On the extremal extensions of a non-negative Jacobi operator

0.17 MB·

by  Aleksandra Ananieva

ON THE EXTREMAL EXTENSIONS OF A NON-NEGATIVE JACOBI
OPERATOR
ALEKSANDRAANANIEVAANDNATALYGOLOSHCHAPOVA
7
Dedicated withdeep respect to Professor Ya.V. Mykytyukon the occasion of his 60th birthday.
1
0
2 Abstract. We consider minimal non-negative Jacobi operator with p×p−matrix

n entries. Usingthe technique ofboundarytriplets andthe correspondingWeyl func-
a tions,wedescribetheFriedrichsandKreinextensionsoftheminimalJacobioperator.
J Moreover,weparameterizethesetofallnon-negativeself-adjointextensionsinterms
ofboundaryconditions.
3
2

]
P
1. Introduction
S
. Let A be a densely defined non-negative symmetric operator in the Hilbert space H.
h
t Since A is non-negative, then by Friedrichs-Krein theorem, it admits non-negative self-
a
adjoint extensions. Qualified description of all non-negative self-adjoint extensions of
m
A and also criterion of uniqueness of non-negative self-adjoint extension of A were first
[
given by Krein in [16]. His results were generalized in numerous papers (see [3, 9, 11]

1 and references therein).
v Among all non-negative self-adjoint extensions of A, two (extremal) extensions are
1 particularlyinteresting and importantenoughto have a name. The Friedrichs extension
7
(so-called”hard”extension) A is the ”greatest”one in the sense of quadratic forms. It
3 F
is given by restriction of A∗ to the domain
6
1.0 dom(AF)=(cid:26) uas∈kd→om∞(Aa∗n)d:∃(Auk(u∈jd−oumk()A,u)jsu−chukt)hHat→ku0−asujk,kkH→→∞0 (cid:27).
0
In other words, A is the self-adjoint operator associated with the closure of the sym-
7 F
1 metric form
: t[u,v]=(Au,v)H, u,v ∈dom(A).
v
i The Krein extension (”soft” extension) AK is defined to be restriction of A∗ to the
X
domain
ar (1) dom(A )= u∈dom(A∗):∃ uk ∈dom(A)such thatkA∗u−AukkH →0 .
K (cid:26) ask→∞and (uj −uk,A(uj −uk))H →0asj,k →∞ (cid:27)
If A is positive definite, A≥εI >0, then (1) takes the form
(2) dom(A )=dom(A)+˙ ker(A∗).
K
Krein proved in [16] that all the non-negative self-adjoint extensions A of A lie between
A and A , i.e.,
F K e
((AF +aI)−1u,u)H ≤((A+aI)−1u,u)H ≤((AK +aI)−1u,u)H, u∈H,
e
2000 Mathematics Subject Classification. 47B36;47B36.
Key words and phrases. Jacobi matrix, non-negative operator, Friedrichs and Krein extensions,
boundarytriplet,Weylfunction.
The authors express their gratitude to Prof. M. Malamud for posing the problem and permanent
attentiontotheworkandtoProf. L.Oridorogaforusefuldiscussions. Thesecondauthorwassupported
byFAPESPgrant2012/50503-6.
1
2 ALEKSANDRAANANIEVAANDNATALYGOLOSHCHAPOVA
for any a>0.
In the present paper we are dealing with the problem of description of the extremal
extensions of non-negative Jacobi operator to be defined below. Let A ,B ∈ Cp×p.
j j
Moreover, we assume that matrices A are self-adjoint and matrices B are invertible
j j
for each j ≥ 0 (see [5, Chapter VII, §2]). We consider semi-infinite Jacobi matrix with
matrix entries
A B O ...
0 0 p
 B0∗ A1 B1 ... 
J= O B∗ A ... ,
 p 1 2 
 ... ... ... ... 
