On Linear Relations Generated by Nonnegative Operator Function and Degenerate Elliptic Differential-Operator Expression

In terms of boundary values, we describe a spectrum of the linear relations indicated in the paper title. We study the invertible restrictions of maximal relation and show that the operators inverse to these restrictions are integral. The criterion of holomorphicity of the family of these operators...

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Datum:	2009
1. Verfasser:	Bruk, V.M.
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Sprache:	English
Veröffentlicht:	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2009
Schriftenreihe:	Журнал математической физики, анализа, геометрии
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spelling	irk-123456789-1065372016-10-01T03:01:50Z On Linear Relations Generated by Nonnegative Operator Function and Degenerate Elliptic Differential-Operator Expression Bruk, V.M. In terms of boundary values, we describe a spectrum of the linear relations indicated in the paper title. We study the invertible restrictions of maximal relation and show that the operators inverse to these restrictions are integral. The criterion of holomorphicity of the family of these operators is determined. Using the results obtained, we show that the minimal relation is symmetric in Hilbert space and describe all generalized resolvents of this relation. 2009 Article On Linear Relations Generated by Nonnegative Operator Function and Degenerate Elliptic Differential-Operator Expression / V.M. Bruk // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 2. — С. 123-144. — Бібліогр.: 25 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106537 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	In terms of boundary values, we describe a spectrum of the linear relations indicated in the paper title. We study the invertible restrictions of maximal relation and show that the operators inverse to these restrictions are integral. The criterion of holomorphicity of the family of these operators is determined. Using the results obtained, we show that the minimal relation is symmetric in Hilbert space and describe all generalized resolvents of this relation.
format	Article
author	Bruk, V.M.
spellingShingle	Bruk, V.M. On Linear Relations Generated by Nonnegative Operator Function and Degenerate Elliptic Differential-Operator Expression Журнал математической физики, анализа, геометрии
author_facet	Bruk, V.M.
author_sort	Bruk, V.M.
title	On Linear Relations Generated by Nonnegative Operator Function and Degenerate Elliptic Differential-Operator Expression
title_short	On Linear Relations Generated by Nonnegative Operator Function and Degenerate Elliptic Differential-Operator Expression
title_full	On Linear Relations Generated by Nonnegative Operator Function and Degenerate Elliptic Differential-Operator Expression
title_fullStr	On Linear Relations Generated by Nonnegative Operator Function and Degenerate Elliptic Differential-Operator Expression
title_full_unstemmed	On Linear Relations Generated by Nonnegative Operator Function and Degenerate Elliptic Differential-Operator Expression
title_sort	on linear relations generated by nonnegative operator function and degenerate elliptic differential-operator expression
publisher	Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate	2009
url	http://dspace.nbuv.gov.ua/handle/123456789/106537
citation_txt	On Linear Relations Generated by Nonnegative Operator Function and Degenerate Elliptic Differential-Operator Expression / V.M. Bruk // Журнал математической физики, анализа, геометрии. — 2009. — Т. 5, № 2. — С. 123-144. — Бібліогр.: 25 назв. — англ.
series	Журнал математической физики, анализа, геометрии
work_keys_str_mv	AT brukvm onlinearrelationsgeneratedbynonnegativeoperatorfunctionanddegenerateellipticdifferentialoperatorexpression
first_indexed	2025-07-07T18:37:07Z
last_indexed	2025-07-07T18:37:07Z
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fulltext	Journal of Mathematical Physics, Analysis, Geometry 2009, vol. 5, No. 2, pp. 123�144 On Linear Relations Generated by Nonnegative Operator Function and Degenerate Elliptic Di�erential-Operator Expression V.M. Bruk Saratov State Technical University 77 Politekhnitcheskaja Str., Saratov, 410054, Russia E-mail:vladislavbruk@mail.ru Received October 28, 2007 In terms of boundary values, we describe a spectrum of the linear relations indicated in the paper title. We study the invertible restrictions of maximal relation and show that the operators inverse to these restrictions are integral. The criterion of holomorphicity of the family of these operators is determined. Using the results obtained, we show that the minimal relation is symmetric in Hilbert space and describe all generalized resolvents of this relation. Key words: linear relation, symmetric relation, spectrum, generalized resolvent, holomorphic operator function, di�erential elliptic-type expres- sion, Green function, Banach space, Hilbert space. Mathematics Subject Classi�cation 2000: 47A06, 47A10, 34B05. 1. Introduction In the present paper, we study the linear relations generated by a weight non- negative operator function and a di�erential expression with variable unbounded positively de�nite operator coe�cient degenerating on one of the ends of the in- terval. For the case when there is no operator weight, the spaces of boundary values (SBV) for maximal operator generated by this di�erential operator expres- sion were constructed in [1�5]. The SBV allows to describe various classes of restrictions of maximal operator. (The results of papers [1�3] can be found in monograph [6].) Di�erential expressions with operator weight generate linear relations that, in general, are not operators. In the present paper we construct the SBV for a maxi- mal relation. We study various restrictions of maximal relation and describe the c V.M. Bruk, 2009 V.M. Bruk spectrum of these restrictions by using SBV. We prove that if the relation (L(�)� �E)�1 is a bounded everywhere de�ned operator, then it is an integral operator. In this case we determine the criterion of holomorphicity for the operator function � ! (L(�) � �E)�1 (here L(�) is a restriction of maximal relation, � 2 C , E is the identity operator). To simplify the proofs the main theorems are proved with abstract spaces of boundary values being used. A description of the generalized resolvents of minimal relation is based on the obtained results. Notice that the formula of generalized resolvents of minimal relation generated by nonnegative operator function and di�erential expression with bounded operator coe�cients was obtained in [7, 8]. Our formula di�ers from that given in [7, 8], because we consider a di�erential elliptic-type expression with unbounded operator coe�cient. One of the di�culties in the studying of operators and relations generated by di�erential operator expression of elliptic-type is the constructing of the Green function in one of the boundary value problems. We construct this function in Sect. 3. 