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In an infinite-dimensional Hilbert space we define and study special unbounded maximal dissipative operators and use them for constructions of abstract examples of families of densely defined closed symmetric operators whose squares have trivial domains. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to study hyponormal and dissipative correct restrictions and extensions as well as their applications to differential operators. [ABSTRACT FROM AUTHOR]

This paper is concerned with a boundary perturbation of m-dissipative linear operators. We provide conditions in such a way that the boundary perturbation preserves the m-dissipativity of operators. The perturbation of m- ω -dissipativity is also given. Our results are illustrated with some examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider a system of first-order equations with the same eigenvalue parameter together with dissipative boundary conditions. Applying Lax-Phillips scattering theory and Sz.-Nagy-Foiaş model operator theory we prove a completeness theorem. [ABSTRACT FROM AUTHOR]

This paper aims to share some completeness theorems related with a boundary value problem generated by a system of equations and non‐self‐adjoint (dissipative) boundary conditions. Indeed, we consider a system of equations that contains a continuous analogous of the orthogonal polynomials on the unit circle. Constructing the characteristic function of the related dissipative operator, we share some completeness theorems. Moreover, we give an explicit form of the self‐adjoint dilation of the dissipative operator. [ABSTRACT FROM AUTHOR]

In [3], Chen, Deng, Ding and Fan proved that the fractional power dissipative operator is bounded on Lebesgue spaces Lp(ℝn), Hardy spaces Hp(ℝn) and general mixed norm spaces, which implies almost everywhere convergence of such operator. In this paper, we study the rate of convergence on fractional power dissipative operator on some sobolev type spaces. [ABSTRACT FROM AUTHOR]

In this paper, we consider applications of the connection between the soliton theory and the commuting nonselfadjoint operator theory, established by Livšic and Avishai. An approach to the inverse scattering problem and to the wave equations is presented, based on the Livšic operator colligation theory (or vessel theory) in the case of commuting bounded nonselfadjoint operators in a Hilbert space, when one of the operators belongs to a larger class of nondissipative operators with asymptotics of the corresponding nondissipative curves. The generalized Gelfand–Levitan–Marchenko equation of the cases of different differential equations (the Korteweg–de Vries equation, the Schrödinger equation, the Sine–Gordon equation, the Davey–Stewartson equation) are derived. Relations between the wave equations of the input and the output of the generalized open systems, corresponding to the Schrödinger equation and the Korteweg–de Vries equation, are obtained. In these two cases, differential equations (the Sturm–Liouville equation and the 3-dimensional differential equation), satisfied by the components of the input and the output of the corresponding generalized open systems, are derived. [ABSTRACT FROM AUTHOR]

We review recent results obtained in the scattering theory of dissipative quantum systems representing the long-time evolution of a system S interacting with another system S ′ and susceptible of being absorbed by S ′ . The effective dynamics of S is generated by an operator of the form H = H 0 + V − i C ∗ C on the Hilbert space of the pure states of S , where H 0 is the self-adjoint generator of the free dynamics of S , V is symmetric and C is bounded. The main example is a neutron interacting with a nucleus in the nuclear optical model. We recall the basic objects of the scattering theory for the pair (H , H 0) , as well as the results, proven in [10, 11], on the spectral singularities of H and the asymptotic completeness of the wave operators. Next, for the nuclear optical model, we show that asymptotic completeness generically holds. [ABSTRACT FROM AUTHOR]

In this article we study a non-self-adjoint eigenparameter dependent singular differential 1D Hamiltonian system with the singular end points a and b in the Hilbert space $L_P^2((a,b);\mathbb{C}^2)$ and we consider that this 1D Hamiltonian system is in the limit-circle cases at a and b. For this purpose we use the maximal dissipative operator associated with the considered problem whose spectral analysis is sufficient for boundary value problem. Self-adjoint dilation theory of Sz.-Nagy-Foias developed for the dissipative operators is used. Moreover we construct incoming and outgoing spectral representations of the self-adjoint dilation. This representations allows us to determine the scattering matrix. Therefore a functional model of the dissipative operator is constructed. Moreover, a functional model of the dissipative operator is constructed and its characteristic function in terms of solutions of the corresponding Hamiltonian system is described. Therefore using the obtained results for the characteristic function theory, theorems on completeness of the system of eigenvectors and associated vectors of the dissipative operator and Hamiltonian boundary value problem have been proved.

