On Linear Relations Generated by a Differential Expression and by a Nevanlinna Operator Function

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1 Journl of Mthemticl Physics, Anlysis, Geometry 2011, vol. 7, No. 2, pp On Liner Reltions Generted by Differentil Expression nd by Nevnlinn Opertor Function V.M. Bruk Srtov Stte Technicl University 77, Politechnichesky Str., Srtov , Russi E-mil: Received April 22, 2010 The fmilies of mximl nd miniml reltions generted by differentil expression with bounded opertor coefficients nd by Nevnlinn opertor function re defined. These fmilies re proved to be holomorphic. In the cse of finite intervl, the spce of boundry vlues is constructed. In terms of boundry conditions, criterion for the restrictions of mximl reltions to be continuously invertible nd criterion for the fmilies of these restrictions to be holomorphic re given. The opertors inverse to these restrictions re stted to be integrl opertors. By using the results obtined, the existence of the chrcteristic opertor on the finite intervl nd the xis is proved. Key words: Hilbert spce, liner reltion, differentil expression, holomorphic fmily of reltions, resolvent, chrcteristic opertor, Nevnlinn function. Mthemtics Subject Clssifiction 2000: 47A06, 47A10, 34B27. Introduction On finite or infinite intervl (, b), we consider differentil expression l λ [y] = l[y] C λ y, where l is formlly self-djoint differentil expression with bounded opertor coefficients in Hilbert spce H, C λ = C λ (t) is Nevnlinn opertor function, i.e., C λ (t) is holomorphic function for Imλ 0 whose vlues re the bounded opertors in H such tht C λ(t) = C λ (t) nd Im C λ (t)/imλ 0. Let A(t)=Im C λ0 (t)/imλ 0. In the spce L 2 (H, A(t);, b), we define the fmilies L 0 (λ), L(λ) of miniml nd mximl reltions, respectively, nd prove tht these fmilies re holomorphic for Imλ 0. In the cse of finite intervl, the spce of boundry vlues for the fmily of mximl reltions L(λ) is constructed. There The work is supported by the Russin Foundtion of Bsic Reserches (grnt ) c V.M. Bruk, 2011

2 V.M. Bruk exists one-to-one correspondence between the reltions L(λ) with the property L 0 (λ) L(λ) L(λ) nd the boundry reltions θ(λ). This correspondence is determined by the equlity γ(λ) L(λ) = θ(λ), where γ(λ) is the opertor which to ech pir from L(λ) ssocites the pir of boundry vlues (in this cse, we denote L(λ) = L θ(λ) (λ) ). We prove tht 1) for fixed λ, the reltion (L θ(λ) (λ)) 1 is bounded everywhere defined opertor if nd only if the reltion θ 1 (λ) is the opertor with the sme property; in this cse the opertor (L θ(λ) (λ)) 1 is integrl; 2) the opertor function (L θ(λ) (λ)) 1 is holomorphic if nd only if θ 1 (λ) is holomorphic. If the function C λ is liner, these results correspond with the description of the generlized resolvents from [1 3]. We pply the obtined results to prove the existence of the chrcteristic opertor on the finite intervl nd the xis. In the ppers by V.I. Khrbustovsky [4 6], the definition of the chrcteristic opertor is given. In these ppers, for differentil opertor of first order the existence of the chrcteristic opertor is estblished nd vrious problems ssocited with the chrcteristic opertor re studied. Bibliogrphy on the chrcteristic opertor is given in [4 6]. 1. Min Assumptions nd Auxiliry Sttements Let H be seprble Hilbert spce with the sclr product (, ) nd the norm. On finite or infinite intervl (, b) we consider differentil expression l of order r n ( 1) k {(p n k (t)y (k) ) (k) i[(q n k (t)y (k) ) k 1 + (q n k (t)y (k 1) ) (k) ]}+ k=1 l[y]= +p n (t)y, n ( 1) k {i[(q n k (t)y (k) ) (k+1) + (q n k (t)y (k+1) ) k ] + (p n k (t)y (k) ) (k) }, k=0 where r = 2n (n = 1, 2,...) or r = 2n + 1 (n = 0, 1, 2,...). The coefficients of l re the bounded self-djoint opertors in H. The leding coefficients, p 0 (t) in the cse r = 2n nd q 0 (t) in the cse r = 2n + 1, hve bounded inverse opertors lmost everywhere. According to [7], l is considered to be qusidifferentil expression. The qusiderivtives for the expression l re defined in [7]. Suppose tht the functions p j (t), q k (t) re strongly continuous, the function q 0 (t) in the cse r = 2n + 1 is strongly differentible, nd the norms of the functions p 1 0 (t), p 1 0 (t)q 0(t), q 0 (t)p 1 0 (t)q 0(t), p 1 (t),..., p n (t), q 0 (t),..., q n 1 (t), in cse r = 2n, q 0(t), q 1 (t),..., q n (t), p 0 (t),..., p n (t), in cse r = 2n + 1, re integrble on every compct intervl [α, β] (, b). 116 Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No. 2

3 On Liner Reltions Generted by Differentil Expression... The boundry is clled regulr boundry if >, nd one cn tke α =, nd the norm C λ (t) is integrble on every intervl [, β) (β < b), where the function C λ (t) is defined bellow. In the contrry cse, the boundry is clled singulr boundry. The regulrity nd singulrity of the boundry b re defined nlogously. For brevity, we ssume tht if the intervl (, b) is finite, then the boundries, b re regulr. The singulr cse is studied on the exmple of the xis (, ). Other cses of singulr boundries re considered nlogously. Let C λ (t) be n opertor function whose vlues re the bounded opertors in H. Suppose C λ (t) stisfies the following conditions [4]: () there exist sets C 0 C \ R 1 nd I 0 (, b) with the following properties: the mesure of the set (, b)\ I 0 equls zero nd every point belonging to C 0 hs neighborhood independent of t I 0 such tht the function C λ (t) is holomorphic with respect to λ in this neighborhood; (b) for ll λ C 0, the function C λ (t) is Bochner loclly integrble in the uniform opertor topology; (c) C λ (t) = C λ (t) s well s Im C λ(t)/imλ is nonnegtive opertor for ll t I 0 nd for ll λ such tht Imλ 0. We denote B λ (t) = Re C λ (t), A λ (t) = Im C λ (t). Using condition (c), we get B λ = B λ, A λ(t) = A λ (t). Let the opertor λ (t) be given by λ (t) = A λ (t)/imλ for λ such tht Imλ 0. It follows from condition (c) tht for ll µ C 0 R there exists lim λ µ±i0 λ (t) = µ (t). The function µ (t) is loclly integrble [4]. Further, we will use the following sttements (see [4]). Sttement 1. If for some element g H there exist numbers t 0 I 0 nd λ 0 C 0 such tht λ0 (t 0 )g = 0, then for ll λ C 0 the equlities λ (t 0 )g = 0 nd (B λ (t 0 ) B λ0 (t 0 ))g = 0 hold. Sttement 2. Suppose t I 0, λ, µ C 0 nd sequence of the elements g n H stisfy the condition ( λ (t)g n, g n ) 0 s n. Then (λ µ) 1 (C λ (t) C µ (t))g n 0. Sttement 3. For ny compct K C 0 there exist constnts k 1, k 2 > 0 such tht for ll h H, t I 0, λ 1, λ 2 K the inequlity k 1 ( λ1 (t)h, h) 1/2 ( λ2 (t)h, h) 1/2 k 2 ( λ1 (t)h, h) 1/2 holds (thus the constnts k 1, k 2 re independent of t I 0, λ K). We denote l λ [y] = l[y] (B λ (t) + ia λ (t))y = l[y] C λ (t)y. Let W j,λ (t) be the opertor solution of the eqution l λ [y] = 0 stisfying the initil conditions W [k 1] j,λ (t 0 ) = δ j,k E, where t 0 (, b), δ j,k is the Kronecker symbol, E is the identity opertor, j, k = 1,..., r. By W λ (t) we denote the opertor one-row mtrix W λ (t) = (W 1,λ (t),..., W r,λ (t)). The opertor W λ (t) is continuous mpping of H r into H. The djoint opertor Wλ (t) mps continuously H into Hr. For fixed t, the function W λ (t) is holomorphic on C 0. Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No