 
where O is zero p×p matrix. Given a sequence u = (u ), u ∈ Cp, Ju is again a
p j j
sequence of column vectors. If we set B =O ,
−1 p
(Ju) =B u +A u +B∗ u , j ≥0.
j j j+1 j j j−1 j−1
The maximal operator T is defined by
max
(T u) =(Ju) , j ≥0
max j j
on the domain
dom(T )={u∈l2 : Ju∈l2}.
max p p
The minimal operator T is the closure in l2 of the preminimal operator T which is
min p
the restriction of T to the domain
max
dom(T)={u∈l2 : u =0 for all but a finite number of values of j}.
p j
It is straightforwardto see that T is a densely defined symmetric operator and
min
T∗ =T , T∗ =T =T .
min max max min
Deficiency indices n (T ) = dim(ker(T ±zI)), z ∈ C , satisfy the inequality 0 ≤
± min max +
n (T ),n (T )≤p(see[5,ChapterVII,§2]). Inthefollowing,weshallassumethat
+ min − min
n (T )= p (completely indefinite case takes place) and T is non-negative. Note that
± min
non-negativity of T implies non-negativity of A , j ≥0.
j
Examples of symmetric block Jacobi matrices generating symmetric operators with
arbitrary possible values of the deficiency numbers were constructed in [13].
The problem of description of the extremal extensions T and T in the scalar case
F K
(p = 1) was studied in the number of papers. Description of the Friedrichs domain in
terms of a weighted Dirichlet sum was obtained in [4].
In [20], Simon showed that certain matrix operators that approximate the Friedrichs
and Krein extensions converge in the strong resolvent norm. In [7] Brown and Chris-
tiansen (assuming that T is positive definite) obtained the description of T and T
F K
using the concept of the so-called minimal solution (see also [18]).
The purpose of this work is to generalize at least partially the results obtained in [7]
to the case of arbitrary p ∈ N and non-negative operator T. We use an abstract de-
scription of extremal non-negative extensions obtained by V. Derkach and M. Malamud
in the framework of boundary triplets and the corresponding Weyl functions approach
(see [9, 14] and also Section 2 for the precise definitions). In particular, we show that
mentioned results from [7] might be expressed in terms of boundary triplets theory.
Notation. In what follows Cp×p denotes the set of p×p complex-valued matrices;
l2 denotes Hilbert space of infinite sequences u = (u ), u ∈ Cp equipped with inner
p j j
ON THE EXTREMAL EXTENSIONS OF A NON-NEGATIVE JACOBI OPERATOR 3
∞
product (u,v)l2p = vj∗uj. The set of closed (bounded) operators in the Hilbert space
jP=0
H is denoted by C(H) (respectively, B(H)).
2. Linear relations, boundary triplets and Weyl functions
Let A be a closed densely defined symmetric operator in the separable Hilbert space
H with equal deficiency indices n (A)=dimker(A∗±iI)≤∞.
±
Definition 1. [14] A triplet Π={H,Γ ,Γ } is called a boundary triplet for the adjoint
0 1
operatorA∗ ofAifHisanauxiliaryHilbertspaceandΓ ,Γ : dom(A∗)→Harelinear
0 1
mappings such that
(i) the second Green identity,
(A∗f,g)H−(f,A∗g)H =(Γ1f,Γ0g)H−(Γ0f,Γ1g)H,
holds for all f,g ∈dom(A∗), and
(ii) the mapping Γ:=(Γ ,Γ )⊤ :dom(A∗)→H⊕H is surjective.
0 1
WitheachboundarytripletΠ={H,Γ ,Γ }oneassociatestwoself-adjointextensions
0 1
A :=A∗ ↾ker(Γ ), j ∈{0,1}.
j j
Definition 2.
(i) A closed linear relation Θ in H is a closed subspace of H⊕H. The domain, the
range, and the multivalued part of Θ are defined as follows
dom(Θ):= f :{f,f′}∈Θ , ran(Θ):= f′ :{f,f′}∈Θ ,
(cid:8) mul(Θ):= (cid:9)f′ :{0,f′}∈Θ(cid:8). (cid:9)
(cid:8) (cid:9)
(ii) A linear relation Θ is symmetric if
(f′,h) −(f,h′) =0 for all {f,f′},{h,h′}∈Θ.