2. Main Assumptions, Notation Let H be a separable Hilbert space with the scalar product (�; �) and the norm k�k. On a compact interval [0; b], we consider the di�erential expression l[y] = �y00 + t�A1(t)y; where � > 0, and the operator function A1(t) satis�es the following conditions: 1) A1(t) is a positively de�nite selfadjoint operator in H for any �xed t 2 [0; b]; 2) the operators A1(t) have the constant domain D(A1(t)) = D(A1); 3) A1(t)x is a function strongly continuously di�erentiable on [0; b] for any x 2 D(A1). We �x a point t0 2 [0; b]. Let fĤ�g, �1 6 � 6 1, be a Hilbert scale of the spaces [6, Ch. 2; 9, Ch. 1] generated by A1(t0). Notice that the de�nition of the Hilbert scale implies Ĥ0 = H. It follows from the properties of A1(t) that the scale fĤ�g does not depend on the choice of point t0 2 [0; b] in the sense below. If t00 2 [0; b] is any other point and fĤ 0 �g is a scale of the spaces generated by operator A1(t 0 0), then the sets Ĥ� and Ĥ 0 � coincide and their norms are equivalent. For �xed t 2 [0; b], the operator A1(t) is a continuous one-to-one mapping of Ĥ+1 onto H. Then its adjoint operator A+ 1 (t) is a continuous one-to-one mapping of H onto Ĥ �1, and A + 1 (t) is an extension of A1(t) [6, Ch. 2; 9, Ch. 1]. Further, we denote l+[y] = �y00 + t�A+ 1 (t)y. Let A(t) be a function strongly measurable on [0; b] whose values are bounded selfadjoint operators in H. Suppose the norm kA(t)k is integrable on [0; b]. More- over, we assume that the inequality (A(t)x; x) > 0 (1) 124 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 On Linear Relations Generated by Nonnegative Operator Function... holds for any x 2 H and for almost all t 2 [0; b]. Generally, it is assumed that a set of points t 2 [0; b] satisfying (1) depends on x. We claim that there exists a set I0 � [0; b] of measure zero such that the set I = [0; b]nI0 has the following property: for all t 2 I and for all x 2 H inequality (1) holds. Indeed, due to separability of the space H there exists a countable set fxng (n 2 N) dense in H. Let In be a set of t 2 [0; b] such that inequality (1) holds, where x is replaced by xn. We denote I0;n = [0; b]nIn, I0 = S n I0;n. Then the measure of the set I0 is equal to zero, and for all t 2 I = [0; b] n I0 and for all n 2 N inequality (1) holds, where x is replaced by xn. Since the operator A(t) is bounded and the set fxng is dense in H, we obtain the desired statement. So, inequality (1) holds on some set I such that I does not depend on x 2 H, and the measure of the set [0; b] n I is equal to zero. Since the norm kA(t)k is integrable on [0; b], we have A1=p(t) 2 Lp(0; b). On the set of functions continuous on the interval [0; b] and ranging in H, we introduce the norm kyk p = 0 @ bZ 0 A1=p(t)y(t) p dt 1 A 1=p ; 1 6 p <1: Identifying the functions y such that kyk p = 0 with zero, then performing the completion, we obtain a Banach space denoted by B=Lp(H;A(t); 0; b). The ele- ments of B are the classes of functions identi�ed with each other in the norm k�k p . In what follows, ~y denotes a class of functions with representative y. To avoid a complicated terminology we say that the function y belongs to B. Let G0(t) be a set of elements x 2 H such that A(t)x = 0, H(t) = H G0(t), and A0(t) be a restriction of A(t) to H(t). Then the operator A0(t) acting in H(t) has the inverse A�1 0 (t) (which, in general, is unbounded). By fH�(t)g, �1 < � <1, we denote a Hilbert scale of spaces generated by operator A�1 0 (t). As known from [6, Ch. 2; 9, Ch. 1], the operator A0(t) can be extended to the operator ~A0(t) = ~A0;�(t) that continuously and bijectively maps H ��(t) onto H1��(t), 0 6 � 6 1. Further, in ~A0;�(t) we will omit the symbol � characterizing the domain of operator ~A0;�(t)). By ~A(t) we denote the operator that is de�ned on H ��(t)�G0(t) and is equal to ~A0(t) on H��(t) and to zero on G0(t). Obviously, the operator ~A(t) is an extension of A(t). The description of the space B for p > 1 was given in [8] and the case of p = 2 was considered in [10]. The space B consists of elements (i.e., function classes) with representatives of the form ~A �1=p 0 (t)P (t)h(t), where P (t) is an orthogonal projection of H onto H(t), h(t) 2 Lp(H; 0; b). Without changing considerably the proof given in [8], we obtain the above statements for p = 1. The space L1(H;A(t); 0; b) is used only when constructing the Green function in Sect. 3. Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 125 V.M. Bruk For p >1, the dual space of B is the space B� = Lq(H;A(t); 0; b)(p �1+q�1 = 1) (see [8]). A sesquilinear form (i.e., the form that is linear in the �rst argument and antilinear in the second one) determined by duality between B and B� is denoted by h�; �i, and the action of the functional ~g 2 B� on the element ~f 2 B is given by the equality h ~f; ~gi = bZ 0 ( ~A(t)f(t); g(t))dt; which is independent of the choice of representatives f 2 ~f , g 2 ~g. 2. The Green Function In this section, we construct the Green function G(t; s; �) of the Neumann problem for the expression l+[y] � �A(t)y. The construction is based on the Green function G(t; s) (see [5]). By [5], the operator function G(t; s) is called the Green function of the Neumann problem for the expression l[y], i.e., of the problem l[y] = �y00 + t�A1(t)y = g(t); (2) y0(0) = y0(b) = 0; (3) if the integral y(t) = Z b 0 G(t; s)g(s)ds is a strong solution (see [11]) of equation (2) and it satis�es conditions (3) for any strongly continuous function g(t) in the space Ĥ+1. By [11], the function y(t) (t 2 [0; b]) is called a strong solution of equation (2) if y(t) 2 D(A1) for any t, and y(t) is twice di�erentiable in H, and y(t) satis�es (2). It was proved in [5] that for su�ciently large k there exists a Green function Gk(t; s) of the Neumann problem for the expression lk[y] = �y00 + t�A1(t)y + k2t�y: Lemma 1. There exists a Green function of problem (2), (3). P r o o f. By L0 (L0 k ) denote an operator generated by the expression l[y] (lk[y]) on the functions y(t) that are strongly continuous in Ĥ+1 on [0; b], twice di�erentiable in H on [0; b] and they satisfy boundary conditions of the Neumann problem (3). Let L (Lk) be a closure of L 0 as well as of L0 k in the space L2(H; 0; b). It was proved in [5] that for su�ciently large k the operator L�1 k exists, it is continuous in L2(H; 0; b) and is an integral operator with the kernel Gk(t; s). Since L di�ers from Lk by a bounded selfadjoint operator and Lk is selfadjoint, we see that L is also selfadjoint. Obviously, L is nonnegative. We claim that the operator L�1 exists and it is bounded in L2(H; 0; b). 