1 Introduction
The relative positions of eigenvalues of two self-adjoint operators that differ by a finite-rank perturbation in their boundary conditions is a question arising in many applications. These include the TE/TM mode analysis in graph-like photonic structures [Reference Kuchment, Bao, Cowsar and Masters40], stability analysis of brake orbits in Lagrangian systems [Reference Hu, Portaluri, Wu and Xing37, Reference Hu, Wu and Yang38], analysis of nodal and Robin domains [Reference Alon, Band and Berkolaiko3, Reference Band, Prokhorova and Sofer7, Reference Berkolaiko13], and spectral minimal partitions [Reference Band, Berkolaiko, Raz and Smilansky6, Reference Hofmann and Kennedy30] on quantum graphs. From an abstract perspective, tracking eigenvalue changes due to variations in boundary conditions is equivalent to comparing self-adjoint extensions of a symmetric operator with finite and equal defect numbers. Such spectral comparison problems for differential operators may be approached via topological indices [Reference Arnold5, Reference Bott18, Reference Conley and Zehnder22, Reference Duistermaat24]. In particular, the Maslov index [Reference Arnold4, Reference Cappell, Lee and Miller19, Reference de Gosson23, Reference Keller39, Reference Leray42, Reference Piccione and Tausk47, Reference Robbin and Salamon48] has played a major role in these investigations and is an active area of research (see [Reference Beck9] for a review and [Reference Agrachev, Baranzini and Beschastnyi1, Reference Band, Prokhorova and Sofer7, Reference Barutello, Offin, Portaluri and Wu8, Reference Beck, Cox, Jones, Latushkin, McQuighan and Sukhtayev10, Reference Beck and Jaquette11, Reference Berkolaiko, Cox, Latushkin and Sukhtaiev15, Reference Chardard, Dias and Bridges20, Reference Chardard, Dias and Bridges21, Reference Garrisi, Portaluri and Wu27, Reference Howard32–Reference Howard and Sukhtayev36, Reference Latushkin and Sukhtaiev41] for a partial list of papers specifically focused on the finite defect setting). The main theme in these papers is to equate the Maslov index to the eigenvalue counting function of some linear differential operator, which is usually obtained by linearizing a dynamical system at an equilibrium solution. However, the Maslov index is difficult to compute in practice. It depends on a path of Lagrangian subspaces, and so without complete knowledge of this path (which amounts to having an explicit solution for the linearized system), it cannot be computed effectively.
The fundamental observation of [Reference Berkolaiko, Cox, Latushkin and Sukhtaiev15, Reference Garrisi, Portaluri and Wu27, Reference Hu, Wu and Yang38] is that the difference of two eigenvalue counting functions can be evaluated in terms of a much simpler index that only depends on the endpoints of the path. The so-called Duistermaat index, introduced in [Reference Duistermaat24], takes three Lagrangian subspaces as its input. In the metric graph setting, these correspond to the two sets of vertex conditions under consideration and also the Dirichlet conditions at the affected vertices. Duistermaat’s definition is somewhat indirect, however, as it involves an arbitrary choice of a fourth Lagrangian plane that is transversal to the three planes of interest. The proof that the resulting index is independent of this choice relies on the Maslov index for paths of Lagrangian subspaces (see [Reference Duistermaat24, Equation (2.16)]). As noted above, the Maslov index is inherently more complicated than an index depending on fixed Lagrangian subspaces.
The principal aim of this note is to define the Duistermaat index directly, without recourse to the Maslov index. We do this by giving a set of axioms (inspired by the approach of [Reference Cappell, Lee and Miller19]) as well as a formula requiring nothing more than linear algebra. It is the set of axioms that turns out to be most useful in practice, allowing us to give short and simple proofs for many known and some unknown properties of the Duistermaat index.
We consider
$\mathbb {C}^n \oplus \mathbb {C}^n$
as a Hermitian symplectic space with the canonical symplectic form
A subspace
$\mathcal {L} \subset \mathbb {C}^n \oplus \mathbb {C}^n$
is Lagrangian if it coincides with its symplectic complement:
The set of Lagrangian subspaces (often called Lagrangian planes) in
$(\mathbb {C}^n \oplus \mathbb {C}^n , \omega )$
is denoted by
$\Lambda (n)$
. Every Lagrangian subspace has dimension n, half of the dimension of the symplectic space, therefore
$\mathcal {L}_1$
and
$\mathcal {L}_2$
are transversal whenever
$\mathcal {L}_1 \cap \mathcal {L}_2 = \{0\}$
.
An important example of a Lagrangian plane is the graph of a Hermitian matrix M:
In particular, the “horizontal” plane
will play an important role in this note. The “vertical” plane
$0\oplus \mathbb {C}^n$
is not the graph of a Hermitian matrix, even though we will abuse the notation in (1.3) to denote it
Being “somewhat vertical” is the only obstacle for a Lagrangian plane to be a graph: a Lagrangian plane
$\mathcal {L}$
is the graph of a Hermitian matrix if and only if
$\mathcal {L}$
is transversal to
${\mathcal {G}_\infty }$
.
The Duistermaat index is an integer symplectic invariant of a triple of Lagrangian planes, describing their relative position. Intuitively, it can be thought of as the maximal dimension of a subspace of
$\mathcal {L}_3$
that lies between
$\mathcal {L}_1$
and
$\mathcal {L}_2$
. The formal definition we propose differs from that of [Reference Duistermaat24] by taking an axiomatic approach.
Theorem 1.1 There is a unique function
$ {\mathrm {\iota }} : \Lambda (n)^3 \to \{0,1,\ldots ,n\}$
, called the Duistermaat index, that satisfies the following properties:
-
(1) (Normalization) For any Hermitian matrix A,
(1.6) $$ \begin{align} {\mathrm{\iota}}({\mathcal{G}_0}, \mathcal{G}_A, {\mathcal{G}_\infty}) = n_-(A), \end{align} $$where $n_-(A)$
denotes the number of negative eigenvalues (also known as the Morse index) of the matrix A.