4 V.M. Bruk If l λ is defined for the function y, then we denote ŷ = (y [0], y [1],..., y [r 1] ) T (T in the upper index denotes trnsposition). Let u = (u 1,..., u n ) be system of functions such tht l λ [u j ] exists for j = 1,..., n. By û we denote the mtrix with the jth column û j (j = 1,..., n). Similr nottions re used for the opertor functions. Note tht ll qusiderivtives up to order r 1 inclusive coincide for the expressions l nd l λ. We consider the opertor mtrices of orders 2n nd 2n + 1 for the expression l in the cses r = 2n nd r = 2n + 1, respectively: E... J 2n (t) = E E,... E J 2n+1 (t) = E... E 2iq 1 0 (t) where ll non indicted elements re equl to zero. (In mtrix J 2n+1 the element 2iq0 1 (t) stnds on the intersection of the row n + 1 nd the column n + 1.) Suppose the expression l is defined for the functions y, z, nd l[y], l[z] re loclly integrble on (, b). Then Lgrnge s formul hs the form β α (l[y], z)dt β α E... E, (y, l[z])dt = (J r (t)ŷ(t), ẑ(t)) β α, < α β < b. (1) In (1), we tke y(t) = W λ (t)c, z(t) = W λ(t)d (c, d H r ). Since l[y] = (B λ +ia λ )y, l[z] = (B λ ia λ )z, we obtin Ŵ λ(t)j r (t)ŵλ(t) = J r (t 0 ). (2) It follows from (2) nd from the method of the vrition of rbitrry constnts tht the generl solution of the eqution l λ [y] = f(t) is represented in the form y λ (t, f) = W λ (t) c + J 1 r (t 0 ) t t 0 W λ(s)f(s)ds, (3) 118 Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No. 2

5 On Liner Reltions Generted by Differentil Expression... where c H r, the function f is loclly integrble on (, b). Consequently, ŷ λ (t, f) = Ŵλ(t) c + J 1 r (t 0 ) t t 0 W λ(s)f(s)ds. (4) We fix some λ 0 C 0. For brevity, we denote A(t) = λ0 (t). On the set of functions continuous nd finite on the intervl (, b) nd rnging in H, we introduce the qusi-sclr product (y, z) A = (A(t)y(t), z(t))dt. (5) We identify the functions such tht (y, y) A = 0 with zero nd perform the completion. Then we obtin Hilbert spce denoted by H = L 2 (H, A(t);, b). The elements of H re the clsses of functions identified with ech order in the norm y A = A 1/2 (t)y(t) 2 dt 1/2. (6) In order not to complicte terminology, we denote the clss of functions with representtive y by the sme symbol. We will lso sy tht the function y belongs to H. We tret the equlities between functions belonging to H s the equlities between corresponding equivlence clsses. It follows from Sttements 1, 3 tht the spce H does not depend on the choice of point λ C 0 in the sense below. If we chnge A(t) = λ0 (t) either to λ (t) (λ C 0 ) or to A λ (t) (Imλ > 0) in (5), we obtin the sme set H supplied with the equivlent norm. According to Sttement 1, the set G(t) = ker λ (t) (t I 0 ) is independent of λ C 0. Let H(t) be the orthogonl complement of G(t) in H, i.e, H(t) = H G(t); nd A 0 (t) be the restriction of A(t) to H(t). By H 1/2 (t) we denote the completion of H(t) with respect to the norm (A 0 (t)x, x) 1/2. The spce H 1/2 (t) cn be treted s spce with negtive norm with regrd to H(t) [8, Ch. 2]. Let H 1/2 (t) denote the corresponding spce with positive norm. It follows from Sttements 1, 3 tht the spces H 1/2 (t), H 1/2 (t) do not depend on the replcement of A(t) = λ0 (t) by λ (t) (λ C 0 ) or by A λ (t) (Imλ > 0) in the sense below. After replcing we obtin the sme sets H 1/2 (t), H 1/2 (t), but the norms on H 1/2 (t), H 1/2 (t) re chnged to the equivlent norms. Moreover, the constnts in the inequlities between the norms do not depend on t I 0 nd λ K, where the compct K C 0. Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No

6 V.M. Bruk Suppose Imλ > 0. Let A 1/2 0,λ (t), A 0,λ(t) be the restrictions respectively of A 1/2 λ (t), A λ(t) to H(t). It follows from the bove rgument nd the definition of positive nd negtive spces tht the opertors A 1/2 0,λ (t), A 0,λ(t) possess the continuous extensions Ã1/2 0,λ (t), à 0,λ (t) to H 1/2 (t). The opertor Ã1/2 0,λ (t) (Ã0,λ(t)) is the continuous nd one-to-one mpping of H 1/2 (t) onto H(t) (onto H 1/2 (t), respectively). Moreover, the equlity Ã0,λ(t) = A 1/2 0,λ (t)ã1/2 0,λ (t) holds. The opertor A 1/2 0,λ (t) is bijection of H(t) onto H 1/2(t). For Imλ<0, we set Ã0,λ(t)= Ã0, λ (t). Let Ãλ(t) (Imλ 0) denote the opertor defined on H 1/2 (t) G(t) which is equl to Ã0,λ(t) on H 1/2 (t) nd to zero on G(t). Obviously, à λ (t) is n extension of A λ (t). We pply the similr nottions for the opertors λ (t) nd A(t). In prticulr, Ã(t) is n extension of A(t) to H 1/2 (t) G(t) obtined in the sme wy s the extension Ãλ(t). The bove rgument nd Sttement 3 imply Ãλ(t)à 1 0,µ (t)x H1/2 k x H (t) 1/2 (t), x H 1/2(t), Ãλ(t)x 1 H1/2 k 1 x 1 H 1/2 (t), x 1 H 1/2 (t), (7) (t) where the constnts k, k 1 re independent of t I 0. (Here nd further, the symbols k, k 1,... denote positive constnts tht re different in vrious inequlities, in generl.) In [1], it ws shown tht the spces H 1/2 (t) re mesurble with respect to the prmeter t [9, Ch. 1] whenever for the fmily of mesurble functions one tkes the fmily of functions of the form à 1 0 (t)a1/2 (t)h(t), where h(t) is mesurble function with the vlues in H. The spce H is mesurble sum of the spces H 1/2 (t), nd H consists of the elements (i.e., clsses of functions) with representtives of the form à 1 0 (t)a1/2 (t)h(t), where h(t) L 2 (H;, b) [1]. We denote B λ,µ (t) = B λ (t) B µ (t), where λ, µ C 0, t I 0. It follows from Sttement 1 tht G(t) = ker λ (t) ker B λ,µ (t). Therefore, R(B λ,µ (t)) R( λ (t)) = H(t). By B (0) λ,µ (t) denote the restriction of B λ,µ(t) to H(t). Lemm 1. For ll t I 0, λ, µ C 0, the opertor B (0) λ,µ (t) possesses the (0) continuous extension B λ,µ (t) onto the spce H (0) 1/2(t). The opertor B λ,µ (t) mps H 1/2 (t) into H 1/2 (t) continuously. P r o o f. Following [4], we represent the function C λ (t) in the form C λ (t) = Re C i (t) + λu(t) + ( 1 τ λ τ ) 1 + τ 2 dσ t (τ), (8) 120 Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No. 2

