In this paper we present a Krein space convexity theorem on the tracial-numerical range of a matrix. This theorem is the analogue of Westwick's theorem.
The proof is an application of Morse theory

Bebiano, Natália

Nakazato, Hiroshi

Providência, João da

Estudo Geral

Pre´-Publicac¸o˜es do Departamento de Matema´tica
Universidade de Coimbra
Preprint Number 09–31
CONVEXITY OF THE KREIN SPACE TRACIAL
NUMERICAL RANGE AND MORSE THEORY
HIROSHI NAKAZATO, NATA´LIA BEBIANO AND JOA˜O DA PROVIDEˆNCIA
Abstract: In this paper we present a Krein space convexity theorem on the tracial-
numerical range of a matrix. This theorem is the analogue of Westwick’s theorem.
The proof is an application of Morse theory.
Keywords: Numerical range, indefinite inner product, J-Hermitian matrix, non-
interlacing.
AMS Subject Classification (2000): 47A12, 46C20.
1. Introduction
Let J denote a Hermitian involutive matrix, that is, J∗ = J , J2 = In,
with signature (r, n − r), 0 < r ≤ n (i.e. with r positive and n− r negative
eigenvalues). We consider Cn endowed with the indefinite inner product
induced by J :
[x, y] = y∗Jx, x, y ∈ Cn.
Let Mn be the algebra of n × n complex matrices. A matrix H ∈ Mn is
J-Hermitian if H# = H, where H# = JH∗J is the J-adjoint of H. A matrix
U ∈ Mn is J-unitary if U#U = UU# = In, the identity matrix of size n.
The J-unitary matrices form a locally compact connected group denoted by
G = U(r, n− r) and called the J-unitary group. For C, T ∈Mn, the J-tracial
numerical range of T is denoted and defined as
(1.1) W JC (T ) = {Tr(CUTU−1) : U ∈ U(r, n− r)}.
This set is a connected set in the Gaussian planeC and satisfies the symmetry
property W JC (T ) = W
J
T (C) (see [1], [2], [4] for other properties). In the
case r = n, the matrix J reduces to the identity matrix and the group
U(n, 0) = U(n) is the unitary group, which is a compact group. In this
case W JC (T ) is simply denoted by WC(T ) and called the C-numerical range
of T . Hausdorff proved the convexity of WC(T ) in the case C is a rank
one orthogonal projection. Using Morse theory, Westwick [8] proved the
Received August 28, 2009.
1
2 H. NAKAZATO, N. BEBIANO AND J. DA PROVIDEˆNCIA
convexity of WC(T ) for C Hermitian and in [5] Poon obtained an alternative
proof by majorization techniques. In [7], Tam generalized Westwick’s result
for certain compact Lie groups. In this paper we obtain a non-compact group
analogue of Westwick’s convexity theorem applying Morse theory (cf. Lemma
4.1). The eigenvalues of a J-unitarily diagonalizable J-Hermitian matrix T
are real and will be denoted as:
σ+(T ) = {λ ∈ C : Tξ = λξ for some ξ ∈ Cn, [ξ, ξ] > 0},
σ−(T ) = {λ ∈ C : Tξ = λξ for some ξ ∈ Cn, [ξ, ξ] < 0}.
The eigenvalues are said to be noninterlacing if
(1.2) max σ−(T ) < minσ+(T ) or max σ+(T )) < min σ−(T ).
Replacing < in (1.2) by ≤ the eigenvalues are said to be weakly noninterlac-
ing. Noninterlacing property plays a key role in the investigation of numerical
ranges associated with non-compact groups.
In the sequel we consider J = Ir⊕(−In−r). Our main result is the following.
Theorem 1.1. Let C be a J-unitarily diagonalizable J-Hermitian matrix
with noninterlacing eigenvalues and let T ∈ Mn be a matrix such that for
some θ ∈ R
Tθ = cos θ(T + T
#)/2 + sin θ(T − T#)/(2i)
is a J-unitarily diagonalizable J-Hermitian matrix with noninterlacing eigen-
values:
σ+(C) = {c1 ≥ · · · ≥ cr}, σ−(C) = {cr+1 ≥ · · · ≥ cn},
σ+(Tθ) = {λ1 ≥ · · · ≥ λr}, σ−(Tθ) = {λr+1 ≥ · · · ≥ λn},
cn > c1 or cr > cr+1, λn > λ1 or λr > λr+1.
