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 description 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)||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 Bibliography 1. 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Math. 137(1998), no.1,182–203. R.Luxemburgstr. 74, 83114,Donetsk, Ukraine E-mail address: ananeva89@gmail.com Ruado Mataõ,1010, 05508-090,SaõPaulo,Brazil E-mail address: nataliia@ime.usp.br

On the extremal extensions of a non-negative Jacobi operator

Similar On the extremal extensions of a non-negative Jacobi operator