7 On Liner Reltions Generted by Differentil Expression... where U(t) re nonnegtive bounded opertors in H, σ t (τ) is nondecresing opertor function for ech fixed t I 0 whose vlues re bounded opertors in H nd d(σ t(τ)g,g) < for ny g H. Then 1+τ 2 λ (t) = U(t) + + Using (8), (9), we obtin (λ µ) 1 ((C λ (t) C µ (t))g, h) U 1/2 (t)g U 1/2 (t)h 1/2 + (dσ t (τ)g, g) (dσ t (τ)h, h) t λ 2 t µ 2 U 1/2 (t)g 2 + (dσ t (τ)g, g) t λ 2 1/2 dσ t (τ) τ λ 2. (9) U 1/2 (t)h 2 + 1/2 (dσ t (τ)h, h) t µ 2 = ( λ (t)g, g) 1/2 ( µ (t)h, h) 1/2 = 1/2 λ (t)g 1/2 µ (t)h for ll g, h H. Hence, (B λ,µ (t)g, h) λ µ 1/2 λ (t)g 1/2 µ (t)h. According to Sttement 3, there exists constnt k > 0 such tht k is independent of t I 0, nd (B λ,µ (t)g, h) k A 1/2 (t)g A 1/2 (t)h (k > 0). (10) It follows from (10) tht the opertor B (0) λ,µ (t) : H(t) H(t) possesses the (0) continuous extension B λ,µ (t) : H 1/2(t) H(t) ( this fct lso follows from Sttement 2). Let B + (0) λ,µ (t) be the djoint opertor for B λ,µ (t). Then B + λ,µ (t) mps H(t) into H 1/2 (t) continuously. We clim tht B + λ,µ (t) = B(0) λ,µ (t). Since the opertor B λ,µ (t) is self-djoint, we hve (B λ,µ (t)x 1, x 2 ) = (x 1, B λ,µ (t)x 2 ) for ll x 1, x 2 H(t). On the other hnd, (B λ,µ (t)x 1, x 2 ) = ( B 0 λ,µ (t)x 1, x 2 ) = (x 1, B + λ,µ (t)x 2). 1/2 Hence, B (0) λ,µ (t) = B + λ,µ (t). Thus, B(0) λ,µ (t) mps H(t) into H 1/2(t) continuously. From this, we obtin tht inequlity (10) holds for ll h H 1/2 (t). Consequently, B (0) λ,µ (t)g 2 = H 1/2 (t) (à 1/2 0 (t)b λ,µ (t)g, à 1/2 0 (t)b λ,µ (t)g) k Ã1/2 0 (t)g à 1/2 0 (t)b (0) λ,µ (t)g = k g H 1/2 (t) B (0) λ,µ (t)g H1/2 (t) Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No

8 V.M. Bruk for ll g H(t). Hence, B (0) λ,µ (t)g H1/2 k g H (t) 1/2 (t), (11) where k > 0 is independent of t I 0. All ssertions of the lemm follow from (11). Lemm 1 is proved. Let B λ,µ (t) denote the opertor defined on H 1/2 (t) G(t) which is equl to B (0) λ,µ (t) on H 1/2(t) nd to zero on G(t). We set C λ,µ (t)= B λ,µ (t)+iãλ(t) iãµ(t). It follows from (7), (11) tht C λ,µ (t), Bλ,µ (t), Ã λ (t) re continuous mppings of H 1/2 (t) G(t) into H 1/2 (t). By C λ,µ, B λ,µ, Ãλ denote the opertors y(t) Ã 1 0 (t) C λ,µ (t)y(t), y(t) Ã 1 0 (t) B λ,µ (t)y(t), y(t) Ã 1 0 (t)ãλ(t)y(t), respectively. The opertors C λ,µ, B λ,µ, Ãλ re considered in the spce H. Thus, with equlities (7), (11) being tken into ccount, we obtin the following corollries from Lemm 1. (Corollry 1 is the generliztion of Sttement 2.) Corollry 1. For ll fixed λ, µ C 0, t I 0, the opertor C λ,µ (t) mps H 1/2 (t) into H 1/2 (t) continuously nd the inequlity C λ,µ (t)x k x H H1/2 (t) 1/2 (t) holds, where x H 1/2 (t), the constnt k is independent of t I 0. Corollry 2. For ll fixed λ, µ C 0, the opertor C λ,µ is bounded in H. For fixed λ or for fixed µ, the opertor function C λ,µ is holomorphic with respect to the other vrible. Corollry 3. For ll fixed λ, µ C 0, the opertors Ãλ, B λ,µ re bounded in H. The opertor functions Ãλ, B λ,µ re continuous with respect to λ, µ C 0 in the uniform opertor topology. 2. Fmilies of Mximl nd Miniml Reltions Let B 1, B 2 be Bnch spces. The liner reltion T is understood s ny liner mnifold T B 1 B 2. Terminology on the liner reltions cn be found, for exmple, in [8], [10], [11]. From now onwrds, the following nottion re used: {, } is n ordered pir; ker T is set of the elements x B 1 such tht {x, 0} T; KerT is set of ordered pirs of the form {x, 0} T; D(T) is the domin of T; R(T) is the rnge of T. The reltion T is clled invertible if T 1 is n opertor nd T is clled continuously invertible if T 1 is bounded everywhere defined opertor. The liner opertors re regrded s liner reltions, nd so the nottions S B 1 B 2, {x 1, x 1 } S re used for the opertors S. Let B 1 = B Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No. 2

9 On Liner Reltions Generted by Differentil Expression... be Hilbert spce. A liner reltion T is clled ccumultive (dissiptive) if for ll pirs {x 1, x 2 } T the inequlity Im(x 2, x 1 ) 0 (Im(x 2, x 1 ) 0) holds. The ccumultive (dissiptive) reltion is clled mximum if it does not hve proper ccumultive (dissiptive) extensions. Since ll the reltions considered further re liner, the word liner will often be omitted. A fmily of the liner mnifolds in Bnch spce B is understood s function λ T (λ), where T (λ) is liner mnifold, T (λ) B. A fmily of (closed) subspces T (λ) is clled holomorphic t the point λ 1 C if there exists Bnch spce B 0 nd fmily of the bounded liner opertors K(λ) : B 0 B such tht the opertor K(λ) bijectively mps B 0 onto T (λ) for ny fixed λ nd the fmily λ K(λ) is holomorphic in some neighborhood of λ 1. A fmily of subspces is clled holomorphic on the domin D if it is holomorphic t ll points belonging to D. Since the closed reltion T(λ) is the subspce in B 1 B 2, the definition of holomorphic fmilies is pplied to the fmilies of liner reltions. This definition generlizes the corresponding definition of holomorphic fmilies of closed opertors [12, Ch. 7]. In [13], [14], the sttement below is proved for the fmilies of closed reltions. However, the proof remins vlid for the fmilies of subspces. Sttement 4. Let T (λ) B be fmily of subspces in Bnch spce B. Suppose there exists subspce N B such tht the decomposition in the direct sum B=T (λ 1 ) N holds for some fixed point λ 1. The fmily T (λ) is holomorphic t λ 1 if nd only if the spce B is decomposed in the direct sum of subspces T (λ) nd N for ll λ belonging to some neighborhood of λ 1 nd the fmily P(λ) is holomorphic t λ 1, where P(λ) is the projection of spce B onto T (λ) in prllel N. The proof of the following sttement is given in [15]. Sttement 5. Let T (λ) be fmily of liner reltions holomorphic t the point λ 1, nd S(λ) be n opertor function holomorphic t the point λ 1 whose vlues re the bounded everywhere defined opertors. Then the fmily of reltions T (λ) + S(λ) is holomorphic t λ 1. Let Q 0 be the set of elements c H r such tht the function W µ (t)c (µ C 0 ) is identified with zero in H, i.e., the equlity A(t)W µ (t)c = 0 holds lmost everywhere. Using (3), we get t + W λ (t)jr 1 (t 0 ) W µ (t)c = W λ (t)c t 0 W λ(s)(b µ (s) B λ (s)+ia µ (s) ia λ (s))w µ (s)cds. (12) It follows from (12) nd Sttement 1 tht the set Q 0 does not chnge if we substitute W λ for W µ (λ, µ C 0 ). Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No

