Special Issue in honor of John W. Neuberger

Electronic Journal of Differential Equations, Special Issue 02 (2023), pp. 11–39.

ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu

FRIEDRICHS EXTENSION OF SINGULAR SYMMETRIC

DIFFERENTIAL OPERATORS

QINGLAN BAO, GUANGSHENG WEI, ANTON ZETTL

Abstract. For singular even order symmetric differential operators we find

the matrices which determine all symmetric extensions of the minimal oper-

ator. And for each of these symmetric operators which is bounded below we
find the boundary condition of its Friedrichs extension. The operators of regu-

lar problems are bounded below and thus each one has a symmetric extension

and thus its symmetric extension has a Friedrichs extension.

1. Introduction

Let I = (a, b), −∞ ≤ a < b ≤ ∞, be an open interval and let n be a positive
integer. Let Mn,m(Ω) denote the n×m matrices with elements from the set Ω for
n,m = 1, 2, 3, 4, . . ., and for m = n we write Mn(Ω). Let C denote the complex
numbers and R the reals.

Let C = (cr,s)1≤r,s≤2n ∈ M2n(C) be a skew diagonal complex matrix with the
following properties:

C−1 = −C = C∗, (1.1)

and let

Z2n(I) := {(qr,s)2nr,s=1 ∈M2n(L1
loc(I)),

qr,r+1 6= 0 a.e. on I, q−1r,r+1 ∈ L1
loc(I), 1 ≤ r ≤ 2n− 1,

qr,s = 0 a.e. on I, 2 ≤ r + 1 < s ≤ 2n;

qr,s ∈ L1
loc(I), s 6= r + 1, 1 ≤ r ≤ 2n− 1}.

For Q ∈ Z2n(I), define the quasi-derivatives y[r](0 ≤ r ≤ 2n) below:

y[0] := y (y ∈ V0),

y[r] = q−1r,r+1[(y[r−1])′ −
r∑
s=1

qr,sy
[s−1]] (y ∈ Vr, r = 1, 2, . . . , 2n),

(1.2)

where

V0 := {y : I → C, y is measurable },

Vr := {y ∈ Vr−1 : y[r−1] ∈ ACloc(I)}, r = 1, 2, . . . , 2n,

2020 Mathematics Subject Classification. 34B24, 34L15, 34B08, 34L05.

Key words and phrases. Friedrichs extension; regular differential expression; boundary matrix.
©2023 This work is licensed under a CC BY 4.0 license.
Published March 27, 2023.

11



12 Q. BAO, G. WEI, A. ZETTL EJDE/SI/02

and q2n,2n+1 = c2n,1. Since the quasi-derivatives depend on Q, we sometimes write

y
[r]
Q instead of y[r], r = 1, 2, . . . , 2n. Finally we set

My = y[2n] = λwy, λ ∈ R, y ∈ V2n, (1.3)

these expressions M = MQ are generated by or associated with Q and the element
of constant matrix C. For V2n we also use the notations D(Q) and V (M). If

Q = −C−1Q∗C, (1.4)

i.e.,

qr,s = cr,2n+1−rq2n+1−s,2n+1−rc2n+1−s,s, (1.5)

thenQ is said to be a C-symmetric matrix. For this caseMQ is called a C-symmetric
quasi-differential expression [2].

Let w ∈ L1
loc(I) be positive a.e. on I. Consider the Hilbert space H = L2

w(I)

with inner product (y, z)w =
∫ b
a
yz̄w and let

DQ = {y ∈ H : y ∈ D(Q) and
1

w
My ∈ H}. (1.6)

This linear manifold DQ is called the maximal domain of M and TQy = 1
wMy = λy

for y ∈ DQ, is called the maximal operator of M . It is well known [1, 7] that DQ

is dense in H. Its uniquely defined adjoint T0 = T ∗Q is called the minimal operator

of M and its domain D0 := D(T0) is called the minimal domain of M . It is well
known that T0 is a densely defined symmetric operator in H [17, 24], and it is the
closure of the pre-minimal operator T ′0 defined by

D′0 := D(T ′0) = {y ∈ DQ : y has compact support in I}.

The operator T ′0 is also a symmetric operator in H and its Friedrichs extension has
the same characterization as the minimal operator T0. Some properties of Zn(I)
and the definition of singular or regular expression can be seen in [28].

For a symmetric densely defined operator S which bounded below, Friedrichs [8]
constructed a self-adjoint extension SF which preserves the lower bound. He also
addressed this question and showed that the Friedrichs extension of the minimal
operator is determined by Dirichlet boundary conditions for a special class of regu-
lar second order Sturm-Liouville (SL) expressions. This result is now known to hold
[14, 18] for all regular SL expressions with minimal conditions on the coefficients
and weight function and for very general classical and quasi-differential expressions
of arbitrary even order. In particular, for a smaller class of symmetric expressions
M Niessen and Zettl [18] presented the boundary conditions of the Friedrichs ex-
tensions for some symmetric extensions with separated boundary conditions. More-
over, for any even order regular Lagrange symmetric differential operators, Möller
and Zettl [15] gave the characterization of the boundary conditions that determine
their Friedrichs extensions.

One can prove that the Friedrichs extension is distinguished in various ways
among other self-adjoint extensions, see [3, 23]. For singular differential operators,
Rellich [20] realized that the principal solution plays a key role in the boundary
conditions of the Friedrichs extension of the second order SL problems. In 1992,
Niessen and Zettl [19] based on the work of Rellich [20], Kalf [10], Rosenberger [23]



EJDE-2023/SI/02 FRIEDRICHS EXTENSIONS 13

and other scholars, used the principal solution to characterize the boundary condi-
tions of the Friedrichs extension of singular second order SL problems. In 2000, for
a smaller class of symmetric operators,
Martletta and Zettl [16] gave the boundary conditions for Friedrichs extension of
some even order singular symmetric differential expressions. In 2015, Yao et al.
[27] used the method of functional analysis to characterize the Friedrichs extension
of the second order SL operators by using the limit-circle (LC) type square inte-
grable solutions of real parameters and trigonometric functions. In 2018, Zheng
and Kong [29] studied singular Hamiltonian operators with intermediate deficiency
indices, and gave a complete characterization of Friedrichs extensions of minimal
Hamiltonian operator. In 2020, Wang and Zettl [26] characterized the two-point
regular and singular boundary conditions which determine the symmetric opera-
tors. In 2022, we [4] found, explicitly, the boundary conditions which determine
the Friedrichs extension of the class of regular even order C-symmetric differen-
tial operators. Furthermore for singular second order SL problems we [5] gave
explicit representations of boundary matrices for any symmetric operators which
are bounded below and the boundary conditions which determine their Friedrichs
extensions.

This article is a follow up of [5] and by using the technique in [5, 16] we
present a new characterization of all symmetric operators generated by (1.3) and
the Friedrichs extension of these symmetric extensions which are bounded below in
terms of boundary conditions for singular or regular general even order C-symmetric
differential expressions. Also some specific examples are given for the boundary
conditions of the Friedrichs extensions.

The rest of this article is organized as follows. We briefly introduce some prop-
erties of Hamiltonian system in Section 2. The relations between the existence
of principal solutions and the boundedness below of the minimal operator are pre-
sented in Section 3. The characterization of symmetric operators is given in Section
4. The Friedrichs extensions for the symmetric extensions which are bounded below
are presented in Section 5 and some special examples of these extensions are given
in Section 6.

2. Hamiltonian system

In this section we briefly discuss properties of Hamitonian systems and give
several lemmas for proving our main results.

Note that C = C2n satisfying (1.1) has the form

C2n =

(
0 Ĉn
−Ĉ∗n 0

)
(2.1)

and Ĉn is a skew-diagonal unitary matrix, that is,

cr,scr,s = 1, for r + s = 2n+ 1, 1 ≤ r ≤ n,
cr,s = 0, otherwise.

(2.2)

For every sufficiently smooth function y ∈ DQ in I, we give a associate vector
function

y(t) =

(
x(t)
u(t)

)
, x(t) = (yy[1] · · · y[n−1])T , u(t) = (y[2n−1]y[2n−2] · · · y[n])T .



14 Q. BAO, G. WEI, A. ZETTL EJDE/SI/02

It is shown that each C-symmetric differential equation (1.3) satisfying (1.4) may
be written in terms of the 2n-dimensional vector function y as

Ly := J̃2ny′(t)−GQ(t)y(t) = λW (t)y(t), (2.3)

where

J̃2n =

(
0 −C̃n
C̃∗n 0

)
, C̃n =


c1,2n 0 · · · 0

0 c2,2n−1 · · · 0
...

. . .
. . .

...
0 · · · · · · cn,n+1

 ,

and

GQ(t) =

(
−C(t) A∗(t)
A(t) B(t)

)
, W (t) =

(
W1(t) 0

0 0

)
,

C(t) =


c1,2nq2n,1 c̄2,2n−1q̄2n−1,1 · · · c̄n,n+1q̄n+1,1

c2,2n−1q2n−1,1 c2,2n−1q2n−1,2 · · · c̄n,n+1q̄n+1,2

...
. . .

. . .
...

cn,n+1qn+1,1 cn,n+1qn+1,2 · · · cn,n+1qn+1,n

 ,

A(t) =



c̄1,2nq1,1 c̄1,2nq1,2 0 · · · 0
c̄2,2n−1q2,1 c̄2,2n−1q2,2 c̄2,2n−1q2,3 · · · 0

...
. . .

. . .
. . .

...

c̄n−1,n+2qn−1,1 c̄n−1,n+2qn−1,2 · · ·
. . . c̄n−1,n+2qn−1,n

c̄n,n+1qn,1 c̄n,n+1qn,2 · · · · · · c̄n,n+1qn,n

 ,

B(t) =


0 0 · · · 0
0 0 · · · 0
...

. . .
. . .

...
0 · · · · · · c̄n,n+1qn,n+1

 , W1(t) =


w 0 · · · 0
0 0 · · · 0
...

. . .
. . .

...
0 · · · · · · 0

 .

