5_837_mamedov.dvi EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 3, No. 5, 2010, 831-838 ISSN 1307-5543 – www.ejpam.com On the Basis Property in Lp (0, 1) of the Root Functions of a Class Non Self Adjoint Sturm-Lioville Operators Khanlar R. Mamedov Mersin University, Science and Arts Faculty, Mathematics Department, 33343 Ciftlikkoy Campus, Mersin-Turkey Abstract. In the present paper, we prove the basisness of the root functions of the non self adjoint Sturm-Liouville operators with periodic and anti-periodic boundary conditions in space Lp(0,1), p > 1. Here we assume that the potential is a complex valued absolutely continuous function in [0,1]. 2000 Mathematics Subject Classifications: 34L10, 34B24, 47E05 Key Words and Phrases: eigenfunctions, basis, periodic and anti-periodic conditions, non self adjoint Sturm-Liouville operator 1. Introduction Consider the eigenvalue problem for the differential equation ℓ(u)≡ u′′ + q(x)u= λu (1) on the interval (0,1) with the periodic u(0) = u(1), u′(0) = u′(1), (2) and antiperiodic u(0) = −u(1), u′(0) = −u′(1), (3) boundary conditions, where the potential q(x) is an arbitrary complex valued function. In this paper we study the basis property of the root functions of boundary value problems (1),(2) and (1),(3). This problem is important for the study of non selfadjoint Sturm-Liouville operators. It is well known that the basisness of the system root functions of linear differential operators depends on regularity of boundary conditions (in Birkhoff sense strongly regular, see.[1], p.71). Email address: hanlar�mersin.edu.tr, hanlarm�yahoo. om http://www.ejpam.com 831 c© 2010 EJPAM All rights reserved. K. Mamedov / Eur. J. Pure Appl. Math, 3 (2010), 831-838 832 In 1962 Mikhailov [17], in 1964 Keselman [9] and in 1971 Dunford and Schwartz [4] it was showed the basisness in L2 (0,1) of the root functions of ordinary linear differential operator with regular boundary conditions. Ionkin [6] in 1976 studied to following boundary value problem u′′ +λu = 0, u′(0)− u′(1) = 0, u(0) = 0 whose boundary conditions are regular, but not strongly regular. All the eigenvalues of this problem starting with the second one are double, the general number of associated that the chosen specially system of root functions form an unconditional basis in L2 (0,1). By Shkalikov [19] in 1979 [see 20], it was proved that the system of root functions of a differential operator with not strongly regular boundary conditions form a Riesz basis with parentheses. Kerimov and Mamedov [8] in 1998 found conditions on the potential q(x) under which the system eigenfunctions of boundary value problems (1),(2) and (1),(3) forms Riesz basis in L2 (0,1). Namely, they proved the following result: Assume that q(x) ∈ C (4)[0,1] is complex- valued functions satisfying the condition q(0) 6= q(1), then the root functions of boundary value problems (1),(2) and (1),(3) form Riesz basis in L2 (0,1). The spectral properties of the boundary value problems (1),(2) and (1),(3) were investigated in [14]. Dernek and Veliev [2] in 2005 established conditions in terms of the Fourier