In thiswork, the general form of maximally dissipative extensions of the minimaloperator generated by first order linear symmetric quasi-differentialexpression in the weighted Hilbert space of vector-functions at rightsemi-infinite interval has been found. Later on, geometry of spectrum of theseextensions is investigated.

This paper deals with the dissipative regular q−Sturm-Liouville problem. We prove that the system of root functions of this operator forms a Bari bases in the space L²q (I) by using the asymptotic behavior at infinity for its eigenvalues. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to investigate the spectral properties of a singular dissipative differential operator with the help of its Cayley transform. It is shown that the Cayley transform of the dissipative differential operator is a completely non-unitary contraction with finite defect indices belonging to the class C 0. Using its characteristic function and the spectral properties of the resolvent operator, the complete spectral analysis of the dissipative differential operator is obtained. Embedding the Cayley transform to its natural unitary colligation, a Carathéodory function is obtained. Moreover, the truncated CMV matrix is established which is unitary equivalent to the Cayley transform of the dissipative differential operator. Furthermore, it is proved that the imaginary part of the inverse operator of the dissipative differential operator is a rank-one operator and the model operator of the associated dissipative integral operator is constructed as a semi-infinite triangular matrix. Using the characteristic function of the dissipative integral operator with rank-one imaginary component, associated Weyl functions are established. [ABSTRACT FROM AUTHOR]

In this paper, using the Calkin-Gorbachuk method, the general form of all maximally dissipative extensions of the minimal operator generated by the first order linear symmetric canonical type quasi-differential expression in the weighted Hilbert space of vector functions has been found. Also, the spectrum set of these extensions has been investigated. [ABSTRACT FROM AUTHOR]

In this article we consider a second-order Sturm-Liouville operator with a spectral parameter in the boundary condition on bounded time scales. We construct a selfadjoint dilation of the dissipative Sturm-Liouville operators. Using by methods of Pavlov [40,41,42], we prove the completeness of the system of eigenvectors and associated vectors of the dissipative Sturm-Liouville operators on bounded time scales.

In this paper we investigate the spectral properties of a third-order differential operator generated by a formally-symmetric differential expression and maximal dissipative boundary conditions. In fact, using the boundary value space of the minimal operator we introduce maximal selfadjoint and maximal non-selfadjoint (dissipative, accumulative) extensions. Using Solomyak's method on characteristic function of the contractive operator associated with a maximal dissipative operator we obtain some results on the root vectors of the dissipative operator. Finally, we introduce the selfadjoint dilation of the maximal dissipative operator and incoming and outgoing eigenfunctions of the dilation.2000 Mathematics Subject Classification: Primary 47A45, 47E05; Secondary 47A20

We consider a quantum system S interacting with another system S ′ and susceptible of being absorbed by S ′. The effective, dissipative dynamics of S is supposed to be generated by an abstract pseudo-Hamiltonian of the form H = H 0 + V − i C ⁎ C. The generator of the free dynamics, H 0 , is self-adjoint, V is symmetric and C is bounded. We study the scattering theory for the pair of operators (H , H 0). We establish a representation formula for the scattering matrices and identify a necessary and sufficient condition to their invertibility. This condition rests on a suitable notion of spectral singularity. Our main application is the nuclear optical model, where H is a dissipative Schrödinger operator and spectral singularities correspond to real resonances. [ABSTRACT FROM AUTHOR]

In this article we study a non-self-adjoint eigenparameter dependent singular differential 1D Hamiltonian system with the singular end points a and b in the Hilbert space LP²((a; b);ℂ²) and we consider that this 1D Hamiltonian system is in the limit-circle cases at a and b. For this purpose we use the maximal dissipative operator associated with the considered problem whose spectral analysis is suffcient for boundary value problem. Self-adjoint dilation theory of Sz.-Nagy-Foiaş developed for the dissipative operators is used. Moreover we construct incoming and outgoing spectral representations of the self-adjoint dilation. This representations allows us to determine the scattering matrix. Therefore a functional model of the dissipative operator is constructed. Moreover, a functional model of the dissipative operator is constructed and its characteristic function in terms of solutions of the corresponding Hamiltonian system is described. Therefore using the obtained results for the characteristic function theory, theorems on completeness of the system of eigenvectors and associated vectors of the dissipative operator and Hamiltonian boundary value problem have been proved. [ABSTRACT FROM AUTHOR]