126 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 On Linear Relations Generated by Nonnegative Operator Function... Indeed, let fyng be a sequence of functions yn from the domain of L0 such that (L0yn; yn)L2(H;0;b) ! 0 as n!1 and kynkL2(H;0;b) = 1. Therefore, (L0yn; yn)L2(H;0;b) = bZ 0 y0n(t) 2 dt+ bZ 0 t�(A1(t)yn(t); yn(t))dt > bZ 0 y0n(t) 2 dt+ c1 bZ 0 t�(yn(t); yn(t))dt! 0 as n!1; where c1 > 0 does not depend on t. (Here and further, the symbols c1; c2; : : : denote positive constants that are di�erent in various inequalities.) Hence, bZ 0 y0n(t) 2 dt! 0 and bZ 0 t�(yn(t); yn(t))dt = (�+ 1)�1b�+1 kyn(b)k 2 � bZ 0 t�+1Re(y0n(t); yn(t))dt! 0 as n ! 1. (Here the formula of integration by parts is used.) This yields that kyn(b)k ! 0: Therefore, as n!1, yn(t) = yn(b)� Z b t y0n(t)dt! 0 uniformly on [0; b]. The above contradicts the equality kynkL2(H;0;b) = 1. Thus the existence and boundedness of the operator L�1 are proved. Consequently, the operator L is positively de�nite in L2(H; 0; b). We denote Gk = L �1 k . As noted above, Gk is an integral operator with the kernel Gk(t; s). By T denote the operator of multiplication on t� in L2(H; 0; b). Suppose G T = T 1=2 GkT 1=2. The operator G T is selfadjoint. Moreover, G T is an integral operator with the kernel t�=2Gk(t; s)s �=2. We will prove that the opera- tor k2G T �E has an everywhere de�ned inverse operator in the space L2(H; 0; b). Let vn 2 L2(H; 0; b), where n 2 N. We denote T 1=2vn = un, Gkun = wn. Then Lkwn = un and T �1=2 Lkwn = vn. It follows from the equality Lk = Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 127 V.M. Bruk L + k2T that Lwn belongs to the domain of operator T �1=2, and T �1=2 Lwn = vn � k2T 1=2wn. Hence, by direct calculation we obtain (vn; (E � k2G T )vn) = (vn; vn)� k2(T 1=2 GkT 1=2vn; vn) = (T �1=2(L+ k2T )wn;T �1=2(L+ k2T )wn)� k2(wn; (L+ k2T )wn) = k2(Lwn; wn) + (T �1=2 Lwn;T �1=2 Lwn) = k2(Lwn; wn) + (vn � k2T 1=2wn; vn � k2T 1=2wn) (in this equality, (�; �) is a scalar product in L2(H; 0; b)). Suppose (vn; (E � k2G T )vn)! 0 as n!1. It follows from the last equalities that (Lwn; wn)! 0; (vn � T 1=2wn; vn � T 1=2wn)! 0: Since L is a positive de�nite operator, we have wn ! 0 in L2(H; 0; b)). Therefore, vn ! 0 in L2(H; 0; b) as n!1. Thus the operator (k2G T �E)�1 exists and it is everywhere de�ned. In the space L2(H; 0; b), we consider the integral equation K(t; s)x = t�=2Gk(t; s)x+ k2 bZ 0 t�=2Gk(t; �)� �=2 K(�; s)xd� (4) with the unknown function K(t; s)x, where x 2 H. Since the operator k2G T �E has the everywhere de�ned inverse operator, we see that the equation (4) is solvable. In [5], it was proved that kGk(t; s)k 6 c1; (5) where c1 does not depend on s, t. Using (4), (5), we obtain bZ 0 kK(t; s)xk2 dt 6 (k2G T �E)�1 2 bZ 0 t�=2Gk(t; s)x 2 dt 6 c2 kxk 2 ; (6) where c2 does not depend on s. Using (4)�(6), we get kK(t; s)k 6 c3; (7) where c3 does not depend on t, s. We de�ne the function G(t; s) by the formula G(t; s)x = Gk(t; s)x+ k2 bZ 0 Gk(t; �)� �=2 K(�; s)xd�: (8) 128 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 On Linear Relations Generated by Nonnegative Operator Function... It follows from (4), (8) that t1=2G(t; s) = K(t; s). Hence, taking into account (8), we obtain G(t; s)x = Gk(t; s)x+ k2 bZ 0 Gk(t; �)� �G(�; s)xd�: (9) The function G(t; s) satis�es the boundary conditions G0 t(0; s) = G0 t(b; s) = 0; s 6= 0; s 6= b; G0 t(0; 0) = �E; G0 t(b; 0) = G0 t(0; b) = 0; G0 t(b; b) = E: (10) These equalities follow from (9) and from the fact that the function Gk(t; s) satis�es the same conditions (see [5]). Formulas (5), (7), (8) imply kG(t; s)k 6 c1; (11) where c1 does not depend on t, s. In [5], the operator Gk(t; s) is proved to extend to Ĝk(t; s) in Ĥ �1 such that it is a continuous mapping of each space Ĥ� , �1 6 � 6 1, of the scale fĤ�g into itself. The operator function Ĝk(t; s) is uniformly bounded on [0; b] � [0; b] with respect to the norm in each space Ĥ� . By the construction, the operator function G(t; s) possesses the same properties. Suppose the function g(t) is strongly continuous in Ĥ+1. We denote z(t) = bZ 0 G(t; s)g(s)ds; zk(t) = bZ 0 Gk(t; s)g(s)ds: It follows from (9), (10) that z(t) takes the values in D(A1), it is twice strongly di�erentiable in H, and z0(0) = z0(b) = 0. Since the function zk(t) is a strong solution of the equation lk[y] = g, we see that (9) implies the equality lk[z] = lk[zk] + k2t�z = g + k2t�z. Hence l[z] = g. Lemma 1 is proved. We notice some more properties of the function G(t; s). Let G be an operator de�ned by the formula Gv = Z b 0 G(t; s)v(s)ds in L2(H; 0; b). Then L �1 = G. Since the operator L is selfadjoint, we have G�(t; s) = G(s; t). The function G(t; s) is strongly continuous with respect to t for each �xed s 2 [0; b] what follows from (7), (8) and the fact that the function Gk(t; s) possesses the same property (see [4, 5]). Lemma 2. Suppose h(t) 2 L1(H; 0; b). Then the function y(t) = bZ 0 G(t; s)h(s)ds (12) Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 129 V.M. Bruk has the following properties: (a) y is continuous on [0; b] in the space H and strongly di�erentiable on [0; b] in the space Ĥ �1; (b) y0 is absolutely continuous in the space Ĥ �1; (c) y satis�es the equation l+[y] = �y00 + t�A+ 1 (t)y = h(t) (13) and boundary conditions (3). P r o o f. We take a sequence of functions hn(t) such that the sequence fhn(t)g converges to h(t) in L1(H; 0; b) as n ! 