-
(2) (Symplectic invariance) For any symplectic transformation $\mathsf {S}$
on
$\mathbb {C}^n \oplus \mathbb {C}^n$
, (1.7) $$ \begin{align} {\mathrm{\iota}}(\mathsf{S}\mathcal{L}_1, \mathsf{S}\mathcal{L}_2, \mathsf{S}\mathcal{L}_3) = {\mathrm{\iota}}(\mathcal{L}_1, \mathcal{L}_2, \mathcal{L}_3). \end{align} $$
-
(3) (Cocycle property) For any Lagrangian planes $\mathcal {L}_1$
,
$\mathcal {L}_2$
,
$\mathcal {L}_3$
, and
$\mathcal {L}_4$
, (1.8) $$ \begin{align} {\mathrm{\iota}}(\mathcal{L}_1,\mathcal{L}_2,\mathcal{L}_3) - {\mathrm{\iota}}(\mathcal{L}_1,\mathcal{L}_2,\mathcal{L}_4) + {\mathrm{\iota}}(\mathcal{L}_1,\mathcal{L}_3,\mathcal{L}_4) - {\mathrm{\iota}}(\mathcal{L}_2,\mathcal{L}_3,\mathcal{L}_4) = 0. \end{align} $$
Remark 1.2. The Duistermaat index as defined in [Reference Duistermaat24, Reference Zhou, Wu and Zhu51] satisfies the axioms in Theorem 1.1 and therefore coincides with our version.
Remark 1.3. To motivate the terminology in (3), we recall the coboundary operator
$\delta $
(see [Reference Hatcher28, Chapter 3]) which maps functions of
$k-1$
arguments to functions of k arguments by
where a hat indicates that the argument has been omitted. Then (1.8) says that the coboundary of
$ {\mathrm {\iota }}$
is zero, that is,
$ {\mathrm {\iota }}$
is a cocycle.
Theorem 1.1 will be established in Section 2 by supplying an explicit and self-contained, if somewhat cumbersome, formula for the index. A more practical formula is the following extension of (1.6).
Theorem 1.4 For any
$n \times n$
Hermitian matrices A, B, and C,
where
$\widetilde {n}_-(X,Y) := n_-(Y-X)$
.
One application of (1.10) is the following curious identity: for any invertible
$n\times n$
Hermitian matrices A and B,
Non-invertible A or B are discussed in Section 5. We also point out that (1.10) can be used for arbitrary
$\mathcal {L}_1$
,
$\mathcal {L}_2$
, and
$\mathcal {L}_3$
using symplectic invariance (1.7) and a symplectic transformation
$\mathsf {S} $
which makes all
$\mathsf {S}\mathcal {L}_j$
transversal to
$\mathcal {G}_\infty $
, for example,
with generic
$\epsilon $
, see the discussion following (2.4).
Example 1.5. In the symplectic space
$\mathbb {C} \oplus \mathbb {C}$
, consider three Lagrangian planes,
$\mathcal {G}_A := \{ (z, Az) \colon z\in \mathbb {C}\}$
and the similarly defined
$\mathcal {G}_B$
and
$\mathcal {G}_C$
, with the “slopes”
${A,B,C \in \mathbb {R}}$
. Theorem 1.4 immediately yields
These results (with distinct A,
$B,$
and C) are illustrated in Figure 1. The index
$ {\mathrm {\iota }}$
can be evaluated by rotating the line
$\mathcal {G}_A$
counterclockwise to
$\mathcal {G}_B$
and counting the number of times (0 or 1 in this low-dimensional case) that it intersects
$\mathcal {G}_C$
. The borderline cases are treated according to (1.13):
$\mathcal {G}_A = \mathcal {G}_C$
counts as an intersection but
$\mathcal {G}_B = \mathcal {G}_C$
does not; if
$\mathcal {G}_A=\mathcal {G}_B$
, the index is 0 for any
$\mathcal {G}_C$
. This “dynamic” perspective readily generalizes to the case when one or more of the Lagrangian planes is vertical, as well as to Lagrangian planes in
$\mathbb {C}^n\oplus \mathbb {C}^n$
where the intersections are counted with their dimension.
Possible transversal configurations of three Lagrangian planes (illustrated in
$\mathbb {R}^2$
) and the corresponding values of the Duistermaat index (see Example 1.5). The directed dashed arcs indicate the non-decreasing paths from A to B, realizing the minimum in (1.14).
Figure 1 Long description
The diagram consists of six coordinate systems arranged in a two-by-three grid. Each panel features three lines passing through the origin: a solid blue line labeled A, a solid black line labeled B, and a dotted red line labeled C. A blue dashed arc with an arrow indicates a counter-clockwise path from line A to line B.
Top row from left to right:
* Panel 1: Line C is the steepest, followed by B, then A. The index iota equals 0.
* Panel 2: Line B is the steepest, followed by C, then A. The index iota equals 1.
* Panel 3: Line B is the steepest, followed by A, then C. The index iota equals 0.
Bottom row from left to right:
* Panel 4: Line C is the steepest, followed by A, then B. The index iota equals 1.
* Panel 5: Line A is the steepest, followed by C, then B. The index iota equals 0.
* Panel 6: Line A is the steepest, followed by B, then C. The index iota equals 1.
In all panels, the dashed arc representing the path from A to B crosses the vertical axis if A is steeper than B, or remains in the first quadrant if B is steeper than A.
The intuition gained in Example 1.5 allows us to interpret the Duistermaat index as the minimal value of the Maslov index over non-decreasing paths. This interpretation highlights the usefulness of the Duistermaat index, since computation of the Maslov index is far more difficult than applying an algebraic identity such as (1.10).
Theorem 1.6 For arbitrary Lagrangian planes
$\mathcal {L}_0$
,
$\mathcal {L}_1$
, and
$\mathcal {M,}$
one has
with the minimum attained by some (non-unique) path
$\mathcal {L}(t)$
. Furthermore, any path
$\mathcal {L}(t)$
that attains the minimum for one
$\mathcal {M}$
will attain the minimum for all
$\mathcal {M}$
.
Here,
$ {\mathrm {Mas}}_{[0,1]}\!\big (\mathcal {L}(t), \mathcal {M}\big )$
is the Maslov index as defined in (4.9); the term non-decreasing is defined in Section 4.2.
The plan for the rest of the note is as follows: in Section 2, we show that an index satisfying the conditions in Theorem 1.1 exists and is unique; in Section 3, we establish some further properties of this index. In Section 4, we prove Theorem 1.6, providing along the way a short proof (see Theorem 4.6) of the Zhou–Wu–Zhu identity [Reference Zhou, Wu and Zhu51, Theorem 1.1], which expresses the so-called Hörmander index as a difference of Duistermaat indices. Finally, in Section 5, we prove (1.11) and its extensions. We also give an elementary derivation of (1.11) using the Haynsworth inertia formula.
Notation
: We use
$x,y$
to denote vectors in
$\mathbb {C}^n$
and
$u,v$
to denote vectors in
$\mathbb {C}^n\oplus \mathbb {C}^n$
. To save space, we write
$u=(x, y)$
for the vector
$u= \left (\begin {smallmatrix} x \\ y \end {smallmatrix} \right ) \in \mathbb {C}^n\oplus \mathbb {C}^n$
with components
$x,y\in \mathbb {C}^n$
. Similarly, we abbreviate
$(X,Y)=\left (\begin {smallmatrix}X\\Y\end {smallmatrix}\right )$
for the
$(2n\times n)$
-matrix with
$(n\times n)$
-matrix components
$X,Y$
. We write
$\ker M=\{x\in \mathbb {C}^n: Mx=0\}$
for the kernel of an
$(n\times n)$
-matrix M and
$\mathcal {X} \simeq \mathcal {Y}$
when subspaces
$\mathcal {X}$
and
$\mathcal {Y}$
are isomorphic.