10 V.M. Bruk R e m r k 1. If we chnge λ to µ nd µ to λ in (12), then we obtin the true equlity (λ, µ C 0 ). Further, in this section nd Section 3, we ssume tht the boundries, b re regulr. Let Q be the orthogonl complement of Q 0 in H r, i.e., Q = H r Q 0. In Q, we introduce the norm c = A 1/2 (s)w µ (s)c 2 ds 1/2 k c, c Q, k > 0. (13) We denote the completion of Q with respect to this norm by Q. By (12), we get A 1/2 (t)w λ (t)c A 1/2 (t)w µ (t)c + k A 1/2 (t) (C λ (s) C µ (s))w µ (s)c ds. We will obtin the nlogous estimte for the norm A 1/2 (t)w µ (t)c if we chnge W λ to W µ nd W µ to W λ. It follows from this nd Sttement 2 tht the replcement of W µ by W λ in (13) leds to the sme set Q with the equivlent norm. Thereby the spce Q is independent of λ C 0. The spce Q cn be treted s spce with negtive norm with respect to Q [8, Ch. 2]. By Q + denote the corresponding spce with positive norm. The opertor c W λ (t)c is continuous one-to-one mpping of Q into H. We denote this opertor by W λ. Its rnge is closed in H. Consequently, the djoint opertor W λ continuously mps H onto Q +. We find the form W λ. For ny elements x Q nd f H, we hve (f, W λ x) A = (Ã(s)f(s), W λ(s)x)ds = (Wλ (s)ã(s)f(s), x)ds = (W λf, x). Hence, tking into ccount tht Q cn be densely embedded in Q, we obtin W λ f = b W λ (s)ã(s)f(s)ds. Here we replce λ by λ nd summrize the properties of opertors W λ, W λ in the following sttement. 124 Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No. 2

11 On Liner Reltions Generted by Differentil Expression... Lemm 2. For fixed λ C 0, the opertor W λ is continuous one-to-one mpping of Q into H, nd the rnge R(W λ ) is closed. The opertor W λ mps continuously H onto Q +. The opertor functions W λ, W λ re holomorphic on C 0. In the sme wy s in [1], [3], we define the miniml nd mximl reltions generted by the expression l λ (λ C 0 ) nd function A(t). Let D λ be set of functions y H stisfying the following conditions: () the qusiderivtives y [k] exist, they re bsolutely continuous up to the order r 1 inclusively, nd l λ [y](t) H 1/2 (t) lmost everywhere; (b) the function à 1 0 (t)l λ[y] H. To ech clss of the functions identified in H with y D λ we ssign the clss of functions identified in H with à 1 0 (t)l λ[y]. Thus, we get liner reltion L (λ) H H. Denote its closure by L(λ) nd cll it mximl reltion. We define the miniml reltion L 0 (λ) s restriction of L(λ) to the set of functions y D λ stisfying the conditions y [k] () = y [k] (b) = 0 (k = 0, 1,..., r 1). The following sttements cn be proved by nlogy with the corresponding ssertions in [3], [16]. Lemm 3. For fixed λ C 0, the reltion L(λ) consists of ll pirs {y, f} H H for which y = W λ (t)c λ + F λ, (14) where c λ Q, F λ (t) = W λ (t)jr 1 (t 0 ) belongs to L 0 (λ) if nd only if c λ = 0 nd t W λ(s)ã(s)f(s)ds. The pir {y, f} L(λ) W λ(s)ã(s)f(s)ds = 0. (15) Corollry 4. The rnge R(L 0 (λ)) is closed nd consists of the elements f H with property (15). The reltion L 1 0 (λ) is bounded opertor on its domin of definition. Corollry 5. The reltion L 0 (λ) is closed. Corollry 6. The opertor W λ is continuous one-to-one mpping of Q onto ker L(λ). R e m r k 2. In equlity (14), the element c λ Q nd the function F λ H re uniquely determined by the pir {y, f} L(λ). Theorem 1. For ll λ, µ C 0, the equlity L(µ)=L(λ) + C λ,µ holds. P r o o f follows from the definition of L(λ) nd from Corollry 2. Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No

12 V.M. Bruk Corollry 7. The fmilies of reltions L(λ), L 0 (λ) re holomorphic on C 0. P r o o f follows from Theorem 1, Corollry 2, Sttement 5. Lemm 4. The equlity (L 0 (λ)) = L( λ) holds for ech λ C 0. P r o o f. Let L(λ) be the mximl reltion nd L 0 (λ) be the miniml reltion generted by the formlly self-djoint expression l[y] B λ (t)y. In [1], [16], it ws shown tht ( L 0 (λ)) = L(λ) = L( λ). Using the equlities L( λ) = L( λ) iã λ, L 0 (λ) = L 0 (λ) iãλ nd Corollry 3, we obtin the desired ssertion. Lemm 4 is proved. R e m r k 3. Tking into ccount Theorem 1 nd the proof of Lemm 4, we get D(L(µ)) = D(L(λ)) = D( L(λ)) nd D(L 0 (µ)) = D(L 0 (λ)) = D( L 0 (λ)) for ll λ, µ C 0. Lemm 5. For fixed λ (Imλ 0) nd for ll pirs {y 0, g} L 0 (λ), the inequlity Imλ 1 Im(g, y 0 ) A k y 0 2 A, k > 0, holds. P r o o f. Suppose Imλ>0. Using (1) nd the definition of L 0 (λ), we get Im(g, y 0 ) A = (A λ (t)y 0 (t), y 0 (t))dt. (16) Now the desired ssertion follows from (16) nd Sttement 3. The cse Imλ < 0 is considered nlogously. Lemm 5 is proved. So, the reltion L 0 (λ) is ccumultive in the upper hlf-plne nd L 0 (λ) is dissiptive in the lower hlf-plne. 3. Holomorphic Fmilies of Invertible Restrictions of Mximl Reltions Let B 1, B 2, B1, B2 be Bnch spces, T B 1 B 2 be closed liner reltion, γ : T B 1 B 2 be liner opertor, γ i = P i γ (i = 1, 2), where P i is the projection B 1 B 2 onto B i, i.e., P i {x 1, x 2 } = x i. The following definition is given in [17] for the opertors nd in [18] for the reltions. Definition 1. The qudruple ( B 1, B 2, γ 1, γ 2 ) is clled spce of boundry vlues (SBV) for closed reltion T if the opertor γ is continuous mpping of T onto B 1 B 2 nd the restriction of the opertor γ 1 to KerT is one-to-one mpping of KerT onto B Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No. 2

13 On Liner Reltions Generted by Differentil Expression... We define n opertor Φ γ : B 1 B 2 by the equlity Φ γ = γ 2 β, where β = (γ 1 KerT ) 1 is the opertor inverse to the restriction of γ 1 to KerT. We denote T 0 = ker γ. It follows from the definition of SBV tht between the reltions θ B 1 B 2 nd the reltions T with property T 0 T T there is one-toone correspondence determined by the equlity γ T = θ. In this cse we denote T = T θ. The reltion θ is clled boundry reltion. The following ssertion is proved in [18]. Sttement 6. The reltion T θ is continuously invertible if nd only if the reltion θ Φ γ is continuously invertible. Suppose {y, f} L(λ) (λ C 0 ). Then (14) holds. To ech pir {y, f} L(λ) we ssign pir of boundry vlues {Y λ, Y λ } Q Q + by the formuls Y λ = W λf = W λ(s)ã(s)f(s)ds, Y λ = c λ + (1/2)J 1 r (t 0 )Y λ. (17) It follows from Remrk 2 tht the pir {Y λ, Y λ } of boundry vlues is uniquely determined for ech pir {y, f} L(λ). Let γ(λ), γ 1 (λ), γ 2 (λ) be the opertors defined by the equlities γ(λ){y, f} = {Y λ, Y λ }, γ 1(λ){y, f} = Y λ, γ 2 (λ){y, f} = Y λ. Lemms 2, 3 nd Corollry 6 imply the following ssertion. Theorem 2. The qudruple {Q, Q +, γ 1 (λ), γ 2 (λ)} is SBV for the reltion L(λ). The corresponding opertor Φ γ(λ) equls zero. Moreover, ker γ(λ) = L 0 (λ). Theorem 3. The rnge R(γ(λ)) of the opertor γ(λ) coincides with Q Q +, nd for ll pirs {y, f} L(λ), {z, g} L( λ) (λ C 0 ) the Green formul is vlid (f, z) A (y, g) A = (Y λ, Z λ ) (Y λ, Z λ), (18) where {Y λ, Y λ } = γ(λ){y, f}, {Z λ, Z λ} = γ( λ){z, g}. P r o o f. Using Lemms 2, 3, we get R(γ(λ)) = Q Q +. According to Lemm 3, y = W λ (t)c λ + F λ, z = W λ(t)d λ + G λ, where c λ, d λ Q, F λ (t) = W λ (t)j 1 r (t 0 ) G λ(t) = W λ(t)j 1 r (t 0 ) t t W λ(s)ã(s)f(s)ds, W λ (s)ã(s)g(s)ds. Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No