It is worthwhile to state that the diagonal elements of C̃n are cr,2n+1−r, r =
1, 2, . . . , n of C2n satisfying (2.1) and GQ(t) is associated with the matrix func-
tion Q(t) satisfying (1.4). Also notice that B(t) = B∗(t), C(t) = C∗(t). The matrix
function GQ(t) ∈M2n(L1

loc(I)) is Hermitian. Observe that (2.3) can be written as

L1{x,u} := −C̃nu′(t) + (C(t)−A∗(t)u(t)− λW1(t))x = 0,

L2{x,u} := C̃∗nx′ −A(t)x−B(t)u = 0.

Because of J̃∗2n = −J̃2n = J̃−12n , and recall that for C̃n = In, the n × n identity
matrix, its occurrence as the canonical form of an accessory system derived from
a variational problem, see [22, Chapter V]. The equation (2.3) may be called a
“Hamiltonian system”, or “Hermitian system”.

The property of disconjugacy is important to system (2.3), see [16, 22].

Definition 2.1. System (2.3) is said to be disconjugate on an interval (α, β) ⊆
I = (a, b) if for every interval (α0, β0) ⊆ (α, β) whose endpoints are regular points
of the differential equation, the boundary value problem

Ly = J̃2ny′(t)−GQ(t)y(t) = λW (t)y(t),x(α0) = 0,x(β0) = 0 (2.4)

the only solution is the trivial solution.



EJDE-2023/SI/02 FRIEDRICHS EXTENSIONS 15

Remark 2.2. The points t0, t1 ∈ I, t0 6= t1, are said to be mutually conjugate

relative to (2.3) if there exists a solution y(t) =

(
x(t)
u(t)

)
of (2.3) such that x(t0) =

0,x(t1) = 0 and x(t) 6= 0 on the subinterval with endpoints t0 and t1. System (2.3)
is disconjugate on a subinterval I0 of I if no two distinct points of I0 are mutually
conjugate relative to (2.3). It is said to be oscillatory if for every t0 ∈ I there exists
a t1 > t0 such that (2.3) is not disconjugate on [t0, t1]. See [13].

We call the differential equation (1.3) disconjugate if the associated system (2.3)
is disconjugate. Next we concentrate on systems which are normal.

Definition 2.3 ([16]). A Hamiltonian system is said to be normal on an interval
(a, b) if it has no solutions with x(t) = 0 for all t ∈ (a, b) and u(t) 6= 0 for some
t ∈ (a, b).

From Definition 2.3 we know that the systems (2.3) arising from differential equa-
tions (1.3) have this property. In addition if (2.3) is normal on every non-degenerate
subinterval of I, then this equation is said to be identically normal (or called com-
plete controllability) on I. Thus in the following discussion of disconjugacy, we
automatically restrict our attention to identically normal systems.

Instead of (2.3), it is more convenient to deal with the matrix equation

LY := J̃2nY
′(t)−GQ(t)Y (t) = λW (t)Y (t). (2.5)

Since C̃∗n = C̃−1n , we obtain by Y (t) =

(
X(t)
U(t)

)
such that

X ′(t) = C̃n(A(t)X(t) +B(t)U(t)), U ′(t)

= C̃∗n(C(t)X(t)−A∗(t)U(t)− λW1(t)X(t)),
(2.6)

where X(t) and U(t) are n× r matrix valued. Note that Y (t) is a matrix solution
of (2.5) if and only if x(t) = X(t)c, u(t) = U(t)c is a vector solution of (2.3)for
every constant vector c ∈ Cr, r ≥ 1. Thus all solutions of (2.3) are determined if

we know two solutions Yj(t) =

(
Xj(t)
Uj(t)

)
∈ M2n,rj (L1

loc(I)), j = 1, 2, of (2.5) such

that the 2n× (r1 + r2) matrix

(
X1(t) X2(t)
U1(t) U2(t)

)
is nonsingular.

If Y1(t) =

(
X1(t)
U1(t)

)
and Y2(t) =

(
X2(t)
U2(t)

)
are solutions of (2.5), then we have

{Y1, Y2} := X∗2 (t)C̃nU1(t)− U∗2 (t)C̃∗nX1(t) = K0, (2.7)

whereK0 ∈Mr2,r1(C) is a constant matrix. This is readily verified by differentiating
(2.7). If K0 = 0 then the solutions Y1(t), Y2(t) are called conjugate solutions of
(2.5). Correspondingly the solutions y1(t),y2(t) are called conjugate solutions of
(2.3). Alternate terminologies are mutually conjoined, or isotropic solutions or
prepared solutions, see [6, 9]. In particular if Y2(t) = Y1(t) then

X∗1 C̃nU1 = U∗1 C̃
∗
nX1; i.e., X∗1 C̃nU1 = (X∗1 C̃nU1)∗ is Hermitian. (2.8)

In this case, the solution Y1 =

(
X1(t)
U1(t)

)
(or y1(t) =

(
x1(t)
u1(t)

)
) is called self-

conjugate. Note that for Y1 if its column vectors are linearly independent solutions
of (2.3) then these solutions form a basis for a conjoined family of solutions of



16 Q. BAO, G. WEI, A. ZETTL EJDE/SI/02

dimension r = r1 and the maximal dimension of a conjoined family of (2.3) is n.
Thus if Y1 satisfies (2.8) with rank(Y1) = n, then for brevity it is referred to as a

conjoined basis for (2.3). A conjoined basis Y (t) =

(
X(t)
U(t)

)
for (2.3) is said to be

oscillatory on I, if det(X(t)) has arbitrarily large zeros; otherwise, Y (t) =

(
X(t)
U(t)

)
is called non-oscillatory. System (2.3) is said to be oscillatory if every conjoined
basis for (2.3) is oscillatory.

Lemma 2.4. If Y =

(
X(t)
U(t)

)
∈ M2n,n(L1

loc(I)) is a conjoined basis for (2.3)

with rank(X(t)) = n on [α, β] ⊂ I, and t0 ∈ [α, β] ⊂ I, then Y0 =

(
X0(t)
U0(t)

)
∈

M2n,n(L1
loc(I)) is a solution of (2.5) on [α, β] ⊂ I if and only if

X0(t) = X(t)H(t), H(t) = K1 −
(∫ t

t0

X−1(r)B(r)X∗−1(r)dr
)
K0. (2.9)

In particular for (2.5) the associated function is

U0(t) = U(t)H(t) + C̃∗nX
∗−1(t)K0,

where K1 = X−1(t0)X0(t0) and K0 = {Y0, Y }.
Furthermore for K1 6= 0, Y0 is also a conjoined basis for (2.3) with rank(X0(t)) =

n if and only if K∗1K0 = K∗0K1. And in this case

X(t) = X0(t)H̃(t), H̃(t) = K−11 +
(∫ t

t0

X−10 (r)B(r)X∗0
−1(r)dr

)
K∗0 . (2.10)

Proof. Assume that Y =

(
X(t)
U(t)

)
is a conjoined basis for (2.3) on I. Let (2.9) hold,

where K0 = {Y0, Y } and K1 = X−1(t0)X0(t0). It is easily verified by differentiation

that Y0 =

(
X0(t)
U0(t)

)
is a solution of (2.5) on [α, β] ⊂ I. By using the assumption

X∗C̃nU = U∗C̃∗nX, we obtain

U0(t) = C̃−1n X∗−1(t)K0 + C̃−1n X∗−1(t)U∗(t)C̃∗nX0(t)

= C̃−1n X∗−1(t)K0 + U(t)X−1(t)X0(t)

= C̃−1n X∗−1(t)K0 + U(t)H(t).

Note that Y0 is a conjoined basis for (2.3) if and only if X∗0 C̃nU0 = U∗0 C̃
∗
nX0 and

rank(Y0(t)) = n. Since {Y, Y0} is independent of t, we obtain

X0(t0) = X(t0)K1, U0(t0) = C̃∗nX
∗−1(t0)K0 + U(t0)K1, t0 ∈ [α, β].

Furthermore for K1 6= 0, since

X∗0 C̃nU0 − U∗0 C̃∗nX0

= K∗1X
∗(t0)C̃n(C̃∗nX

∗−1(t0)K0 + U(t0)K1)

− (K∗0X
−1(t0)C̃n +K∗1U

∗(t0))C̃∗nX(t0)K1

= K∗1K0 −K∗0K1 +K∗1X
∗(t0)C̃nU(t0)K1 −K∗1U∗(t0)C̃∗nX(t0)K1,



EJDE-2023/SI/02 FRIEDRICHS EXTENSIONS 17

we obtain that Y0 is a conjoined basis for (2.3) with rank(X0(t)) = n if and only
if K∗1K0 = K∗0K1. Therefore for K1 6= 0, if Y0 is a conjoined basis for (2.3) with
rank(X0(t)) = n, then from (2.9), Y (t) can be written by Y0(t) as follows:

X(t) = X0(t)H̃(t), H̃(t) = K̃1 −
(∫ t

t0

X−10 (r)B(r)X∗0
−1(r)dr

)
K̃0,

U(t) = U0(t)H̃(t) + C̃∗nX
∗−1(t)K̃0,

where K̃1 = X−10 (t0)X(t0) and K̃0 = {Y, Y0}.
Note that H̃(t)H(t) = In. For t = t0 we have K̃1K1 = In. Also from {Y0, Y } =

−{Y, Y0}∗, it follows that K̃0 = −K∗0 . Thus (2.10) holds. This completes the
proof. �

Lemma 2.5. Suppose that B(t) is semi-positive definite a.e. on [α, β]. Then (2.3)
is disconjugate or non-oscillatory on [α, β] ⊂ I if and only if there exists a conjoined

basis Y =

(
X(t)
U(t)

)
∈M2n,n(L1

loc(I)) for (2.3) with rank(X(t)) = n on [α, β] ⊂ I.

Proof. Let Yj =

(
Xj

Uj

)
∈ M2n,n(L1

loc(I)), j = 1, 2 are solutions of (2.5) satisfying

X1(α) = 0 = X2(β) and U1(α) = In, U2(β) = C̃∗nX
∗
1
−1(β). From Definition 2.1 we

know that (2.3) is disconjugate on [α, β] (or on (α, β)) if and only if rank(X1(t)) = n
for t ∈ (α, β]. Also (2.3) is disconjugate on [α, β] (or on (α, β)) if and only if
rank(X2(t)) = n on [α, β). Thus {Y1, Y2} = −In with X1 6≡ 0 for subinterval of
(α, β). Note that matrix solutions Y1, Y2 both are self-conjugate. By (2.9) with
K1 = X−12 (α)X1(α) = 0, K0 = {Y1, Y2} = −In in Lemma 2.4, we have the relation

X1(t) = X2(t)

∫ t

α

X−12 (r)B(r)X∗2
−1(r)dr, t ∈ [α, β).

