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Автор: Grafiati

Опубліковано: 3 червня 2023

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1

Kukushkin, M. V. "Замечание о спектральной теореме для неограниченных несамосопряженных операторов". Вестник КРАУНЦ. Физико-математические науки, № 2 (25 вересня 2022): 42–61. http://dx.doi.org/10.26117/2079-6641-2022-39-2-42-61.

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Анотація:

In this paper, we deal with non-selfadjoint operators with the compact resolvent. Having been inspired by the Lidskii idea involving a notion of convergence of a series on the root vectors of the operator in a weaker – Abel-Lidskii sense, we proceed constructing theory in the direction. The main concept of the paper is a generalization of the spectral theorem for a non-selfadjoint operator. In this way, we come to the definition of the operator function of an unbounded non-selfadjoint operator. As an application, we notice some approaches allowing us to principally broaden conditions imposed on the right-hand side of the evolution equation in the abstract Hilbert space. В данной работе, дав определение сходимости ряда по корневым векторам в смысле Абеля-Лидского, мы представляем актуальное приложение в теории эволюционных уравнений. Основной целью является подход, позволяющий нам принципиально расширить условия, налагаемые на правую часть эволюционного уравнения в абстрактном гильбертовом пространстве. Таким образом, мы приходим копределению функции неограниченного не самосопряженно- го оператора. Между тем, мы вовлекаем дополнительную концепцию, которая является обобщением спектральной теоремы для не самосопряженного оператора.

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2

Donsig, Allan P., and S. C. Power. "The Failure of Approximate Inner Conjugacy for Standard Diagonals in Regular Limit Algebras." Canadian Mathematical Bulletin 39, no. 4 (December 1, 1996): 420–28. http://dx.doi.org/10.4153/cmb-1996-050-5.

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Анотація:

AbstractAF C*-algebras contain natural AF masas which, here, we call standard diagonals. Standard diagonals are unique, in the sense that two standard diagonals in an AF C*-algebra are conjugate by an approximately inner automorphism. We show that this uniqueness fails for non-selfadjoint AF operator algebras. Precisely, we construct two standard diagonals in a particular non-selfadjoint AF operator algebra which are not conjugate by an approximately inner automorphism of the non-selfadjoint algebra.

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3

Pelloni, B., and D. A. Smith. "Spectral theory of some non-selfadjoint linear differential operators." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, no. 2154 (June 8, 2013): 20130019. http://dx.doi.org/10.1098/rspa.2013.0019.

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Анотація:

We give a characterization of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and determined by general linear boundary conditions. The boundary conditions may be such that the resulting operator is not selfadjoint. We associate the spectral properties of such an operator S with the properties of the solution of a corresponding boundary value problem for the partial differential equation ∂ t q ±i Sq =0. Namely, we are able to establish an explicit correspondence between the properties of the family of eigenfunctions of the operator, and in particular, whether this family is a basis, and the existence and properties of the unique solution of the associated boundary value problem. When such a unique solution exists, we consider its representation as a complex contour integral that is obtained using a transform method recently proposed by Fokas and one of the authors. The analyticity properties of the integrand in this representation are crucial for studying the spectral theory of the associated operator.

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4

Kukushkin, Maksim V. "On One Method of Studying Spectral Properties of Non-selfadjoint Operators." Abstract and Applied Analysis 2020 (September 1, 2020): 1–13. http://dx.doi.org/10.1155/2020/1461647.

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Анотація:

In this paper, we explore a certain class of Non-selfadjoint operators acting on a complex separable Hilbert space. We consider a perturbation of a nonselfadjoint operator by an operator that is also nonselfadjoint. Our consideration is based on known spectral properties of the real component of a nonselfadjoint compact operator. Using a technique of the sesquilinear forms theory, we establish the compactness property of the resolvent and obtain the asymptotic equivalence between the real component of the resolvent and the resolvent of the real component for some class of nonselfadjoint operators. We obtain a classification of nonselfadjoint operators in accordance with belonging their resolvent to the Schatten-von Neumann class and formulate a sufficient condition of completeness of the root vector system. Finally, we obtain an asymptotic formula for the eigenvalues.

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5

Dong, Aiju, and Chengjun Hou. "On some maximal non-selfadjoint operator algebras." Expositiones Mathematicae 30, no. 3 (2012): 309–17. http://dx.doi.org/10.1016/j.exmath.2012.08.001.

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6

Zhao, Junxi. "On invertibility in non-selfadjoint operator algebras." Proceedings of the American Mathematical Society 125, no. 1 (1997): 101–9. http://dx.doi.org/10.1090/s0002-9939-97-03645-9.

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7

Holubová, G., and P. Nečesal. "Nontrivial Fučík spectrum of one non-selfadjoint operator." Nonlinear Analysis: Theory, Methods & Applications 69, no. 9 (November 2008): 2930–41. http://dx.doi.org/10.1016/j.na.2007.08.066.

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8

Ryzhov, Vladimir. "Functional Model of a Closed Non-Selfadjoint Operator." Integral Equations and Operator Theory 60, no. 4 (March 13, 2008): 539–71. http://dx.doi.org/10.1007/s00020-008-1574-9.

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9

Hou, Chengjun, and Cuiping Wei. "Completely bounded cohomology of non-selfadjoint operator algebras." Acta Mathematica Scientia 27, no. 1 (January 2007): 25–33. http://dx.doi.org/10.1016/s0252-9602(07)60003-4.