H H
(iii) The adjoint relation Θ∗ is defined by
Θ∗ = {h,h′}:(f′,h) =(f,h′) for all {f,f′}∈Θ .
H H
(iv) A closed linea(cid:8)r relation Θ is called self-adjoint if both Θ and(cid:9)Θ∗ are maximal
symmetric, i.e., they do not admit symmetric extensions.
For the symmetric relation Θ ⊆ Θ∗ in H the multivalued part mul(Θ) is orthogonal
to dom(Θ) in H. Setting H := dom(Θ) and H := mul(Θ), one verifies that Θ can
op ∞
be rewritten as the direct orthogonal sum of a self-adjoint operator Θ (operator part
op
of Θ) in the subspace H and a “pure” relation Θ = {0,f′} : f′ ∈ mul(Θ) in the
op ∞
subspace H∞. (cid:8) (cid:9)
Proposition 1. [9, 14] Let Π = {H,Γ ,Γ } be a boundary triplet for A∗. Then the
0 1
mapping
(3) Ext ∋A:=A →Θ:=Γ(dom(A))= {Γ f,Γ f}: f ∈dom(A)
A Θ 0 1
(cid:8) (cid:9)
establishes a bijecteive correspondence betweeen the set of all closed propeer extensions
Ext ofAandthesetofallclosedlinearrelationsC(H)inH. Furthermore,thefollowing
A
assertions hold.
e
(i) The equality (AΘ)∗ =AΘ∗ holds for any Θ∈C(H).
(ii) TheextensionA in(3)is symmetric(self-adjoint)ifandonly ifΘissymmetric
Θ e
(self-adjoint).
(iii) If, in addition, extensions A and A are disjoint, i.e., dom(A )∩dom(A ) =
Θ 0 Θ 0
dom(A), then (3) takes the form
A =A =A∗↾ker Γ −BΓ , B ∈C(H).
Θ B 1 0
(cid:0) (cid:1)
4 ALEKSANDRAANANIEVAANDNATALYGOLOSHCHAPOVA
Remark 1. In the case n (A) = n < ∞, any proper extension A of the operator A
± Θ
admits representation(see [12])
(4) A :=A =A∗ ↾ker DΓ −CΓ , C,D ∈B(H).
Θ C,D 1 0
(cid:0) (cid:1)
Moreover, according to the Rofe-Beketov theorem [19] (see also [2, Theorem 125.4]),
A is self-adjoint if and only if C,D, satisfy the following conditions
C,D
(5) CD∗ =DC∗ and 0∈ρ(CC∗+DD∗).
Definition3. [9]LetΠ={H,Γ ,Γ }beaboundarytripletforA∗. Theoperator-valued
0 1
function M(·):ρ(A )→B(H) defined by
0
Γ f =M(z)Γ f , for all f ∈ker(A∗−zI), z ∈ρ(A ),
1 z 0 z z 0
is called the Weyl function, corresponding to the triplet Π.
Proposition 2. [9,11]LetAbeadenselydefinednonnegativesymmetricoperatorwith
finite deficiency indices in H, and let Π={H,Γ ,Γ } be a boundary triplet for A∗. Let
0 1
also M(·) be the corresponding Weyl function. Then the following assertions hold.
(i) There exist strong resolvent limits
(6) M(0):=s−R−limM(x), M(−∞):=s−R− lim M(x).
x↑0 x↓−∞
(ii) M(0)andM(−∞)areself-adjointlinearrelationsinHassociatedwiththesemi-
bounded below (above) quadratic forms
t [f]=lim(M(x)f,f)≥β||f||2, t [f]= lim (M(x)f,f)≤α||f||2,
0 −∞
x↑0 x↓−∞
and
dom(t )= f ∈H: lim|(M(x)f,f)|<∞ =dom((M(0) −β)1/2),
0 op
x↑0
(cid:8) (cid:9)
dom(t )= f ∈H: lim |(M(x)f,f)|<∞ =dom((α−M(−∞) )1/2).
−∞ op
x↓−∞
(cid:8) (cid:9)
Moreover,
dom(A )={f ∈dom(A∗): {Γ f,Γ f}∈M(0)},
K 0 1
dom(A )={f ∈dom(A∗): {Γ f,Γ f}∈M(−∞)}.
