${\mathcal{F}}(\Phi)$ in (5.15).

######  Remark 5.7.

Consider the continuous Lagrangian paths $Y_{a},Y_{b}\in{\mathcal{F}}(\Phi)$,
which are associated in (5.15) with the matrices $C_{a}:=\Phi^{-1}(a)\hskip
1.0ptE$ and $C_{b}:=\Phi^{-1}(b)\hskip 1.0ptE$, i.e.,

$Y_{a}(t)=\Phi(t)\,C_{a},\quad Y_{b}(t)=\Phi(t)\,C_{b},\quad t\in[a,b],\quad
Y_{a}(a)=E=Y_{b}(b).$ (5.18)

Since for any Lagrangian path $Y\in{\mathcal{F}}(\Phi)$ we have
$\mu(Y(a),E)=0$ and $\mu^{*}(Y(b),E)=0$ by (2.1), it follows from (5.16) with
$\hat{Y}:=Y_{a}$ and from (5.17) with $\hat{Y}:=Y_{b}$ that

$\displaystyle{\mathcal{N}}(Y,[a,b])$
$\displaystyle={\mathcal{N}}(Y_{a},[a,b])+\mu(Y(b),Y_{a}(b)),$ (5.19)
$\displaystyle{\mathcal{N}}^{*}(Y,[a,b])$
$\displaystyle={\mathcal{N}}^{*}(Y_{b},[a,b])+\mu^{*}(Y(a),Y_{b}(a)).$ (5.20)

In addition, in the same spirit as in [21, Theorem 4.5] we obtain from (5.19)
and (5.20) for any Lagrangian path $Y\in{\mathcal{F}}(\Phi)$ the estimates

$\displaystyle{\mathcal{N}}(Y_{a},[a,b])$
$\displaystyle\leq{\mathcal{N}}(Y,[a,b])\leq{\mathcal{N}}(Y_{b},[a,b]),$
(5.21) $\displaystyle{\mathcal{N}}^{*}(Y_{b},[a,b])$
$\displaystyle\leq{\mathcal{N}}^{*}(Y,[a,b])\leq{\mathcal{N}}^{*}(Y_{a},[a,b]).$
(5.22)

Moreover, with the special choices of $Y:=Y_{b}$ in (5.19) and $Y:=Y_{a}$ in
(5.20) we deduce that

$\displaystyle{\mathcal{N}}(Y_{a},[a,b])={\mathcal{N}}^{*}(Y_{b},[a,b]),\quad{\mathcal{N}}(Y_{b},[a,b])={\mathcal{N}}^{*}(Y_{a},[a,b]),$
(5.23)
$\displaystyle{\mathcal{N}}(Y_{b},[a,b])-{\mathcal{N}}(Y_{a},[a,b])=\operatorname{\text{\rm
rank}}W(Y_{a},Y_{b})={\mathcal{N}}^{*}(Y_{a},[a,b])-{\mathcal{N}}^{*}(Y_{b},[a,b]),$
(5.24)

compare with [36, Corollary 5.4 and Theorem 5.6]. Note that in order to derive
(5.23) and (5.24) from (5.19) and (5.20) we used that $\mu(E,Y_{a}(b))$ and
$\mu^{*}(E,Y_{b}(a))$ have the same value, since the Wronskian
$W(Y_{a}(t),Y_{b}(t))\equiv W(Y_{a},Y_{b})$ is in this case constant on
$[a,b]$.

Based on the above separation theorem we are able to answer the question about
the existence of a continuous Lagrangian path $Y$ in the given set
${\mathcal{F}}(\Phi)$, whose oscillation number and dual oscillation number
attain prescribed values satisfying estimates (5.21) and (5.22). This result
generalizes [40, Theorem 1.1] to arbitrary continuous Lagrangian paths or even
to conjoined bases of system (H) without assuming the Legendre condition
(1.3).