ThenW JC (T ) is a closed convex set in C contained in a closed cone {z0+reiη :
r ≥ 0, θ1 ≤ η ≤ θ2} with 0 ≤ θ2 − θ1 < pi.
Denote by C∞(G) the commutative algebra of all C∞-differentiable complex-
valued functions on G = U(r, n− r). We consider the even dimensional coset
space M = G/D, where
D = {diag(d1, . . . , dn) : d1, . . . , dn ∈ C, |d1| = · · · = |dn| = 1}.
CONVEXITY OF THE KREIN SPACE TRACIAL NUMERICAL RANGE 3
Let the real valued function f0 on M be defined by
(1.3) f0(g) =
n∑
h,j=1
chλj|uhj|2²h²j = Tr (CgTθg−1),
where ²h = 1 for 1 ≤ h ≤ r and ²h = −1 for r + 1 ≤ h ≤ n.
This paper is organized as follows. The critical points and the Morse indices
of the function f0 are investigated in Section 2. Global properties of f0 and
of W JC (T ) are studied in Section 3. The proof of Theorem 1.1 is presented in
Section 4 using the results in Sections 2, 3.
2. Local properties of the function f0
To treat local properties of the function f0, we introduce some notation
and prerequisites. The Lie algebra of G = U(r, n− r) is given as
(2.1) G = {X ∈Mn : X# = −X}.
For 1 ≤ k0 < l0 ≤ r or r + 1 ≤ k0 < l0 ≤ n, let
Xk0,l0 = (δkk0δll0 − δkl0δlk0), Yk0,l0 = i(δkk0δll0 + δkl0δlk0),
where δkl is the Kronecker symbol. For 1 ≤ k0 ≤ r, r + 1 ≤ l0 ≤ n, let
Vk0,l0 = (δkk0δll0 + δkl0δlk0), Wk0,l0 = i(δkk0δll0 − δkl0δlk0).
Throughout, we consider the basis of the Lie algebra G constituted by
the matrices Xk0,l0, Yk0,l0, Vk0,l0, Wk0,l0 and the diagonal matrices D1 =
diag(i, 0, . . . , 0), · · · , Dn = diag(0, . . . , 0, i), which generate the Lie algebra
D. We shall use a canonical coordinate system of second kind
exp(t1X1) exp(t2X2) · · · exp(tn2Xn2)
in a neighborhood of the identity In, where {Xn2−n+1, . . . , Xn2} is the above
basis of D, and {X1, X2, . . . , Xn2−n} is the remaining system of vectors of the
basis of G.
For f ∈ C∞(G), consider a representation α of G on C∞(G) given by
(2.2) α(X)(f)(g) = − lim
t→0
f(etXg)− f(g)
t
= lim
t→0
f(e−tXg)− f(g)
t
.
The function space C∞(G/D) is identified with the space
(2.3) {f ∈ C∞(G) : f(gd) = f(g) for d ∈ D}.
4 H. NAKAZATO, N. BEBIANO AND J. DA PROVIDEˆNCIA
For Sn the symmetric group of degree n, let
Sr,n−r = {σ1σ2 : σ1(j) = j (r + 1 ≤ j ≤ n), σ2(j) = j (1 ≤ j ≤ r)}.
Proposition 2.1. Let H = diag (λ1, · · · , λn) and C = diag (c1, · · · , cn) be
such that ch 6= cj, λh 6= λj for 1 ≤ h 6= j ≤ n. A point g = (uhj) of G is
a critical point of f0 if and only if (uhj) is a permutation matrix associated
with a permutation in Sr,n−r.
Proof. We take an arbitrary critical point g. A straightforward computation
shows that
−α(X)(f0)(g) = Tr (X[gHg−1, C]) = 0,
where [X,Y ] = XY − Y X stands for the commutator. Thus, gHg−1 com-
mutes with the diagonal matrix C, whose diagonal entries are pairwise dis-
tinct, and so
gHg−1 = diag (λσ(1), . . . , λσ(n)),
for some σ ∈ Sn. For g = (uhj), we have uhjλj = λσ(h)uhj and so uhj = 0
unless λj = λσ(h). Therefore, uhj = ηjδjσ(h), where |ηj| = 1. We notice that
gHg−1 = dgHg−1d−1 for any d ∈ D. Moreover, σ belongs to Sr,n−r because
g−1 = Jg∗J. The converse is clear. ¤
Proposition 2.2. Let H = diag (λ1, · · · , λn) and C = diag (c1, · · · , cn) be
such that ch 6= cj, λh 6= λj for 1 ≤ h 6= j ≤ n. The numbers of positive
and negative eigenvalues of the Hessian matrix of f0 at each critical point
are even.