F 0 1
(iii) Extensions A and A are disjoint (A and A are disjoint) if and only if
0 K 0 F
M(0)∈C(H) (M(−∞)∈C(H), respectively).
Moreover,
dom(A )=dom(A∗)↾ker(Γ −M(0)Γ )
K 1 0
(dom(A )=dom(A∗)↾ker(Γ −M(−∞)Γ ), respectively).
F 1 0
(iv) A =A (A =A ) if and only if
0 K 0 F
lim(M(x)f,f)=+∞, f ∈H\{0}
x↑0
( lim (M(x)f,f)=−∞, f ∈H\{0}, respectively).
x↓−∞
(v) If, in addition, A =A (A =A ), then the set ofall non-negativeself-adjoint
0 F 0 K
extensions of A admits parametrization(3), where Θ satisfies
(7) Θ−M(0)≥0 (Θ−M(−∞)≤0, respectively).
Moreover, if (7) does not hold, the number of negative eigenvalues of arbitrary
self-adjoint extension κ (A ) is given by
− Θ
κ (A )=κ (Θ−M(0)) (κ (A )=κ (M(−∞)−Θ), respectively).
− Θ − − Θ −
ON THE EXTREMAL EXTENSIONS OF A NON-NEGATIVE JACOBI OPERATOR 5
Remark 2. We should mention that the existence of the limits in (6) follows from
finiteness of deficiency indices of A.
Remark 3. Note that if the lover bound of A is zero and the spectrum of A is purely
F
discrete,thenA andA arenotdisjoint. InthiscaseM(0)isalinearrelationifA ≥0.
F K 0
Corollary 1. Let assumptions of Proposition 2 hold and A = A . Let also A be
0 K C,D
self-adjoint extension of A defined by (4). Then A is non-negative if and only if
C,D
(8) CD∗−DM(−∞)D∗ ≤0.
Moreover, if (8) does not hold, the number of negative eigenvalues of A (counting
C,D
multiplicities) coincides with the number of positive eigenvalues of the linear relation
CD∗−DM(−∞)D∗, i.e.,
κ (A )=κ (CD∗−DM(−∞)D∗).
− C,D +
Corollary 2. [11] Suppose that A has purely discrete spectrum. Then the Krein exten-
F
sion A is given by
K
dom(A )=dom(A)∔ker(A∗).
K
Moreover, the spectrum of A ↾ker(A∗)⊥ is purely discrete.
K
3. Extremal extensions of T
min
As usual, we denote by (P (z)) the solution to the matrix equation
j
(JU) =zU , j ≥0
j j
with the initial conditions P (z) = I , P (z) = B−1(zI −A ). Here I ∈ Cp×p is the
0 p 1 0 p 0 p
identity matrix. Furthermore, we denote by (Q (z)) the solution to
j
(JU) =zU , j ≥1
j j
withQ (z)=O andQ (z)=B−1. ThematrixfunctionsP (z)andQ (z)arepolynomi-
0 p 1 0 j j
als inthe complex variablez ofdegreej andj−1,respectively,with matrixcoefficients.
We mention that (P (z)) and (Q (z)) are called matrix polynomials of the first and
j j
second kind, respectively.
Following [10], we define a boundary triplet Π={H,Γ ,Γ } for T by setting
0 1 max
(9) H=Cp, Γ u=(Q(0))∗T u−P u, Γ u=(P(0))∗T u,
1 max 0 0 max
∞
where u∈dom(T ) and P is orthoprojection in H onto H , in which H =Cp.
max 0 j 0 j
jL=0
Weshouldnotethatthemappings(P(0))∗and(Q(0))∗actasinfinite”p×∞”matrices,
i.e., (P(0))∗,(Q(0))∗ : l2 → Cp. Each their row is ”constructed” by the corresponding
p
rows of P∗(0) and Q∗(0), respectively.
j j
It is easily seen that
∞ ∞
Γ (P (z))=z Q∗(0)P (z)−I , Γ (P (z))=z P∗(0)P (z),
1 j j j p 0 j j j
Xj=1 Xj=0
and, by Definition 3, we get
M(z)=Γ (P (z))(Γ (P (z)))−1
1 j 0 j
−1
∞ ∞
=z Q∗j(0)Pj(z)−Ip·z Pj∗(0)Pj(z) .