coefficients qn = (q(x), exp i2nπx) of the potential q(x) proved the following result: Assume that the conditions lim n→∞ ln|n| nq2n = 0, q2n ∼ q−2n for (1),(2) (and lim n→∞ ln|n| nq2n+1 = 0, q2n+1 ∼ q−2n−1 for (1),(3)) hold, where an ∼ bn weans that c1 � �bn � � < � �an � � < c2 � �bn � � . Then the root functions of the boundary problem (1),(2) (and (1),(3)) form a Riesz basis in L2 (0,1) . Makin [12] in 2005 obtained the following result: Suppose that q(x) ∈W m 1 [0,1], q(l)(0) = q(l)(1), for all l = 0,1, ..., m − 1 and q2n > c0n−m−1, 0 < c1 < |α2n| |β2n| < c2 for (1),(2) (and q2n−1 > c0n−m−1, 0 < c1 < |α2n−1| |β2n−1| < c2, c0 > 0 for (1),(3)) for all n > N1, then the root functions of boundary value problem (1),(2) (and (1),(3)) form a Riesz basis in L2 (0,1). Moreover, some theorems for determining whether the root functions form a Riesz basis in L2 (0,1) or not were given in[12]. A classification on the boundary conditions for Sturm- Lionville operator under which the root functions form a Riesz basis in L2 (0,1) is established in [13]. Shkalikov and Veliev [21] in 2008 obtained similar result in terms of the Fourier coeffi- cients qn on the potential q(x) ∈W p 1 [0,1] (q(ℓ)(0) = q(ℓ)(1) = 0,0 ≤ ℓ ≤ s− 1, where s ≤ p) under which the system of root functions form a Riesz basis in L2 (0,1). Mamedov and Menken [15] in 2008 showed that the root functions of the boundary-value problems (1),(2) and (1),(3) form a Riesz basis in L2 (0,1) when q(x) ∈ C (4)[0,1] is complex- valued functions satisfying the condition the conditions q(0) = q(1) and q′(0) 6= q′(1). In 2008 Djakov and Mitjagin [3] obtained some result on the absence of the Riesz basis property. In 2010 Kıraç [10] showed that the root functions of the boundary-value problems (1),(2) and (1),(3) form a Riesz basis in L2 (0,1) when q(x) is absolutely continuous function in K. Mamedov / Eur. J. Pure Appl. Math, 3 (2010), 831-838 833 [0,1] and q(0) 6= q(1). By Kurbanov [11] in 2006 it was showed that the root functions of the boundary value problems (1),(2) and (1),(3) forms a basis in Lp (0,1) , p > 1 when q(x) ∈ C (4)[0,1] and q(0) 6= q(1). In the case q(x) ∈ C (4)[0,1], q(0) = q(1) and q′(0) 6= q′(1) Menken and Mamedov [16] in 2010 proved the basisness in Lp (0,1). The purpose of this paper is to find the weaker conditions on the potential q(x) under which the system of root functions form basis in space Lp (0,1) , p > 1. The following results plays an important role in the proof of main results. Theorem 1 ([1, 5]). The following assertions are equivalent: 1) sequence ¦ ϕ j ©∞ j=1 forms a Riesz basis in H; 2) The sequence ¦ ϕ j ©∞ j=1 is complete in the Hilbert space H, there corresponds to it a com- plete biorthogonal sequence ¦ ψ j ©∞ j=1 , and for any f ∈ H one has ∞ ∑ j=1 � �( f ,ϕ j) � � < ∞, ∞ ∑ j=1 � �( f ,ψ j) � � 2 <∞. Theorem 2 ([7]). A system ¦ ϕ j ©∞ j=1 is a basis in the Banach space X if only if the following conditions are satisfied: a) ¦ ϕ j ©∞ j=1 is complete in X , b) ¦ ϕ j ©∞ j=1 is minimal, c) there exists a number M > 0 such that for each f ∈ X , the inequality N ∑ j=1 ( f ,ψ j)ϕ j ≤ M ‖x‖ , N = 