Non self-adjoint correct restrictions and extensions with real spectrum
aa r X i v : . [ m a t h . SP ] F e b Non self-adjoint correct restrictions andextensions with real spectrum
B.N. Biyarov, Z.A. Zakarieva, G.K. AbdrashevaFebruary 2, 2021
Key words: maximal (minimal) operator, correct restriction, correct extension, realspectrum, non self-adjoint operator.
AMS Mathematics Subject Classification:
Abstract.
The work is devoted to the study of the similarity of a correct restrictionto some self-adjoint operator in the case when the minimal operator is symmetric.The resulting theorem was applied to the Sturm-Liouville operator and the Laplaceoperator. It is shown that the spectrum of a non self-adjoint singularly perturbedoperator is real and the corresponding system of eigenvectors forms a Riesz basis. Introduction
Let a linear operator L be given in a Hilbert space H . The linear equation Lu = f (1.1)is said to be correctly solvable on R ( L ) if k u k ≤ C k Lu k for all u ⊂ D ( L ) (where C > does not depend on u ) and everywhere solvable if R ( L ) = H . If (1.1) is simultaneouslycorrect and solvable everywhere, then we say that L is a correct operator . A correctlysolvable operator L is said to be minimal if R ( L ) = H . A closed operator b L is calleda maximal operator if R ( b L ) = H and Ker b L = { } . An operator A is called a restriction of an operator B and B is said to be an extension of A if D ( A ) ⊂ D ( B ) and Au = Bu for all u ∈ D ( A ) .Note that if a correct restriction L of a maximal operator b L is known, then theinverses of all correct restrictions of b L have in the form [1] L − K f = L − f + Kf, (1.2)where K is an arbitrary bounded linear operator from H into Ker b L .Let L be some minimal operator, and let M be another minimal operator relatedto L by the equation ( L u, v ) = ( u, M v ) for all u ∈ D ( L ) and v ∈ D ( M ) . Then1 B.N. Biyarov, Z.A. Zakarieva, G.K. Abdrasheva b L = M ∗ and c M = L ∗ are maximal operators such that L ⊂ b L and M ⊂ c M . A correctrestriction L of a maximal operator b L such that L is simultaneously a correct extensionof the minimal operator L is called a boundary correct extension . The existence of atleast one boundary correct extension L was proved by Vishik in [2], that is, L ⊂ L ⊂ b L .The inverse operators to all possible correct restrictions L K of the maximal operator b L have the form (1.2), then D ( L K ) is dense in H if and only if Ker ( I + K ∗ L ∗ ) = { } .All possible correct extensions M K of M have inverses of the form M − K f = ( L ∗ K ) − f = ( L ∗ ) − f + K ∗ f, where K is an arbitrary bounded linear operator in H with R ( K ) ⊂ Ker b L such thatKer ( I + K ∗ L ∗ ) = { } . Lemma 1.1 (Hamburger [3, p. 269]) . Let A be a bounded linear transformation in H and N a linear manifold. If we write A ( N ) = M then A ∗ ( M ⊥ ) = N ⊥ ∩ R ( A ∗ ) . Proposition 1.1 ([4, p. 1863]) . A correct restrictions L K of the maximal operator b L are correct extensions of the minimal operator L if and only if R ( K ) ⊂ Ker b L and R ( M ) ⊂ Ker K ∗ . The main result of this work is the following.
Theorem 1.2.
Let L be symmetric minimal operator in a Hilbert space H , L beself-adjoint correct extension of the L , and L K be correct restriction of the maximaloperator b L ( b L = L ∗ ) . If R ( K ∗ ) ⊂ D ( L ) , I + KL ≥ , and I + KL is invertible, where L and K are the operators in representation (1.2) ,then L K similar to a self-adjoint operator. Corollary 1.1. If K satisfies the assumtions of Theorem 1.2, then the spectrum of L K is real, that is, σ ( L K ) ⊂ R . Corollary 1.2. If K satisfies the assumtions of Theorem 1.2 and L − is the compactoperator, then the system of eigenvectors of L K forms a Riesz basis in H . Corollary 1.3.
The results of Theorem 1.2 are also valid if conditions “ I + KL ≥ and I + KL is invertible” replase to condition “ KL ≥ ” . Corollary 1.4.