Proof. We analyze the Hessian matrix of the function f0 at a critical point.
For f ∈ C∞(G), the representation α of G satisfies
(2.4) α(X)α(Y )(f)(g) =
d
dt
d
ds
f(e−sY e−tXg
)∣∣∣∣
s,t=0
.
For X = −X#, consider the expansion of f(e−tXg) in powers of t
f0(e
−tXg) = Tr(CgHg−1)−t Tr(C[X, gHg−1])+t
2
2
Tr(C[X, [X, gHg−1]])+O(t3).
Assuming g is critical, we get
f0(e
−tXg) = Tr(CgHg−1) +
t2
2
Tr(C[X, [X, gHg−1]]) +O(t3).
CONVEXITY OF THE KREIN SPACE TRACIAL NUMERICAL RANGE 5
Let M(X, Y ) = α(X)α(Y )(f0)(g) = Tr(C[Y, [X, gHg−1]]). Having in mind
Jacobi identity and performing some computations, we find
(2.5) α(X)α(Y )(f0)(g) = Tr(C[Y, [X, gHg
−1]]) = Tr([C, Y ][X, gHg−1]),
and so M(X, Y ) = M(Y,X). For X = (xkl), Y = (ykl), we have [C, Y ] =
(yhj(ch− cj)) and [X, gHg−1] = (xhj(λσ(j)− λσ(h))). By straightforward com-
putations we obtain
M(X,Y ) = Tr(C[Y, [X, gHg−1]]) = Tr ((yhj(ch − cj)) (xhj(λσ(j) − λσ(h))))
=
n∑
k,l=1
yklxlk(ck − cl)(λσ(k) − λσ(l)).
The matrix representation of M(X, Y ), relative to the fixed basis in G/D,
is diagonal. In fact, for 1 ≤ k0 < l0 ≤ r or r + 1 ≤ k0 < l0 ≤ n, an easy
computation leads to
M(Xk0l0, Xk′0l′0) =M(Yk0l0, Yk′0l′0) = −2δk0k′0δl0l′0(ck0 − cl0)(λσ(k0) − λσ(l0)).
For 1 ≤ k0, k′0 ≤ r, r + 1 ≤ l0, l′0 ≤ n, we analogously have
M(Vk0l0, Vk′0l′0) =M(Wk0l0,Wk′0l′0) = 2δk0k′0δl0l′0(ck0 − cl0)(λσ(k0) − λσ(l0)).
The off-diagonal entries of M(X, Y ) vanish. Thus, the (n2 − n) × (n2 −
n)-Hessian matrix of the function f0 at every critical point is nonsingular.
Moreover, the numbers of positive and negative eigenvalues of the Hessian
matrix are even. ¤
We recall, in passing, that the number of negative eigenvalues of the Hessian
is called the Morse index.
3. Global properties of the function f0 and W
J
C (H)
Proposition 3.1. Let C and H be J-unitarily diagonalizable J-Hermitian
matrices with respective eigenvalues σ+(C) = {c1, . . . , cr}, σ−(C) = {cr+1, . . . , cn},
σ+(H) = {λ1, . . . , λr}, σ−(H) = {λr+1, . . . , λn} satisfying the conditions
(3.1) cr+1 ≥ · · · ≥ cn > c1 ≥ · · · ≥ cr,
(3.2) λ1 ≥ · · · ≥ λr > λr+1 ≥ · · · ≥ λn.
Then the superlevel set {U ∈ U(r, n − r) : a ≤ f0(U)} is compact for an
arbitrary real number a.