1 and the functions hn(t) are strongly continuous in the space Ĥ+1. Then, by Lemma 1, the functions yn(t) =Z b 0 G(t; s)hn(s)ds are strong solutions of the problem (13), (3), where h(t) is replaced by hn(t). Thus the equality �y00n(t) + t�A+ 1 (t)yn(t) = hn(t) (14) holds. From (11), (12), it follows that the sequence fyn(t)g converges to y(t) uniformly in H. Therefore the sequence fA+ 1 (t)yn(t)g uniformly converges to A + 1 (t)y(t) in the space Ĥ �1. Then (14) implies the convergence of the sequence fy00n(t)g in L1(Ĥ�1; 0; b). From this and (3) it follows that fy0n(t)g converges uniformly in Ĥ �1. Now all assertions of Lemma 2 are obtained from the above in a standard way. The proof of Lemma 2 is complete. Lemma 3. For any function h(t) 2 L1(H; 0; b) and any elements x1; x2 2 H there exists a unique solution y(t) of equation (13) such that y(t) has the properties (a), (b) of Lemma 2 and satis�es the boundary conditions y0(0) = �x1; y0(b) = x2: (15) This solution has the form y(t) = G(t; 0)x1 +G(t; b)x2 + Z b 0 G(t; s)h(s)ds. P r o o f. First, we notice that the invertibility of operator L yields the unique- ness of solution. Further, as follows from [5], the function zk(t) = Gk(t; 0)x1 + Gk(t; b)x2 has the properties (a), (b) and satis�es the equation l+ k [y] = 0 and conditions (15). Hence, taking into account (9), (10), we obtain that the function z(t) = G(t; 0)x1 + G(t; b)x2 has the properties (a), (b), it is a solution of the equation l+[y] = 0 and it satis�es the conditions (15). Now, applying Lemma 2, we complete the proof of Lemma 3. To construct the Green function G(t; s; �) we consider the equation l+[y]� �A(t)y = �y00 + t�A+ 1 (t)y(t)� �A(t)y(t) = ~A(t)f(t): (16) 130 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 On Linear Relations Generated by Nonnegative Operator Function... Let G(t; s; �) be an operator function whose values are bounded operators in H. We say that G(t; s; �) is the Green function of problem (16), (3) if for any function f 2 L1(H;A(t); 0; b) the integral z1(t) = bZ 0 G(t; s; �) ~A(s)f(s)ds possesses the properties (a), (b) of Lemma 2 and satis�es equation (16) and the boundary conditions (3). As shown in the proof of Lemma 1, the operator L is positively de�nite in L2(H; 0; b). From the equality G = L �1 it follows that G is a positively de�nite operator. Consequently, the kernel A1=2(t)G(t; s)A1=2(s) determines the bounded nonnegative operator GAv = bZ 0 A1=2(t)G(t; s)A1=2(s)v(s)ds (v 2 L2(H; 0; b)) in the space L2(H; 0; b). By �0(GA) we denote a set � 2 C such that the operator �GA � E has a bounded everywhere de�ned inverse operator. The set �0(GA) contains all nonreal numbers, the negative ones and zero. Further, we will assume that � 2 �0(GA). Theorem 1. For any � 2 �0(GA), there exists a Green function G(t; s; �) of problem (16), (3). P r o o f. We consider the integral equation K(t; s; �)x = A1=2(t)G(t; s)x+ � bZ 0 A1=2(t)G(t; �)A1=2(�)K(�; s; �)xd� (17) with the unknown function K(t; s; �)x, where x 2 H. Equation (17) can be solved in L2(H; 0; b) for � 2 �0(GA). We introduce the function G(t; s; �) by the equality G(t; s; �)x = G(t; s)x+ � bZ 0 G(t; �)A1=2(�)K(�; s; �)xd�: (18) For �xed s 2 [0; b], the function A1=2(t)K(t; s; �)x (x 2 H) belongs to L1(H; 0; b). Consequently, G(t; s; �) is a strongly continuous function with respect to t in the space H. It follows from (17), (18) that A1=2(t)G(t; s; �) = K(t; s; �): Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 131 V.M. Bruk Hence, using (18), we get G(t; s; �)x = G(t; s)x+ � bZ 0 G(t; �)A(�)G(�; s; �)xd�: (19) Moreover, by (10), it follows that G0 t(0; s; �)x = G0 t(b; s; �)x = 0; s 6= 0; s 6= b; G0 t(0; 0; �)x = �x; G0 t(b; 0; �)x = G0 t(0; b; �)x = 0; G0 t(b; b; �)x = x: (20) Further proof is done analogously to that of [12], where the case of � = 0 was considered. In particular, similarly as in [12], we obtain that for any element d1 2 H �1(s)�G0(s) the equality G�(s; t; ��) ~A(s)d1 = G(t; s; �) ~A(s)d1 holds. Therefore, G�(s; t; ��) ~A(s)f(s) = G(t; s; �) ~A(s)f(s) (21) for any function f 2 L1(H;A(t); 0; b). For � 2 �0(GA), the function G(t; s; �) is bounded with respect to the �rst argument. Therefore the function G�(t; s; �) has the same property. From this and from (21) there follows the equality bZ 0 G(t; s; �) ~A(s)f(s)ds = bZ 0 G�(s; t; ��) ~A(s)f(s)ds and the existence of integrals in it. Using (19), (20) and the properties of function G(t; s), we complete the proof. In [12], the Green function for the expression l+[y] � �A(t) was constructed in the case of � = 0. If � = 0 and � = 0, then G(t; s; 0) = G(t; s) coincides with the Green function constructed in [13]. By U(t; �), denote the operator one-row matrix U(t; �) = (U1(t; �); U2(t; �)), where U1(t; �) = G(t; 0; �); U2(t; �) = G(t; b; �): (22) Lemma 4. Let � 2 �0(GA). For any elements x1; x2 2 H and any function f 2 L1(H;A(t); 0; b) there exists a unique function y having the properties (a), (b) of Lemma 2 and satisfying equation (16) and boundary conditions (15). This function has the form y(t) = U1(t; �)x1 + U2(t; �)x2 + F (t); (23) 132 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 On Linear Relations Generated by Nonnegative Operator Function... where F (t) = bZ 0 G(t; s; �) ~A(s)f(s)ds = bZ 0 G�(s; t; ��) ~A(s)f(s)ds: (24) P r o o f. It follows from Lemma 3 and equalities (19), (20), (22) that the function y0(t) = U1(t; �)x1 + U2(t; �)x2 has the properties (a), (b) of Lemma 2, and y0(t) satis�es the boundary conditions (15) and the equation �y00 + t�A+ 1 (t)y � �A(t)y = 0: (25) Hence, taking into account Theorem 1, we obtain that (23) has all the properties indicated in the lemma. To prove that problem (25), (15) has a unique solution is to prove the uniqueness of the solution of problem (16), (15). Let the function u0, having the properties (a), (b), be a solution of equation (25) with homogeneous conditions (3). We put u(t) = � Z b 0 G(t; s)A(s)y0(s)ds. Using Lemma 3, we get u0(t) = u(t). Hence, A1=2(t)u0(t) = � bZ 0 A1=2(t)G(t; s)A(s)y0(s)ds: Since � 2 �0(GA), we have A1=2(t)u0(t) = 0 for almost all t 2 [0; b]. Therefore, u0(t) = u(t) = 0 for all t 2 [0; b]. So, the uniqueness of the solution of problem (25), (15) is established. Lemma 4 is proved. R e m a r k 1. Suppose the function y has the properties (a), (b), and it satis�es equation (16) and boundary conditions (15), where x1; x2 2 H. Then y0(t) 2 H for all t 2 [0; b]. Indeed, the function y is a solution of nondegenerate equation on each interval [�; b] (� > 0). Consequently, y0(t) 2 H for all t 2 [�; b] (see [12]). Hence, taking into account (15), we obtain the desired statement. Lemma 5. Suppose F is de�ned by equality (24); then the operator ~f ! F = F (t; ~f; �) is a continuous mapping of the space B into the space C(H; 0; b). P r o o f coincides with that of the analogous lemma in [12]. Corollary 1. The operator ~f ! ~F = ~F (t; ~f; �) is continuous in B. 4. Maximal and Minimal Relations In this section, the maximal and minimal relations generated by expression l+[y] and operator function A(t) in the space B = Lp(H;A(t); 0; b) are de�ned and the properties of these relations are studied. Everywhere below we will assume that p > 1. Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 133 V.M. Bruk Terminology concerning linear relations can be found, for example, in [6, 14, 15]. The linear relation T in the Banach space B is understood as a linear manifold T � B � B. Further the following notations are used: f�; �g is an ordered pair; KerT is a set of ordered pairs of the form fz; 0g 2 T ; ker T is a set of elements z such that fz; 0g 2 T ; D(T ) is a domain of T ; R(T ) is a range of values; �(T ) is a resolvent set of the relation T , i.e., a set of points � 2 C such that the relation (T � �E)�1 is a bounded everywhere de�ned operator; �c(T ) (�r(T )) is a continuous spectrum (residual spectrum) of the relation T , i.e., a set of points � 2 C such that the relation (T ��E)�1 is a densely de�ned and unbounded (not densely de�ned) operator; �p(T ) is the point spectrum of T , i.e., a set of points � 2 C such that the relation (T � �E)�1 is not an operator. Since all relations considered are linear, the word "linear" will often be omitted. By D0 we denote a set of functions y(t) 2 B satisfying the following conditions: i) y is strongly continuous on [0; b] in the space H and strongly di�erentiable in the space Ĥ �1, and y0(t) 2 H for all t 2 [0; b]; (ii) y0 is absolutely continuous in Ĥ �1; iii) l +[y](t) 2 H1=q(t) for almost all t, and the function ~A�1 0 (t)l+[y] 2 B (p�1 + q�1 = 1). To each class of functions identi�ed with y 2 D0 in B we assign the class of functions identi�ed with ~A�1 0 (t)l+[y] in B. In general, this correspondence is not an operator as the function y may be identi�ed with zero in B and ~A�1 0 (t)l+[y] may be nonzero. Thus, in the space B we obtain a linear relation L0. Denote its closure by L and call it a maximal relation. We de�ne the minimal relation L0 as a restriction of L to the set of elements ~y 2 B that have representatives y 2 D0 with the property y(0) = y0(0) = y(b) = y0(b) = 0. Let Q0 be a set of elements x 2 H�H for which the equality A(t)U(t; �)x = 0 holds almost everywhere. Using Theorem 1, we get U(t; 0)x = U(t; �)x� � bZ 0 G�(s; t; ��) ~A(s)U(s; 0)xds; (26) U(t; �)x = U(t; 0)x+ � bZ 0 G�(s; t) ~A(s)U(s; �)xds: (27) By (26), (27), it follows that Q0 does not depend on �. By Q we denote an orthogonal complement of Q0 in H �H. In Q we introduce the norm kxk r = 0 @ bZ 0 A1=r(s)U(s; 0)x r ds 1 A 1=r 6 k kxk ; r > 1; x 2 Q: (28) We denote the completion of Q with respect to the norm k�k r by Q � (r). It follows from (26), (27) that the replacement of U(s; 0) by U(s; �) in (28) leads to the 134 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 On Linear Relations Generated by Nonnegative Operator Function... same set Q � (r) with the equivalent norm. Let the symbol ~U(s; �)x (x 2 Q � (r)) denote a class of functions to which the sequence f ~U(t; �)xng (xn 2 Q) converges whenever fxng converges to x in the space Q � (r). We introduce the operator Vr(�) : Q� (r)! Lr(H;A(t); 0; b) by the formula Vr(�)x = ~U(t; �)x. It follows from (28) that the operator Vr(�) is continuous, the range R(Vr(�)) is closed, and kerVr(�) = f0g. Hence the range of the adjoint operator V � r (�) : Lr1(H;A(t); 0; b) ! Q� � (r) � Q� = Q coincides with Q� � (r) (here r�1 + r�1 1 = 1). We �nd the form of V � r (�). For any elements x 2 Q and ~f 2 Lr1(H;A(t); 0; b), we have h ~f; Vr(�)xi = bZ 0 ( ~A(s)f(s); U(s; �)x)ds = ( bZ 0 U�(s; �) ~A(s)f(s)ds; x) = (V � r (�) ~f; x): (29) Here (V � r (�) ~f; x) is a scalar product of the elements V � r (�) ~f 2 Q� � (r) � Q and x 2 Q in Q. For x+ 2 Q� � (r), this scalar product (x+; x) is extended by continuity to the sesquilinear form (x+; x�) determined by the duality between Q� � (r) and Q � (r). Taking into account (29) and that Q can be densely embedded in Q � (r), we obtain V � r (�) ~f = bZ 0 U�(s; �) ~A(s)f(s)ds: (30) Further, to avoid complicated notation, we denote Q � = Q � (p), ~Q+ = Q� � (q), where p�1 + q�1 = 1. Thus the following lemma is proved. Lemma 6. The operator V � q ( ��) is a continuous mapping of B onto ~Q+. Lemma 7. For any � 2 �0(GA), the relation L � �E consists of the pairs f~y; ~fg 2 B� B such that ~y = ~U(t; �)x+ ~F ; (31) where x 2 Q � and ~F are a class of functions identi�ed in B with the function (24). P r o o f. It follows from Theorem 1, Lemma 1 and the de�nition of the space Q � that a pair f~y; ~fg 2 B � B satisfying (31) belongs to L � �E. Now let f~y; ~fg 2 L � �E. Then there exists a sequence of pairs f~yn; ~fng 2 L0 � �E converging to the pair f~y; ~fg in B � B. Using Lemma 4, we obtain that the Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 135 V.M. Bruk function yn can be represented in the form yn(t) = U(t; �)xn + bZ 0 G�(s; t; ��) ~A(s)fn(s)ds; (32) where xn 2 Q. From the convergence of the sequence of pairs f~yn; ~fng in B � B there follows the convergence of the sequence f ~U(t; �)xng in B. When passing to (32) to the limit as n ! 1, we �nd that ~y admits the form (31). The proof of Lemma 7 is complete. Corollary 2. The operator Vp(�) is a continuous one-to-one mapping of Q � onto ker(L� �E). R e m a r k 2. In equality (31), the element x 2 Q � and the function F are uniquely determined by the pair f~y; ~fg 2 L � �E. The pair f~y; ~fg 2 L0 � �E if and only if x 2 Q and in this case x = f�y0(0); y0(b)g. R e m a r k 3. It follows from (22), (24), (30) that V � q ( ��) ~f = fF (0); F (b)g. R e m a r k 4. When p = 2 and there is no operator weight (i.e., A(t) = E), the equality Q � = Ĥ �3=2(�+2) � Ĥ �3=4 is valid (see [5]). Lemma 8. For any � 2 �0(GA) the relation L0 � �E is closed. P r o o f. Suppose the sequence of pairs f~yn; ~fng 2 L0 � �E converges to the pair f~y; ~fg in the space B�B. It follows from the de�nition of L0 and Remark 2 that we can choose representatives yn, fn of the classes of functions ~yn, ~fn such that they satisfy (32), where xn 2 Q and yn(0) = yn(b) = y0n(0) = y0n(b) = 0. Using Remark 3 and Lemma 4, we get xn = 0 and V � q ( ��) ~fn = 0. Passing to the limit as n ! 1 in the last equality and in (32), we obtain that x = 0 and V � q ( ��) ~f = 0 in (31). Therefore f~y; ~fg 2 L0 � �E. Lemma 8 is proved. R e m a r k 5. It follows from the proof of Lemma 8 that R(L0 � �E) = ker V � q ( ��). 5. Spectrum of Restrictions of the Maximal Relation L In this section, we introduce an abstract space of boundary values (SBV). By means of SBV we describe the spectrum of restrictions of the relation L and study the bounded operators (L(�)� �E)�1, where L0 � L(�) � L. Suppose B1, B2, ~B1, ~B2 are Banach spaces, T � B1 � B2 is a closed relation, and Æ : T ! ~B1 � ~B2 is a linear operator. We denote Æi = PiÆ, i = 1; 2, where Pi is the projection ~B1 � ~B2 onto ~Bi, i.e., Pifx1; x2g = xi (the similar notation will be used in the analogous cases below). The following de�nition is given in [16] for operators and in [17] for relations. 