The results of Theorem 1.2, Corollary 1.1-1.3 are also valid for the L ∗ K . on self-adjoint correct restrictions and extensions with real spectrum Preliminaries
In this section, we present some results for correct restrictions and extensions whichare used in Section 3.If A is bounded linear transformation from a complex Hilbert space H into itself,then the numerical range of A is by definition the set W ( A ) = { ( Ax, x ) : x ∈ H, k x k = 1 } . It is well known and easy to prove that if σ ( A ) denotes the spectrum of A , then σ p ( A ) ⊂ W ( A ) , σ ( A ) ⊂ W ( A ) , for the point spectrum σ p ( A ) and the spectrum σ ( A ) of A , where the bar indicatesclosure. The numerical range of an unbounded operator A in a Hilbert space H isdefined as W ( A ) = { ( Ax, x ) : x ∈ D ( A ) , k x k = 1 } , and similarly to the bounded case, W ( A ) is convex and satisfies σ p ( A ) ⊂ W ( A ) . Ingeneral, the conclusion σ ( A ) ⊂ W ( A ) does not surely hold for unbounded operators A (see [5]). Theorem 2.1 (Theorem 2 in [6, p. 181]) . The following are equivalent conditions onan operator T : (1) T is similar to a self-adjoint operator. (2) T = P A , where P is positive and invertible and A is self-adjoint. (3) S − T S = T ∗ and = W ( S ) . Theorem 2.2 (Theorem 1 in [7, p. 215]) . Let A and B operators on the complex Hilbertspace H . If / ∈ W ( A ) then σ ( A − B ) ⊂ W ( B ) /W ( A ) . Corollary 2.1 (Corollary in [7, p. 218]) . If A > , B ≥ and C = C ∗ , then σ ( AB ) is positive and σ ( AC ) is real. Theorem 2.3 (Theorem A in [8, p. 508]) . The numerical range W ( T ) of T is convexand W ( aT + b ) = aW ( T ) + b for all complex numbers a and b . Proof of Theorem 1.2
We transform (1.2) to the form L − K = L − + K = ( I + KL ) L − . (3.1)Then L K is defined as the restriction of the maximal operator b L on the domain D ( L K ) = { u ∈ D ( b L ) : ( I − K b L ) u ∈ D ( L ) } . B.N. Biyarov, Z.A. Zakarieva, G.K. Abdrasheva
Now let us prove Theorem 1.2. It was proved in [9, p. 27] that KL is bounded on D ( L ) (that is, KL ∈ B ( H ) ) if and only if R ( K ∗ ) ⊂ D ( L ∗ ) . It follows from D ( L ) = H that KL is bounded on H . In the future, instead of KL , wewill write KL . Then, by virtue of Theorem 2.1 and taking into account the conditionsof Theorem 1.2 that I + KL ≥ and I + KL is invertible, we obtain proof of Theorem1.2.The proof of Corollary 1.1 follows from Corollary 2.1. Corollary 1.2 is easy to obtainfrom the fact that the operator C = ( I + KL ) / L − ( I + KL ) / is self-adjoint and L − K = ( I + KL ) / C ( I + KL ) − / = ( I + KL ) L − . (3.2)Let us proof Corollary 1.3. By Theorem 2.3, we get that / ∈ W ( I + KL ) . Then I + KL ≥ and I + KL is invertible.The proof of Corollary 1.4 follows from (3.2), since C is a self-adjoint operator andin the case Corollary 1.2 the self-adjoint operator C is compact. Non self-adjoint perturbations for some differential opera-tors
Example 1.