14 V.M. Bruk The opertions l λ, l λ cn be pplied to the functions F λ, G λ nd l λ [F λ ] = Ã(t)f, l λ[g λ] = Ã(t)g. From this, (1), (2), nd (17), we obtin (f, G λ) A (F λ, g) A =(J r (b)ŵλ(b)jr 1 (t 0 )Y λ 1, Ŵ λ (b)jr (t 0 )Z λ)=(y λ, J r 1 (t 0 )Z λ). (19) We tke two sequences {c λ,n }, {d λ,n } such tht c λ,n, d λ,n Q nd {c λ,n }, {d λ,n } converge to c λ Q nd d λ Q in Q, respectively. We denote v n (t) = W λ (t)c λ,n, u n (t) = W λ(t)d λ,n. Then the sequences {v n (t)}, {u n (t)} converge to v(t) = W λ (t)c λ, u(t) = W λ(t)d λ in H, respectively. Using (1), (2), (17), we obtin (f, u n ) A =(f, u n ) A (F λ, 0) A =(J r (t 0 )Ŵλ(b)Jr 1 (t 0 )Y λ, Ŵ λ (b)d λ,n)=(y λ, d λ,n ), (v n, g) A =(v n, g) A (0, G λ) A =(J r (t 0 )Ŵλ(b)c λ,n, In these equlities, we pss to the limit s n. We get 1 Ŵ λ (b)jr (t 0 )Z λ)= (c λ,n, Z λ). (f, u) A = (Y λ, d λ ), (v, g) A = (c λ, Z λ). (20) Using (19), (20), nd the equlity J r (t 0 ) = J r (t 0 ), we obtin (f, z) A (y, g) A = (Y λ, d λ ) (c λ, Z λ) + (Y λ, J r 1 (t 0 )Z λ) = = (Y λ 1, d λ +(1/2)Jr (t 0 )Z λ) (c λ +(1/2)Jr 1 (t 0 )Y λ, Z λ) = (Y λ, Z λ ) (Y λ, Z λ). Theorem 3 is proved. Notice tht for the first time the liner reltions were pplied to describe selfdjoint extensions of differentil opertors in [19]. Further bibliogrphy cn, for exmple, be found in [8], [10]. The boundry vlues in form (17) re suggested in [16]. For fixed λ C 0, between the reltions L(λ) with the property L 0 (λ) L(λ) L(λ) nd reltions θ Q Q + there is one-to-one correspondence determined by the equlity γ(λ) L(λ) = θ. In this cse we denote L(λ) = L θ (λ). Thus, pir {y, f} L θ (λ) if nd only if {y, f} L(λ) nd {Y λ, Y λ } θ, where {Y λ, Y λ } = γ(λ){y, f}. Using Lemm 4, we get L (λ) = L 0 ( λ), (L 0 (λ)) = L( λ). Suppose L 0 (λ) L θ (λ) L(λ). Then L 0 ( λ) (L θ (λ)) L( λ). Hence, tking (18) into ccount, we obtin (L θ (λ)) = L θ ( λ), λ C 0. (21) Lemm 6. Let θ(λ) Q Q + be fmily of liner reltions defined on set D 1 C 0 symmetric with respect to the rel xis. Then the equlity θ (λ) = θ( λ) holds if nd only if the equlity (L θ(λ) (λ)) =L θ( λ) ( λ) holds. P r o o f follows from equlity (21). 128 Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No. 2

15 On Liner Reltions Generted by Differentil Expression... Theorem 4. For fixed λ C 0, the reltion L θ(λ) (λ) is continuously invertible if nd only if so is the reltion θ(λ). In this cse, the opertor R λ = L 1 θ(λ) (λ) is integrl R λ g = K λ (t, s)ã(s)g(s)ds, g H, where K λ (t, s) = W λ (t)(θ 1 (λ) (1/2)sgn(s t)jr 1 (t 0 ))W λ(s). Proof. The first prt of the theorem follows from Sttement 6 nd Theorem 2. Let us prove the second prt of the theorem. For ny pir {y, g} L(λ), we trnsform equlity (14) to the form y(t) = W λ (t) c λ + (1/2)J 1 r (t 0 ) t W λ(s)ã(s)g(s)ds (1/2)J 1 r (t 0 ) t W λ(s)ã(s)g(s)ds, (22) where c λ = c λ + (1/2)Jr 1 (t 0 ) W λ(s)ã(s)g(s)ds = Y λ = γ 1 (λ){y, g}. Using (17), (22), we get y(t) = W λ (t) θ 1 (λ)y λ + (1/2)J 1 r (t 0 ) t W λ(s)ã(s)g(s)ds (1/2)J 1 r (t 0 ) This implies the desired ssertion. Theorem 4 is proved. t W λ(s)ã(s)g(s)ds. Corollry 8. The equlities θ (λ)= θ( λ) nd Rλ = R λ hold together. Theorem 5. The function R λ is holomorphic t the point λ 1 C 0 if nd only if the function θ 1 (λ) is holomorphic t the sme point. P r o o f. According to Lemm 2 nd Theorem 4, if the function θ 1 (λ) is holomorphic t λ 1, then the function R λ hs the sme property. Now suppose tht the function R λ is holomorphic t λ 1. Using Lemm 2 nd Theorem 4, we obtin tht the function W λ θ 1 (λ)w λg is holomorphic for every g H. It follows from Remrk 1 nd equlity (12) with the element c replced by θ 1 (λ)w λg tht the functions W λ θ 1 (λ)w λg nd W µ θ 1 (λ)w λg re holomorphic together for Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No

16 V.M. Bruk fixed µ C 0. From Corollry 6 we conclude tht the function θ 1 (λ)w λg is holomorphic t λ 1 for ny g H. Now the property tht the function θ 1 (λ) is holomorphic t λ 1 follows from Sttement 7 proved in [20]. In this sttement it should be tken tht B 1 = H, B 2 = Q +, B 3 = Q, S 1 (λ) = W λ, S 2 (λ) = θ 1 (λ), S 3 (λ) = θ 1 (λ)w λ. Theorem 5 is proved. Sttement 7. Let B 1, B 2, B 3 be Bnch spces. Suppose the bounded opertors S 3 (λ) : B 1 B 3, S 1 (λ) : B 1 B 2, S 2 (λ) : B 2 B 3 stisfy the equlity S 3 (λ) = S 2 (λ)s 1 (λ) for every fixed λ belonging to some neighborhood of point λ 1 ; moreover, R(S 1 (λ 1 )) = B 2. If the functions S 1 (λ), S 3 (λ) re strongly differentible t the point λ 1, then t the sme point the function S 2 (λ) is strongly differentible. Let N µ =H R(L 0 (µ)), where Imµ > 0. Using Lemms 3, 4 nd Corollry 6, we get N µ =ker L( µ). It follows from Theorem 1 tht N µ D(L(λ)) nd the pir {w(t), à 1 0 (t) C µ,λ (t)w(t)} L(λ), (23) where w(t) N µ. The set D(L 0 (λ)) = D(L 0 ( µ)) nd N µ re linerly independent in H. For Imλ > 0, we define the reltion L(λ) s the restriction of L(λ) to D(L 0 (λ)) N µ. By (23) nd the equlity A µ = A µ, it follows tht the reltion L(λ) consists of ll pirs of the form {y 0,λ (t) + w(t), y 1,λ (t) + à 1 0 (t)( B µ,λ (t) i(ãλ(t) + õ(t)))w(t)}, (24) where {y 0,λ, y 1,λ } L 0 (λ), w(t) N µ. For Imλ < 0, we set L(λ) = L ( λ). Theorem 6. The fmily of reltions L(λ) is holomorphic for Imλ 0. P r o o f. First, we consider the cse Imλ > 0. By Corollry 7, the fmily of reltions L 0 (λ) is holomorphic t every point λ 1 (Imλ 1 > 0). It follows from the definition of holomorphic fmily tht there exists Bnch spce B nd fmily of bounded liner opertors K 0 (λ):b H such tht the opertor K 0 (λ) bijectively mps B onto L 0 (λ) for ny fixed λ nd the fmily λ K 0 (λ) is holomorphic in some neighborhood of λ 1. By K 1 (λ) we denote n opertor tking every element w N µ to the pir {w, C µ,λ w} H H. By Corollry 2, the opertor K 1 (λ) is bounded for fixed λ C 0 nd the fmily of opertors K 1 (λ) is holomorphic on C 0. Consider the spce B = B N µ nd define the opertor K(λ) : B H H by the formul K(λ){u, w} = K 0 (λ)u + K 1 (λ)w, where u B, w N µ. For fixed λ, the opertor K(λ) is continuous one-to-one mpping of B onto L(λ) nd the fmily λ K(λ) is holomorphic in some neighborhood of λ 1. So, the theorem is vlid for Imλ > Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No. 2