Xj=1 Xj=0
   
Applying Proposition 2 to the operator T , we obtain the following result.
min
6 ALEKSANDRAANANIEVAANDNATALYGOLOSHCHAPOVA
Theorem 1. Let Π= {H,Γ ,Γ } be the boundary triplet for T given by (9) and let
0 1 max
M(·) be the corresponding Weyl function. Then the following assertions hold.
(i) The Krein extension T coincides with T , i.e.,
K 0
dom(T )=dom(T )↾ker(Γ )={u∈T : (P(0))∗T u=0}.
K max 0 max max
(ii) Self-adjoint extension T is non-negative if and only if it admits representation
Θ
T =T =T ↾ker(DΓ −CΓ ),
Θ C,D max 1 0
where C,D satisfy (5) and (8).
(iii) The number of negative eigenvalues of T = T∗ (defined by (4)) coincides
C,D C,D
withthenumberofpositiveeigenvaluesofthelinearrelationCD∗−DM(−∞)D∗,
i.e.,
κ (T )=κ (CD∗−DM(−∞)D∗).
− C,D +
Proof. (ii) The statement might be proven at least in two different ways.
1. Since M(x) is holomorphic and increasing in (−ε,0) (see [11]), the strong limit
s−lim(M(x)+γ)−1 exists for any γ >0. Namely,
x↑0
∞
(10) Mγ(0):=s−lim M(x)+γ −1 =s−limx Pj∗(0)Pj(x)
x↑0(cid:0) (cid:1) x↑0 Xj=0
 
−1
∞ ∞
×x Q∗j(0)Pj(x)−Ip+γx Pj∗(0)Pj(x) =Op.
Xj=1 Xj=0
 
∞ ∞
Indeed, since n (T )=p (see [15]), the series ||P (z)||2 and ||Q (z)||2 converge
± min j j
jP=0 jP=0
uniformly on each bounded subset C (see, for instance, [15, Theorem 1]). Therefore, we
can pass to the limit in (10) under the sum sign as x → 0. Hence M−1(0) = {0,H} =
γ
{{0,f}:f ∈H}. Since
dom(T )={u∈dom(T ): Γ u=0}={u∈dom(T ):{Γ u,Γ u}∈{0,H}},
0 max 0 max 0 1
by Proposition 2 (ii), we arrive at T =T .
K 0
2. Suppose, in addition, that T ≥ εI > 0. Using the equality dom(T ) = dom(T ) ↾
0 max
ker(Γ ), we easily get from (9) that ker(T ) ⊂ dom(T ). Therefore, (2) implies the
0 max 0
inclusion dom(T )⊃dom(T ). Since T and T are self-adjoint, we arrive at T =T .
0 K 0 K K 0
(ii) and (iii) easily follow from Corollary 1 and assertion (i). (cid:3)
Assume now that p = 1 and A ,B are positive real numbers. It is known that T
j j min
is connected with some Stiltjes moment problem, see [1]. Briefly, a Stiltjes moment
problem has a following description. Given a sequence γ ,γ ,γ ,... of reals. When is
0 1 2
there a measure, dµ on [0,∞) so that
∞
γ = xndµ(x)
n
Z
0
and if such a µ exists, is it unique?
The operator T is self-adjoint if and only if associated Stiltjes moment problem is
min
determinate, i.e., it has unique solution. Since n (T ) = 1, the determinacy does not
± min
take place and, therefore, sequence Qj(0) converges (see [1, Theorem 0.4, p. 293] or [6,
Pj(0)
Section 3]).