2 Existence and uniqueness of the Duistermaat index
In Theorem 2.1, we define a function
$ {\mathrm {\iota }}$
that satisfies the conditions of Theorem 1.1. Uniqueness then follows from the observation that these conditions provide a prescription for the computation of
$ {\mathrm {\iota }}$
(see Section 2.2 for details).
2.1 Existence of the index
Theorem 2.1 Given a triple
$(\mathcal {L}_1, \mathcal {L}_2, \mathcal {L}_3)$
of Lagrangian planes, define a Hermitian form
$\Omega $
on
$\mathcal {L}_1\oplus \mathcal {L}_2\oplus \mathcal {L}_3$
by
Then the index
$ {\mathrm {\iota }}(\mathcal {L}_1, \mathcal {L}_2, \mathcal {L}_3)$
defined by
satisfies the conditions of Theorem 1.1.
Remark 2.2. This is more direct than the definition of
$ {\mathrm {\iota }}$
given in [Reference Duistermaat24, Reference Zhou, Wu and Zhu51], which requires choosing an arbitrary fourth Lagrangian plane. One benefit of equation (2.2) is that
$ {\mathrm {\iota }}$
is evidently a symplectic invariant. This definition is not optimal for practical computations, however, since
$\Omega $
is defined on a
$3n$
-dimensional space. One also notices a lack of symmetry caused by the
$\dim \mathcal {L}_1\cap \mathcal {L}_3$
term. Both of these concerns are addressed by a more practical (and more symmetric) formula (2.7) for the index given in Proposition 2.3 below.
Before proving the theorem, we recall that any n-dimensional subspace
${\mathcal {M} \subset \mathbb {C}^n \oplus \mathbb {C}^n}$
can be described by a frame, which is an injective linear operator
whose range is
$\mathcal {M}=\{(Xx,Yx): x\in \mathbb {C}^n\}$
. Moreover,
$\mathcal {M}$
is Lagrangian if and only if
$X^*Y = Y^*X$
, as easily seen from (1.1) and (1.2).
This description is not unique, but it is easy to see that frames Z and
$\tilde Z$
describe the same subspace if and only if
$\tilde Z = ZC$
for some invertible
$C \colon \mathbb {C}^n \to \mathbb {C}^n$
. Therefore, the set
and the so-called
$\epsilon $
-Robin map
$R^\epsilon :\mathbb {C}^n \to \mathbb {C}^n$
,
are independent of the choice of frame. When
$\mathcal {M}$
is Lagrangian, the set
$E(\mathcal {M})$
is finite, since
$\det (X + \epsilon Y)$
is a polynomial in
$\epsilon $
that is not identically zero.Footnote
1
The Lagrangian condition
$X^*Y = Y^*X$
also implies that
is Hermitian, therefore
$R^\epsilon $
is Hermitian for all
$\epsilon \in \mathbb {R} \backslash E(\mathcal {M})$
.
Proposition 2.3 Let
$\mathcal {L}_1$
,
$\mathcal {L}_2$
, and
$\mathcal {L}_3$
be Lagrangian planes and
$R^\epsilon _1$
,
$R^\epsilon _2$
, and
$R^\epsilon _3$
the corresponding
$\epsilon $
-Robin maps, for any real
$\epsilon $
outside the finite set
$E_{123} := E(\mathcal {L}_1)\cup E(\mathcal {L}_2)\cup E(\mathcal {L}_3)$
. Then, for the index
$ {\mathrm {\iota }}$
defined in (2.2) and any
$\epsilon \in \mathbb {R}\backslash E_{123}$
, we have
where
$\widetilde {n}_-(X,Y) := n_-(Y-X)$
. In particular, the right-hand side of (2.7), when defined, is independent of
$\epsilon $
.
Proof We follow the proof of a similar identity in [Reference Meinrenken46, Lemma 8.3] (attributed therein to Brian Feldstein). We first compute the form
$\Omega $
defined in (2.1) as a Hermitian form on
$\mathbb {C}^n\oplus \mathbb {C}^n\oplus \mathbb {C}^n$
, where each
$\mathbb {C}^n$
parameterizes the corresponding Lagrangian plane
$\mathcal {L}_j$
via a frame
$(X_j, Y_j)$
. If, for instance,
then
Similar calculations for the other terms in (2.1) result in
$\Omega $
being represented by the formula
where
$u_j=(X_jx_j,Y_jx_j), v_j=(X_jy_j,Y_jy_j)$
,
$j\in \{1,2,3\}$
, with the Hermitian matrix
We now note that for
$\epsilon \in \mathbb {R} \backslash E_{123}$
and
$j,k\in \{1,2,3\}$
,
where we used the fact that
$R^\epsilon $
is Hermitian. We now verify by direct computation that
where the matrices
are invertible for
$\epsilon \in \mathbb {R} \backslash E_{123}$
.
We next observe that, by (2.12) and invertibility of
$X_k+\epsilon Y_k$
,
We then have
where we used (2.10) to get to the second line and the definition of a Lagrangian plane in (1.2) to get to the third. We can now conclude that
where
$n_0(A) = \dim \ker A$
.
Finally, applying Sylvester’s law of inertia to (2.13), we use the identity
along with (2.17), to obtain
which is equivalent to (2.7).
Remark 2.4. For future use, we observe that (2.13) together with (2.17) gives
Proof of Theorem 2.1
To check the normalization condition (1.6), we apply (2.7) with
$\mathcal {L}_1 = {\mathcal {G}_0}$
,
$\mathcal {L}_2 = \mathcal {G}_A$
, and
$\mathcal {L}_3 = {\mathcal {G}_\infty }$
. For small positive
$\epsilon ,$
we have
$R_1^\epsilon = 0$
,
$R_2^\epsilon =A(I+\epsilon A)^{-1} = A + o(1),$
and
$R_3^\epsilon = \epsilon ^{-1} I$
, therefore
$R_3^\epsilon -R_2^\epsilon =\epsilon ^{-1}(I-\epsilon A+o(\epsilon ))$
and so
The normalization condition then follows from (2.7).
Symplectic invariance of
$ {\mathrm {\iota }}$
follows from the symplectic invariance of every term in (2.1), and the cocycle property
$\delta {\mathrm {\iota }}=0$
follows from (2.8) by the well-known identity
$\delta ^2 \equiv 0$
for the coboundary operator, which is easily verified from (1.9).