######  Theorem 5.8.

Let $\Phi(t)$ be a continuous symplectic matrix on $[a,b]$ and let
$Y_{a},Y_{b}\in{\mathcal{F}}(\Phi)$ be the unique continuous Lagrangian paths
determined by formula (5.18) in Remark 5.7. Then for any integers $\ell$ and
$r$ satisfying

${\mathcal{N}}(Y_{a},[a,b])\leq\ell\leq{\mathcal{N}}(Y_{b},[a,b])\quad\text{
and }\quad{\mathcal{N}}^{*}(Y_{b},[a,b])\leq
r\leq{\mathcal{N}}^{*}(Y_{a},[a,b])$ (5.25)

there exists a continuous Lagrangian path $Y\in{\mathcal{F}}(\Phi)$ such that

${\mathcal{N}}(Y,[a,b])=\ell\quad\text{ and
}\quad{\mathcal{N}}^{*}(Y,[a,b])=r.$ (5.26)

Moreover, if $\ell\geq r$, then the Lagrangian path $Y$ can be chosen with
$X(a)=I$, and if $\ell\leq r$, then the Lagrangian path $Y$ can be chosen with
$X(b)=I$. In particular, when $\ell=r$ the Lagrangian path $Y$ may be chosen
with both $X(a)$ and $X(b)$ invertible.

###### Proof.

Let $\ell$ and $r$ be given integers satisfying (5.25). Define the integers

$p:=\ell-{\mathcal{N}}(Y_{a},[a,b]),\quad
q:=r-{\mathcal{N}}^{*}(Y_{b},[a,b]).$ (5.27)

Then from (5.25) and (5.24) it follows that $\max\\{p,q\\}\leq
w:=\operatorname{\text{\rm rank}}W(Y_{a},Y_{b})$ holds, where the Wronskian is
constant on $[a,b]$ by the last part of Remark 5.7. We will apply the ideas of
[40, Theorem 2.1], which we adopt to the setting of continuous Lagrangian
paths in the set ${\mathcal{F}}(\Phi)$. We partition the Lagrangian paths
$Y_{a}=(X_{a}^{T},U_{a}^{T})^{T}$ and $Y_{b}=(X_{b}^{T},U_{b}^{T})^{T}$
according to notation (1.2). First we assume that $\ell\geq r$ holds. In view
of (5.23) we then obtain that $p\geq q$. In the spirit of the proof of [40,
Theorem 2.1] we construct the Lagrangian path $Y\in{\mathcal{F}}(\Phi)$ by

$Y(t)=\left(\begin{matrix}X(t)\\\ U(t)\end{matrix}\right):=\Phi(t)\,C,\quad
t\in[a,b],\quad C:=\Phi^{-1}(a)\left(\begin{matrix}I\\\ D+R_{b}(a)\hskip
1.0ptU_{b}(a)\hskip 1.0ptX_{b}^{\dagger}(a)\end{matrix}\right),$ (5.28)

where $R_{b}(a):=X_{b}(a)\hskip 1.0ptX_{b}^{\dagger}(a)$ is the orthogonal
projector onto $\operatorname{\text{\rm Im}}X_{b}(a)$ and where the symmetric
$n\times n$ matrix $D$ has $q$ negative eigenvalues $\lambda_{j}=-1$ and $w-p$
positive eigenvalues $\lambda_{j}=1$. More precisely, see [40, Eqs.
(2.29)–(2.31)], we take

$D:=L\operatorname{\text{\rm diag}}\\{-I_{q},I_{w-p},0_{n-w+p-q}\\}\hskip
1.0ptL^{T},$ (5.29)

where $L$ is an orthogonal matrix satisfying

$R_{b}(a)=L\operatorname{\text{\rm diag}}\\{I_{w},0_{n-w}\\}\hskip
1.0ptL^{T},\quad R_{b}(a)\hskip 1.0ptDR_{b}(a)=D.$ (5.30)

Equations (5.29) and (5.30) imply, compare with [40, Lemma 2.2], that

$\displaystyle\mu(Y(b),Y_{a}(b))$ $\displaystyle=w-\operatorname{\text{\rm
ind}}\hskip 1.0pt[\hskip 1.0pt-R_{b}(a)\hskip
1.0ptDR_{b}(a)]\overset{\eqref{E:main.number.pq.hlp2}}{=}w-\operatorname{\text{\rm
ind}}\hskip 1.0pt(-D)\overset{\eqref{E:main.number.pq.hlp4}}{=}w-(w-p)=p,$
(5.31) $\displaystyle\mu^{*}(Y(a),Y_{b}(a))$
$\displaystyle=\operatorname{\text{\rm ind}}\hskip 1.0pt[\hskip
1.0ptR_{b}(a)\hskip
1.0ptDR_{b}(a)]\overset{\eqref{E:main.number.pq.hlp2}}{=}\operatorname{\text{\rm
ind}}D\overset{\eqref{E:main.number.pq.hlp4}}{=}q.$ (5.32)