1,2, . . . where the sequence ¦ ψ j ©∞ j=1 is the biorthogonal adjoint system to ¦ ϕ j ©∞ j=1 . Assume that q(x) ∈ AC[0,1] (absolutely continuous function in [0,1]) and q(0) 6= q(1). Under these conditions it is known ([10]) that: a) all eigenvalues of problem (1),(2) starting from number N are simple and form two infinite sequences. λ2n = −(2nπ)2+ q(0)− q(1) 4nπ + o( 1 n ), λ2n+1 = −(2nπ)2+ q(1)− q(0) 4nπ + o( 1 n ), n= N , N + 1, . . . , K. Mamedov / Eur. J. Pure Appl. Math, 3 (2010), 831-838 834 and the corresponding eigenfunctions are of the form u2n(x) = p 2 sin(2nπx − π 4 ) +O( 1 n ), (4) u2n+1(x) = p 2 cos(2nπx − π 4 ) +O( 1 n ), n= N , N + 1, . . . ; (5) b) all eigenvalues of the boundary value problem (1),(3) starting from number N are sim- ple and form two infinite sequences. λ2n = − [(2n+ 1)π]2 + q(0)− q(1) 4nπ + o( 1 n ), λ2n+1 = − [(2n+ 1)π]2 + q(1)− q(0) 4nπ + o( 1 n ), n= N , N + 1, . . . , and the corresponding eigenfunctions are of the form u2n(x) = p 2sin((2n+ 1)πx − π 4 ) +O( 1 n ), (6) u2n+1(x) = p 2cos((2n+ 1)πx − π 4 ) +O( 1 n ), n= N , N + 1, . . . . (7) In the present paper, in Section 2 we using the asymptotic formulas (4)-(7) of eigenfunctions of the boundary problems (1), (2) and Theorem 1 shown Riesz basisness in L2 (0,1). In Section 3, using Theorem 2 and F.Riesz Theorem (see [22], p.154), we prove the basisness in Lp(0,1) of the root functions of the periodic and anti-periodic boundary value problems. 2. The Riesz Basisness in L2(0, 1) Firstly, by a different method, using the asymptotic formulas (4)-(7) and the Theorem 1 we will show the Riesz basisness in L2 (0,1) of the root functions of the boundary problems (1),(2) and (1),(3). Theorem 3. Suppose that q(x) ∈ AC[0,1] and q(0) 6= q(1). Then the system of root functions of the boundary value problems (1),(2) and (1),(3) forms a Riesz basis in L2 (0,1) . Proof. It is well known that the system of eigenfunctions and associated eigenfunctions of problem (1),(2) is complete in L2(0,1). The system of the root functions is minimal in L2(0,1). The minimality of this system follows from the fact that this system has a biorthogo- nal system consisting of the root functions of the adjoint operator l∗(υ) = υ′′ + q(x)υ, υ(1) = υ(0), υ′(1) = υ′(0). K. Mamedov / Eur. J. Pure Appl. Math, 3 (2010), 831-838 835 For any f ∈ L2(0,1), with a direct computation we have that ∞ ∑ n=N � �( f ,u2n) � � 2 < ∞, ∞ ∑ n=N � �( f ,u2n+1) � � 2 < ∞. On the other hand, the eigenfunctions of the adjoint operator have of the form υ2n(x) = p 2 sin((2n+ 1)πx − π 4 ) +O( 1 n ), (8) υ2n+1(x) = p 2 cos((2n+ 1)πx − π 4 ) +O( 1 n ), n= N , N + 1, . . . . (9) and the inequalities ∞ ∑ n=N � �( f ,υ2n) � � 2 <∞ and ∞ ∑ n=N � �( f ,υ2n+1) � � 2 <∞ hold. According to Theo- rem 1, the root functions of the boundary problem (1),(2) form a Riesz basis in L2(0,1). Similarly, is proved the basisness in L2(0,1) of root functions of boundary problem (1),(3). This completes the proof. 