6 H. NAKAZATO, N. BEBIANO AND J. DA PROVIDEˆNCIA
Proof. Analogously to (1.3) we define a function g0 for the diagonal J-
Hermitian matrices C ′ = diag(c′1, . . . , c
′
n), H
′ = diag(λ′1, . . . , λ
′
n) with eigen-
values c′1 = · · · = c′r = c1, c′r+1 = · · · = c′n = cn and λ′1 = · · · = λ′r =
λr, λ
′
r+1 = · · · = λ′n = λr+1. We show that the functions f0, g0 satisfy the
inequality f0(g) ≤ g0(g) for g ∈ G. To prove this inequality, we may assume
that cn > 0 > c1, λr > 0 > λr+1 by adding suitable scalar operators to C and
H. Observing that c1λr − chλj = (c1 − ch)λr − ch(λj − λr) (1 ≤ h, j ≤ r),
the function g0(U)−f0(U) can be expressed as a linear combination of |uh,j|2
(1 ≤ h, j ≤ n) with nonnegative coefficients. Henceforth the inclusion holds
(3.3) {[U ] ∈M : f0([U ]) ≥ a} ⊂ {[U ] ∈M : g0(U) ≥ a},
and the left-hand side of (3.3) is a closed subset of the set in the right-hand
side. It can be easily checked that
g0(U) = rc1λr + (n− r)cnλr+1 − cn − c1
4
(λr − λr+1)
 n∑
h,j=1
|uh,j|2 − n
 ,
so the right-hand side of (3.3) is compact for an arbitrary a and so is the
left-hand side. ¤
Proposition 3.2. Let C be a J-unitarily diagonalizable J-Hermitian matrix
with noninterlacing eigenvalues σ+(C) = {c1 ≥ · · · ≥ cr}, σ−(C) = {cr+1 ≥
. . . ≥ cn}, cn > c1. Let H be a J-Hermitian matrix such that W JC (H) =
(−∞, a0]. Then the matrix H is J-unitarily diagonalizable and its eigenvalues
are weakly noninterlacing.
Proof. We assume that the maximum value a0 is attained at Tr(CgHg
−1).
This assumption implies that Tr(X[gHg−1, C]) = 0 for an arbitrary X ∈ G,
so that [gHg−1, C] = 0. The condition cn > c1 implies that gHg−1 = H1⊕H2,
where H1, H2 are Hermitian block matrices of sizes r and n− r, respectively.
Thus, gHg−1 is J-unitarily diagonalizable as V gHg−1V −1 = diag(λ1, . . . , λn),
where V is a J-unitary matrix of the form V = V1 ⊕ V2, the blocks V1, V2
being unitary of sizes r and n−r, respectively. Weak noninterlacing property
of the eigenvalues follows from Theorem 1.1 (iii) in [1]. ¤
Proposition 3.3. Let C = diag(c1, . . . , cn) with σ+(C) = {c1 ≥ · · · ≥ cr},
σ−(C) = {cr+1 ≥ · · · ≥ cr}, cn > c1, and let H = diag(λ1, · · · , λn) with
CONVEXITY OF THE KREIN SPACE TRACIAL NUMERICAL RANGE 7
σ+(H) = {λ1 ≥ · · · ≥ λr}, σ−(H) = {λr+1 ≥ · · · ≥ λn}, λr ≥ λr+1, so that
{
Tr(CgHg−1) : g ∈ U(r, n− r)} = (−∞, n∑
j=1
cjλj
]
.
Then, for K J-Hermitian, the set
W =
{
Tr(CgKg−1) : g ∈ Ur,n−r,Tr(CgHg−1) =
n∑
j=1
cjλj
}
is connected.
Proof. Let H = (λ′1In1)⊕· · ·⊕(λ′sIns), where the λ′j are all distinct. Consider
C = C1 ⊕ · · · ⊕ Cs, J = J1 ⊕ · · · ⊕ Js, and let K be the J-Hermitian block
matrix
K =
K11 . . . K1s. . . . . . . . .
Ks1 . . . Kss
 ,
where Cj, Jj, Kjj ∈ Mnj , j = 1, . . . , s. If Tr(CgHg−1) =
∑n
j=1 cjλj, then
g = WV , where W,V ∈ U(r, n − r), WCW−1 = C and V HV −1 = H.
Obviously, V = V1 ⊕ · · · ⊕ Vs, Vj ∈Mnj , and VjJjV ∗j = Jj. Moreover,
V KV −1 =
V1K11V −11 . . . V1K1sV −1s. . . . . . . . .