Hence we know that X1, X2 have the same algebraic sign on [α, β). Set

Y =

(
X(t)
U(t)

)
=

(
X1(t) +X2(t)
U1(t) + U2(t)

)
.

By taking β = 0 we find that Y is a self-conjugate solution of (2.5) and X(α) =
X2(α). Moreover note that {Y2, Y } = In. By Lemma 2.4 we have

X(t) = X2(t)(In +

∫ t

α

X−12 (r)B(r)X∗2
−1(r)dr), t ∈ [α, β).

Since B(t) is semi-positive definite a.e. on [α, β], it follows that X(t) is nonsingular
for t ∈ [α, β). Also note that X(β) = X1(β). We obtain rank(X(t)) = n on
[α, β] ⊂ I. This completes the proof. �

Note that the property of disconjugacy is λ-dependent. See also the next section.

3. Boundedness below and principal solutions

In this section we discuss the essential relation between the boundedness below of
minimal operators associated with the differential equation (1.3) and the existence
of principal solutions of system (2.5). For an excellent presentation of the relation
between the associated quadratic form and boundedness below also see [21, 22].



18 Q. BAO, G. WEI, A. ZETTL EJDE/SI/02

We now consider the functional for [α, β] ⊂ I

J(x1,x2;α, β) :=

∫ β

α

u∗2B(t)u1dt+

∫ β

α

x∗2C(t)x1dt−
∫ β

α

x∗2λW1(t)x1dt, (3.1)

where n-dimensional vector functions xj ,uj , j = 1, 2 belong locally to class L∞(I).
If these vector functions satisfy

C̃∗nx′j −A(t)xj −B(t)uj = 0, j = 1, 2, (3.2)

then we have the relation

J(x1,x2;α, β) = x∗2C̃nu1|βα +

∫ β

α

x∗2L1{x1,u1}dt. (3.3)

Moreover note that for J(x1,x2;α, β) = J∗(x2,x1;α, β) we have∫ β

α

y2My1dt−
∫ β

α

y1My2dt

= x∗2(β)C̃nu1(β)− u∗2(β)C̃∗nx1(β)− x∗2(α)C̃nu1(α) + u∗2(α)C̃∗nx1(α),

(3.4)

where

xj(t) = (yjy
[1]
j · · · y

[n−1]
j )T , uj(t) = (y

[2n−1]
j y

[2n−2]
j · · · y[n]j )T , j = 1, 2.

Thus we obtain

[y1, y2]βα = x∗2(β)C̃nu1(β)− u∗2(β)C̃∗nx1(β)− x∗2(α)C̃nu1(α) + u∗2(α)C̃∗nx1(α),

where

[y1, y2](t) = y∗2J̃2ny1, y1, y2 ∈ DQ (3.5)

is a Lagrange sesquilinear form.
We abbreviate J(x1,x1;α, β) to J(x1;α, β) in the following assertions. Define

X−1xj := zj , j = 1, 2, where zj belongs to locally class of L∞(I) and Y (t) =(
X(t)
U(t)

)
is a conjoined basis for (2.3) with rank(X) = n. In this case we have

J(x1,x2;α, β)

=

∫ β

α

(u2 − Uz2)∗B(t)(u1 − Uz1)dt+

∫ β

α

x∗2(C(t)− λW1(t))x1dt

+

∫ β

α

u∗2B(t)Uz1dt+

∫ β

α

z∗2U
∗B(t)u1dt−

∫ β

α

z∗2U
∗B(t)Uz1dt

=

∫ β

α

(u2 − Uz2)∗B(t)(u1 − Uz1)dt+

∫ β

α

x∗2(C(t)− λW1(t))x1dt

+

∫ β

α

(C̃nx′2 −A(t)x2)∗Uz1dt+

∫ β

α

z∗2U
∗(C̃nx′1 −A(t)x1)dt−

∫ β

α

z∗2U
∗BUz1dt

=

∫ β

α

(u2 − Uz2)∗B(t)(u1 − Uz1)dt+

∫ β

α

x∗2(C(t)− λW1(t))x1dt

+

∫ β

α

z∗2
′X∗C̃nUz1dt+

∫ β

α

z∗2U
∗C̃∗nXz′1dt+

∫ β

α

z∗2U
∗BUz1dt

=

∫ β

α

(u2 − Uz2)∗B(t)(u1 − Uz1)dt+

∫ β

α

x∗2(C(t)− λW1(t))x1dt



EJDE-2023/SI/02 FRIEDRICHS EXTENSIONS 19

+

∫ β

α

z∗2
′X∗C̃nUz1dt+

∫ β

α

z∗2U
∗C̃∗nXz′1dt+

∫ β

α

z∗2(C̃∗nX
′∗Uz1dt

= z∗2X
∗C̃nUz1|βα +

∫ β

α

(u2 − Uz2)∗B(t)(u1 − Uz1)dt,

where we used the condition X∗C̃nU = U∗C̃∗nX, (2.6) and (3.2).
Therefore if y ∈ D(T ′0), then for x(t) = (yy[1] · · · y[n−1])T , we obtain

J(x;α, β) =

∫ β

α

(u− Uz)∗B(t)(u− Uz)dt, x = Xz. (3.6)

We can easily obtain the following Theorem.

Theorem 3.1. Suppose that B(t) is semi-positive definite a.e. on [α, β] ⊂ I. Then
the following statements are equivalent:

(i) The pre-minimal operator T ′0 is bounded below with a bound γ.

(ii) there exists a value λ̂ such that

J(x;α, β) =

∫ β

α

(My − λ̂wy)y > 0,x(t) = (yy[1] · · · y[n−1])T , y ∈ D(T ′0). (3.7)

(iii) there exists a self-conjugate solution y(t) =

(
x(t)
u(t)

)
of (2.3) with x(t) 6≡ 0

for all t ∈ [α, β] ⊂ I.
(iv) (1.3) is disconjugate or non-oscillatory on [α, β] for all λ < γ, where γ is

a lower bound of T ′0.

Proof. (i)⇒(ii). If (i) holds, i.e., (T ′0y, y)w ≥ γ(y, y), y ∈ D(T ′0) for some γ ∈ R, then
(( 1
wM−γ)y, y)w ≥ 0 for all y ∈ D(T ′0). Hence for λ < γ we have (( 1

wM−λ)y, y)w >

(( 1
wM − γ)y, y)w ≥ 0. Then there exists a value λ̂, for λ̂ < λ we know that

J(x;α, β) =
∫ β
α

x∗L1{x,u} =
∫ β
α

(My − λ̂wy)y > 0 holds. Therefore (ii) holds.
(ii)⇒(iii). Let (ii) hold. Assume that x = x1 + x2, u = u1 + u2, where yj =(

xj
uj

)
, j = 1, 2 are solutions of (2.3) satisfying x1(α) = x2(β) = 0 and associated

function u1(α) = 1, u2(β) = −C̃∗nx∗1
−1(β). Clearly {y1,y2} = 1 for subinterval of

(α, β) throughout x1 6≡ 0. Since y1,y2 are self-conjugate solutions, y =

(
x
u

)
=(

x1 + x1

u1 + u2

)
is a self-conjugate solution of (2.3). Also note that Y (t) =

(
X(t)
U(t)

)
∈

M2n,n(L1
loc(I)) is a matrix solution of (2.5) if and only if x(t) = Xc,u(t) = Uc is

a vector solution of (2.3) for every nonzero constant vector c ∈ Cn. By Lemma
2.5 we know that x(t) = Xc is invertible on [α, β] ⊂ I. Thus we have x(t) 6≡ 0 for
t ∈ [α, β]. Therefore (iii) holds.

(iii)⇒(iv). Let (iii) hold. It is clear from Lemma 2.5 that system (2.3) is discon-
jugate. Hence (1.3) is disconjugate. Therefore (iv) holds.

(iv)⇒(i). Let (iv) hold. Since (1.3) can be transformed into the Hamiltonian
system (2.3), there exists a conjoined basis for (2.3) with rank(X) = n and thus
(3.6) holds. By the assumption B(t) ≥ 0 a.e. on [α, β] ⊂ I we have J(x;α, β) > 0,
i.e., (ii) holds. Suppose now that (( 1

wM −λ)y, y)w < 0 for y ∈ D(T ′0), i.e., (( 1
wM −

γ)y, y)w < 0. From (3.3) we have J(x;α, β) =
∫ β
α

x∗L1{x,u} =
∫ β
α
y(My−λwy) <



20 Q. BAO, G. WEI, A. ZETTL EJDE/SI/02

0 for y ∈ D(T ′0) on [α, β]. This reaches a contradiction. Therefore (i) holds. This
completes the proof. �

Remark 3.2. It is easy to show that if y1 =

(
x1

u1

)
is a solution of (2.3) and

y2 ∈ D(T ′0) on [α, β], then J(x1,x2;α, β) = 0. From Theorem 3.1 and Definition
2.1 we know that if (2.3) is disconjugate on I, then

J(x; a, b) ≥ 0, x(t) = (yy[1] · · · y[n−1])T , y ∈ D(T ′0),

and the equality sign holds if and only if there exists a n-dimensional vector

function u1 such that

(
x1

u1

)
is a solution of (2.3) and x = kx1, k ∈ C,x1(t) =

(y1y
[1]
1 · · · y1[n−1])T , y1(a) = 0 = y1(b), and y1(t) 6≡ 0, t ∈ I.

Let B(t) ≥ 0 a.e. on I. Suppose that Y =

(
X(t)
U(t)

)
is a conjoined basis for (2.3)

with rank(X) = n on s ≤ t < b, s ∈ I. Clearly

Ss(t) :=

∫ t

s

X−1(r)B(r)X∗−1(r)dr (3.8)

is positive for s < t < b and

lim
t→b

S−1s (t) = Φ exists, (3.9)

where Φ depends on s and the matrix function X(t).
If Φ = 0, then Y (t) is called a principal solution (or recessive solution) of (2.5).