17 On Liner Reltions Generted by Differentil Expression... Suppose Imλ < 0. Let the opertor J : H H H H be given by the formul J {x, x 1 } = { x 1, x}. Then L ( λ)=j L ( λ) (Here nd further, we denote N =H N, where N is subspce of Hilbert spce H.) The fmily of subspces J L ( λ) is holomorphic if nd only if the fmily L ( λ) is holomorphic. Now the desired ssertion is obtined from Lemm 7 below. Theorem 6 is proved. Corollry 9. The fmily of reltions L 1 (λ) is holomorphic for Imλ 0. Lemm 7. Let H be Hilbert spce. If the fmily of subspces T (λ) H is holomorphic t the point λ 1, then the fmily T ( λ) is holomorphic t λ 1. Proof. By Sttement 4, H=T (λ) N nd the fmily P(λ) is holomorphic t λ 1, where P(λ) is the projection of spce H onto T (λ) in prllel N. Then Q(λ) = E P(λ) is the projection H onto N in prllel T (λ). Therefore, ker Q (λ) = N, R(Q (λ)) = T (λ), nd (Q (λ)) 2 = Q (λ). Consequently, Q (λ) is the projection of H onto T (λ) in prllel N. The corresponding decomposition hs the form H = T (λ) N. Since the fmily Q ( λ) is holomorphic, the fmily of subspces T ( λ) is holomorphic t the point λ 1. Lemm 7 is proved. Theorem 7. For fixed λ, the reltion L(λ) is mximl ccumultive if Imλ > 0 nd it is mximl dissiptive if Imλ < 0. The reltion L 1 (λ) is bounded everywhere defined opertor. P r o o f. Let Imλ > 0, {z, z 1 } L(λ). Then {z, z 1 } hs form (24). Using (1), (24) nd the equlities l µ [w] = 0, l[y 0,λ ] = Ã0y 1,λ + ia λ y 0,λ + B λ y 0,λ, we get (z 1, z) A = (y 1,λ, y 0,λ ) A + (y 1,λ, w) A + ( B µ,λ w, y 0,λ ) A + ( B µ,λ w, w) A i((ãλ + õ)w, y 0,λ ) A i((ãλ + õ)w, w) A = (y 1,λ, y 0,λ ) A + (y 1,λ, w) A + (w, y 1,λ ) A + ( B µ,λ w, w) A Hence, tking (16) into ccount, we obtin 2i(Ãλw, y 0,λ ) A i((ãλ + õ)w, w) A. Im(z 1, z) A = Im(y 1,λ, y 0,λ ) A 2Re(Ãλw, y 0,λ ) A ((Ãλ + õ)w, w) A = (Ãλ(y 0,λ + w), y 0,λ + w) A (õw, w) A = (Ãλz, z) A (õw, w) A Therefore, Im(z 1, z) A = (Ãλ(t)z(t), z(t))dt (õ(t)w(t), w(t))dt. (Ãλ(t)z(t), z(t))dt k(z, z) A (k > 0). (25) Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No

18 V.M. Bruk Thus, the reltion L(λ) is ccumultive. It follows from (25) tht the rnge R(L(λ)) is closed nd the reltion L 1 (λ) is the opertor. Moreover, R(L(µ)) = H. The equlity R(L(λ)) = H is obtined from Lemm 8 nd Corollry 10 below. Consequently, the reltion L(λ) is mximl ccumultive if Imλ > 0. For Imλ < 0, the desired ssertion is obtined from the equlity L(λ)=L ( λ). Theorem 7 is proved. Lemm 8. Let H be Hilbert spce nd S(λ) H H be fmily of liner reltions holomorphic t the point µ. Suppose tht the reltion S(µ) hs the following properties: R(S(µ)) is closed subspces in H nd the reltion S 1 (µ) is n opertor. Then there exists neighborhood of µ such tht the reltions S(λ) hve the sme properties for ll points λ belonging to this neighborhood. Moreover, dim H R(S(λ)) = dim H R(S(µ)). P r o o f. We denote H 0 = H R(S(µ)), N = H H 0. We clim tht the decomposition in the direct sum H H = S(µ) N holds. Indeed, ny pir {x 1, x 2 } H H is uniquely produced by the form {x 1, x 2 } = {S 1 (µ)p x 2, P x 2 } + {x 1 S 1 (µ)p x 2, P x 2 }, where P is the orthogonl projection H onto R(S(µ)), P = E P. According to Sttement 4, the decomposition in the direct sum H H = S(λ) N (26) holds for ll λ belonging to some neighborhood of µ. Using (26), we obtin {x 1, x 2 }={f 1, f 2 }+{x 1 f 1, x 2 f 2 } for ll {x 1, x 2 } H H, where {f 1, f 2 } S(λ), {x 1 f 1, x 2 f 2 } N. If f 2 =0, then x 2 H 0. Hence, {x 1, x 2 } N. The uniqueness of decomposition (26) implies f 1 = 0. So, S 1 (λ) is the opertor. Let us prove tht R(S(λ)) is closed. For this purpose, we consider the opertor U(λ) : R(S(µ)) H defined by the equlity U(λ)x = P 2 P(λ){S 1 (µ)x, x}, where x R(S(µ)), P(λ) is the projection of spce H H onto S(λ) in prllel N corresponding to (26), P 2 is the projection H H onto the second fctor, i.e., P 2 {x 1, x 2 } = x 2. For fixed λ, the opertor U(λ) is continuous one-to-one mpping of R(S(µ)) onto R(S(λ)). The opertor function U(λ) is holomorphic in some neighborhood of µ nd U(µ)x = x. Therefore the set R(S(λ)) is closed for ll λ from some neighborhood of µ. Lemm 8 is proved. Corollry 10. Suppose the fmily of reltions S(λ) is holomorphic in connected domin D, nd S(µ) stisfies the ssumptions of Lemm 8 for ll points µ D. Then dim H R(S(λ)) is constnt in D. Proof follows from the fct tht this dimension is constnt in neighborhood of every point belonging to D. 132 Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No. 2