VsKs1V
−1
1 . . . VsKssV
−1
s

and
Tr(CgKg−1) =
n∑
j=1
Tr(CjVjKjjV
−1
j ).
Thus
(3.4) W = W J1C (K11) +W J2C2(K22) + · · ·+W JsCs(Kss).
Since each summand in the Minkowski sum in the right hand side of (3.4) is
connected, the result follows. ¤
8 H. NAKAZATO, N. BEBIANO AND J. DA PROVIDEˆNCIA
4. Proof of Theorem 1.1
Lemma 4.1. (cf. [3, 6]). Let M be an m-dimensional connected C∞-
manifold (m ≥ 2) and let f be a real-valued C∞-function on M . Assume
that the function f satisfies the following conditions:
(i) The function f attains a minimum value;
(ii) The set {x ∈ M : a ≤ f(x) ≤ b} is compact for every −∞ < a < b <
+∞;
(iii) The number of critical points x1, x2, . . . , xn of f is finite, and the crit-
ical values f(x1), f(x2), . . . , f(xn) are all distinct;
(iv) The Hessian matrix of f at each critical point xj is non-singular;
(v) The number of the negative eigenvalues of the Hessian matrix of f at
each critical point xj is neither 1 nor m− 1.
Then the level set {x ∈M : f(x) = a} is connected for every a ∈ R.
Some critical values of the function f0 given by (1.3) at distinct critical
points may coincide. Then we apply a perturbative method. We choose a
vector (y1, . . . , yn) satisfying
∑n
k=1 yk(cσ(k) − cτ(k)) 6= 0 for any distinct pair
σ, τ ∈ Sr,n−r. We can find t0 ∈ R with sufficiently small modulus such that
λk + ykt0 satisfy the condition (iii) of Lemma 4.1.
Now, we prove Theorem 1.1. First we show the simple connectedness of
W JC (T ) under the assumptions of the theorem. Consider the case cn > c1,
λr > λr+1. The remaining cases reduce to this one by replacing C by −C
and/or Tθ by Tθ+pi. By a rotation in the Gaussian plane C, we assume
that θ = 0 and so H = T0 = (T + T
#)/2 is a J-unitarily diagonalizable
J-Hermitian matrix with noninterlacing eigenvalues λr > λr+1. Define the
following sequences of J-Hermitian matrices
Cm = diag(c
(m)
1 , . . . , c
(m)
n ), Hm = diag(λ
(m)
1 , . . . , λ
(m)
n ),
where c
(m)
n = cn > c1 = c
(m)
1 , λ
(m)
r = λr > λr+1 = λ
(m)
r+1 for m = 1, 2, 3, . . ..
Suppose that |c(m)h − ch| → 0, |λ(m)h − λh| → 0 as m → ∞; c(m)h 6= c(m)j ,
λ
(m)
h 6= λ(m)j for each m, 1 ≤ h < j ≤ n and assume that the critical values∑n
h=1 c
(m)
h λ
(m)
σ(h) for any distinct permutations σ ∈ Sr,n−r are distinct. By
Proposition 3.1 the set
{g ∈ U(r, n− r) : Tr(CmgHmg−1) ≥ b}∪{g ∈ U(r, n− r) : Tr(CgHg−1) ≥ b}
CONVEXITY OF THE KREIN SPACE TRACIAL NUMERICAL RANGE 9
is contained in a compact set B for any b ∈ R. The pairs {Cm, Hm} satisfy
the conditions of Propositions 2.1 and 2.2. It follows that these pairs satisfy
the conditions of Lemma 4.1. By Lemma 4.1, each set {g ∈ U(r, n − r) :
Tr(CmgHmg
−1) = a − 1/2m} is a compact connected set. Hence for Tm =
Hm + i(T − T#)/(2i) ∈ Mn and a ∈ R, Im = {z ∈ W JCm(Tm) : <(z) =
a − 1/2m} is a closed interval. For every a ≤ ∑nh=1 chλh, the set {z ∈
W JC (T ) : <(z) = a} is compact. Let
b0 = min{y ∈ R : a+ iy ∈ W JC (T )}, c0 = max{y ∈ R : a+ iy ∈ W JC (T )},
bm = min{y ∈ R : a− 1
2m
+ iy ∈ W JCm(Tm)},
cm = max{y ∈ R : a− 1
2m
+ iy ∈ W JCm(Tm)}.