See [6]. In this case Φ = 0 can be expressed as∫ t

s

X−1(r)B(r)X∗−1(r)dr → +∞ as t→ b, (3.10)

in the sense that

‖
∫ t

s

X−1(r)B(r)X∗−1(r)drc‖ → +∞ as t→ b,

uniformly for all nonzero constant vectors c ∈ Cn. For n = 1 this is corresponding
to [5, Lemma 4] with B = −c̄12q12 and [18, Thereom 2.2] with B = p−1.

Similar with Reid’s concept [22] on the principal solution of Hamiltonian system,
we have the following Theorem.

Theorem 3.3. Suppose that for some λ ∈ R and some e ∈ (a, b) system (2.5) is

disconjugate on an interval (e, b). Then for each s ∈ (e, b), let Ys =

(
Xs

Us

)
∈

M2n,n(L1
loc(I)) be a matrix solution of (2.5) satisfying the boundary condition

Xs(e) = C̃n, Xs(s) = 0. Moreover

Yb(t) := lim
s→b

Ys(t) :=

(
Xb

Ub

)
(3.11)

exists, uniformly for t in compact subset of [e, b), and is a principal solution of
system (2.5) at b. In particular, Yb(t) is a conjoined basis for (2.3) on (e, b). The
principal solution at a is described in a similar manner.

It is readily known that the following Theorem holds.



EJDE-2023/SI/02 FRIEDRICHS EXTENSIONS 21

Theorem 3.4. Suppose that B(t) is semi-positive definite a.e. on I = (a, b), and
(2.3) is disconjugate on I. Then

(i) (2.5) possesses a principal solution Y =

(
X(t)
U(t)

)
at b.

(ii) Another solution Y0 =

(
X0(t)
U0(t)

)
is also a principal solution at b if and only

if

X0 = XK1, U0 = UK1,

where K1 is a constant nonsingular matrix.

(iii) Let Y =

(
X(t)
U(t)

)
be a conjoined basis for (2.3) with det(X(t)) 6= 0 for

t near b. Assume that Y1 =

(
X1(t)
U1(t)

)
is any solution of (2.5), which is

not multiple of Y and det(X1(t)) 6= 0 for t near b. Then Y is a principal
solution at b if and only if K0 = {Y1, Y } is nonsingular and

X−11 (t)X(t)→ 0, as t→ b. (3.12)

Samiolar argument holds for the principal solution at a.

Proof. (i). Since B(t) is semi-positive definite a.e. on I = (a, b) and (2.3) is
disconjugate on I, from Lemma 2.5 we know that there exists a conjoined basis

Y0 =

(
X0(t)
U0(t)

)
such that rank(X0(t)) = n on I and the symmetric matrix

S0(t) :=

∫ t

s

X−10 (r)B(r)X∗0
−1(r)dr

is positive increasing function, and hence invertible for t > s. Thus it follows that
its inverse is a decreasing function of t and there exists a symmetric matrix Φ0 ≥ 0
such that

S−10 (t)→ Φ0, s→ b.

Set

X(t) = X0(t)(In − S0(t)Φ0),

U(t) = U0(t)(In − S0(t)Φ0)− C̃∗nX∗0
−1(t)Φ0.

From Lemma 2.4 we know that Y =

(
X(t)
U(t)

)
is a conjoined basis for (2.3). Moreover

rank(X) = n for every t ≥ s, since X(s) = X0(s) and for t > s,

X(t) = X0(t)S0(t)(S−10 (t)− Φ0)

is a product of nonsingular matrices. Note that

S−1s (t) = S−10 (t)− Φ0.

We obtain that as s→ b, S−1s (t)→ 0.
From (3.10) we know that Y (t) is a principal solution at b, i.e., (i) holds. And

for any constant matrix K, note that

(Ss(t) +K)−1 = S−1s (t)(In +KS−1s (t))−1 → 0.

We know that there still exists a principal solution y at b if we change the lower

limit integration in
∫ t
s
X−1(r)B(r)X∗−1(r)dr from s to t0 ∈ I.



22 Q. BAO, G. WEI, A. ZETTL EJDE/SI/02

(ii) Suppose that

S−10 (t) =
(∫ t

s

X−10 (r)B(r)X∗0
−1(r)dr

)−1
→ Φ0, s→ b.

In accord with (3.10), Y0 is a principal solution if and only if Φ0 = 0. From (2.9)
in Lemma 2.4 we know that

X0(t) = X(t)H(t), H(t) = K1 − Ss(t)K0,

U0(t) = C̃∗nX
∗−1K0 + U(t)H(t),

where Y =

(
X(t)
U(t)

)
is a principal solution, Ss(t) is described by (3.8) and the

constants K1,K0 are the same as in Lemma 2.4. If Y0 is a principal solution, then
Y is also given by Y0, i.e., (2.10)) holds. Thus we have

(K1 − Ss(t)K0)(K−11 + S0(t)K∗0 ) = In,

i.e.,

(K1 − Ss(t)K0)S0(t)K∗0 = Ss(t)K0K
−1
1 .

Also from Lemma 2.4 we have K∗0K1 = K0K
∗
1 , i.e., K1K

−1
0 = K∗0

−1K∗1 . Hence

(K1 − Ss(t)K0)S0(t) = Ss(t)K
∗
1
−1,

viz.,

S−10 (t) = K∗1 (S−1s (t)K1 −K0).

From this and the fact that S−1s (t)→ 0, s→ b, we obtain Φ0 = −K∗1K0. Note that
K1 is nonsingular. Φ0 = 0 if and only if K0 = 0, i.e., if and only if

X0(t) = X(t)K1, U0(t) = U(t)K1.

So result (ii) follows.

(iii) Assume that Y1 =

(
X1(t)
U1(t)

)
is any solution of (2.5), which is not multiple

of Y and det(X1(t)) 6= 0 for t near b. Suppose that (3.12) holds. From (2.9) in
Lemma 2.4 we know that

(K1 − Ss(t)K0)−1 = X−11 (t)X(t)→ 0, s→ b,

where K0 = {Y1, Y } and K1 = X−1(s)X1(s) is invertible if s is sufficiently near
b. Together with (3.10) we obtain that Y is a principal solution at b. Conversely,
suppose that Y is a principal solution at b and K0 6= 0. From (3.10) it follows that

lim
s→b

S−1s (t) = 0.

This and (2.9) in Lemma 2.4 it yield

X−11 (t)X(t) = (K1 − Ss(t)K0)−1 = (S−1s (t)K1 −K0)−1S−1s (t)→ 0, s→ b.

The same argument holds for principal solution at a. This completes the proof. �

We can now assert that there are precisely n principal solutions yj , j = 1, . . . , n,
at the endpoints for the differential equation (1.3) by the Hamiltonian system (2.3).
For example, the j-th principal solution yj at b of (1.3) is the j-th column of
principal solution Yb for (2.5), it is given by

yj(t) = lim
s→b

yjs(t), (3.13)



EJDE-2023/SI/02 FRIEDRICHS EXTENSIONS 23

and yjs(t) is the solution of boundary value problem

My = λwy, t ∈ (e, s), x(e) = (0 · · · 0 cj,2n+1−j︸ ︷︷ ︸
j−th

0 · · · 0)T , x(s) = 0,

where cj,2n+1−j is the j row 2n+ 1− j column element of C2n = (cr,s)
2n
r,s=1 and x

belongs to y, i.e., y =

(
x
u

)
, which is a solution of Hamiltonian system (2.3).

4. Singular symmetric differential operators

In this section we present the explicit characterization of symmetric operators
generated by the differential expression (1.3). Assume that both endpoints a, b
of I are singular for (1.3). Assume that da, db denote the deficiency indices of
the minimal operator generated by the equation (1.3) on (a, e) and (s, b), respec-
tively. Clearly da is also equivalent to the deficiency indices of the minimal operator
generated by the equation (1.3) on (a, s). Suppose that d denotes the deficiency
indices of the minimal operator generated by the equation (1.3) on I. Recall that

d = da + db − 2n. Now we fix a number λ̂ ∈ R and assume that there exists a
λa < λ̂ ∈ R such that (1.3) has da linearly independent solutions near a in H. We
denote them as

y11, y12, . . . , y1,da−n, y1,da−n+1, . . . , y1,n, y1,n+1, . . . , y1,da . (4.1)

Now suppose that in (4.1) the functions y11, y12, . . . , y1,da−n, y1,n+1, . . . , y1,da are
LC type solutions on (a, e) and these ma := 2da − 2n solutions can contribute to
the symmetric and self-adjoint boundary conditions. For the details of LC type
solutions of differential equations, see [11, 25, 26, 30]. Assume that [y11, y11](a) · · · [y11, y1,ma

](a)
... · · ·

...
[y1,ma

, y11](a) · · · [y1,ma
, y1,ma

](a)

 = Cma
, (4.2)

where y1,da−n+j = y1,n+j , 1 ≤ j ≤ da − n, and

Cma
=

(
0 Ĉda−n

−Ĉ∗da−n 0

)
(4.3)

satisfies the properties (1.1), that is,

crscrs = 1, for r + s = ma + 1, 1 ≤ r ≤ da − n,
crs = 0, otherwise.

(4.4)

Therefore the matrix ([y1i, y1j ](a))1≤i,j≤ma
is nonsingular.

Similarly, assume that there exists a λb < λ̂ ∈ R such that (1.3) has db linearly
independent solutions near b in H. We denote them as

y21, y22, . . . , y2,db−n, y2,db−n+1, . . . , y2,n, y2,n+1, . . . , y2,db . (4.5)

Suppose that in (4.5) the functions y21, y22, . . . , y2,db−n, y2,n+1, . . . , y2,db are LC
type solutions on (e, b) and these mb := 2db − 2n LC type solutions at b satisfy [y21, y21](b) · · · [y21, y2,mb

](b)
... · · ·

...
[y2,mb

, y21](b) · · · [y2,mb
, y2,mb

](b)

 = Cmb
, (4.6)



24 Q. BAO, G. WEI, A. ZETTL EJDE/SI/02

where y2,db−n+j = y2,n+j , 1 ≤ j ≤ db − n, and

Cmb
=

(
0 Ĉdb−n

−Ĉ∗db−n 0

)
. (4.7)

Here Ĉdb−n ∈Mdb−n(C) is a skew-diagonal unitary matrix defined by (4.4) in simi-
lar way, i.e., Cmb

satisfies properties (1.1), and thus the matrix ([y2i, y2j ](b))1≤i,j≤mb

is nonsingular. Therefore these mb solutions can contribute to the symmetric and
self-adjoint boundary conditions at b.