19 On Liner Reltions Generted by Differentil Expression On the Chrcteristic Opertor In the spce H = L 2 (H, A(t);, b), the norm nd the sclr product re defined by equlities (5), (6). According to Sttements 1, 3, fter replcing A(t) = λ0 (t) by λ (t), we obtin spce with the equivlent norm. By λ ( (, ) λ ) we denote the norm (the sclr product, respectively) in the spce L 2 (H, λ (t);, b). We consider the eqution l[y] B λ (t)y ia λ (t)y = λ (t)f(t) (f H) (27) on the finite or infinite intervl (, b) nd use the following definition (see [4], [5]). Definition 2. Let M(λ) = M ( λ) be function holomorphic for Imλ 0 whose vlues re bounded liner opertors such tht D(M(λ)) = Q + nd R(M(λ)) Q. This function M(λ) is clled chrcteristic opertor of eqution (27) if for ny function f H with compct support the corresponding solution y λ (t) of (27) = stisfies the inequlity y λ (t) = (R λ f)(t) W λ (t)(m(λ) (1/2)sgn(s t)j 1 r (t 0 ))W λ(s) λ (s)f(s)ds (28) R λ f 2 λ Im(R λ f, f) λ /Imλ, Imλ 0. (29) (For the cse of the singulr boundry or b, the spces Q, Q + re defined bellow.) In [4], [5], the existence of the chrcteristic opertor is proved. In the cse of the xis, the chrcteristic opertors pper tht re unbounded in Q [4, pp. 167, 172]. In [5, p. 209], there is proposed nother method llowing to connect holomorphic fmily of dissiptive opertors with the expression l λ. (In [4], [5], the cse r = 1 is considered.) Suppose tht the boundries, b re regulr. Let L 0 (λ) L(λ) L(λ) for Imλ 0. We sy tht fmily of liner reltions L(λ) genertes chrcteristic opertor M(λ) of eqution (27) if γ(λ)l(λ) = M 1 (λ). Theorem 8. The fmily of closed reltions L(λ) (Imλ 0) genertes the chrcteristic opertor of eqution (27) if nd only if L(λ) stisfies the following conditions: 1) L( λ) = L (λ); 2) the fmily L(λ) is holomorphic; 3) for fixed λ, Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No

20 V.M. Bruk the reltion L 1 (λ) is n everywhere defined opertor (L 1 (λ) is bounded since L 1 (λ) is closed); 4) the inequlity Im(g λ, z λ ) A (Ã 1 0 Ã λ z λ, z λ ) A = holds for ll pirs {z λ, g λ } L(λ) nd ll λ such tht Imλ > 0. (Ãλ(t)z λ (t), z λ (t))dt (30) P r o o f. Assume tht the fmily L(λ) possesses the properties 1) 4). It follows from Theorems 4, 5 nd Lemm 6 tht there exists holomorphic function M(λ) = M ( λ) (Imλ 0) whose vlues re the bounded opertors such tht D(M(λ)) = Q +, R(M(λ)) Q, nd the equlity = z λ (t) = (L 1 (λ)g)(t) W λ (t)(m(λ) (1/2)sgn(s t)jr 1 (t 0 ))W λ(s)ã(s)g(s)ds (31) holds for ll g H. The function M(λ) is defined by the equlity γ(λ)l(λ) = M 1 (λ). (32) We set f(t) = (Imλ)Ã 1 0,λ (t)ã(t)g(t). It follows from the rguments to the definition of the spces H 1/2 (t), H 1/2 (t) nd Corollry 3 tht the functions f nd g belong or do not belong to the spce H together. Moreover, the equlities g(t) = Ã 1 0 (t) λ(t)f(t), λ (t)f(t) = Ã(t)g(t) (33) hold. We fix the function f H nd denote g λ (t) = Ã 1 0 (t) λ(t)f(t), y λ = L 1 (λ)g λ = R λ f. By (31), (33), it follows tht y λ is the solution of eqution (27) nd y λ hs the form (28). Now we clim tht inequlity (29) holds. Indeed, (30) implies Im(f, R λ f) λ = Im(g λ, y λ ) A Imλ y λ 2 λ = Imλ R λ f 2 λ for Imλ > 0. If Imλ < 0, then (29) follows from the equlity L( λ) = L (λ). Thus, if the fmily L(λ) stisfies the conditions 1) 4), then the function M(λ) defined by (32) is the chrcteristic opertor. The proof is convertible. From this there follows the converse. Theorem 8 is proved. Using Theorems 6, 7, 8, formul (25) nd the equlity L( λ)=l (λ), we obtin Corollry 11. The fmily M(λ) = (γ(λ)l(λ)) 1 is chrcteristic opertor. 134 Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No. 2

21 On Liner Reltions Generted by Differentil Expression... R e m r k 4. In (14), suppose tht c λ Q. By Lemm 2, y is the function rnging in H. Using (14), (4), we get ŷ() = Ŵλ()c λ, ŷ(b) = Ŵλ(b)c λ + F λ (b). From this nd (17), by strightforwrd clcultions, we obtin Y λ =J r(t 0 )(Ŵ 1 1 (b)ŷ(b) Ŵ ()ŷ()), Y λ =2 1 (Ŵ 1 1 (b)ŷ(b)+ŵ ()ŷ()). λ λ λ These equlities correspond to Remrk 1.1 in [4]. For L(λ), the vlues ŷ(), ŷ(b) re clculted by formul (12), where µ is chnged to µ. R e m r k 5. Let L(µ 0 ) be the mximl reltion nd L 0 (µ 0 ) be the miniml reltion generted by the formlly self-djoint expression l[y] B µ0 (t)y in H, where µ 0 C 0 is fixed. It follows from Theorem 8 tht the fmily L(λ) genertes chrcteristic opertor if nd only if the fmily Λ(λ) = L(λ) + B λ,µ0 + iãλ hs the properties: 1) L 0 (µ 0 ) Λ(λ) L(µ 0 ); 2) Λ (λ) = Λ( λ); 3) the fmily Λ(λ) is holomorphic for Imλ 0; 4) Λ(λ) is mximl ccumultive reltion for Imλ > 0. Thus, (Λ(λ) λe) 1 is generlized resolvent of L 0 (µ 0 ). This sttement is estblished by the other method in [4]. Below we ssume tht =, b =. So, H = L 2 (H, A(t);, ). We define mximl nd miniml reltions in the following wy. Let D 0,λ (λ C 0) be set of finite functions y H stisfying the conditions: () the qusiderivtives y [k] exist, they re bsolutely continuous up to the order r 1 inclusively; (b) l λ [y](t) H 1/2 (t) lmost everywhere; (c) à 1 0 (t)l λ[y] H. We introduce correspondence between ech clss of functions identified with y D 0,λ in H nd the clss of functions identified with à 1 0 (t)l λ[y] in H. Thus, in the spce H, we obtin liner reltion L 0 (λ), denote its closure by L 0(λ), nd cll L 0 (λ) the miniml reltion. (In the regulr cse, one cn define the miniml reltion in the sme wy. One cn show tht this definition nd the definition of miniml reltion from Section 2 re equivlent. We do not use this fct.) We denote L(λ) = (L 0 ( λ)). The reltion L(λ) is clled mximl. (According to Lemm 4, in the regulr cse the mximl reltion cn be defined in the sme wy.) Using the equlity L 0 (µ) = L 0 (λ) + C λ,µ (λ, µ C 0 ) nd Corollry 2, we get L 0 (µ) = L 0 (λ) + C λ,µ. This implies tht the fmily L 0 (λ) is holomorphic on C 0. By Lemm 7, it follows tht the fmily L(λ) is holomorphic on C 0. Using Corollry 2, we get L(µ) = L(λ) + C λ,µ. Thus, Theorem 1 nd Corollry 7 re vlid for the reltions L(λ) nd L 0 (λ). For the reltion L 0 (λ) (Imλ 0), the proof of Lemm 5 is the sme s the bove proof for the regulr cse. By the limit pssge, we obtin tht Lemm 5 is vlid for reltion L 0 (λ). Consequently, the rnge R(L 0 (λ)) is closed in H nd the reltion L 1 0 (λ) is bounded everywhere defined opertor on R(L 0 (λ)) (Imλ 0). In Corollry 10, we set S(λ) = L 0 (λ). Then we obtin tht the dimension dim H R(L 0 (λ)) = dim ker L( λ) is constnt in the hlf-plnes Imλ > 0, Imλ < 0. In [4, 6, 7, 21 23] this ssertion ws proved by vrious methods for vrious cses. λ Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No