We consider the Sturm-Liouville equation on the interval (0 , b Ly = − y ′′ + q ( x ) y = f, (4.1)where q ( x ) is the real-valued function of L (0 , . We denote by L the minimal operatorand by b L the maximal operator generated by the differential equation (4.1) in the space L (0 , . It’s clear that D ( L ) = ˚W (0 , and D ( b L ) = { y ∈ L (0 ,
1) : y, y ′ ∈ AC [0 , , y ′′ − q ( x ) y ∈ L (0 , } . Then Ker b L = { a c ( x ) + a s ( x ) } , where a , a are arbitrary constants, and thefunctions c ( x ) and s ( x ) are defined as follows c ( x ) = 1 + Z x K ( x, t ; 0) dt, s ( x ) = x + Z x K ( x, t ; ∞ ) t dt, where K ( x, t ; 0) = K ( x, t ) + K ( x, − t ) , K ( x, t ; ∞ ) = K ( x, t ) − K ( x, − t ) , on self-adjoint correct restrictions and extensions with real spectrum K ( x, t ) is the solution of the following Goursat problem ∂ K ( x, t ) ∂x − ∂ K ( x, t ) ∂t = q ( x ) K ( x, t ) , K ( x, − x ) = 0 , K ( x, x ) = 12 Z x q ( t ) dt, in the domain Ω = (cid:8) ( x, t ) : 0 < x < , − x < t < x (cid:9) . Note that c (0) = s ′ (0) = 1 , c ′ (0) = s (0) = 0 and Wronskian W ( c, s ) ≡ c ( x ) s ′ ( x ) − c ′ ( x ) s ( x ) = 1 . As a fixed boundary correct extension L we take the operator corresponding to theDirichlet problem for equation (4.1) on (0 , . Then D ( L ) = (cid:8) y ∈ W (0 ,
1) : y (0) = 0 , y (1) = 0 (cid:9) . Therefore the description of the inverse of all correct restrictions L K of the maximaloperator b L has the form y ≡ L − K f = Z x (cid:2) c ( x ) s ( t ) − s ( x ) c ( t ) (cid:3) f ( t ) dt − s ( x ) s (1) Z (cid:2) c (1) s ( t ) − s (1) c ( t ) (cid:3) f ( t ) dt + c ( x ) Z f ( t ) σ ( t ) dt + s ( x ) Z f ( t ) σ ( t ) dt, where σ ( x ) , σ ( x ) ∈ L (0 , which uniquely determine the operator K from (1.2) inthe following form Kf = c ( x ) Z f ( t ) σ ( t ) dt + s ( x ) Z f ( t ) σ ( t ) dt, for all f ∈ L (0 , .K is a bounded operator in L (0 , acting from L (0 , to Ker b L . The operator L K isthe restriction of b L on the domain D ( L K ) = (cid:26) y ∈ W (0 ,
1) : y (0) = Z (cid:2) − y ′′ ( t ) + q ( t ) y ( t ) (cid:3) σ ( t ) dt ; y (1) = c (1) y (0) + s (1) Z (cid:2) − y ′′ ( t ) + q ( t ) y ( t ) (cid:3) σ ( t ) dt (cid:27) . From the condition R ( K ∗ ) ⊂ D ( L ∗ ) = D ( L ) we have that KLy = c ( x ) Z y ( t )[ − σ ′′ ( t ) + q ( t ) σ ( t )] dt + s ( x ) Z y ( t )[ − σ ′′ ( t ) + q ( t ) σ ( t )] dt, B.N. Biyarov, Z.A. Zakarieva, G.K. Abdrasheva where y ∈ D ( L ) , σ , σ ∈ W (0 , , σ (0) = σ (1) = σ (0) = σ (1) = 0 . If I + KL ≥ and I + KL is invertible, then the spectrum of the operator L K consistsonly of real eigenvalues { λ k } ∞ k =1 and the corresponding eigenfunctions { ϕ k } ∞ k =1 formsa Riesz basis in L (0 , , since L − is a compact self-adjoint positive operator. Inparticular, if σ ( x ) = α ( L − c )( x ) , σ ( x ) = β ( L − s )( x ) , α, β ≥ , then KL ≥ . Therefore, by Corollary 1.3, the results of Theorem 1.2 are valid for L K .In this case, L − K has the form y = L − K f = L − f + c ( x ) Z f ( t )( L − c )( t ) dt + s ( x ) Z f ( t )( L − s )( t ) dt. Then ( L − K ) ∗ = ( L ∗ K ) − has form v ( x ) = ( L − f )( x ) + α ( L − c )( x ) Z f ( t ) c ( t ) dt + β ( L − s )( x ) Z f ( t ) s ( t ) dt. Thus, we have ( L ∗ K v )( x ) = − v ′′ ( x ) + q ( x ) v ( x ) + a ( x ) v ′ (0) + b ( x ) v ′ (1) = f ( x ) ,D ( L ∗ K ) = { v ∈ W (0 ,
1) : v (0) = v (1) = 0 } , where a ( x ) = αβ ( c, s ) s ( x ) − α (1 + β k s k ) c ( x )(1 + α k c k )(1 + β k s k ) − αβ | ( c, s ) | ,b ( x ) = α [ c (1)(1 + β k s k ) − βs (1)( s, c )] c ( x ) − β [ αc (1)( c, s ) − s (1)(1 + α k c k )] s ( x )(1 + α k c k )(1 + β k s k ) − αβ | ( c, s ) | ,a ( x ) , b ( x ) ∈ Ker b L and ( · , · ) is scalar product in L (0 , . The operator L ∗ K acts as L ∗ K = L ∗ + Q, where L ∗ = − d dx + q ( x ) , ( Qv )( x ) = a ( x ) < δ ′ ( x ) , v ( x ) > + b ( x ) < δ ′ ( x − , v ( x ) > = a ( x ) v ′ (0) + b ( x ) v ′ (1) , that is, the function Q ∈ W − (0 , . Thus, we have constructed an example of a nonself-adjoint singularly perturbed Sturm-Liouville operator with a real spectrum andthe system of eigenvectors that forms a Riesz basis in L (0 , . on self-adjoint correct restrictions and extensions with real spectrum Example 2.