Notice that we have the decomposition

DQ = D0 ⊕ span{y11, y12, . . . , y1,da−n, y1,n+1, . . . , y1,da}
⊕ span{y21, y22, . . . , y2,db−n, y2,n+1, . . . , y2,db}.

We introduce the following notation:

Yda−n(a) = ([y, y11](a) · · · [y, y1,da−n](a))T ∈ Cda−n,

Ŷda−n(a) = ([y, y1,da−n+1](a) · · · [y, y1,ma ](a))T ∈ Cda−n,

Ydb−n(b) = ([y, y21](b) · · · [y, y2,db−n](b))T ∈ Cdb−n,

Ŷdb−n(b) = ([y, y2,db−n+1](b) · · · [y, y2,mb
](b))T ∈ Cdb−n.

Assume that

Ya,b =


Yda−n(a)

Ŷda−n(a)
Ydb−n(b)

Ŷdb−n(b)

 =

(
Ya
Yb

)
, Ŷa,b =


Yda−n(a)
Ydb−n(b)

Ŷda−n(a)

Ŷdb−n(b)

 ∈ C2d. (4.8)

It is known [26] that Ya,b runs through the entire space C2d as y runs through
the maximal domain DQ. For a given matrix U ∈ Ml,2d(C) with rank(U) = l, 0 ≤
l ≤ 2d, if an operator T = T (U) is defined by

T (U)y = TQy,

y ∈ D(T (U)) = {y ∈ DQ : UYa,b = 0},
(4.9)

then U is a boundary matrix of the operator T and UYa,b = 0 is its boundary
condition. It is obvious that when l = 0 we have rank(U) = 0 and T = TQ and
when l = 2d we have rank(U) = 2d and T = T0. Also it is clear that, for any
nonsingular matrix G of order l, the operators T (M) and T (GM), corresponding
to the boundary matrices M and GM respectively, are the same.

For which matrices U is T = T (U) a symmetric operator in L2
w(I)? This question

is answered by the next Theorem.

Theorem 4.1. Suppose that M = MQ, Q ∈ Z2n(I), I = (a, b), −∞ ≤ a < b ≤ ∞,
is C-symmetric, w is a weight function. Let U = (AB), A ∈ Ml,ma

(C), B ∈
Ml,mb

(C) be a boundary condition matrix with rank(U) = l, 0 ≤ l ≤ 2d of (4.7))
and let

R = ACma
A∗ −BCmb

B∗, r = rank(R).

Then we have

(1) If l < d, then T (U) is not symmetric.
(2) If l = d, then T (U) is self-adjoint (and hence also symmetric) if and only

if r = 0.
(3) Let l = d+ s, 0 < s ≤ d. Then T (U) is symmetric if and only if r = 2s.



EJDE-2023/SI/02 FRIEDRICHS EXTENSIONS 25

For a proof of the above theorem see [26, Theorems 6 and 11]. From the above
Theorem if U satisfies rank(UGU∗) = 2(l − d), d < l ≤ 2d with

G =

(
Cma

0
0 −Cmb

)
, (4.10)

then the operator T = T (U) defined by (4.9) is symmetric. Now we write

U := V J, V = (V1V2), Vj ∈Ml,d(C), j = 1, 2,

in (4.9), and set

Ĝ = JGJ∗, (4.11)

with

J =


Ida−n 0 0 0

0 0 Idb−n 0
0 Ida−n 0 0
0 0 0 Idb−n

 ,

where Id−n denotes the (d− n)× (d− n) identity matrix. Also N (V ) denotes the
null space of the matrix V and R(V ) denotes the range of the matrix V .

Note that Cma
and Cmb

have the form (4.3) and (4.7), respectively. We have

Ĝ =

(
0 G1

−G∗1 0

)
, G1 =

(
Ĉda−n 0

0 −Ĉdb−n

)
.

Remark 4.2. Here Ĝ satisfies Ĝ−1 = −Ĝ = Ĝ∗.

Following are our new characterizations of symmetric operators for symmetric
differential equation (1.3) we have the following result.

Lemma 4.3. Suppose d ≤ l ≤ 2d, where d denotes the deficiency indices of the
minimal operator on I = (a, b). Then the operator T which is defined on

D(T ) = {y ∈ DQ : V Ŷa,b = 0, V ∈Ml,2d(C)}, (4.12)

is a symmetric operator with l dimensional restriction of TQ if and only if there
exists a matrix N ∈M(2d−l),2d(C) satisfying

rank(N) = 2d− l, NĜN∗ = 0, (4.13)

and V is a complete solution of the matrix equation

NV ∗ = 0, (4.14)

i.e., V satisfies the equation (4.14) with rank(V ) = l. Moreover, the domain of its
adjoint operator T ∗ is characterized by

D(T ∗) = {y ∈ DQ : NĜŶa,b = 0}, (4.15)

where Ĝ ∈M2d(C) is defined by (4.11).

Proof. Assume that N ∈ M(2d−l),2d(C) satisfies (4.13) and (4.14). Now we show
that T defined by (4.12) is symmetric. Note that T (U) = T (V J) with U = V J
and JN (U) = N (V ), from [26, Lemma 14] and Theorem 4.1 we only need to prove

that N (V ) ⊂ R(ĜV ∗), i.e.,

Ẑ∗a,bĜŶa,b = 0



26 Q. BAO, G. WEI, A. ZETTL EJDE/SI/02

for all y, z ∈ D(T ) with

Ŷa,b =


Yda−n(a)
Ydb−n(b)

Ŷda−n(a)

Ŷdb−n(b)

 , Ẑa,b =


Zda−n(a)
Zdb−n(b)

Ẑda−n(a)

Ẑdb−n(b)

 .

If y, z ∈ D(T ), in view of (4.14) there exists a column vector c ∈ C2d−l such that

Ẑa,b = N∗c and a column vector ĉ ∈ C2d−l such that Ŷa,b = N∗ĉ. This yields

Ẑ∗a,bĜŶa,b = c∗(NĜN∗)ĉ = 0. (4.16)

Since V is a complete solution of matrix equation (4.14), it follows that rank(V ) = l.
From Von Neumann Theorem in [26] we also see that T is a l dimensional restriction
of the maximal operator TQ. Clearly the converse also holds. In fact, if T defined by
(4.12) is symmetric, then from Theorem 4.1 we obtain that rank(UGU∗) = 2(l−d),
d ≤ l ≤ 2d, with U = V J . From Naimark Patching Lemma [26, Lemma 6] and
(4.16) we know that there exists a matrix N ∈ M(2d−l),2d(C) such that (4.13) is
establish. Moreover 4.14 also holds.

Next we prove that (4.15) holds. Note that

D0 ⊆ D(T ) ⊆ D(T ∗) ⊆ DQ.

since T is an l dimensional restriction of the maximal operator TQ, this shows
that the deficiency index of T is (l − d) and, therefore, T ∗ is a 2d − l dimensional
restriction of the maximal operator TQ. On the other hand, we obtain

0 = (Ty, z)w − (y, T ∗z)w = Z∗a,bGYa,b = Ẑ∗a,bĜŶa,b,

where y ∈ D(T ) and z ∈ D(T ∗). It should be noted that, for any c ∈ C2d−l, there

exists a function y ∈ D(T ) such that Ŷa,b = N∗c. It leads to (NĜ∗)Ẑ∗a,b = 0 if

z ∈ D(T ∗). By the fact rank(NĜ∗) = 2d − l, we know that the dimension of the

space solutions of equation (NĜ∗)Ẑ∗a,b = 0 is l. Therefore, combining this with

Ĝ∗ = −Ĝ, we obtain that (4.15) holds. The proof is complete. �

Remark 4.4. Suppose that B(t) is semi-positive definite a.e. on I and system (2.3)
is disconjugate on I for some λ < γ, where γ is a lower bound of T ′0. Let

xj(t) = (y1jy
[1]
1j · · · y

[n−1]
1j )T , uj(t) = (y

[2n−1]
1j y

[2n−2]
1j · · · y[n]1j )T , 1 ≤ j ≤ da.

Thus system (2.3) has da linearly independent solutions y1,y2, . . . ,yda near a with

yj =

(
xj
uj

)
, 1 ≤ j ≤ da in L2

W (I). Note that a principal solution is an LC type

solution of Hamiltonian system. (See also [29] and the references therein.) Set

Y (t) =
(
Ỹa(t) Ŷ (t)

)
∈M2n,da(L1

loc(I)),

where Ỹa(t) = (y1y2 · · ·yn) is a principal solution at a of (2.5) such that Ỹ (s) =

Ỹe(s) =

(
C̃n
0

)
, X̃e(e) = 0, and lime→a Ỹe(t) = Ỹa(t). Furthermore Ŷ (t) =

(yn+1yn+2 · · ·yda) ∈ M2n,(da−n)(L
1
loc(I)) satisfies Ŷ (s) =

(
0

Ûn(s)

)
with da − n



EJDE-2023/SI/02 FRIEDRICHS EXTENSIONS 27

rows of Ûn(e) forming a nonsingular (da − n) × (da − n) sub-matrix. We may as-

sume Û(s) =

(
Ĉda−n
Ǔn(s)

)
with Ǔn a (2n− da)× (da − n) matrix. Hence we see that

for 1 ≤ i ≤ da − n,yi(s) =

(
xi(s)

0

)
with xi(s) = (0 · · · 0 ci,2n+1−i︸ ︷︷ ︸

i−th

0 · · · 0)T and for

1 ≤ j ≤ da − n,yn+j(s) =

(
0

un+j(s)

)
with

un+j(s) = (0 . . . 0 cj,ma+1−j︸ ︷︷ ︸
j−th

0 · · · 0uj,1 · · ·uj,(2n−d))T . Thus from (3.5) it follows

that

y∗n+j J̃2nyi = u∗n+jC̃
∗
nxi = ci,ma+1−iδi,da−n+1−j ,

where δ is the Kronecker delta and

y∗n+iJ̃2nyn+j = 0 = y∗i J̃2nyj .

Therefore there are da−n principal solutions which are contributed to the symmetric
and self-adjoint boundary conditions at a and da − n non-principal solutions are
contributed to the symmetric and self-adjoint boundary conditions at a. Similar
assertions hold at endpoint b. Otherwise the Wronskian matrices (4.2) and (4.6)
may be singular.