22 V.M. Bruk We fix some µ (Imµ > 0) nd denote N µ = H R(L 0 (µ)) (it is not inconceivble tht N µ = {0}). Obviously, N µ = ker(l 0 (µ)) = ker L( µ). The definition of the reltion L(λ) is the sme s the bove for the regulr cse, i.e, L(λ) is the restriction L(λ) to D(L 0 (λ)) N µ. The reltion L(λ) consists of the ordered pirs of the form (24). We set L(λ) = L ( λ) for Imλ < 0. Theorems 6, 7 re obtined by the literl repetition of the proof used for the regulr cse. We construct the domin nd the rnge of the chrcteristic opertor. By Q 0 denote set of elements c H r such tht the norm A(t)W µ (t)c = 0 lmost everywhere on (, ), Q = H r Q 0. The sets Q 0, Q do not depend on µ C 0. (This ssertion cn be proved by nlogy with the corresponding ssertion for the regulr cse.) Let [α n, β n ] be sequence of intervls such tht [α n, β n ] (α n+1, β n+1 ), nd α n, β n s n (n N). In Q, we introduce the system of seminorms β n p n (x) = A 1/2 (s)w µ (s)x 2 ds α n 1/2, µ C 0, x Q. (34) We denote the completion of Q with respect to this system by Q. The spce Q is loclly convex. Arguing s bove, we see tht the replcement of µ by λ C 0 leds to n isomorphic spce [25, Ch. 1]. The spce Q is isomorphic to projective limit of fmily of Bnch spces Q (n) (see [25, Ch. 2, proof 5.4]). The spces Q (n) re constructed bellow. Let Q + (n) be the spce djoint to Q (n), Q + be n inductive limit of the spces Q + (n). We clim tht D(M(λ)) = Q +, R(M(λ)) Q. We describe this construction more detil for the proof of properties of the opertor M(λ). This construction is little distinguished from the corresponding construction in [24]. Therefore we omit the detils of rgumenttion. We denote H n = L 2 (H, A(t); α n, β n ), n N. Let Q 0 (n) be set of elements c Q such tht the function W λ (t)c is identified with zero in the spce H n, Q(n) = Q Q 0 (n). Then Q 0 (n) Q 0 (m) nd Q(n) Q(m) for n > m. In Q(n), we introduce the seminorm p n (x) by formul (34). This seminorm is the norm on Q(n). We denote it by (n). If m < n, c Q(m), then c (m) Let Q (n) be the completion of Q(n) with respect to the norm (n) mppings h mn : Q (n) Q (m) (n > m) in the following wy. Let Q (n) c (n).. We define (n, m) be the completion of Q(n, m)=q(n) Q(m) with respect to the norm (n) nd Q (n) (m) be the completion of Q(m) with respect to this norm. Then Q (n) = Q (n) (m) Q(n) (n, m). We define h m,nc=0 whenever c Q (n) (n, m), nd we define h m,n c = j mn c whenever c Q (n) (m), where j mn is the inclusion mp of Q (n) (m) into Q (m). We set h nn c=c for m=n. The mppings h mn re continuous. 136 Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No. 2

23 On Liner Reltions Generted by Differentil Expression... We consider the projective limit lim(pr)h mn Q (n) of the fmily of the spces {Q (n); n N} with respect to the mppings h mn (m, n N, m n) [25, Ch. 2]. By repeting the corresponding rguments from [25, Ch. 2], one cn show tht this projective limit is isomorphic to the spce Q introduced fter formul (34). We do not use this fct nd thereby omit its proof. Further, by Q we denote the projective limit lim(pr)h mn Q (n). It follows from the definition of projective limit [25, Ch. 2] tht Q is subspce of the product n Q (n) nd Q consists of ll elements c = {c n } such tht c m = h mn c n for ll m, n, where m n. The spce Q (n) cn be treted s negtive one with respect to Q(n) [8, Ch. 2]. By Q + (n) we denote the corresponding spce with positive norm. Then Q + (m) Q + (n) for m n, nd the inclusion mp of Q + (m) into Q + (n) is continuous. This inclusion mp coincides with h + mn, where h + mn :Q + (m) Q + (n) is the djoint mpping of h mn. By Q + we denote the inductive limit [25, Ch. 2] of the fmily {Q + (n); n N} with respect to the mppings h + mn, i.e., Q + = lim(ind)h + mnq + (n). According to [25, Ch. 4], Q + is the djoint spce of Q. The spce Q + cn be treted s the union Q + = n Q + (n) with the strongest topology such tht ll inclusion mps of Q + (n) into Q + re continuous [25, Ch. 2]. By Lemm 2, the opertor c n W λ (t)c n is continuous one-to-one mpping of Q (n) into H n nd it hs the closed rnge. Denote this opertor by W λ (n). It follows from Lemm 2 tht the djoint opertor Wλ (n) mps continuously H n onto Q + (n), nd Wλ (n)f = β n α n Wλ (s)ã(s)f(s)ds. Hence, for ech function f H nd ll finite α, β, we hve β α W λ (s)ã(s)f(s)ds Q +. Let c={c n } Q. Then c m =h mn c n (m n). Consequently, the restriction of the function Ã1/2 (t)w λ (t)c n to the segment [α m, β m ] coincides with the function à 1/2 (t)w λ (t)c m in the spce L 2 (H; α m, β m ). By Ã1/2 (t)w λ (t)c we denote the function tht is equl to Ã1/2 (t)w λ (t)c n on ech segment [α n, β n ]. Now by W λ (t)c denote the function rnging in H 1/2 (t) G(t) nd coinciding with W λ (t)c n in the spce H n for ech n N. So, for ll m, n (m n), the equlity W λ (t)c n = W λ (t)c m holds in the spce H m. Lemm 9. If the pir {y, f} L(λ), then y hs the form where c Q. y(t) = W λ (t)c + W λ (t)j 1 r (t 0 ) t t 0 W λ(s)ã(s)f(s)ds, P r o o f follows from Lemm 3 nd the definition of the function W λ (t)c. Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No

24 V.M. Bruk Theorem 9. For fixed λ (Imλ 0), the opertor L 1 (λ) is integrl L 1 (λ)g = K λ (t, s)ã(s)g(s)ds, g H, where K λ (t, s)=w λ (t)(m(λ) (1/2)sgn(s t)jr 1 (t 0 ))W λ(s) nd M(λ):Q + Q is continuous opertor such tht M( λ) = M (λ). The opertor function M(λ)x is holomorphic for Imλ 0 nd ech x Q +. R e m r k 6. In Theorem 9, the integrl converges t lest wekly in H. P r o o f. As estblished bove, the reltion L 1 (λ) is bounded everywhere defined opertor in the spce H nd the opertor function L 1 (λ) is holomorphic for Imλ 0. By this fct, the further proof of the theorem nd the remrk repets word-for-word the proof of the min theorem in [24]. Theorem 9 is proved. Theorem 10. In Theorem 9, the opertor function M(λ) is the chrcteristic opertor of eqution (27) on the xis (, ). P r o o f. It follows from Theorem 9 tht for ny finite function g H the equlity (31) holds with the replcement of by nd of b by +. The fmily of reltions L(λ) stisfies ll conditions of Theorem 8. We obtin the desired ssertion by repeting the proof of this theorem. Theorem 10 is proved. The uthor wishes to express grtitude to the referee for useful remrks. References [1] V.M. Bruk, On Liner Reltions in Spce of Vector Functions. Mt. Zmetki 24 (1978), No. 4, (Russin) [2] V.I. Khrbustovsky, Spectrl Anlysis of Periodic Systems with the Degenerting Weight on the Axis nd Semixis. Teor. Funkts. Anl. Prilozh. 44 (1985), (Russin) [3] V.M. Bruk, On Invertible Restrictions of Reltions Generted by Differentil Expression nd by Nonnegtive Opertor Function. Mt. Zmetki 82 (2007), No. 5, (Russin) [4] V.I. Khrbustovsky, On the Chrcteristic Opertors nd Projections nd on the Solutions of Weil Type of Dissiptive nd Accumultive Opertor Systems. 1. Generl cse; 2. Abstrct Theory; 3. Seprted Boundry Conditions. J. Mth. Phys., Anl., Geom. 2 (2006), No. 2, No. 3, No. 4, , , [5] V.I. Khrbustovsky, On Chrcteristic Mtrix of Weil-Titchmrsh Type for Differentil-Opertor Equtions which Contins the Spectrl Prmeter in Linerly or Nevnlinn s Mnner. Mt. Fiz, Anliz, Geom. 10 (2003), No. 2, (Russin) 138 Journl of Mthemticl Physics, Anlysis, Geometry, 2011, vol. 7, No. 2