Finally, to estimate the range of the possible values of the index, we use a symplectic transformation
$\mathsf {S}$
such that
$\mathsf {S}\mathcal {L}_3 = \mathcal {G}_\infty $
to reduce the question to bounding the range of
$ {\mathrm {\iota }}(\mathcal {L}_1,\mathcal {L}_2,\mathcal {G}_\infty )$
. That the latter is between
$0$
and n follows from a version of equation (2.7):
Equation (2.22) was already established in [Reference Berkolaiko, Cox, Latushkin and Sukhtaiev15, Equation (3.32)], but we repeat the proof here for completeness. We use equation (2.7) with
$R^\epsilon _3 = \epsilon ^{-1} I$
, and the desired result follows once we establish that
$n_-\left ({R_{3}^\epsilon-R_{1}^\epsilon}\right )=0=n_-\left ({R_{3}^\epsilon-R_{2}^\epsilon}\right )$
. In other words, we claim that for any Lagrangian frame
$(X, Y)^T$
, the Hermitian operator
$\epsilon ^{-1} I - Y(X+\epsilon Y)^{-1}$
is nonnegative definite for small enough
$\epsilon>0$
. It is equivalent to consider
The right-hand side is identically zero on
$\ker (X^*X) = \ker X$
, while on
$\ker (X^*X)^\perp $
, it is a Hermitian perturbation of a positive-definite matrix.
2.2 Proofs of Theorems 1.1 and 1.4
Proof of Theorem 1.1
Theorem 2.1 proves the existence of a function
$ {\mathrm {\iota }}$
satisfying the axioms in Theorem 1.1. Uniqueness follows from the observation that these axioms provide a prescription for the computation of
$ {\mathrm {\iota }}$
. First, we reduce the computation to the case when
$\mathcal {L}_3$
is transversal to both
$\mathcal {L}_1$
and
$\mathcal {L}_2$
by using the cocycle identity with some
$\mathcal {L}_4$
that is transversal to the other three. We can then find a symplectic transformation that maps
$\mathcal {L}_3$
to
${\mathcal {G}_\infty }$
and
$\mathcal {L}_1$
to
${\mathcal {G}_0}$
(which is well-known to be possible, see, e.g., [Reference Meinrenken46, Exercise 8.1]). Since this transformation preserves transversality, the transformed
$\mathcal {L}_2$
can be represented as the graph of a Hermitian matrix, and (1.6) can be used.
3 Further properties of the Duistermaat index
We now derive useful properties of the Duistermaat index, some of which involve algebraic operations that can be defined when Lagrangian subspaces are viewed as self-adjoint linear relations.
3.1 General identities
Theorem 3.1 The Duistermaat index has the following properties:
-
(1) (Special values) For any Lagrangian $\mathcal {L}$
and
$\mathcal {M}$
, (3.1) $$ \begin{align} {\mathrm{\iota}}(\mathcal{L}, \mathcal{L}, \mathcal{M}) = {\mathrm{\iota}}(\mathcal{M}, \mathcal{L}, \mathcal{L}) = 0, \qquad {\mathrm{\iota}}(\mathcal{L}, \mathcal{M}, \mathcal{L}) = n - \dim\mathcal{L}\cap\mathcal{M}. \end{align} $$
-
(2) (Permuting the arguments)
(3.2) $$ \begin{align} {\mathrm{\iota}}(\mathcal{L}_1,\mathcal{L}_2,\mathcal{L}_3) + {\mathrm{\iota}}(\mathcal{L}_2,\mathcal{L}_1,\mathcal{L}_3) &= n - \dim \mathcal{L}_1 \cap \mathcal{L}_2, \end{align} $$
(3.3) $$ \begin{align} {\mathrm{\iota}}(\mathcal{L}_1,\mathcal{L}_2,\mathcal{L}_3) + {\mathrm{\iota}}(\mathcal{L}_1,\mathcal{L}_3,\mathcal{L}_2) &= n - \dim \mathcal{L}_2 \cap \mathcal{L}_3, \end{align} $$
(3.4) $$ \begin{align} {\mathrm{\iota}}(\mathcal{L}_1,\mathcal{L}_2,\mathcal{L}_3) + {\mathrm{\iota}}(\mathcal{L}_3,\mathcal{L}_2,\mathcal{L}_1) &= n - \dim \mathcal{L}_1 \cap \mathcal{L}_2 \nonumber\\ &\qquad - \dim \mathcal{L}_2 \cap \mathcal{L}_3 + \dim\mathcal{L}_1\cap\mathcal{L}_3, \end{align} $$
(3.5) $$ \begin{align} {\mathrm{\iota}}(\mathcal{L}_1,\mathcal{L}_2,\mathcal{L}_3) - \dim\mathcal{L}_1\cap\mathcal{L}_3 &= {\mathrm{\iota}}(\mathcal{L}_3,\mathcal{L}_1,\mathcal{L}_2) - \dim\mathcal{L}_2\cap\mathcal{L}_3 \end{align} $$
(3.6) $$ \begin{align} &= {\mathrm{\iota}}(\mathcal{L}_2,\mathcal{L}_3,\mathcal{L}_1) - \dim\mathcal{L}_1\cap\mathcal{L}_2. \end{align} $$
-
(3) (Symplectic additivity) For any three Lagrangians $\mathcal {L}_1$
,
$\mathcal {L}_2$
, and
$\mathcal {L}_3$
in a symplectic space V and any three Lagrangians
$\mathcal {M}_1$
,
$\mathcal {M}_2$
, and
$\mathcal {M}_3$
in a symplectic space W, (3.7) $$ \begin{align} {\mathrm{\iota}}(\mathcal{L}_1\oplus\mathcal{M}_1, \mathcal{L}_2\oplus\mathcal{M}_2, \mathcal{L}_3\oplus\mathcal{M}_3) = {\mathrm{\iota}}(\mathcal{L}_1,\mathcal{L}_2,\mathcal{L}_3) + {\mathrm{\iota}}(\mathcal{M}_1,\mathcal{M}_2,\mathcal{M}_3). \end{align} $$
-
(4) (Anti-symplectic transformation) If $\mathsf {A}$
is an anti-symplectic linear transformation on
$(\mathbb {C}^n\oplus \mathbb {C}^n, \omega )$
, that is, a bijection that satisfies
$\omega (\mathsf {A}u, \mathsf {A}v) = -\omega (u,v)$
, then (3.8) $$ \begin{align} {\mathrm{\iota}}(\mathsf{A}\mathcal{L}_1, \mathsf{A}\mathcal{L}_2, \mathsf{A}\mathcal{L}_3) = {\mathrm{\iota}}(\mathcal{L}_3,\mathcal{L}_2,\mathcal{L}_1). \end{align} $$
Proof The first two equalities in (3.1) follow directly from (2.7); for the third, we also use (2.16) and (2.18).