136 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 On Linear Relations Generated by Nonnegative Operator Function... De�nition. The quadruple ( ~B1; ~B2; Æ1; Æ2) is called a space of boundary values (SBV) for a closed relation T if the operator Æ is a continuous mapping of T onto ~B1 � ~B2, and the restriction of the operator Æ1 to KerT is a one-to-one mapping of KerT onto ~B1. We de�ne an operator �Æ : ~B1 ! ~B2 by the equality �Æ = Æ2�, where � = (Æ1jKerT ) �1 is the operator inverse to the restriction of Æ1 to KerT . We denote T0 = ker Æ, T1 = ker Æ1. Then T0 � T1 � T , R(T1) = R(T ), and the relation T�1 1 is an operator (see [16, 17]). From the de�nition of SBV, it follows that between the relations � � ~B1� ~B2 and ~T with the property T0 � ~T � T there is a one-to-one correspondence determined by the equality Æ ~T = �. In this case we denote ~T = T�. Lemma 9. T� = T � . Corollary 3. The relation T� is closed if and only if � is closed. R e m a r k 6. By the continuity of operator �Æ the relation � is closed if and only if the relation � � �Æ is closed. Lemma 10. Let R(T ) = B2. Then the following statements are valid: 1) the range R(T�) is closed if and only if the range R(� � �Æ) is closed; 2) dimB2=R(T�) = dim ~B2=R(� � �Æ); 3) dimker(T�) = dimker(� � �Æ). The proves of Lemmas 9, 10 are based on the following statement, that might be known. Lemma 11. Suppose B1, B2 are Banach spaces, � : B1 ! B2 is a bounded linear operator with the range R(�) = B2, X � B1 is a linear manifold such that ker� � X. Then �X = �X and dimB1=X = dimB2=�X. P r o o f of Lemma 11. The continuity of operator � implies �X � �X. We prove the inverse inclusion. Let B (0) 1 = B1= ker� be a quotient space and � be a canonical mapping of B1 onto B 0 1. We de�ne an operator �0 by the equality � = �0�. Then �0 is a continuous one-to-one mapping of B0 1 onto B2. Let a 2 �X, an 2 �X, where n 2 N. If a sequence fang converges to a, then the sequence f��1 0 ang converges to ��1 0 a in the space B (0) 1 . Since ker� � X, we see that all elements of the classes of adjacency ��1 0 an belong to X. Let b 2 ��1 0 a. Then we can choose a sequence fbng such that bn 2��1 0 an�X and fbng converges to b. Therefore b 2 X . Since �b = a, we have �X � �X. The equality �X = �X is proved. Let �1 be an operator de�ned by the equality �1�1 = �2�, where �1, �2 are canonical mappings of B1, B2 onto quotient spaces B1=X , B2=�X , respectively. Since �1 is a continuous one-to-one mapping of B1=X onto B2=�X, we have dimB1=X = dimB2=�X. The proof of Lemma 11 is complete. Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 137 V.M. Bruk P r o o f of Lemma 9. In Lemma 11 we take B1 = B1 � B2, B2 = ~B1 � ~B2, � = Æ, X = T�. Then �X = ÆT� = �, and �X = ÆT� = �. Hence, T� = T � . Lemma 9 is proved. P r o o f of Lemma 10. We de�ne an operator W : B2 ! ~B2 by the equality Wf = Æ2fT �1 1 f; fg, where f 2 B2. From the continuity of T�1 1 (see [16, 17]) and the properties of operators Æ1, Æ2 it follows that W is a continuous mapping of B2 onto ~B2. Moreover, using the de�nition of the relations T0, T1, we get kerW = R(T0). Any pair fy; fg 2 T is uniquely represented in the form fy; fg = m0+m, where m0 2 KerT , m 2 T1, namely, fy; fg = fy� T�1 1 f; 0g+ fT�1 1 f; fg. Hence, (Æ2��ÆÆ1)fy; fg = Æ2fT �1 1 f; fg =Wf . Therefore, WR(T�) = R(��Æ). In Lemma 11 we take B1 = B2, B2 = ~B2, � = W , X = R(T�). Then we obtain the �rst and the second statements of Lemma 10. An element u 2 T has the form u = u0 + v, where u0 2 KerT , v 2 T0, if and only if Æ2u � �ÆÆ1u = 0. Hence the restriction of the operator Æ1 to KerT� is a one-to-one mapping of KerT� onto ker(��Æ). From the above the third statement of the lemma follows. Lemma 10 is proved. Let B1 = B2 = B0 and let the quadruple ( ~B1; ~B2; Æ1; Æ2) be an SBV for a closed relation T � B0 � B0. A pair fy1; y2g 2 T if and only if the pair fy1; y2 � �y1g 2 T � �E. For any pair fy1; y2 � �y1g 2 T � �E we put Æ(�)fy1; y2 � �y1g = Æfy1; y2g. As proved in [17], � 2 �(T1) if and only if the quadruple ( ~B1; ~B2; Æ1(�); Æ2(�)) is an SBV for the relation T � �E. As above, we denote �Æ(�) = Æ2(�)(Æ1(�) jKer(T��E)) �1 : ~B1 ! ~B2. Lemma 10 implies the following assertion. Theorem 2. Let � 2 �(T1). Then the following statements are valid: 1) the range R(T� � �E) is closed if and only if the range R(� � �Æ(�)) is closed; 2) dimB0=R(T� � �E) = dim ~B2=R(� � �Æ(�)); 3) dimker(T� � �E) = dimker(� � �Æ(�)). Corollary 4. Suppose that the relation � is closed. A point � 2 �(T1) belongs to the point spectrum �p(T�) of the relation T� if and only if ker(��Æ(�)) 6= f0g. A point � 2 �(T1) belongs to the residual spectrum �r(T�) (to the continuous spectrum �c(T�)) if and only if the relation (��Æ(�)) �1 is a non-densely de�ned (densely de�ned and unbounded) operator. A point � 2 �(T1) belongs to the resolvent set �(T�) if and only if (� � �Æ(�)) �1 is a bounded everywhere de�ned operator. Notice that for abstract SBV introduced in [20, 21] the statements similar to Cor. 4 were obtained in [18, 19]. In view of Lemma 10 and Theorem 2, we recall the following de�nitions (see [22] for relations and [23, Ch. 4] for operators). Let S � B1 � B2 be a closed linear relation. The quantity �(S) = dimkerS � dimB2=R(S) is called an index 138 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 On Linear Relations Generated by Nonnegative Operator Function... of S if one of the subspaces kerS or B2=R(S) is �nite-dimensional. The relation S is called normal solvable if R(S) is closed; it is called semi-Fredholm if it is normal solvable and kerS or B2=R(S) is �nite-dimensional; it is called a Fredholm relation if it is semi-Fredholm, and the subspaces kerS and B2=R(S) are �nite- dimensional; it is called regular solvable if it is a Fredholm relation and �(S) = 0; it is called solvable if R(S) = B2 and kerS = f0g. Theorem 2 implies that the relations T� � �E and � � �Æ(�) simultaneously possess or do not possess the properties listed in this de�nition. We apply the obtained results to the relation L generated by the expression l+[y] and the operator function A(t). We de�ne the boundary operators 1 : L! Q � , 2 : L! ~Q+ for the relation L in the following way. Let a pair f~y; ~fg 2 L. Then ~y has form (31) for � = 0. By (31), to