In the Hilbert space L (Ω) , where is a bounded domain in R m withan infinitely smooth boundary ∂ Ω , let us consider the minimal L and maximal b L operators generated by the Laplace operator − ∆ u = − (cid:18) ∂ u∂x + ∂ u∂x + · · · + ∂ u∂x m (cid:19) . (4.2)The closure L , in the space L (Ω) of Laplace operator (4.2) with the domain C ∞ (Ω) ,is the minimal operator corresponding to the Laplace operator . The operator b L , adjointto the minimal operator L corresponding to Laplace operator, is the maximal operatorcorresponding to the Laplace operator . Then D ( b L ) = { u ∈ L (Ω) : b Lu = − ∆ u ∈ L (Ω) } . Denote by L the operator, corresponding to the Dirichlet problem with the domain D ( L ) = { u ∈ W (Ω) : u | ∂ Ω = 0 } . We have (1.2), where K is an arbitrary linear operator bounded in L (Ω) with R ( K ) ⊂ Ker b L = { u ∈ L (Ω) : − ∆ u = 0 } . Then the operator L K is defined by b Lu = − ∆ u, on D ( L K ) = { u ∈ D ( b L ) : [( I − K b L ) u ] | ∂ Ω = 0 } , where I is the identity operator in L (Ω) . Note that L − is a self-adjoint compact oper-ator. If K satisfies the conditions of Theorem 1.2, then L K is non self-adjoint operatorwith a real positive spectrum (i.e., σ ( L K ) ⊂ R + ), and the system of eigenvectors L K forms a Riesz basis in L (Ω) . In particular, if Kf = ϕ ( x ) Z Ω f ( t ) ψ ( t ) dt, where ϕ ∈ W ,loc (Ω) ∩ L (Ω) is a harmonic function and ψ ∈ L (Ω) , then K ∈ B ( L (Ω)) and R ( K ) ⊂ Ker b L . From R ( K ∗ ) ⊂ D ( L ) it follows that ψ ∈ W (Ω) and ψ | ∂ Ω = 0 .From the condition KL ≥ we have that ψ ( x ) = α ( L − ϕ )( x ) , α ∈ R + . Hence theoperator L K is the restriction of b L to the domain D ( L K ) = n u ∈ D ( b L ) : (cid:16) u − ϕ k ϕ k Z Ω u ( y ) ϕ ( y ) dy (cid:17)(cid:12)(cid:12)(cid:12) ∂ Ω = 0 o . The inverse of L − K has the form u = L − K f = L − f + ϕ Z Ω f ( y )( L − ϕ )( y ) dy. (4.3) B.N. Biyarov, Z.A. Zakarieva, G.K. Abdrasheva
We find the adjoint operator L ∗ K . From (4.3) we have v = ( L − K ) ∗ g = L − g + L − ϕ Z Ω g ( y ) ϕ ( y ) dy, for all g ∈ L (Ω) . Then L ∗ K v = − ∆ v + ϕ k ϕ k Z Ω (∆ v )( y ) ϕ ( y ) dy = g,D ( L ∗ K ) = D ( L ) = (cid:8) v ∈ W (Ω) : v (cid:12)(cid:12) ∂ Ω = 0 (cid:9) . By virtue of Corollary 1.4, the spectrum of the operator L ∗ K consists only of real positiveeigenvalues and the corresponding eigenfunctions forms a Riesz basis in L (Ω) . Notethat ( L ∗ K v )( x ) = − (∆ v )( x ) + ϕ ( x )1 + k ϕ k F ( u ) = g ( x ) , where F ∈ W − (Ω) , since F ( u ) = Z Ω (∆ v )( y ) ϕ ( y ) dy. This is understood in the sense of the definition of the space H − s (Ω) , s > as inTheorem 12.1 (see [10, p. 71]) . Thus, we have shown the examples of a non self-adjoint singularly perturbed op-erator with a real spectrum. Moreover, the corresponding eigenvectors forms a Rieszbasis in L (Ω) . References [1] B.K. Kokebaev, M. Otelbaev, A.N. Shynibekov,
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