Then by (5.19) and (5.20) we obtain that

$\displaystyle{\mathcal{N}}(Y,[a,b])$
$\displaystyle\overset{\eqref{E:separ.osc.number.Ya}}{=}{\mathcal{N}}(Y_{a},[a,b])+\mu(Y(b),Y_{a}(b))\overset{\eqref{E:distrib.osc.number.hlp3}}{=}{\mathcal{N}}(Y_{a},[a,b])+p\overset{\eqref{E:main.number.lr.hlp1}}{=}\ell,$
$\displaystyle{\mathcal{N}}^{*}(Y,[a,b])$
$\displaystyle\overset{\eqref{E:separ.osc.number*.Yb}}{=}{\mathcal{N}}^{*}(Y_{b},[a,b])+\mu^{*}(Y(a),Y_{b}(a))\overset{\eqref{E:distrib.osc.number.hlp4}}{=}{\mathcal{N}}^{*}(Y_{b},[a,b])+q\overset{\eqref{E:main.number.lr.hlp1}}{=}r.$

This completes the proof of (5.26) for the case of $\ell\geq r$. Moreover,
from (5.28) with $t=a$ we can see that $X(a)=I$ holds. Next we suppose that
$\ell\leq r$, so that $p\leq q$ in view of (5.23). We follow the second part
of the proof of [40, Theorem 2.1]. Hence, we construct the Lagrangian path
$Y\in{\mathcal{F}}(\Phi)$ by the formula

$Y(t)=\left(\begin{matrix}X(t)\\\ U(t)\end{matrix}\right):=\Phi(t)\,C,\quad
t\in[a,b],\quad C:=\Phi^{-1}(b)\left(\begin{matrix}I\\\ D+R_{a}(b)\hskip
1.0ptU_{a}(b)\hskip 1.0ptX_{a}^{\dagger}(b)\end{matrix}\right),$ (5.33)

where $R_{a}(b):=X_{a}(b)\hskip 1.0ptX_{a}^{\dagger}(b)$ is the orthogonal
projector onto $\operatorname{\text{\rm Im}}X_{a}(b)$ and where as in [40, Eq.
(2.33)] the symmetric $n\times n$ matrix $D$ has the form

$D:=L\operatorname{\text{\rm diag}}\\{-I_{w-q},I_{p},0_{n-w+q-p}\\}\hskip
1.0ptL^{T}$ (5.34)

with an orthogonal matrix $L$ satisfying

$R_{a}(b)=L\operatorname{\text{\rm diag}}\\{I_{w},0_{n-w}\\}\hskip
1.0ptL^{T},\quad R_{a}(b)\hskip 1.0ptDR_{a}(b)=D.$ (5.35)

Equations (5.34) and (5.35) imply, compare with [40, Lemma 2.3], that

$\displaystyle\mu(Y(b),Y_{a}(b))$ $\displaystyle=\operatorname{\text{\rm
ind}}\hskip 1.0pt[\hskip 1.0pt-R_{a}(b)\hskip
1.0ptDR_{a}(b)]\overset{\eqref{E:main.number.pq.hlp7}}{=}\operatorname{\text{\rm
ind}}\hskip 1.0pt(-D)\overset{\eqref{E:main.number.pq.hlp6}}{=}p,$ (5.36)
$\displaystyle\mu^{*}(Y(a),Y_{b}(a))$ $\displaystyle=w-\operatorname{\text{\rm
ind}}\hskip 1.0pt[\hskip 1.0ptR_{a}(b)\hskip
1.0ptDR_{a}(b)]\overset{\eqref{E:main.number.pq.hlp7}}{=}w-\operatorname{\text{\rm
ind}}D\overset{\eqref{E:main.number.pq.hlp6}}{=}w-(w-q)=q.$ (5.37)

Then by (5.19) and (5.20) we obtain that

$\displaystyle{\mathcal{N}}(Y,[a,b])$
$\displaystyle\overset{\eqref{E:separ.osc.number.Ya}}{=}{\mathcal{N}}(Y_{a},[a,b])+\mu(Y(b),Y_{a}(b))\overset{\eqref{E:distrib.osc.number.hlp8}}{=}{\mathcal{N}}(Y_{a},[a,b])+p\overset{\eqref{E:main.number.lr.hlp1}}{=}\ell,$
$\displaystyle{\mathcal{N}}^{*}(Y,[a,b])$
$\displaystyle\overset{\eqref{E:separ.osc.number*.Yb}}{=}{\mathcal{N}}^{*}(Y_{b},[a,b])+\mu^{*}(Y(a),Y_{b}(a))\overset{\eqref{E:distrib.osc.number.hlp9}}{=}{\mathcal{N}}^{*}(Y_{b},[a,b])+q\overset{\eqref{E:main.number.lr.hlp1}}{=}r.$