each pair f~y; ~fg 2 L we assign a pair of boundary values by the formulas 1f~y; ~fg = x; 2f~y; ~fg = V � q (0) ~f = bZ 0 U�(s; 0) ~A(s)f(s)ds: (33) It follows from Remark 2 that the pair f 1f~y; ~fg; 2f~y; ~fgg of boundary values is uniquely determined for each pair f~y; ~fg 2 L. By Lemmas 6, 7 and Corollary 2, for each � 2 �0(GA) the quadruple (Q� ; ~Q+; 1; 2) is the space of boundary values for the relation L. As above, by we denote the operator de�ned by the equality f~y; ~fg = f 1f~y; ~fg; 2f~y; ~fgg. The operator is a continuous mapping of L onto Q � � ~Q+. It follows from Remark 3 and the proof of Lemma 8 that ker = L0. Analogously as above, for any pair fy1; y2g 2 L we put (�)fy1; y2 � �y1g = fy1; y2g. Using Lemma 7, we get �0(GA) � �(L1), where L1 = ker 1. Hence, for all � 2 �0(GA) the quadruple (Q� ; ~Q+; 1(�); 2(�)) is an SBV for the relation L � �E. By �(�) we denote the corresponding operator � (�). Using (33), we obtain �(�) = � bZ 0 U�(s; 0) ~A(s)U(s; �)ds: Let � � Q � � ~Q+ be a linear relation and L� � L be a linear relation such that L� = �. From Theorem 2, we get the following statement. Theorem 3. Let � 2 �0(GA). Then the following statements are valid: 1) the range R(L� � �E) is closed if and only if the range R(� � �(�)) is closed; 2) dimB=R(L� � �E) = dim ~Q+=R(� � �(�); 3) dimker(L� � �E) = dimker(� � �(�)). Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 139 V.M. Bruk Corollary 4 with T� replaced by L� and �Æ(�) replaced by �(�) holds for the relation L. Suppose �(�) � Q � � ~Q+ and L�(�) � L are the families of linear relations such that L�(�) = �(�). By Lemma 4, the relation R(�) = (L�(�) � �E)�1 is a bounded everywhere de�ned operator if and only if (�(�)��(�))�1 is bounded everywhere de�ned. The following two theorems can be proved in view of Lemma 7 and Corollary 4 by analogy with the corresponding assertions in [12], where the case of p = 2, � = 0, was considered. Theorem 4. Suppose R(�) = (L�(�)��E) �1 (or (�(�)��(�))�1) is a bounded everywhere de�ned operator. Then R(�) is an integral operator of the form R(�) ~f = bZ 0 ( ~U(t; �)(�(�) � �(�))�1U�(s; ��) +G�(s; t; ��)) ~A(s))f(s)ds: (34) Theorem 5. Suppose the relation R(�0) (or (�(�0)� �(�0) �1) is a bounded everywhere de�ned operator. Then the family R(�) is holomorphic in the point �0 if and only if the family (�(�)� �(�))�1 is holomorphic in �0. R e m a r k 7. If the relation T (�0) is a bounded everywhere de�ned operator and the family of relations T (�) is holomorphic in the point �0, then the relations T (�) are bounded everywhere de�ned operators in some neighborhood of �0 (see [23, Ch. 7; 24]). 6. Maximal and Minimal Relations in L2(H;A(t); 0; b). Description of Generalized Resolvents In this section, we prove that the minimal relation L0 is symmetric in the space L2(H;A(t); 0; b) and describe the generalized resolvents of the relation L0. Further, we will consider the case of B = L2(H;A(t); 0; b), i.e., p = 2. Notice that the norm in B is generated by the scalar product ( ~f; ~g)B = bZ 0 ( ~A(t)f(t); g(t))dt: The space Q � is a Hilbert space with the scalar product (x1; x2)� = ( ~U (�; 0)x1; ~U(�; 0)x2)B: This scalar product generates the norm (28) under r = 2. The space Q � can be treated as a space with the negative norm with respect to Q [6, Ch. 2; 9, 140 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 On Linear Relations Generated by Nonnegative Operator Function... Ch. 1]. By Q+, denote the corresponding space with the positive norm. Using the de�nitions of positive and negative spaces, we get Q+ = ~Q+. Lemma 12. Let the pairs f~y; ~fg; f~z; ~gg 2 L0. Then there exist such represen- tatives y 2 ~y, z 2 ~z that the following equality holds: ( ~f; ~z)B� (~y; ~g)B = �(y0(b); z(b))+(y0(0); z(0))+(y(b); z0(b))� (y(0); z0(0)): (35) P r o o f. It follows from Lemma 7 and Remark 2 that there exist such representatives y 2 ~y, z 2 ~z that y(t) = U(t; 0)v + bZ 0 G�(s; t) ~A(s)f(s)ds; z(t) = U(t; 0)w + bZ 0 G�(s; t) ~A(s)g(s)ds; where v; w 2 Q. Since ~f; ~g 2 B, we obtain that the functions ~A(s)f(s), ~A(s)g(s) belong to L1(H; 0; b). We chose two sequences ffng and fgng of functions such that fn, gn are strongly continuous functions in the space Ĥ+1 and the sequences ffng, fgng converge to the functions ~A(s)f(s) and ~A(s)g(s), respectively, in the space L1(H; 0; b). Moreover, we take two sequences fvng, fwng, where vn; wn 2 Ĥ+1, such that fvng, fwng converge to v, w, respectively, in the space H. Then the functions yn(t) = U(t; 0)vn + bZ 0 G�(s; t)fn(s)ds; zn(t) = U(t; 0)wn + bZ 0 G�(s; t)gn(s)ds are strong solutions [11] of equation (2) with the right parts fn, gn, respectively. Hence, yn(t); zn(t) 2 D(A1) for each t 2 [0; b]. Therefore, bZ 0 (l[yn]; zn)dt� bZ 0 (yn; l[zn])dt = bZ 0 (�y00n(t) +A1(t)yn(t); zn(t))dt � bZ 0 (yn(t);�z 00 n(t) +A1(t)zn(t))dt = � bZ 0 (y00n(t); zn(t))dt+ bZ 0 (yn(t); z 00 n(t))dt = �(y0n(b); zn(b)) + (y0n(0); zn(0)) + (yn(b); z 0 n(b))� (yn(0); z 0 n(0)): (36) It follows from (11) that yn(0), yn(b), zn(0), zn(b) converge to y(0), y(b), z(0), z(b), respectively, in the space H. Since vn = f�y0n(0); y 0 n(b)g, wn = Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 141 V.M. Bruk f�z0n(0); z 0 n(b)g, v = f�y0(0); y0(b)g, w = f�z0(0); z0(b)g, we have y0n(0), y 0 n(b), z0n(0), z 0 n(b) converge to y0(0), y0(b), z0(0), z0(b), respectively, in the space H. In (36), we pass to the limit as n!1 and obtain (35). The proof of Lemma 12 is complete. Corollary 5. The relation L0 is symmetric. P r o o f follows from Remark 2, Lemma 12 and the de�nition of L0. Lemma 13. L� 0 = L. In view of Lemma 7 and Corollary 5, the proof of Lemma 13 is the same as that of the similar assertion in [12]. Theorem 6. The range R( ) of the operator coincides with Q � �Q+, and for any pairs f~y; ~fg; f~z; ~gg 2 L �the Green formula� is valid: ( ~f; ~z)B � (~y; ~g)B = (Y2; Z1)� (Y1; Z2); (37) where fY1; Y2g = f~y; ~fg, fZ1; Z2g = f~z; ~gg. P r o o f. The equalityR( ) = Q � �Q+ follows from Lemmas 6, 7, Corollary 2 and equalities (33). In view of Lemma 12, formula (37) is proved in the same way as the similar one in [12]. Theorem 6 is proved. In a particular case of Q � = Q+ = Q, Theorem 6 implies that the ordered triple (Q; 1; 2) is a space of boundary values in the sense of papers [20, 21]. Using the argumentation of [20, 21], we obtain the following assertion. Lemma 14. For �xed �, the relations L�(�) and �(�) are or are not simul- taneously accumulative (dissipative, symmetric, maximal accumulative, maximal dissipative, maximal symmetric, selfadjoint). When