To obtain (3.2), we apply the cocycle property
and use
$ {\mathrm {\iota }}(\mathcal {L}_2, \mathcal {L}_1,\mathcal {L}_2) = n - \dim \mathcal {L}_1\cap \mathcal {L}_2$
and
$ {\mathrm {\iota }}(\mathcal {L}_2, \mathcal {L}_2,\mathcal {L}_3) = 0$
. Equation (3.3) follows by a similar computation of
$(\delta {\mathrm {\iota }})(\mathcal {L}_1, \mathcal {L}_2,\mathcal {L}_3, \mathcal {L}_2)$
; other permutations are obtained by composition of the established two.
Symplectic additivity (3.7) follows by noting that a possible choice of frame for
$\mathcal {L}\oplus \mathcal {M}$
is the direct sum of frames. Therefore, for all but finitely many
$\epsilon $
, one has
$R^\epsilon _{\mathcal {L}\oplus \mathcal {M}} = R^\epsilon _{\mathcal {L}} \oplus R^\epsilon _{\mathcal {M}}$
. Equation (3.7) then follows from (2.7) and the additivity of
$n_-$
under direct sums.
Finally, to obtain (3.8), we note that the form
$\Omega $
of (2.1) changes sign under an anti-symplectic transformation. Therefore, by (2.2),
We now use
$n_-(-\Omega ) = 3n - n_0(\Omega ) - n_-(\Omega )$
, as well as (2.20) and (3.4), to get the result.
3.2 Self-adjoint linear relations
For a given splitting of the symplectic space (in our case,
$\mathbb {C}^n\oplus \mathbb {C}^n$
), every Lagrangian plane can be viewed as a self-adjoint linear relation (see [Reference Booss-Bavnbek and Zhu17, Section 4.2] or [Reference Schmüdgen49, Section 14.1]). In this context, the difference of two Lagrangian planes,
$\mathcal {L}$
and
$\mathcal {M}$
, is the Lagrangian plane
This definition is chosen so that
for any
$n\times n$
Hermitian matrices A and B. Another important special case is
for any Lagrangian plane
$\mathcal {L}$
.
The inverse of a Lagrangian plane
$\mathcal {L}$
is
We define by
the domain and the multivalued part of the linear relation
$\mathcal {L}$
. For instance, if M is an
$(n\times n)$
matrix, then
$ {\mathrm {dom}}\ \mathcal {G}_M=\mathbb {C}^n$
,
$ {\mathrm {mul}}\mathcal {G}_M = \{0\},$
and
$ {\mathrm {mul}}\mathcal {G}_M^{-1}=\ker M$
; on the other hand,
$ {\mathrm {dom}}\ \mathcal {G}_\infty =\{0\}$
and
$ {\mathrm {mul}}\mathcal {G}_\infty =\mathbb {C}^n$
. In general, we have
$ {\mathrm {mul}}\mathcal {L} = \left ( {\mathrm {dom}}\ \mathcal {L}\right )^\perp $
and
$\mathcal {L}$
can be represented as
for some Hermitian matrix
$L \colon {\mathrm {dom}}\ \mathcal {L} \to {\mathrm {dom}}\ \mathcal {L}$
, as in [Reference Behrndt, Hassi and de Snoo12, Theorem 1.5.1] or [Reference Schmüdgen49, Proposition 14.2].
Corollary 3.2 (1)
The Duistermaat index enjoys the following further properties:
-
(1) For any Hermitian matrix A,
(3.18) $$ \begin{align} {\mathrm{\iota}}(\mathcal{L}_1,\mathcal{L}_2,\mathcal{L}_3) = {\mathrm{\iota}}(\mathcal{L}_1-\mathcal{G}_A,\mathcal{L}_2-\mathcal{G}_A,\mathcal{L}_3-\mathcal{G}_A). \end{align} $$
-
(2) For any Lagrangian planes $\mathcal {L}_1$
,
$\mathcal {L}_2$
, and
$\mathcal {L}_3$
, (3.19) $$ \begin{align} {\mathrm{\iota}}(\mathcal{L}_1,\mathcal{L}_2,\mathcal{L}_3) = {\mathrm{\iota}}(\mathcal{L}_3^{-1},\mathcal{L}_2^{-1},\mathcal{L}_1^{-1}). \end{align} $$
-
(3) For any Hermitian A and Lagrangian $\mathcal {L}$
, represented as in (3.17), (3.20) $$ \begin{align} &{\mathrm{\iota}}(\mathcal{G}_A, \mathcal{L}, {\mathcal{G}_\infty}) = n_-\big(L - A_{{\mathrm{dom}}\ \mathcal{L}}\big), \end{align} $$
(3.21) $$ \begin{align} &{\mathrm{\iota}}(\mathcal{L}, \mathcal{G}_A, {\mathcal{G}_\infty}) = n_-\big(A_{{\mathrm{dom}}\ \mathcal{L}} - L\big) + \dim \mathcal{L} \cap \mathcal{G}_\infty, \end{align} $$
where $A_{ {\mathrm {dom}}\ \mathcal {L}}$
denotes the compression of A to
$ {\mathrm {dom}}\ \mathcal {L}$
, namely, (3.22) $$ \begin{align} A_{{\mathrm{dom}}\ \mathcal{L}} := P A|_{{\mathrm{dom}}\ \mathcal{L}}, \end{align} $$where P is the orthogonal projector in $\mathbb {C}^n$
onto
$ {\mathrm {dom}}\ \mathcal {L}$
.
Proof Part (1) follows from symplectic invariance with
which maps
$\mathcal {L} - \mathcal {G}_A \mapsto \mathcal {L}$
.
For Part (2), we note that the mapping
$(x,y) \mapsto (y,x)$
on
$\mathbb {C}^n\oplus \mathbb {C}^n$
, used in the definition of
$\mathcal {L}^{-1}$
, is anti-symplectic.