3. The Basisness in Lp (0, 1) , p > 1 Theorem 4. Suppose that q(x) ∈ AC[0,1], q(0) 6= q(1). Then the system of the root functions of the boundary problems (1),(2) and (1),(3) forms a basis in the space Lp(0,1) (1< p <∞). Proof. Let ψ1(x) = 1, ψ2n(x) = p 2 sin(2nπx − π 4 ), ψ2n+1(x) = p 2 cos(2nπx − π 4 ), n = 1,2, . . .. The systems � ψn(x) ∞ n=1 = � ψ1(x),ψ2n(x),ψ2n+1(x) ∞ n=1 is a basis in the space Lp (0,1). Moreover, if p = 2, this basis is orthonormal. We introduce notation: � un(x) ∞ n=1 = � u1(x),u2n(x),u2n+1(x) ∞ n=1 , � υn(x) ∞ n=1 = � υ1(x),υ2n(x),υ2n+1(x) ∞ n=1 . It follows from the asymptotic formulas (4),(5) and (8),(9) that un(x) =ψn(x)+O( 1 n ), υn(x) =ψn(x)+O( 1 n ) (10) for sufficiently large n. Let 1 < p < 2. By Theorem 3, the system of the root functions � un(x) ∞ n=1 is basis in L2 (0,1) and � υn(x) ∞ n=1 is biorthogonal to the system � un(x) ∞ n=1. Thus, this system is complete in Lp (0,1). Therefore, by Theorem 2, to prove the basis property of � un(x) ∞ n=1 in Lp (0,1) , it is necessary and sufficient to prove that existence of a constant M > 0 such that N ∑ n=1 ( f ,υn)un p ≤ M f p , N = 1,2, . . . , (11) for all f ∈ Lp(0,1), where ‖·‖p denotes the norm in Lp(0,1). K. Mamedov / Eur. J. Pure Appl. Math, 3 (2010), 831-838 836 By (10) we obtain N ∑ n=1 ( f ,υn)un p ≤ N ∑ n=1 ( f ,ψn)ψn p + N ∑ n=1 ( f ,ψn)O( 1 n ) p + N ∑ n=1 ( f ,O( 1 n ))ψn p + N ∑ n=1 ( f ,O( 1 n ))O( 1 n ) p . (12) We shall now prove that all the summands on the right side of (12) are bounded from above by constant f p . Since the system � ψn(x) ∞ n=1 is a basis in the space Lp (0,1), applying Theorem 2, we have N ∑ n=1 ( f ,ψn)ψn p ≤ C f p , C = const (N = 1,2, · · · ) (13) for arbitrary for all f ∈ Lp(0,1). By F.Riesz Theorem (see [22], p.154) for an arbitrary function f ∈ Lp(0,1) one has the estimate N ∑ n=1 ( f ,ψn)O( 1 n ) p ≤ C N ∑ n=1 ( f ,ψn) 1 n ≤ C N ∑ n=1 � �( f ,ψn) � � q ! 1 q · N ∑ n=1 1 np ! 1 p ≤ C f p , (14) where 1 p + 1 q = 1. Using the Parseval’s equality we have N ∑ n=1 ( f ,O( 1 n ))ψn p ≤ N ∑ n=1 ( f ,O( 1 n ))ψn 2 = N ∑ n=1 � � � � ( f ,O( 1 n )) � � � � 2 ! 1 2 ≤ C f 1 N ∑ n=1 1 n2 ! 1 2 ≤ C f p , (15) N ∑ n=1 ( f ,O( 1 n ))O( 1 n ) p ≤ C f 1 N ∑ n=1 1 n2 ! 1 2 ≤ C f p . (16) Using the inequalities (13)-(16) in the estimate (12) we have (11). Thus the proof of assertion (11) is complete. Consequently, the system � un(x) ∞ n=1 is a basis in the space Lp (0,1) , 1 < p < 2. Now assume that 2< p <∞. It is clear that the system � un(x) ∞ n=1 is a basis in Lq (0,1) . By Corollary 2 in [7] (in Section I) it follows that the system � υn(x) ∞ n=1 is a basis in the REFERENCES 837 space Lp (0,1) , where 1 p + 1 q = 1. Note that 1< q < 2. Using the discussions introduced above entirely analogously it is proved that the system � υn(x) ∞ n=1 is a basis in Lq (0,1) . It follows that � un(x) ∞ n=1 is a basis in the space Lp (0,1) , p > 2. Entirely analogously it is not difficult to prove that the system of the root functions of the boundary problem (1),(3) form a basis in the space Lp(0,1) (1 < p < ∞). The proof of the theorem is complete. References [1] N K Bari. Biorthogonal systems and bases in Hilbert spaces. Uchen. Zap. Moskov. Gos. Univ. 148(4):68-107, 1951 (Russian). [2] N Dernek and O A Veliev. On the Riesz basisness of the root