there is no operator weight (i.e., (A(t) = E), the relation L is an ope- rator, and in this case Theorem 6 was proved in [1] for the expression l[y] with a constant operator coe�cient A1(t) = A1, and in [2, 3] for l[y] with a variable operator coe�cient A1(t) satisfying the conditions listed in Sect. 2. The case of � = 0 was considered in these papers. In [1], the boundary values did not contain the Green function. In [3], for the variable operator coe�cient A1(t), the boundary values were constructed so that they did not contain the Green function. Moreover, additional conditions were imposed on the function A1(t), and the example proving the necessity of these conditions was given. The boundary values containing the Green function were constructed in [2] as � = 0 and in [4, 5] as � > 0, and they di�er from the boundary values (33) introduced in the present paper. The papers [1�3, 20, 21] are reviewed in the monograph [6]. Notice that for the �rst time linear relations were applied to the description of extensions of di�erential operators in [25] (see also [14]), where the di�erential expressions with bounded operator coe�cients were considered. 142 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 On Linear Relations Generated by Nonnegative Operator Function... We recall a de�nition of the generalized resolvent. Suppose that B is a Hilbert space, L0 is a closed symmetric relation, L0 � B �B. The operator function R�, Im� 6= 0, is called a generalized resolvent of the relation L0 if there exists the Hilbert space ~B � B and the selfadjoint relation ~L � L0, ~L � ~B� ~B such that the condition R� = P ( ~L � �E)�1 j B , where P is an orthogonal projection of ~B onto B, is satis�ed. Detailed bibliography on generalized resolvents is given in the monograph [14]. In view of Theorems 4, 5 and Lemma 14, the proof of the following theorem is the same as that of the similar assertion in [12], where the case of � = 0 was considered. Theorem 7. Any generalized resolvent R� (Im� 6= 0) of the relation L0 is the integral operator (34), where �(�) � Q � �Q+, and �(�) is a holomorphic family, the values of which �(�) are maximal accumulative relations in the case of Im� > 0 and maximal dissipative relations in the case of Im� < 0, with ��(�) = �(��). Conversely, if �(�) is a family of the linear relations with the mentioned above properties, then the family of operators R� of form (34) is a generalized resolvent of the relation L0. References [1] M.L. Gorbatchuk, Selfadjoint Boundary Problems for a Di�erential Equation of Second Order with an Unbounded Operator Coe�cient. � Funct. Anal. Appl. 5 (1971), No. 1, 10�21. (Russian) [2] L.I. Vainerman, Selfadjoint Boundary Problems for Strongly Elliptic and Hyperbolic Equations of Second Order in a Hilbert Space. � Dokl. Akad. Nauk USSR 218 (1974), 345-348. (Russian) [3] V.M. Bruk, Dissipative Extensions of a Di�erential Elliptic-type Operator. � Funct. Anal., Ulyanovsk 3 (1974), 35�43. (Russian) [4] L.I. Vainerman, A Degenerating Elliptic Equation of Second Order in a Hilbert Space. � Di�. Eq. 14 (1978), 482�491. (Russian) [5] L.I. Vainerman, A Degenerating Elliptic Equation with a Variable Operator Coef- �cient. � Ukr. Mat. Zh. 31 (1979), 247�255. (Russian) [6] V.I Gorbatchuk and M.L. Gorbatchuk, Boundary Value Problems for Di�erential- Operator Equations. Kluwer Acad. Publ., Dordrecht, Boston, London, 1991. [7] V.I.Khrabustovsky, On the Characteristic Operators and Projections and on the So- lutions of Weil Type of Dissipative and Accumulative Operator Systems. 1. General Case; 2. Abstract Theory; 3. Separated Boundary Conditions. � J. Math. Phys., Anal., Geom. 2 (2006), 149�175; 299�317; 449�473. [8] V.M. Bruk, On Invertible Restrictions of Relations Generated by a Di�erential Expression and by a Nonnegative Operator Function. � Mat. Zametki 82 (2007), 652�664. (Russian) Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2 143 V.M. Bruk [9] Yu. M. Berezanski, Expansions in Eigenfunctions of Selfadjoint Operators. Naukova Dumka, Kiev, 1965. (Russian) [10] V.M. Bruk, On Linear Relations in a Space of Vector Functions. � Mat. Zametki 24 (1978), 499�511. (Russian) [11] V.I Gorbatchuk and M.L. Gorbatchuk, Some Questions of Spectral Theory of a Linear Di�erential Equation of Second Order with Unbounded Operator Coe�cients. � Ukr. Mat. Zh. 23 (1971), 3�14. (Russian) [12] V.M. Bruk, On Generalized Resolvents of Linear Relations Generated by a Nonneg- ative Operator Function and a Di�erential Elliptic-Type Expression. � Izv. Vuz. Mat. 11 (2008), 12�26. (Russian) [13] G.I. Laptev, Strong Elliptic Di�erential Equations of Second Order in a Hilbert Space. � Lit. Mat. Sb. 8 (1968), No. 1, 87�99. (Russian). [14] F.S. Rofe-Beketov and A.M. Khol'kin, Spectral Anaysis of Di�erential Operators. Interplay between Spectral and Oscillatory Properties. World Sci. Monogr. Ser. Math., Singapore 7 (2005), XXII. [15] A.G. Baskakov and K.I. Chernyshov, Spectral Analysis of Linear Relations, and Degenerate Semigroups of Operators. � Mat. Sb. 193 (2002), 1573�1610. (Russian) [16] V.M. Bruk, On Invertible Restrictions of Closed Operators in Banach Spaces. � Funct. Anal., Ulyanovsk 28 (1988), 17�22. (Russian) [17] V.M. Bruk, On the Spectrum of Linear Relations Associated with Uniformly Well- Posed Problems. � Di�. Eq. 43 (2007), 21�27. (Russian) [18] A.N. Kochubei, On the Spectrum of Selfadjoint Extensions of the Symmetric Ope- rator. � Mat. Zametki 19 (1976), 429�434. (Russian) [19] V.A. Derkach and M.M. Malamud, Generalized Resolvents and the Boundary Value Problems for Hermitian Operators with Gaps. � J. Funct. Anal. 95 (1991), No. 1, 1�95. [20] A.N. Kochubei, On Extensions of Symmetric Operators and Symmetric Binary Relations. � Mat. Zametki 17 (1975), 41�48. (Russian) [21] V.M. Bruk, On One Class of Boundary Value Problems with a Spectral Parameter in the Boundary Condition. � Mat. Sb. 100 (1976), 210�215. (Russian) [22] L.I. Vainerman, On Extensions of Closed Operators in a Hilbert Space. � Mat. Zametki 28 (1980), 833�842. (Russian) [23] T. Kato, Perturbation Theory for Linear Operators. � Springer�Verlag, Berlin, Heidelberg, New York, 1966. [24] V.M. Bruk, On Holomorphic Families of Linear Relations. � Funct. Anal., Ulyanovsk 33 (1992), 24�28. (Russian) [25] F.S. Rofe-Beketov, Selfadjoint Extensions of Di�erential Operators in a Space of Vector Functions. � Dokl. Akad. Nauk USSR 164 (1969), 1034�1037. (Russian) 144 Journal of Mathematical Physics, Analysis, Geometry, 2009, vol. 5, No. 2

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