Almost invariant spectral properties of a contraction and multiplicative properties of analytic operator-functions

The paper is concerned with the completeness problem of root functions of general boundary value problems for first order systems of ordinary differential equations. We introduce and investigate the class of weakly regular boundary conditions. We show that this class is much broader than the class of regular boundary conditions introduced by G.D. Birkhoff and R.E. Langer. Our main result states that the system of root functions of a boundary value problem is complete and minimal provided that the boundary conditions are weakly regular. Moreover, we show that in some cases the weak regularity of boundary conditions is also necessary for the completeness. Also we investigate the completeness for 2 × 2 Dirac type equations subject to irregular boundary conditions. Emphasize that our results are the first results on the completeness for general first order systems even in the case of regular boundary conditions.

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“…Marchenko [35, Exercise 1.3.3] (2 × 2 Dirac system, B = diag(−1, 1)) and Yu.P. Ginzburg [18] (B = I n , Q = 0) (see Remark 4.5 below).…”

Section: It Is Clear That T a (C D) = T −A (D C)mentioning

confidence: 99%

“…(ii) In the case of the simplest operator L C,D = −iI n ⊗ d dx (B = I n , Q = 0), another (and rather complicated) proof of Corollary 4.3 was obtained in [18].…”

Section: Weakly Regular Bvp For Dirac Type Systemsmentioning

confidence: 99%

On the completeness of root subspaces of boundary value problems for first order systems of ordinary differential equations

Malamud

Оридорога

2012

Journal of Functional Analysis

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“…Marchenko [46, §1.3] (2 × 2 Dirac system) and V.P. Ginzburg [26] (B = I n , Q = 0). Our first main result (Theorem 4.1) states the completeness property for the general BVP (1.2)-(1.4) with non-weakly regular boundary conditions.…”

mentioning

confidence: 99%

“…26) and dom( L) = U dom(L) = y = col(y 1 , y 2 , y 3 , y 4 ) ∈ W 1,1 ([0, ℓ]; C 4 ) :Ly ∈ L 2 ([0, ℓ]; C 4 ), y 2 (0) = y 4 (0) = 0,y 1 (ℓ) + α 1 y 2 (ℓ) + β 1 y 4 (ℓ) = 0, y 3 (ℓ) + α 2 y 4 (ℓ) + β 2 y 2 (ℓ) = 0 . (7.27) Thus, the operator L is similar to the operator L, Ly = −i B(x)y ′ + Q(x)y (7.28)with the domain dom( L) given by(7.27), and the matrix functions B(·), Q(·), given byB(x) Q ∈ L 1 ([0, ℓ]; C 4×4) in view of conditions (7.6)-(7.7).…”

mentioning

confidence: 99%

On the completeness and Riesz basis property of root subspaces of boundary value problems for first order systems and applications

Lunyov¹,

Malamud²

2015

J. Spectr. Theory

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Abstract. The paper is concerned with the completeness property of root functions of general boundary value problems for n × n first order systems of ordinary differential equations on a finite interval. In comparison with the recent paper [45] we substantially relax the assumptions on boundary conditions guarantying the completeness of root vectors, allowing them to be nonweakly regular and even degenerate. Emphasize that in this case the completeness property substantially depends on the values of a potential matrix at the endpoints of the interval.It is also shown that the system of root vectors of the general n × n Dirac type system subject to certain boundary conditions forms a Riesz basis with parentheses. We also show that arbitrary complete dissipative boundary value problem for Dirac type operator with a summable potential matrix admits the spectral synthesis in L 2 ([0, 1]; C n ). Finally, we apply our results to investigate completeness and the Riesz basis property of the dynamic generator of spatially non-homogenous damped Timoshenko beam model.

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“…(ii) In case of the simplest operator L C,D = −iI n ⊗ d dx (B = I n , Q = 0), another (and rather complicated) proof of Corollary 4.3 was obtained in [17].…”

Section: 2mentioning

confidence: 99%

On the completeness of the root vectors of first-order systems

Malamud

Оридорога

2010

Dokl. Math.

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The paper is concerned with the completeness problem of root functions of general boundary value problems for first order systems of ordinary differential equations. Namely, we introduce and investigate the class of weakly regular boundary conditions. We show that this class is much broader than the class of regular boundary conditions introduced by G.D. Birkhoff and R.E. Langer. Our main result states that the system of root functions of a boundary value problem is complete and minimal provided that the boundary conditions are weakly regular. Moreover, we show that in some cases the weak regularity of boundary conditions is also necessary for the completeness. Also we investigate the completeness for 2 × 2 Dirac and Dirac type equations subject to irregular or even to degenerate boundary conditions.We emphasize that our results are the first results on the completeness problem for general first order systems even in the case of regular boundary conditions.

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Almost invariant spectral properties of a contraction and multiplicative properties of analytic operator-functions

Cited by 8 publications

References 1 publication

On the completeness of root subspaces of boundary value problems for first order systems of ordinary differential equations

On the completeness of root subspaces of boundary value problems for first order systems of ordinary differential equations

On the completeness and Riesz basis property of root subspaces of boundary value problems for first order systems and applications

On the completeness of the root vectors of first-order systems

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