For Part (3), we use (3.18) to get
We observe that
This shows that it is enough to establish the
$A=0$
versions of (3.20) and (3.21), namely,
We now partition the symplectic space as
It is straightforward to check that the spaces
$V'$
and
$V"$
are symplecticFootnote
2
with the forms
$\omega ' := \omega |_{V'}$
and
$\omega " := \omega |_{V"}$
, and that
$\omega = \omega '\oplus \omega "$
. With respect to this partition, we have
${\mathcal {G}_0} = \mathcal{G}_0^{\prime} \oplus \mathcal{G}_0^{\prime\prime} $
,
$\mathcal {L} = \mathcal{G}_L^{\prime} \oplus \mathcal{G}_{\infty}^{\prime\prime} $
, and
${\mathcal {G}_\infty } = \mathcal{G}_{\infty}^{\prime} \oplus \mathcal{G}_{\infty}^{\prime\prime} $
. We then use symplectic additivity (3.7), as well as the special values (3.1) and normalization (1.6), to compute
Similarly, using the modified normalization
$ {\mathrm {\iota }}(\mathcal {G}_A, {\mathcal {G}_0}, {\mathcal {G}_\infty }) = {\mathrm {\iota }}({\mathcal {G}_0}, \mathcal {G}_{-A}, {\mathcal {G}_\infty }) = n_-(-A)$
, obtained from (1.6) and (3.18), we obtain
which completes the proof.
4 Relationship with other indices
4.1 The Kashiwara index
Another commonly used index of a triple of Lagrangian planes arose independently in the work of Hörmander, Kashiwara, and Wall (see, for instance, [Reference Cappell, Lee and Miller19, Section 8], [Reference Furutani26, Section 4.2], [Reference Hörmander31, Equation (3.3.15)], [Reference Lion and Vergne43, Definition 1.5.1], and [Reference Wall50]). The Hörmander–Kashiwara–Wall index
$s(\mathcal {L}_1, \mathcal {L}_2, \mathcal {L}_3)$
is definedFootnote
3
as the signature of the form
$\Omega $
from Theorem 2.1:
The advantage of dealing with
$ {\mathrm {Sig}}(A)$
is its symmetry property
$ {\mathrm {Sig}}(-A) = - {\mathrm {Sig}}(A)$
, in contrast to (2.18). However, our motivation comes, ultimately, from operators that are bounded only from below; in this case,
$n_-$
may be well-defined while
$n_+$
will not be.
The two indices are connected through the following formulas.
Lemma 4.1 For any Lagrangian planes
$\mathcal {L}_1$
,
$\mathcal {L}_2$
, and
$\mathcal {L}_3$
,
or, equivalently,
Proof Applying (2.7) to
$ {\mathrm {\iota }}(\mathcal {L}_2, \mathcal {L}_1, \mathcal {L}_3)$
and
$ {\mathrm {\iota }}(\mathcal {L}_3, \mathcal {L}_1, \mathcal {L}_2)$
and recalling that
$n_-(A) = n_+(-A)$
, we get
which is equal to
$s(\mathcal {L}_1, \mathcal {L}_2, \mathcal {L}_3)$
by (4.1) and (2.13). Using Theorem 3.1(2), we obtain (4.3) .
It follows that
$ {\mathrm {\iota }}(\mathcal {L}_1, \mathcal {L}_2, \mathcal {L}_3)$
is the only nontrivial invariant of a triple of Lagrangian planes, in the following sense.
Proposition 4.2 For any two triples
$(\mathcal {L}_1, \mathcal {L}_2, \mathcal {L}_3)$
and
$(\mathcal {M}_1, \mathcal {M}_2, \mathcal {M}_3)$
of Lagrangian planes, there is a linear symplectic transformation
$\mathsf {S}$
satisfying
$\mathsf {S} \mathcal {L}_j = \mathcal {M}_j$
for all
$j\in \{1,2,3\}$
if and only if
Proof Proposition 4.2 is well known when (4.7) is replaced by
$s(\mathcal {L}_1, \mathcal {L}_2, \mathcal {L}_3) = s(\mathcal {M}_1, \mathcal {M}_2, \mathcal {M}_3)$
(see [Reference Agrachev and Gamkrelidze2, Proposition 4.4] or [Reference Meinrenken46, Theorem 8.4(d)]). By Lemma 4.1, the indices s and
$ {\mathrm {\iota }}$
are related by an invertible transformation involving the invariants in (4.5) and so, assuming (4.5) holds,
$s(\mathcal {L}_1, \mathcal {L}_2, \mathcal {L}_3) = s(\mathcal {M}_1, \mathcal {M}_2, \mathcal {M}_3)$
if and only if
$ {\mathrm {\iota }}(\mathcal {L}_1, \mathcal {L}_2, \mathcal {L}_3) = {\mathrm {\iota }}(\mathcal {M}_1, \mathcal {M}_2, \mathcal {M}_3)$
.
4.2 The Maslov index
Suppose
$\mathcal {L} \colon [0,1] \to \Lambda (n)$
is a differentiable path of Lagrangian planes, given by a frame
$\big (X(t), Y(t)\big )$
. For
$t_0 \in [0,1]$
and
$v \in \mathcal {L}(t_0),$
we define theFootnote
4
crossing form
$\mathfrak q$
on
$\mathcal {L}(t_0)$
by
We say that the path
$\mathcal {L}(\cdot )$
is non-decreasing if
$\mathfrak q$
is positive semidefinite at all t. We say that it is regular, with respect to a given Lagrangian plane
$\mathcal {M}$
, if it is continuously differentiable and the restricted quadratic form
$\mathfrak q|_{\mathcal {L}(t_0)\cap \mathcal {M}}$
is non-degenerate whenever
$\mathcal {L}(t_0)\cap \mathcal {M}$
is nontrivial. Such
$t_0$
are referred to as crossings or conjugate points.
For a regular path
$\mathcal {L}(\cdot ),$
the conjugate points are isolated, and its Maslov index with respect to a Lagrangian reference plane
$\mathcal {M}$
is given by
Remark 4.3. There are two ways of viewing the formula (4.9), depending on how one defines the Maslov index. One approach is to define the Maslov index for any continuous path via the spectral flow of a certain family of unitary operators. For a regular path, one then recovers (4.9) as a special case (see [Reference Booss-Bavnbek and Furutani16, Reference Furutani26] for details). Alternately, one can take (4.9) as the definition of the Maslov index for a regular path, then define the Maslov index of a continuous path by first deforming it to a regular path and then arguing that the resulting index is independent of the choice of deformation, as in [Reference Robbin and Salamon48].
Lemma 4.4 For any Lagrangian planes
$\mathcal {L}_0$
and
$\mathcal {L}_1$
, there exists a smooth, non-decreasing path
$\mathcal {L} \colon [0,1] \to \Lambda (n)$
between
$\mathcal {L}_0$
and
$\mathcal {L}_1$
such that
for any Lagrangian plane
$\mathcal {M}$
.