This completes the proof of (5.26) for the case of $\ell\leq r$. Moreover,
from (5.33) with $t=b$ we can see that $X(b)=I$ holds. Finally, if $\ell=r$,
then from (5.26) we have ${\mathcal{N}}(Y,[a,b])={\mathcal{N}}^{*}(Y,[a,b])$.
By formula (3.36) in Theorem 3.13 it follows that $\operatorname{\text{\rm
rank}}X(a)=\operatorname{\text{\rm rank}}X(b)$. This means that if $X(a)=I$,
then $X(b)$ is nonsingular, while if $X(b)=I$, then $X(a)$ is nonsingular. In
conclusion, if $\ell=r$, then the Lagrangian path $Y$ can be chosen with both
$X(a)$ and $X(b)$ invertible. The proof is complete. ∎

######  Remark 5.9.

The proof of Theorem 5.8 shows that the Lagrangian path
$Y\in{\mathcal{F}}(\Phi)$ satisfying condition (5.26) is constructed on
$[a,b]$ as a constant multiple of the matrix $\Phi(t)$ by prescribing its
initial condition at $a$ in (5.28) if $\ell\geq r$, or at $b$ in (5.33) if
$\ell\leq r$.

##  6\. Conclusions

In this section we make comments about the main results of this paper and
their relationship with some related mathematical problems. The main purpose
of this paper was to study the oscillation number ${\mathcal{N}}(Y,[a,b])$ and
the dual oscillation number ${\mathcal{N}}^{*}(Y,[a,b])$ for a continuous
Lagrangian path $Y$ on $[a,b]$. These are integer quantities defined in an
algebraic way through the comparative index and the dual comparative index by
using a certain partition of the interval $[a,b]$. Here we use a more general
definition than in [18, 21] in the sense that we use nonconstant symplectic
matrices in the partition. As the main results (Theorem 3.8 and the subsequent
results in Section 5) we express the quantities ${\mathcal{N}}(Y,[a,b])$ and
${\mathcal{N}}^{*}(Y,[a,b])$ in terms of the total changes in the interval
$[a,b]$ of the integers $q_{j}(t)$ and $q_{j}^{*}(t)$, which are associated
through (3.12) and (3.13) with the continuous Lidskii angles $\varphi_{j}(t)$
of the symplectic and orthogonal matrix $Z_{Y}(t)$ on $[a,b]$ defined in
(1.4). The methods, which were used for the above analysis, are based on the
Lidskii angles of symplectic matrices and their relationship with the
comparative index obtained recently in [42]. This approach allowed us to
connect the oscillation numbers with the Maslov index (Theorems 4.2 and 4.6,
Corollary 4.4, and Remarks 4.5 and 5.5). In addition, we derive a general
comparison theorem (Theorem 5.1) for the oscillation numbers and the dual
oscillation numbers of two arbitrary continuous Lagrangian paths $Y$ and
$\hat{Y}$ on $[a,b]$, as well as general separation theorems for the case when
$Y$ and $\hat{Y}$ are constant multiples of a given continuous symplectic
matrix (Theorems 5.6 and 5.8).

The theory presented in this paper is based on detailed matrix analysis with
motivations coming from the theory of differential equations, resp. from the
oscillation theory of linear Hamiltonian systems (H). It can be further
developed in the direction of the oscillation theory on discrete time domains
[10, 37] or in the direction of singular comparison theorems for the
oscillation numbers, as we recently presented in [38, 39] for conjoined bases
of nonoscillatory linear Hamiltonian systems. Such results may have
fundamental applications in the theory of Maslov index on unbounded intervals,
or in the study of the rotation number of a family of linear Hamiltonian
systems, such as in [28, 29]. We also expect related research activity in the
theory of matrices in general, for example in the limit theorems for symmetric
matrix valued functions generalizing the results in [30, 34] or [31, Theorem
3.3.7].

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