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In this paper, we investigate the spectral properties of a boundary value transmission problem generated by a dynamic equation on the union of two time scales. For such an analysis we assign a suitable dynamic operator which is in limit-circle case at infinity. We also show that this operator is a simple maximal dissipative operator. Constructing the inverse operator we obtain some information about the spectrum of the dissipative operator. Moreover, using the Cayley transform of the dissipative operator we pass to the contractive operator which is of the class C0. With the aid of the minimal function we obtain more information on the dissipative operator. Finally, we investigate other properties of the contraction such that multiplicity of the contraction, unitary colligation with basic operator and CMV matrix representation associated with the contraction. [ABSTRACT FROM AUTHOR]

In this paper we prove that for an arbitrary pair { T , T } of contractions on Hilbert space with trace class difference, there exists a function ξ in L (T) (called a spectral shift function for the pair { T , T }) such that the trace formula trace( f( T ) − f( T )) = ∫ f′( ζ) ξ( ζ) dζ holds for an arbitrary operator Lipschitz function f analytic in the unit disk. [ABSTRACT FROM AUTHOR]

In this article we consider a second-order Sturm-Liouville operator with a spectral parameter in the boundary condition on bounded time scales. We construct a selfadjoint dilation of the dissipative Sturm-Liouville operators. Using by methods of Pavlov [40, 41, 42], we prove the completeness of the system of eigenvectors and associated vectors of the dissipative Sturm-Liouville operators on bounded time scales. [ABSTRACT FROM AUTHOR]

In this paper, maximal dissipative fourth order operators with equal deficiency indices are investigated. We construct a self adjoint dilation of such operators. We also construct a functional model of the maximal dissipative operator which based on the method of Pavlov and define its characteristic function. We prove theorems on the completeness of the system of eigenvalues and eigenvectors of the maximal dissipative fourth order operators.

This article concerns the generation of analytic semigroups by an operator matrix in the space $L^2(Omega)imes L^2(Omega)$. We assume that one of the diagonal elements is strongly elliptic and the other is weakly elliptic, while the sum of the non-diagonal elements is weakly elliptic.

In this note, a sufficient condition guaranteing the existence of fixed points in a nonempty, closed convex K for T+S is given, where $T:Ksubset Eo E$ is dissipative and $S:Ko E$ is condensing. This may indicate a new direction of the Krasnoselskii type fixed point theorem.

In this paper, we consider the one-dimensional Schrödinger operator on bounded time scales. We construct a space of boundary values of the minimal operator and describe all maximal dissipative, maximal accretive, self-adjoint, and other extensions of the dissipative Schrödinger operators in terms of boundary conditions. In particular, using Lidskii's theorem, we prove a theorem on completeness of the system of root vectors of the dissipative Schrödinger operators on bounded time scales. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the spectral properties of singular eigenparameter dependent dissipative problems in Weyl's limit-circle case with finite transmission conditions. In particular, these transmission conditions are assumed to be regular and singular. To analyze these problems we construct suitable Hilbert spaces with special inner products and linear operators associated with these problems. Using the equivalence of the Lax-Phillips scattering function and Sz-Nagy-Foiaş characteristic functions we prove that all root vectors of these dissipative operators are complete in Hilbert spaces. [ABSTRACT FROM AUTHOR]

In this paper we investigate the deficiency indices theory and the selfad-joint and nonselfadjoint (dissipative, accumulative) extensions of the minimal symmetric direct sum Hamiltonian operators. In particular using the equivalence of the Lax-Phillips scattering matrix and the Sz.-Nagy-Foia¸s characteristic function, we prove that all root (eigen and associated) vectors of the maximal dissipative extensions of the minimal symmetric direct sum Hamiltonian operators are complete in the Hilbert spaces. [ABSTRACT FROM AUTHOR]

In this paper we consider a second-order Sturm-Liouville operator of the form on bounded time scales. In this study, we construct a space of boundary values of the minimal operator and describe all maximal dissipative, maximal accretive, self-adjoint and other extensions of the dissipative Sturm-Liouville operators in terms of boundary conditions. Using Krein's theorem, we proved a theorem on completeness of the system of eigenvectors and associated vectors of the dissipative Sturm-Liouville operators on bounded time scales. [ABSTRACT FROM AUTHOR]