We decompose N = (N1N2) with matrices N1, N2 ∈M(2d−l),d(C). Then

NĜ = (−N2G
∗
1N1G1).

Note that D(T ) ⊂ D(T ∗). This implies that NĜ can be represented by a linear
combination of row vectors of V . By transformation of rows of V , we can rewrite
V as

V =

(
V11 V12
−N2G

∗
1 N1G1

)
, (4.17)

where V11, V12 ∈M(2l−2d),d(C).

Theorem 4.5. Let T be a symmetric operator as stated in Lemma (4.3). Then V
can be represented as

V =


V̂11 V̂12
V̂21 0

−N̂12G
∗
1 0

−N̂22G
∗
1 N̂21G1

 , V̂rs ∈M(l−d),d(C), r, s = 1, 2; (4.18)

with rank(V̂12) = rank(V̂21) = l − d, and N̂21, N̂22 ∈ Mr1,d(C) with rank(N̂21) =

r1 = rank(N̂22), r1 ≤ 2n − l. Here 0 is a (2d − l − r1) × d zero matrix and

N̂12 ∈M(2d−l−r1),d(C).

Proof. For N = (N1N2) with N1, N2 ∈ M(2d−l),d(C) being given, if rank(N1) =
r1(≤ (2d− l)), and B ∈M(d−r1),d(C) is a complete solution of the matrix equation
N1B

∗ = 0, it is easy to see that rank(B) = d− r1. Since V is a complete solution
of matrix equation NV ∗ = 0, it follows that the row vectors of (B0) can be linear



28 Q. BAO, G. WEI, A. ZETTL EJDE/SI/02

expressed by the row vectors of V . This implies that there exists a matrix (Ṽ11Ṽ12)
and a nonsingular matrix of order l such that

V →
(
Ṽ11 Ṽ12
B 0

)
, (4.19)

with 0 ∈M(d−r1),d and rank(Ṽ11) = rank(Ṽ12) = l− d+ r1. Here the notation “→”
denotes the left multiplication process of a nonsingular matrix.

On the other hand, in view of rank(N1) = r1, we obtain by elementary matrix
transformation of the rows that

N →
(

0 N̂12

N̂21 N̂22

)
= (N1N2),

where rank(N̂12) = 2d − l − r1, N̂21, N̂22 ∈ Mr1,d(C), 0 is a (2d − l − r1) × d zero
matrix. Moreover, it is easy to see that

(−N2G
∗
1N1G1) =

(
−N̂12G

∗
1 0

−N̂22G
∗
1 N̂21G1

)
.

Since NĜN∗ = 0 and rank(N̂21) = r1, we obtain that the row vectors of (−N̂12G
∗
10)

and (−N̂22G
∗
1N̂21G1) can be linear expressed by the row vectors of (B0) and

(Ṽ11Ṽ12) in (4.19), respectively. This together with (4.17) and (4.19) yields (4.18)

and rank(V̂12) = l − d. This also shows that rank(V̂21) = l − d. The proof is
complete. �

Remark 4.6. Let l = 2d in Theorem 4.5. In this case N is singular, T = T0, and
(4.18) can be reduced to

V =

(
V̂11 V̂12
V̂21 0

)
, V̂rs ∈Md(C), r, s = 1, 2; (4.20)

with rank(V̂12) = rank(V̂21) = d. Moreover by transformation of rows (4.20) can
be transformed into

V →
(
Id 0
0 Id

)
→
(
Ima 0

0 Imb

)
→ J.

Thus in this case Theorem 4.5 can be reduced to [26, Theorem 4.4.1].
Furthermore if ma = 2n = mb, i.e., l = 4n, then we have

D(T0) = {y ∈ DQ : Yn(a) = Yn(b) = 0, Ŷn(a) = Ŷn(b) = 0},

which is the characterization of domain of the minimal operator of (1.3) when a, b
are LC endpoints or regular endpoints. In this case from Remark 4.4 we know that
all principal solutions at endpoints a, b can contribute to the characterization of
boundary conditions of the minimal symmetric operator.

Note that if B(t) be semi-positive definite a.e. on I = (a, b) and (2.3) is discon-
jugate on I, then Theorem 3.1 and Remark 3.2 ensure that the minimal operator
associated with (1.3) is bounded below. So we infer every symmetric extension T
of T0 is also bounded below and thus the Friedrichs extension TF of T exists. We
fix a value of λ, say λ̂. Let γ̂ > 0 be a lower bound of symmetric operator T . Then(

(T − λ̂)y, y
)
w
≥ γ̂(y, y)w, γ̂ ≤ γ, (4.21)



EJDE-2023/SI/02 FRIEDRICHS EXTENSIONS 29

where γ is a lower bound of T0. Define a semi-bounded sesquilinear form on D(T ):

l(y, z) = ((T − λ̂)y, z)w, y, z ∈ D(T ). (4.22)

If

〈·, ·〉s =
(
(T − λ̂)y, z

)
w

+ (1− γ̂)(y, z)w, y, z ∈ D(T ),

then (D(T ), 〈·, ·〉s) is a pre-Hilbert space. Let Hs be a ‖ · ‖s completion of D(T ),
with ‖ · ‖s denotes the norm induced by the inner product 〈·, ·〉s on H. Then Hs is
a Hilbert space. Let

l̄(y, z) = ((T − λ̂)y, z)w, y, z ∈ Hs.

Consequently l̄(y, z) = l(y, z) for y, z ∈ D(T ) and l̄ is the closure of l. Combined
with (4.21) we infer that l and l̄ are both bounded below by γ̂. Thus according to
Kato’s result in [12, p. 352] we have y ∈ D(TF ) ⊆ D(l̄). In particular this implies
that for any y ∈ D(TF ) there exists a sequence {ym}∞m=1 in D(T ) such that

lim
m→∞

l̄(y − ym, y − ym) = 0. (4.23)

In the next section we characterize explicitly the boundary conditions of Friedrichs
extensions TF for any symmetric operators T which are bounded below for (1.3)
applying the property (4.23).

5. Characterization of Friedrichs extensions

In this section we find, explicitly, a boundary condition which determines the
Friedrichs extension of every symmetric operator which is bounded below. All the
symbols and notation in this section are the same as those in Section 4. Next we
state and prove our main results.

Theorem 5.1. Assume that B(t) is semi-positive definite a.e. on I = (a, b) and
system (2.3) is disconjugate on I for some λ. Let y1a, y2a, . . . , yda−n,a be the prin-
cipal solutions at a of (1.3) and v1a, v2a, . . . , vda−n,a be the non-principal solutions
at a of (1.3); Let y1b, y2b, . . . , ydb−n,b be the principal solutions at b of (1.3) and
v1b, v2b, . . . , vdb−n,b be the non-principal solutions at b of (1.3). Assume that T
defined by

D(T ) = {y ∈ DQ : V Ŷa,b = 0, V ∈Ml,2d(C)}, quadd < l ≤ 2d, (5.1)

is a symmetric extension of T0, where Ŷa,b = (Yda−n(a)Ydb−n(b)Ŷda−n(a)Ŷdb−n(b))T

with

Yda−n(a) = ([y, y1a](a) · · · [y, yda−n,a](a))T ,

Ŷda−n(a) = ([y, v1a](a) · · · [y, vda−n,a](a))T ,

Ydb−n(b) = ([y, y1b](b) · · · [y, ydb−n,b](b))T ,

Ŷdb−n(b) = ([y, v1b](b) · · · [y, vdb−n,b](b))T .

Then V has the form (4.18) in Theorem 4.5 and T is bounded below. Furthermore
the Friedrichs extension TF of T is characterized by

D(TF ) =
{
y ∈ D(T ∗) : V̂21

(
Yda−n(a)
Ydb−n(b)

)
= 0
}

(5.2)



30 Q. BAO, G. WEI, A. ZETTL EJDE/SI/02

=
{
y ∈ DQ : V̂21

(
Yda−n(a)
Ydb−n(b)

)
= 0, NĜŶa,b = 0

}
, (5.3)

where N ∈M(2d−l),2d(C) satisfying NĜN∗ = 0 is a complete solution of the matrix
equation NV ∗ = 0.

Proof. It is clear that T is bounded below from Theorem 3.1. From Theorem 4.5
we obtain the boundary matrix V of T can be rewritten as

V =


V̂11 V̂12
V̂21 0

−N̂12G
∗
1 0

−N̂22G
∗
1 N̂21G1

 , V̂rs ∈M(l−d),d(C), r, s = 1, 2.

with rank(V̂12) = rank(V̂21) = l − d, and(
0 N̂12

N̂21 N̂22

)
= (N1N2) = N,

where rank(N̂12) = 2d − l − r1, N̂21, N̂22 ∈ Mr1,d(C), 0 is a (2d − l − r1) × d zero
matrix. We define an operator Ts with domain D(Ts):

Ds :=
{
y ∈ D(T ∗) : V̂21

(
Yda−n(a)
Ydb−n(b)

)
= 0
}
.

It is easy to ensure that D(T ) ⊂ Ds ⊂ DQ. Denote by

A =

(
V̂21 0
−N2G

∗
1 N1G1

)
.

Thus Ds = {y ∈ DQ : AŶa,b = 0}. Since T is symmetric, from Lemma 4.3 we have

NV ∗ = 0. It follows from (4.18) that N1V̂
∗
21 = 0. Furthermore, by Theorem 4.5 we

obtain rank(A) = d. Together with G1G
∗
1 = Id we have

AĜA∗ =

(
V̂21 0
−N2G

∗
1 N1G1

)(
0 G1

−G∗1 0

)(
V̂ ∗21 −G1N

∗
2

0 G∗1N
∗
1

)
=

(
0 V̂21G1

−N1G1G
∗
1 −N2G

∗
1G1

)(
V̂ ∗21 −G1N

∗
2

0 G∗1N
∗
1

)
=

(
0 V̂21G1G

∗
1N
∗
1

−N1G1G
∗
1V̂
∗
21 N1G1G

∗
1G1N

∗
2 −N2G

∗
1G1G

∗
1N
∗
1

)
=

(
0 V̂21N

∗
1

−N1V̂
∗
21 NĜN∗

)
= 0.