functions of the nonself- adjoint Sturm-Liouville operators. Israel Journal of Mathematics, 145:113-123, 2005. [3] P Djakov and B S Mitjagin. Instability Zones of Periodic 1-dimensional Schrodinger and Dirac Operators. Uspekhi Mat. Nauk, 61:4, 77-182, 2006. English Transl. in Russian Math. Surves, 61:4, 663-776, 2006. [4] N Dunford and J T Schwartz. Linear Operators, Prt.3 Spectral Operators. Wiley, New York, 1970. [5] I C Gohberg and M G Krein. Introduction to the Theory of Linear Nonselfadjoint Operators. American Math. Soc., Providence, Rhode Island, 1969. [6] N I Ionkin. The solution of a boundary-value problem in heat conduction with a non- classical boundary condition. Differ. Equations, 13(2): 294-304, 1977. [7] B S Kashin and A A Saakyan. Orthogonal Series. American Mathematical Society, 1989. [8] N B Kerimov and Kh R Mamedov. On the Riesz basis property of the root functions in certain regular boundary value problems. Math. Notes, 64(4): 483-487, 1998. [9] G M Kesel’man. On the unconditional convergence of expansions in the eigenfunctions of some differential operators, Izv. Vyssh. Uchebn. Zaved. Mat. [Soviet Math. (Iz. VUZ)], 2: 82-93, 1964. [10] A A Kıraç. Riesz Basis Property of the Root functions of non-selfadjoint operators wit regular boundary conditions, Int.Journal of Math.Analysis. 3(22):1101-1109, 2009. [11] V M Kurbanov. A theorem on equivalent bases for a differential operator. Dokl. Akad. Nauk, 406(1):17-20, 2006. [12] A S Makin. Convergence of expansions in the root functions of periodic boundary value problems. Doklady Math., 73(1):71-76, 2006. REFERENCES 838 [13] A S Makin. On spectral decompositions corresponding to non-self-adjoint Sturm-Lioville operators. Doklady Math., 73(1):15-18, 2006. [14] Kh R Mamedov. On spectrally of differential operator of second order. Proceeding of Institute of Mathematics and Mechanics, Acad, Sci. Azer. Repub. 5:179-181, 1996. [15] Kh R Mamedov and H Menken. On the basisness in L2(0,1) of the root functions in not strongly regular boundary value problems. European Journal of Pure and Applied Math.,1(2):51-60, 2008. [16] H Menken and Kh R Mamedov. Basis property in Lp(0,1) of the root functıons corre- spondıng to a boundary-value problem. Journal of Applied Functional Analysis, 5(4):351- 356, 2010. [17] V P Mikhailov. On the bases in L2(0,1). Dokl. Akad. Nauk SSSR [Soviet Math. Dokl.], 144(5): 981-984, 1962. [18] M A Naimark. Linear Differential Operators, Part I. Frederick Ungar Pub. Co., New York, 1967. [19] A A Shkalikov. On the Riesz basis property of the root vectors of ordinary differential operators. Russian Math. Surveys, 34(5):249-250, 1979. [20] A A Shkalikov. On the basis property of the eigenfunctions of ordinary differential op- erators with integral boundary conditions. Vestnik Moscow University, Ser. Mat. Mekh., 37(6):12-21, 1982. [21] O A Veliev and A A Shkalikov. On the Riesz Basis Property of the Eigen- and Associ- ated Functions of Periodic and Antiperiodic Sturm–Liouville Problems. Mat. Zametki, 85(5):671–686, 2009. [22] A Zygmund. Trigonometric Series, Vol. 2. Cambridge Univ. Press, Cambridge, 1959.