ON THE EXTREMAL EXTENSIONS OF A NON-NEGATIVE JACOBI OPERATOR 7
TheexistenceofthislimitisthekeyfactforthedescriptionoftheFriedrichsextension
which we are going to present. In particular, the limit
Q (0)
j
(11) α:= lim
j→∞ Pj(0)
is negative. Indeed, since all the zeros of the polynomials P (x) and Q (x) lie in the
j j
interval [0,∞) (see, [8, Chapter I]), P (·) and Q (·) do not change the sign in (−∞,0).
j j
Noticing that P (x)/Q (x) < 0 for x < 0 large enough (the degrees of P (·) and Q (·)
j j j j
are j and j −1, respectively, and leading coefficients equal B−1 ·..·B−1 > 0), we get
j−1 0
the negativity of α.
Theorem 2. Assume p = 1. Let also Π = {H,Γ ,Γ } be the boundary triplet for
0 1
T defined by (9) and M(·) be the corresponding Weyl function. The domain of the
max
Friedrichs extension is given by
(12) dom(T )={u∈dom(T ): (Γ −αΓ )u=0},
F max 1 0
where α is defined by (11).
Proof. To provethe statementwe use Proposition2(iii). Namely, it is sufficient to show
that
∞
x Q (0)P (x)−1
j j
(13) M(−∞)= lim M(x)= lim jP=1 =α.
∞
x↓−∞ x↓−∞
x P (0)P (x)
j j
jP=0
Since the orthogonal polynomials do not change the sign in (−∞,0),
(14) P (0)P (x)=|P (0)P (x)| and Q (0)P (x)=−|Q (0)P (x)|.
j j j j j j j j
n n
Thus,thesequence P (0)P (x)= |P (0)P (x)|increasesasn→∞and,therefore,
j j j j
jP=0 jP=0
∞ ∞
x Q (0)P (x)−1 |Q (0)P (x)|
j j j j
lim M(x)= lim jP=1 =− lim jP=1 .
∞ ∞
x↓−∞ x↓−∞ x↓−∞
x P (0)P (x) |P (0)P (x)|
j j j j
jP=0 jP=0
Itfollowsfrom(11)thatforanysmallδ >0thereexistsN =N(δ)suchthatestimate
Q (0)
j
α−δ < <α+δ
P (0)
j
holds for j >N(δ). Combining this inequality with (14), we get
Q (0)P (x)
j j
α−δ < <α+δ, x∈(−∞,0), j ≥N(δ).
|P (0)P (x)|
j j
The latter is equivalent to
(α−δ)|P (0)P (x)|<Q (0)P (x)<(α+δ)|P (0)P (x)|, x∈(−∞,0), j ≥N(δ).
j j j j j j
8 ALEKSANDRAANANIEVAANDNATALYGOLOSHCHAPOVA
Hence
(15)
∞ ∞ N(δ)−1
|Pj(0)Pj(x)| Qj(0)Pj(x) Qj(0)Pj(x)
(α−δ)j=PN(δ) < jP=1 − jP=1
∞ ∞ ∞
|P (0)P (x)| |P (0)P (x)| |P (0)P (x)|
j j j j j j
jP=0 jP=0 jP=0
∞
|P (0)P (x)|
j j
<(α+δ)j=PN(δ) , x∈(−∞,0), j ≥N(δ).
∞
|P (0)P (x)|
j j
jP=0
Since
∞ ∞
P (x)P (0)= |P (x)P (0)|>|P (x)P (0)|,
j j j j N(δ) N(δ)
Xj=0 Xj=0
and P is polynomial of degree j, we get
j
N(δ)−1 N(δ)−1
|P (0)P (x)| |P (0)P (x)|
j j j j
(16) 0≤ lim jP=1 < lim jP=1 =0.
x→−∞ ∞ |P (0)P (x)| x→−∞ |PN(δ)(x)PN(δ)(0)|
j j
jP=0
Similarly we obtain
N(δ)−1
Q (0)P (x)
j j
(17) lim jP=1 =0.
∞
x→−∞
|P (0)P (x)|
j j
jP=0
Taking into account that
∞ ∞ N(δ)−1
|P (0)P (x)|= |P (0)P (x)|+ |P (0)P (x)|,
j j j j j j
Xj=0 j=XN(δ) Xj=0
we get from (15)–(17) the following inequality
∞
Q (0)P (x)
α−δ ≤ lim j=1 j j ≤α+δ
x→−∞ P∞j=0|Pj(0)Pj(x)|
P
for any arbitrary small δ. Thus, equality (13) takes place.