Thus we know that the operator Ts is a self-adjoint extension of T .
Next we prove that Ts is the Friedrichs extension of T . Let y ∈ D(TF ). From

result (4.23) with [24, Theorem 5.38] we see that y ∈ D(T ∗) and there exists a
sequence {ym} ⊂ D(T ) such that

lim
m→∞

l̄(y − ym, y − ym) = 0. (5.4)

Since all the functions ym belong to D(T ), it follows that V Ŷm,a,b = 0. This implies

0 = (V̂210)Ŷm,a,b = V̂21

(
Ym,da−n(a)
Ym,db−n(b)

)
.



EJDE-2023/SI/02 FRIEDRICHS EXTENSIONS 31

Note that on [α, β] ⊂ I we have

l̄(y, y) = −x∗C̃nu|βα +

∫ β

α

u∗B(t)udt+

∫ β

α

x∗C(t)xdt−
∫ β

α

x∗λW1(t)xdt,

with u = B−1(t)(C̃∗nx′ −A(t)x), where

x(t) = (yy[1] · · · y[n−1])T , u(t) = (y[2n−1]y[2n−2] · · · y[n])T .

We infer that ∫ β

α

u∗B(t)udt ≤ C0((l̄ +K)y, y), (5.5)

where C0,K ∈ C. By applying (5.5) to ym − y instead of y, we obtain∫ β

α

(um − u)∗B(t)(um − u)dt ≤ C0((l̄ +K)(ym − y), ym − y),

where um = B−1(t)(C̃∗nx′m −A(t)xm).
Since B(t) ≥ 0 a.e. on I, from (5.4) we obtain that

lim
m→∞

∫ β

α

(um − u)∗B(t)(um − u)dt = 0.

Thus we have um → u as m→∞ for all t ∈ [α, β] ⊂ I on D(T ∗) ⊂ DQ. Therefore
limm→∞ um(α) = u(α) on D(T ∗) ⊂ DQ. Moreover

[ym − y, y1i](α) = {ym − y,y1i}(α)

= x∗1i(α)C̃n(um − u)(α)− (um − u)∗(α)C̃∗nx1i(α).

Thus we infer that

lim
m→∞

[ym − y, y1i](α) = 0.

Hence we have limm→∞[y−ym, y1i](a) = 0, i = 1, 2, . . . , da−n. Similarly, we obtain
limm→∞[y − ym, y2j ](b) = 0, j = 1, 2, . . . , db − n. Therefore

lim
m→∞

(
Ym,da−n(a)
Ym,db−n(b)

)(
Yda−n(a)
Ydb−n(b)

)
.

Then V̂21

(
Yda−n(a)
Ydb−n(b)

)
= 0 shows that y ∈ Ds and so D(TF ) ⊂ Ds. On the

other hand, we have proved that Ds is the domain of a self-adjoint extension Ts
of T . Consequently, the self-adjointness of TF leads to Ts = TF . The proof is
complete. �

Remark 5.2. From Theorem 5.1 we know that there are da−n principal solutions
at a and db−n principal solutions at b which contribute to the Friedrichs extension
of any symmetric operator which is bounded below. Moreover, if the differential
equation (1.3) is regular, then da = 2n = db in Theorem 5.1 i.e., d = 2n. In this
case the result of Theorem 5.1 is reduced to the Friedrichs extension of any even
order regular symmetric differential operators, see [4, Theorem 4.7].

Corllary 5.3. Let B(t) be semi-positive definite a.e. on I = (a, b). Suppose that
system (2.3) is disconjugate on I for some λ < γ, where γ is a lower bound of T ′0.
Then the Friedrichs extension T0,F of the minimal operator T0 is characterized by

D(T0,F ) = {y ∈ DQ : Yda−n(a) = 0 = Ydb−n(b)}, (5.6)



32 Q. BAO, G. WEI, A. ZETTL EJDE/SI/02

where

Yda−n(a) = ([y, y1a](a) · · · [y, yda−n,a](a))T ,

Ydb−n(b) = ([y, y1b](b) · · · [y, ydb−n,b](b))T

and yi,a, i = 1, . . . , da − n, and yj,b, j = 1, . . . , db − n are principal solutions at a
and b respectively.

Proof. Let l = 2d in Theorem 5.1. In this case the operator T defined by (5.1) is
the minimal operator T0, and the domain of the Friedrichs extension T0,F of T0 is
obtained easily by (5.2), i.e. (5.6) holds. This completes the proof. �

Remark 5.4. Here the principal solutions yi,a, i = 1, . . . , da−n define independent
boundary conditions at t = a and the principal solutions yj,b, j = 1, . . . , db−n define
independent boundary conditions at t = b. For instance, if a is regular and b is
singular, i.e., da = 2n and d = db in Corollary 5.3, then (5.6) can be reduced to

D(T0,F ) = {y ∈ DQ : y[n−i](a) = 0, Yd−n(b) = 0, i = 1, 2, . . . , n}. (5.7)

For C2n = E2n = ((−1)rδr,2n+1−s)
2n
r,s=1, Cmb

= Emb
= ((−1)rδr,mb+1−s)

mb
r,s=1 we

know that (5.7) can be reduced to the result in [16, Theorem 12]. However, for
d < 2n, the characterization of the Friedrichs extension of the minimal operator
in [16, Theorem 12] did not define independent boundary conditions at t = b.
Furthermore if both a, b are regular, then da = 2n = db, and (5.7) can be reduced
to

D(T0,F ) = {y ∈ DQ : y[n−i](a) = 0 = y[n−i](b) = 0, i = 1, 2, . . . , n}. (5.8)

This is an extension of a result of [14, Theorem 8.1] from E2n to C2n.

Theorem 5.5. Let the assumptions in Theorem 5.1 hold and let the symmetric
operator T be given by (5.1). Let V = (V1V2), Vj ∈ Ml,d(C), j = 1, 2, d < l ≤ 2d.
Then the domain D(TF ) of its Friedrichs extension TF is characterized by

D(TF ) =
{
y ∈ D(TQ) : Ŷa,b ∈ R(ĜV ∗),

(
Yda−n(a)
Ydb−n(b)

)
∈ G1V

∗
2 (N (V ĜV ∗))

}
(5.9)

Proof. Observe that V = UJ in (4.7) and JGU∗(N (UGU∗)) = ĜV ∗(N (V ĜV ∗)),
Then (5.9) is obtained easily from [26, Lemma 14 property (7)] and the proof of
Theorem 5.1. �

Remark 5.6. Under the assumptions of Theorem 5.5 we have

D(TF ) =
{
y ∈ D(TQ) : Ŷa,b ∈ R(ĜV ∗),

(
Yda−n(a)
Ydb−n(b)

)
∈ V −11 R(V2)

}
.

Remark 5.7. Let both endpoints a, b of (1.3) be regular, i.e., da = db = 2n and
C2n = E2n in Theorem 5.5 and Remark 5.6, i.e.,

Ĝ =

(
0 G1

−G∗1 0

)
, with G1 =

(
−En 0

0 En

)
.

In this case we obtain that the Möller-Zettl results in [15] is a special case. Moreover
in this case if l = 2n, then we obtain the characterization of Friedrichs extension
of the minimal operator in [14, Theorem 8.1]. Also Niessen-Zettl [18, Theorem 2.1]
found a special case for E2n and a certain matrix Q.



EJDE-2023/SI/02 FRIEDRICHS EXTENSIONS 33

Remark 5.8. Comparing Theorem 5.1 and Theorem 5.5 although they are equiv-
alent to each other Theorem 5.1 is more explicit than the result in Theorem 5.5.
For a better understanding of our main results we give some simple examples for
the special cases in the next section.

6. Examples

In this section we consider the Friedrichs extension TF of the symmetric operator
T for some special cases. We consider the real symmetric differential equation

My =

n∑
j=0

(−1)j(pjy
(j))(j) = λwy, pn, w > 0 a.e. on I = (0, b), b ≤ ∞, (6.1)

where y(j) denotes the classical derivatives. In this case, M = MQ is generated by
Q = (qr,s)

2n
r,s=1 ∈ Z2n(I) whose components are

qr,r+1 = 1, qr+n,r+n+1 = −1, r = 1, 2, . . . , n− 1;

qn,n+1 = p−1n , qr,s = ps−1, s 6= n, r + s = 2n+ 1, 1 ≤ r ≤ 2n

and Q satisfies Q = CQ∗C with

C =

(
0 Ĉn
−Ĉn 0

)
, Ĉn =

(
δr,n+1−s

)n
r,s=1

.

Thus the quasi-derivatives are as follows:

y[r] = y(r), r = 1, 2, . . . , n− 1,

y[n] = pny
(n),

y[n+r] = (−1)r(pny
(n))(r) + (−1)r−1(pn−1y

(n−1))(r−1) + · · ·+ pn−ry
(n−r),

r = 1, 2, . . . , n.

Note that (6.1) is equivalent to Hamiltonian system

Ly = J̃2ny′ −GQ(t)y = λWy, (6.2)

where

J̃2n =

(
0 −In
In 0

)
, GQ(t) =

(
−C(t) A∗(t)
A(t) B(t)

)
, W (t) =

(
W1(t) 0

0 0

)
with

A(t) =



0 1 0 · · · 0
0 0 1 · · · 0
...

. . .
. . .

. . .
...

0 0 · · ·
. . . 1

0 0 · · · · · · 0

 , C(t) =


p0 0 · · · 0
0 p1 · · · 0
...

. . .
. . .

...
0 · · · · · · pn−1

 ,

B(t) =


0 0 · · · 0
0 0 · · · 0
...

. . .
. . .

...
0 · · · · · · p−1n

 , W1(t) =


w 0 · · · 0
0 0 · · · 0
...

. . .
. . .

...
0 · · · · · · 0

 .

Assume that (6.2) is disconjugate on I for some λ < γ, where γ is a lower bound
of T ′0. Notice that B(t) ≥ 0 a.e. on I (For n = 1 B(t) > 0 a.e. on I). From



34 Q. BAO, G. WEI, A. ZETTL EJDE/SI/02

Theorem 3.1 we know that the minimal operator T0 generated by the equation (6.1)
is bounded below with γ̂ ≤ γ. Assume that d0 and db denote the deficiency indices
of T0 on (0, e) and (s, b), respectively. Let y10, . . . , yd0−n,0 and v10, . . . , vd0−n,0 be
principal and non-principal solutions of (6.1) at 0, respectively; y1b, . . . , ydb−n,b and
v1b, . . . , vdb−n,b be principal and non-principal solutions of (6.1) at b, respectively.