Proof Let
$Q \colon \mathbb {C}^n \to \mathbb {C}^n$
be an arbitrary orthogonal projector with
$\dim \ker Q = \dim \mathcal {L}_0 \cap \mathcal {L}_1$
. Since
$\mathcal {G}_0 \cap \mathcal {G}_Q$
and
$\mathcal {L}_0 \cap \mathcal {L}_1$
have the same dimension, there exists a symplectic transformation
$\mathsf {S}$
so that
$\mathsf {S}\mathcal {L}_0 = \mathcal {G}_0$
and
$\mathsf {S}\mathcal {L}_1 = \mathcal {G}_Q$
(see, e.g., [Reference Meinrenken46, Exercise 8.1]). Since both quantities in (4.10) are symplectic invariants, without loss of generality, we can assume that
$\mathsf {S}$
is the identity. That is, it suffices to find a non-decreasing path
$\mathcal {L}(\cdot )$
with
$\mathcal {L}(0) = \mathcal {G}_0$
,
$\mathcal {L}(1) = \mathcal {G}_Q$
, and
$ {\mathrm {\iota }}(\mathcal {G}_0,\mathcal {G}_Q,\mathcal {M}) = {\mathrm {Mas}}\big (\mathcal {L}(t), \mathcal {M}\big )$
for any Lagrangian plane
$\mathcal {M}$
. We claim that
$\mathcal {L}(t) = \mathcal {G}_{tQ}$
has this property.
Representing
$\mathcal {M}$
using a Hermitian matrix M on
$ {\mathrm {dom}}\ \mathcal {M}$
, as in (3.17), and letting
$Q_{ {\mathrm {dom}}\ \mathcal {M}}$
denote the compression of Q to
$ {\mathrm {dom}}\ \mathcal {M}$
(see (3.22)), we use the cocycle property (1.8) to obtain
where we used (1.6) for the first term on the right-hand side and (3.20) for the remaining terms. Since
$n_-(Q) = 0$
, the proof of (4.10) will be complete once we show that
To calculate this Maslov index, we define the two-parameter family
$\mathcal {L}(s,t) = \mathcal {G}_{tQ - s I}$
. It is clear that
$\mathcal {L}(t) = \mathcal {L}(0,t)$
is homotopic, with fixed endpoints, to the concatenation of
$\mathcal {L}(s,0)|_{s\in [0,s_*]}$
,
$\mathcal {L}(s_*,t)|_{t\in [0,1]}$
, and
$\mathcal {L}(s_* - s,1)|_{s\in [0,s_*]}$
, so the index of
$\mathcal {L}(t)$
is the sum of these three indices. We choose the parameter
$s_*> 0$
to be small enough to guarantee that neither M nor
$M-Q_{ {\mathrm {dom}}\ \mathcal {M}}$
has any spectrum in the interval
$[-s_*,0)$
.
Recalling the representation of
$\mathcal {M}$
in (3.17) and computing as in (3.25), we observe that
Our condition on
$s_*$
means that this intersection is trivial for
$t=0$
and
$s \in (0,s_*]$
, as well as for
$t=1$
and
$s \in (0,s_*]$
.
The path
$\mathcal {L}(s,0)|_{[0,s_*]}$
may still have a conjugate point at
$s = 0$
. However, the s-crossing form, computed via (4.8), is
$-\left <x, x\right>_{\mathbb {C}^n}$
, which is negative definite and hence generates no contribution to (4.9) at the initial point of a path. We conclude that
$ {\mathrm {Mas}}_{[0,s_*]}\!\big (\mathcal {L}(s,0), \mathcal {M}\big ) = 0$
and, by similar reasoning,
$ {\mathrm {Mas}}_{[0,s_*]}\!\big (\mathcal {L}(s_* - s,1), \mathcal {M}\big ) = 0$
.
We now evaluate the remaining term,
$ {\mathrm {Mas}}_{[0,1]}\!\big (\mathcal {L}(s_*,t), \mathcal {M}\big )$
. The t-crossing form evaluates to
$\left <x, Qx\right>_{\mathbb {C}^n}$
, which we claim to be strictly positive on
$\mathcal {L}(s_*,t) \cap \mathcal {M}$
whenever this intersection is nontrivial. Indeed, the form is obviously nonnegative and if
$Qx = 0$
for some nonzero
$x \in {\mathrm {dom}}\ \mathcal {M}$
, then (4.14) implies
$Mx=-s_*x$
, which is excluded by our condition that M has no spectrum in
$[-s_*,0)$
, thus proving the claim. The same argument shows that any eigenvalue curve
$\lambda (t)$
for the matrix
$M - t Q_{ {\mathrm {dom}}\ \mathcal {M}} + s_*I_{ {\mathrm {dom}}\ \mathcal {M}}$
has
$\lambda '(t) \leq 0$
in general and
$\lambda '(t_0) < 0$
at any time
$t_0$
for which
$\lambda (t_0) = 0$
.
It follows that each
$n_+$
term in (4.9) equals the dimension of the space
${\mathcal {L}(s_*,t) \cap \mathcal {M}}$
, so
Since the eigenvalues of
$M - t Q_{ {\mathrm {dom}}\ \mathcal {M}} + s_*I_{ {\mathrm {dom}}\ \mathcal {M}}$
are strictly decreasing in t when they equal zero, the sum in (4.15) is equal to the number of eigenvalues of
$M - t Q_{ {\mathrm {dom}}\ \mathcal {M}} + s_* I_{ {\mathrm {dom}}\ \mathcal {M}}$
that pass through zero (and hence become negative) at some
$t \in [0,1)$
, thus
Finally, we observe that our conditions on the smallness of
$s_*$
imply
establishing (4.13).
Remark 4.5. We note that
$\ker M \cap \ker Q\not =\{0\}$
implies
$\mathcal {L}(0,t) \cap \mathcal {M}\not =\{0\}$
for all t, because
$(\ker M \cap \ker Q)\oplus \{0\}\subset \mathcal {L}(0,t) \cap \mathcal {M}$
. This means that the crossings are not isolated and the crossing form is degenerate, so (4.9) does not apply. This is the reason for the introduction of s in the proof above. A similar homotopy argument was used in [Reference Robbin and Salamon48] to prove the “zero axiom” of the Maslov index, which says that a path
$\mathcal {L}(t)$
for which
$\dim (\mathcal {L}(t) \cap \mathcal {M})$
is constant has
$ {\mathrm {Mas}}\big (\mathcal {L}(t), \mathcal {M}\big ) = 0$
.
As a corollary of Lemma 4.4, we obtain the Zhou–Wu–Zhu identity [Reference Zhou, Wu and Zhu51, Theorem 1.1], which has numerous applications in the analysis of differential equations [Reference Barutello, Offin, Portaluri and Wu8, Reference Berkolaiko, Cox, Latushkin and Sukhtaiev15, Reference Elyseeva, Šepitka and Hilscher25, Reference Howard32, Reference Liu, Zhang and Zhao44].