(cid:3)
Remark 4. We should mention that the description of the Krein and Friedrichs exten-
sionsgiveninTheorems1 and2in the scalarcasecoincides withone obtainedearlierby
Brown and Christiansen in [7].
Remark 5.
(i) The condition B > 0 can be dropped. Indeed, it is easy to show that scalar
j
Jacobimatrix with arbitraryreal B is unitarily equivalent to the Jacobimatrix
j
with positive B . The unitary equivalence is established by the diagonal matrix
j
with 1 and −1 on the diagonal. Besides, if B <0, then 1 and −1 have to stand
j
next to each other in the same rows as B .
j
(ii) The fact that all zeroes of P (·) belong to [0,∞) might be also derived from the
j
holomorphicity of the Weyl function, corresponding to another boundary triplet
(see [11, Proposition 10.1(2)]), on (−∞,0).
ON THE EXTREMAL EXTENSIONS OF A NON-NEGATIVE JACOBI OPERATOR 9
(iii) In [7], authors obtained the description (12) by a different method. Namely,
they essentially used the fact that the minimal (or principal) solution u = (u )
j
of the equation (Ju) =0, j ≥1, has the formu=(u )=(P (0)−αQ (0)) and
j j j j
belongs to the domain dom(T ).
F
(iv) In [17, Chapter 5, §3] (see also [7]), it was noted that all solutions µ of the
Stieltjes momentproblemassociatedwith T lie betweenthe solutionsµ and
min K
µ coming from the Friedrichs and Krein extensions in the following senese
F
∞ dµ (t) ∞ dµ(t) ∞ dµ (t)
K F
≤ ≤ , x<0.
Z x−t Z x−t Z x−t
0 0 e 0
Proposition 2(v) leads to the description of all non-negative self-adjoint extensions of T
in the scalar case.
Corollary 3. Assumep=1. Let Π={H,Γ ,Γ } be the boundary triplet for T given
0 1 max
by (9) and let M(·) be the corresponding Weyl function. The set of all non-negative
self-adjoint extensions T of the operator T is parameterized as follows
h min
(18) dom(T )= u∈dom(T ):(Γ −hΓ )u=0 ,
h max 1 0
(cid:8) (cid:9)
h∈[−∞;α] where α is defined by (11). In particular,
dom(T )= u∈dom(T ):Γ u=0 .
−∞ max 0
(cid:8) (cid:9)
Proof. First note that for h=α and h=−∞ the statement was proved above. Indeed,
dom(T )={u∈dom(T ):(Γ −αΓ )u=0}=dom(T )
α max 1 0 F
and
dom(T )=dom(T )↾ker(Γ )=dom(T ).
−∞ max 0 K
Thus, it remains to prove that for h < α formula (18) defines non-negative self-adjoint
extension. The result is implied by combining Proposition 2(v) with Theorem 2.
Indeed, consider a new boundary triplet Π={H,Γ ,Γ }
0 1
H=C, Γ f =Γ f −eαΓ f,e f Γff =−Γ f,
0 1 0 1 0
where Γ0,Γ1 are giveen by (9). Ofne easily obtains that tfhe corresponding Weyl function
is M(z) = (α − M(z))−1. Taking into account the above information about ”limit
values” of the Weyl function M(·), we get that M(−∞) is ”pure” linear relation, i.e.,
f
M(−∞) = {0,H}, and M(0) = 0. Hence, by Proposition 2(ii), T := T ↾ ker(Γ ) =
f 0 max 0
T . Equation (18) in terms of the new boundary triplet takes the form
fF f f
1
dom(T )= u∈dom(T ): Γ − Γ u=0 .
h max 1 0
α−h
n (cid:16) (cid:17) o
e e
Applying Proposition 2(v), we get that T ≥ 0 if and only if 1 > M(0) = 0 or
h α−h
h<α. (cid:3)
f
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