Suppose that d denotes the deficiency indices of the minimal operator generated
by (6.1) on I = (0, b), b ≤ ∞. Recall that d = d0 + db − 2n. Let

Ŷ0,b =


Yd0−n(0)
Ydb−n(b)

Ŷd0−n(0)

Ŷdb−n(b)

 ∈ C2d,

where

Yd0−n(0) = ([y, y10](0) · · · [y, yd0−n,0](0))T ,

Ŷd0−n(0) = ([y, v10](0) · · · [y, vd0−n,0](0))T ,

Ydb−n(b) = ([y, y1b](b) · · · [y, ydb−n,b](b))T

Ŷdb−n(b) = ([y, v1b](b) · · · [y, vdb−n,b](b))T .

For a given matrix N ∈M(2d−l),2d(C) satisfying

rank(N) = 2d− l, NĜN∗ = 0, d ≤ l ≤ 2d, (6.3)

where

Ĝ = (−1)n
(

0 G1

−G1 0

)
, G1 =

(
Ĉd0−n 0

0 −Ĉdb−n

)
(6.4)

with

Ĉd0−n = (δr,d0−n+1−s)
d0−n
r,s=1, Ĉdb−n = (δr,db−n+1−s)

db−n
r,s=1.

If V ∈Ml,2d(C) is a complete solution of the matrix equation

NV ∗ = 0, (6.5)

then by Lemma 4.3, T generated by (6.1) is a symmetric operator with an l dimen-
sional restriction of the associated maximal operator TQ for (6.1) and

D(T ∗) = {y ∈ DQ : NĜŶ0,b = 0}. (6.6)

Moreover by Theorem 4.5 we obtain that the domain of T is equivalent to

D(T ) = {y ∈ DQ : V Ŷ0,b = 0}, (6.7)

with the boundary matrix

V =


V̂11 V̂12
V̂21 0

−N12G1 0
−N22G1 N11G1

 , V̂rs ∈M(l−d),d(C), r, s = 1, 2 (6.8)

with rank(V̂12) = rank(V̂21) = l − d for d ≤ l ≤ 2d, and(
0 N̂12

N̂21 N̂22

)
= (N1N2) = N.

Thus from Theorem 5.1 we infer the following result.



EJDE-2023/SI/02 FRIEDRICHS EXTENSIONS 35

Corllary 6.1. Assume that both endpoints of (6.1) are limit-circle (LC). Let 2n <
l ≤ 4n and T defined by (6.7) be a symmetric operator with an l dimensional
restriction of the associated maximal operator TQ for (6.1). Then the boundary
conditions of its Friedrichs extensions TF can be characterized by

V̂21

(
Yn(0)
Yn(b)

)
= 0,

(−N2G1N1G1)Ŷ0,b = 0.

(6.9)

Here V̂21 is a submatrix of (6.8).

Example 6.2. We consider a differential equation

My = (−1)ny(2n) + qy, q ∈ L1
loc(I,R), I = (0, b), (6.10)

where both 0, b are LC endpoints. Let A ∈ Mm,2n(C) and B ∈ M(2n−m),2n(C)
satisfy

rank(A) = m,AB∗ = 0, n ≤ m ≤ 2n,

with rank(B) = 2n−m. Define

V =

(
A 0
0 I2n

)
.

By setting N = (B0), N satisfies (6.3) and (6.5), where

Ĝ = (−1)n
(

0 G1

−G1 0

)
, G1 =

(
Ĉn 0

0 −Ĉn

)
, Ĉn = (δr,n+1−s)

n
r,s=1.

Accordingly we know that an operator T defined on

D(T ) =
{
y ∈ DQ : V Ŷ0,b = 0

}
, (6.11)

is a symmetric operator with a 2n + m dimensional restriction of the associated
maximal operator TQ for (6.10). Note that T is bounded below. By using Corollary
6.1, we obtain that the characterization TF of T is characterized by

D(TF ) =
{
y ∈ DQ : A

(
Yn(0)
Yn(b)

)
= 0, Ŷ2n−m(b) = 0

}
.

Here when m = 2n we obtain T = T0 and TF = T0,F .

Remark 6.3. In Example 6.2 if we set

V =

A1 0 A2 0
0 In 0 0
0 0 0 In

 , rank(A1A2) = m,A1, A2 ∈Mm,n(C)

in (6.11), then by setting N = (B10B20), rank(B1B2) = 2n − m, B1, B2 ∈
M(2n−m),n(C) and

B1ĈnB
∗
2 −B2ĈnB

∗
1 = 0,

B1A
∗
1 +B2A

∗
2 = 0

(6.12)

we obtain that

D(T ) =
{
y ∈ DQ : (A1A2)

(
Y (0)

Ŷ (0)

)
= 0,

(
Y (b)

Ŷ (b)

)
= 0
}

is symmetric. Notice that from (6.12) we have A2 = −A1B
∗
1B2(B∗2B2)−1 and

−B2Ĉn(A∗1A1)−1A∗1A2 = B2ĈnB
∗
1B2(B∗2B2)−1 = B1Ĉn.



36 Q. BAO, G. WEI, A. ZETTL EJDE/SI/02

By transformation of rows V can be rewritten as

V →

 0 0 0 Im
0 In 0 0
In 0 (A∗1A1)−1A∗1A2 0

→
 0 0 0 Im

0 Im 0 0

−B2Ĉn 0 B1Ĉn 0

 .

Thus from Corollary 6.1 we infer that its Friedrichs extension is

D(TF ) =
{
y ∈ DQ :

(
0 Im

)(Y (0)
Y (b)

)
= 0,

(
−B2Ĉn B1Ĉn

)(Y (0)

Ŷ (0)

)
= 0
}

=
{
y ∈ DQ : ImY (b) = 0,

(
A1 A2

)(Y (0)

Ŷ (0)

)
= 0
}
.

Clearly when m = n and rank(A1) = n (6.12) is equivalent to(
A1 A2

)( 0 Ĉn
−Ĉn 0

)(
A∗1
A∗2

)
= 0.

Moreover in this case we have

D(TF ) =
{
y ∈ DQ : (A1A2)

(
Y (0)

Ŷ (0)

)
= 0, Y (b) = 0

}
.

This result was obtained in the regular case with Ĉn = ((−1)rδr,n+1−s)
n
r,s=1 in [15,

Theorem 4.3].

Since there is no boundary condition is required or allowed at the limit-point
endpoints, if the both endpoints 0, b for (6.1) are limit-point, then the associated
minimal operator is self-adjoint. We now assume that one endpoint of I for (6.1)
is limit-point, i.e., d = n. We give some special examples in n = 2 case.

Example 6.4. Consider the differential equation

My = (p2y
′′)′′ − (p1y

′)′ + p0y, p2 > 0, pj(t) ∈ L1
loc(I,R), j = 0, 1, 2, I = (0, b),

(6.13)
where the endpoint 0 is limit-circle and the endpoint b is limit-point. Assume that
T generated by (6.13) is defined on

D(T ) =
{
y ∈ DQ : V


[y, y10](0)
[y, y20](0)
[y, v10](0)
[y, v20](0)

 = 0
}

with

V =

0 1 1 1
1 0 1 1
1 1 0 1

 .

Here y10, y20 and v10, v20 are principal and non-principal solutions of (6.13) at

endpoint 0, respectively. Note that there exists a matrix N =
(

1 1 1 − 2
)

such

that
NV ∗ = 0, NĜ4N

∗ = 0,

where

Ĝ4 =


0 0 0 1
0 0 1 0
0 −1 0 0
−1 0 0 0

 .



EJDE-2023/SI/02 FRIEDRICHS EXTENSIONS 37

Thus T is a symmetric operator with 3 dimensional restriction of TQ. By elementary
matrix transformations of the rows V can be transformed into

V =

2 0 0 1
1 −1 0 0
2 −1 1 1

 .

Since T is bounded below, from Corollary 6.1 we obtain that the boundary condi-
tions of Friedrichs extension of T are:

[y, y10](0)− [y, y20](0) = 0,

2[y, y10](0)− [y, y20](0) + [y, v10](0) + [y, v20](0) = 0.

Example 6.5. Consider the differential equation (6.13) with the endpoint 0 is
regular and the endpoint b is limit-point. Let T generated by (6.13) be defined on

D(T ) =
{
y ∈ DQ :

0 2 3 4
1 0 3 4
1 2 3 4




y(0)
y′(0)

(p2y
′′)(0)

(p1y
′)(0)− (p2y

′′)′(0)

 = 0
}
.

Let

V =

0 2 3 4
1 0 3 4
1 2 3 4

 .

Note that there exists a matrix N =
(

0 0 4 − 3
)

such that NV ∗ = 0 and

NĜ4N
∗ = 0, where

Ĝ4 =


0 0 0 1
0 0 1 0
0 −1 0 0
−1 0 0 0

 .

Thus T is a symmetric operator with 3 dimensional restriction of TQ. By transfor-
mations of rows V can be transformed into

V =

0 0 3 4
0 1 0 0
3 −4 0 0

 .

Note that T is bounded below. From Corollary 6.1 we obtain that the Friedrichs
extension of T is characterized as

y′(0) = 0,

3y(0)− 4y′(0) = 0,

i.e., y(0) = y′(0) = 0.

Acknowledgements. This Project was supported by the National Nature Sci-
ence Foundation of China (No. 11971284, 11801286), Natural Science Foundation
of Shaanxi Province (Nos. 2021JQ-300), and by the Inner Mongolia Autonomous
Region University Scientific Research Project (No. NJZY21570).



38 Q. BAO, G. WEI, A. ZETTL EJDE/SI/02

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Qinglan Bao

School of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710062, China
Email address: baoqinglan19@163.com

Guangsheng Wei
School of Mathematics and Statistics, Shaanxi Normal University, Xi’an 710062, China

Email address: weimath@vip.sina.com

Anton Zettl

Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115,

USA
Email address: zettl@msn.com


	1. Introduction
	2. Hamiltonian system
	3. Boundedness below and principal solutions
	4. Singular symmetric differential operators
	5. Characterization of Friedrichs extensions
	6. Examples
	Acknowledgements

	References