Let us take an arbitrary 0 < t0 < 1 and choose a point Um in B satisfying
Tr(CmUmTmU
−1
m ) = a−
1
2m
+ i(t0bm + (1− t0)cm)
for each m. Since the set B is compact, we consider a subsequence Umk of
Um converging to a point U0 of B. Then as k →∞
Tr(CmkUmkTmkU
−1
mk
)→ Tr(CU0TU−10 ) = a+ i(t0b0 + (1− t0)c0).
Hence the set
(4.1) {z ∈ W JC (T ) : <(z) = a},
is also a closed interval. We set a0 =
∑n
h=1 chλh. Since the set (4.1) depends
continuously on a, then
W JC (T ) = {x+ iy : −∞ < x ≤ a0, φ1(x) ≤ y ≤ φ2(x)}
for some continuous real-valued functions φ1(x) ≤ φ2(x) defined on the half-
line (−∞, a0], and so it is simply connected.
Now we prove that the boundary of W JC (T ) , ∂W
J
C (T ), is convex. The set
W JC (T ) has a point z0 with <(z0) = a0 and the line <(z) = a0 is a support
line of W JC (T ). If ∂W
J
C (T ) is not convex, there exists a tangent line at some
point z1 ∈ ∂W JC (T ) expressed as <(ze−iθ1) = b0 for some θ1, b0 ∈ R satisfying
<(z2e−iθ1) < b0, <(z3e−iθ1) > b0, where z2, z3 ∈ W JC (T ). Then we find a
point z4 on the arc of ∂W
J
C (T ) joining z0, z1 and a support line `θ passing
through z4, <(ze−iθ0) = aθ, such that W JC (T )∩ `θ0 is not connected. However
the last condition of `θ contradicts Proposition 3.3. So ∂W
J
C (T ) is convex.
10 H. NAKAZATO, N. BEBIANO AND J. DA PROVIDEˆNCIA
We take a point x0 < a0 at which φ1, φ2 are differentiable. Then we have
φ′2(x0) ≤ φ′1(x0) and the cone {x + iy ∈ C : φ′1(x0)(x − x0) + φ1(x0) ≤ y ≤
φ′2(x0)(x − x0) + φ2(x0)} satisfies the last assertion of Theorem 1.1. This
completes the proof of the theorem. ¤
Acknowledgement. The authors would like to express their thanks to
Professor Osamu Saeki for useful suggestions.
References
[1] N. Bebiano, H. Nakazato, J. da Provideˆncia, R. Lemos and G. Soares, Inequalities for J-
Hermitian matrices, Linear Algebra and Its Applications, 407 (2005), 125-139.
[2] C. K. Li and L. Rodman, Remarks of numerical ranges of operators in spaces with an indefinite
inner product, Proc. Amer. Math. Soc. 126 (1998), 973-982.
[3] Y. Matsumoto, An introduction to Morse theory, (Translations of Mathematical Monographs,
208), American Mathematical Society, Providence, 2002.
[4] H. Nakazato, N. Bebiano and J. da Provideˆncia, The numerical range of 2-dimensional Krein
space operators, Canadian Mathematical Bulletin 51 (2008), 86-99.
[5] Y. T. Poon, Another proof of a result of Westwick, Linear and Multilinear Algebra 9 (1980),
35-37.
[6] O. Saeki, Private communication, November 2008.
[7] T. Y. Tam, Convexity of generalized numerical range associated with a compact Lie group, J.
Aust. Math. Soc. 72 (2002), 57-66.
[8] R. Westwick, A theorem on numerical range, Linear and Multilinear Algebra 2 (1975), 311-315.
Hiroshi Nakazato
Department of Mathematical Sciences, 036-8561 Hirosaki, Japan
E-mail address: nakahr@cc.hirosaki-u.ac.jp
Nata´lia Bebiano
CMUC, Department of Mathematics, University of Coimbra, 3001-454 Coimbra, Portugal
E-mail address: bebiano@mat.uc.pt
Joa˜o da Provideˆncia
Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal
E-mail address: providencia@teor.fis.uc.pt


Convexity of the Krein space tracial numerical range and Morse theory

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Convexity of the Krein space tracial numerical range and Morse theory

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