Theorem 4.6 (Zhou–Wu–Zhu identity)
For any path
$\mathcal {L} \colon [0,1] \to \Lambda (n)$
and any Lagrangian planes
$\mathcal {M}_1$
and
$\mathcal {M}_2$
,
Proof Use Lemma 4.4 to choose a path
$\widetilde {\mathcal {L}} \colon [1,2] \to \Lambda (n)$
from
$\mathcal {L}(1)$
to
$\mathcal {L}(0)$
such that
The concatenation of
$\mathcal {L}$
and
$\widetilde {\mathcal {L}}$
is a loop, therefore its Maslov index does not depend on the reference plane
$\mathcal {M}_j$
. Using the additivity of the Maslov index under concatenation, we get
Equation (4.18) now follows from (3.2), and equation (4.19) follows from the cocycle property (1.8).
We are now ready to prove the relationship (1.14) between the Duistermaat and Maslov indices that was claimed in the introduction.
Proof of Theorem 1.6
For any non-decreasing path
$\mathcal {L}(t)$
from
$\mathcal {L}_0$
to
$\mathcal {L}_1$
, we use (4.18) with
$\mathcal {M}_1=\mathcal {M}$
and
$\mathcal {M}_2 = \mathcal {L}(1)$
and note that
$ {\mathrm {\iota }}\big (\mathcal {L}(0),\mathcal {L}(1),\mathcal {L}(1)\big )=0$
by (3.1). We obtain
because the path
$\mathcal {L}(\cdot )$
is non-decreasing. Lemma 4.4 shows that the upper bound is achievable. Finally, once the path
$\mathcal {L}(t)$
achieves equality for some
$\mathcal {M}$
, that is,
the same equality holds for any other
$\mathcal {M}$
by (4.18).
5 The Morse index of a difference of Hermitian matrices
We are now ready to prove equation (1.11) and its variants. For a Hermitian matrix X, we let
$X^+$
denote the Moore–Penrose pseudoinverse and let
$P_X$
denote the orthogonal projector onto
$ {\mathrm {Ran}} X$
.
Proposition 5.1 Let A and B be
$n \times n$
Hermitian matrices.
-
(1) If $\ker A \subseteq \ker B$
, then (5.1) $$ \begin{align} n_-\big(A-B\big) = n_-(A) - n_-(B) + n_-\big(B^+ - P_B A^+ P_B\big). \end{align} $$
-
(2) If $\ker B \subseteq \ker A$
, then (5.2) $$ \begin{align} n_-\big(A-B\big) = n_-(A) - n_-(B) + n_-\big(P_A B^+ P_A - A^+\big) + n_0(A) - n_0(B). \end{align} $$
We note that equation (1.11) follows from either one of (5.1) and (5.2): for an invertible X,
$P_X$
is the identity and
$X^+=X^{-1}$
.
Proof In the case when
$\ker A \subseteq \ker B$
, the Hermitian matrices A, B,
$A-B$
, and
$B^+ - P_B A^+ P_B$
are all identically zero on
$\ker A$
. This means that we can restrict A and B (and thus
$A-B$
and
$B^+ - P_B A^+ P_B$
) to the subspace
$(\ker A)^\perp $
, on which A is invertible, without changing their Morse indices. It thus suffices to prove (5.1) when A is invertible.
Using Theorem 1.4, we just need to show that
$ {\mathrm {\iota }}(\mathcal {G}_0, \mathcal {G}_B, \mathcal {G}_A) = n_-\big (B^+ - P_B A^+ P_B\big )$
. We use (3.19) to get
and observe that, for invertible
$A,$
we have
(cf. (3.17)). The desired conclusion now follows from equation (3.20).
The proof of (5.2) proceeds similarly, by restricting all matrices in the equation to
$(\ker B)^\perp $
and observing that
To establish (5.2) under the assumption that B is invertible, we use Theorem 1.4, equation (5.3), an analog of (5.4) with A and B swapped and, finally, (3.21).
Corollary 5.2 For any invertible
$n\times n$
Hermitian matrices A and B,
Proof Equation (1.11) with
$-A$
in place of A becomes
We then use
$n_+(X) = n-n_-(X)-n_0(X)$
to obtain (5.6);
$n_0(A)$
disappears because A is invertible.
Similar computations lead to a much simpler proof of [Reference Berkolaiko, Cox, Latushkin and Sukhtaiev15, Proposition 6.1], which we give below.
Proposition 5.3 (Proposition 6.1 of [Reference Berkolaiko, Cox, Latushkin and Sukhtaiev15])
Assume
$\mathcal {L}_1$
,
$\mathcal {L}_2$
, and
$\mathcal {L}_3$
are Lagrangian planes such that
$\mathcal {L}_3$
is transversal to
$\mathcal {L}_1$
,
$\mathcal {L}_2$
, and
${\mathcal {G}_\infty }$
. Then
the relation in the right-hand side is the graph of a matrix and
$n_-$
denotes its Morse index.
Proof We have the following chain of equalities:
In the first step, we applied (3.5) and used
$\mathcal {L}_1\cap \mathcal {L}_3 = \mathcal {L}_2\cap \mathcal {L}_3 = \{0\}$
. In the second step, we used (3.18), which is allowed because
$\mathcal {L}_3$
is transversal to
${\mathcal {G}_\infty }$
and therefore is the graph of a matrix. In the third step, we used (3.19). We remark that because
$\mathcal {L}_3$
is transversal to
$\mathcal {L}_1$
,
$\mathcal {L}_1-\mathcal {L}_3$
is transversal to
${\mathcal {G}_0}$
, and thus
$(\mathcal {L}_1 - \mathcal {L}_3)^{-1}$
is transversal to
${\mathcal {G}_\infty }$
. Similarly,
$(\mathcal {L}_2 - \mathcal {L}_3)^{-1}$
is also a graph. Therefore, in the last step, we used (3.18) to subtract
$(\mathcal {L}_2 - \mathcal {L}_3)^{-1}$
and used the normalization condition (1.6).
Remark 5.4. The identity (1.11) can also be obtained using Haynsworth’s inertia additivity formula. For a block Hermitian matrix
$H = \left (\begin {smallmatrix} A & B \\ B^* & D \end {smallmatrix}\right )$
with invertible A, the Haynsworth formula [Reference Haynsworth29] states
A similar identity holds if D is invertible. Applying this formula in two possible ways to the matrix
$\left (\begin {smallmatrix} A & I \\ I & B^{-1} \end {smallmatrix}\right )$
yields
which establishes (1.11). Proposition 5.1 can be proved analogously; this requires a suitable extension of the Haynsworth formula, such as [Reference Berkolaiko, Canzani, Cox and Marzuola14, Theorem A.1] or [Reference Maddocks45, Theorem 6.1].
Acknowledgements
The authors are extremely grateful to an anonymous referee who read the article thoroughly and suggested numerous corrections and clarifications, substantially improving the manuscript.
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