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10

Bairamov, E., E. K. Arpat, and G. Mutlu. "Spectral properties of non-selfadjoint Sturm–Liouville operator with operator coefficient." Journal of Mathematical Analysis and Applications 456, no. 1 (December 2017): 293–306. http://dx.doi.org/10.1016/j.jmaa.2017.07.001.

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11

Saito, Kichi-Suke. "Generalized interpolation in finite maximal subdiagonal algebras." Mathematical Proceedings of the Cambridge Philosophical Society 117, no. 1 (January 1995): 11–20. http://dx.doi.org/10.1017/s0305004100072893.

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Анотація:

Non-selfadjoint operator algebras have been studied since the paper of Kadison and Singer in 1960. In [1], Arveson introduced the notion of subdiagonal algebras as the generalization of weak *-Dirichlet algebras and studied the analyticity of operator algebras. After that, we have many papers about non-selfadjoint algebras in this direction: nest algebras, CSL algebras, reflexive algebras, analytic operator algebras, analytic crossed products and so on. Since the notion of subdiagonal algebras is the analogue of weak *-Dirichlet algebras, subdiagonal algebras have many fruitful properties from the theory of function algebras. Thus, we have several attempts in this direction: Beurling–Lax–Halmos theorem for invariant subspaces, maximality, factorization theorem and so on.

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12

Kukushkin, Maksim V. "Abstract Evolution Equations with an Operator Function in the Second Term." Axioms 11, no. 9 (August 26, 2022): 434. http://dx.doi.org/10.3390/axioms11090434.

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Анотація:

In this paper, having introduced a convergence of a series on the root vectors in the Abel-Lidskii sense, we present a valuable application to the evolution equations. The main issue of the paper is an approach allowing us to principally broaden conditions imposed upon the second term of the evolution equation in the abstract Hilbert space. In this way, we come to the definition of the function of an unbounded non-selfadjoint operator. Meanwhile, considering the main issue we involve an additional concept that is a generalization of the spectral theorem for a non-selfadjoint operator.

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13

Janas, Jan, Maria Malejki, and Yaroslav Mykytyuk. "SIMILARITY AND THE POINT SPECTRUM OF SOME NON-SELFADJOINT JACOBI MATRICES." Proceedings of the Edinburgh Mathematical Society 46, no. 3 (October 2003): 575–95. http://dx.doi.org/10.1017/s0013091502000925.

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Анотація:

AbstractIn this paper spectral properties of non-selfadjoint Jacobi operators $J$ which are compact perturbations of the operator $J_0=S+\rho S^*$, where $\rho\in(0,1)$ and $S$ is the unilateral shift operator in $\ell^2$, are studied. In the case where $J-J_0$ is in the trace class, Friedrichs’s ideas are used to prove similarity of $J$ to the rank one perturbation $T$ of $J_0$, i.e. $T=J_0+(\cdot,p)e_1$. Moreover, it is shown that the perturbation is of ‘smooth type’, i.e. $p\in\ell^2$ and$$ \varlimsup_{n\rightarrow\infty}|p(n)|^{1/n}\le\rho^{1/2}. $$When $J-J_0$ is not in the trace class, the Friedrichs method does not work and the transfer matrix approach is used. Finally, the point spectrum of a special class of Jacobi matrices (introduced by Atzmon and Sodin) is investigated.AMS 2000 Mathematics subject classification: Primary 47B36. Secondary 47B37

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14

Belinschi, Serban T., Tobias Mai, and Roland Speicher. "Analytic subordination theory of operator-valued free additive convolution and the solution of a general random matrix problem." Journal für die reine und angewandte Mathematik (Crelles Journal) 2017, no. 732 (November 1, 2017): 21–53. http://dx.doi.org/10.1515/crelle-2014-0138.

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Анотація:

Abstract We develop an analytic theory of operator-valued additive free convolution in terms of subordination functions. In contrast to earlier investigations our functions are not just given by power series expansions, but are defined as Fréchet analytic functions in all of the operator upper half plane. Furthermore, we do not have to assume that our state is tracial. Combining this new analytic theory of operator-valued free convolution with Anderson’s selfadjoint version of the linearization trick we are able to provide a solution to the following general random matrix problem: Let {X_{1}^{(N)},\dots,X_{n}^{(N)}} be selfadjoint {N\times N} random matrices which are, for {N\to\infty} , asymptotically free. Consider a selfadjoint polynomial p in n non-commuting variables and let {P^{(N)}} be the element {P^{(N)}=p(X_{1}^{(N)},\dots,X_{n}^{(N)})} . How can we calculate the asymptotic eigenvalue distribution of {P^{(N)}} out of the asymptotic eigenvalue distributions of {X_{1}^{(N)},\dots,X_{n}^{(N)}} ?

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15

Esina, Anna I., and Andrei I. Shafarevich. "SEMICLASSICAL ASYMPTOTICS OF EIGENVALUES FOR NON-SELFADJOINT OPERATORS AND QUANTIZATION CONDITIONS ON RIEMANN SURFACES." Acta Polytechnica 54, no. 2 (April 30, 2014): 101–5. http://dx.doi.org/10.14311/ap.2014.54.0101.

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Анотація:

This paper reports a study of the semiclassical asymptotic behavior of the eigenvalues of some nonself-adjoint operators that are important for applications. These operators are the Schrödinger operator with complex periodic potential and the operator of induction. It turns out that the asymptotics of the spectrum can be calculated using the quantization conditions. These can be represented as the condition that the integrals of a holomorphic form over the cycles on the corresponding complex Lagrangian manifold, which is a Riemann surface of constant energy, are integers. In contrast to the real case (the Bohr–Sommerfeld–Maslov formulas), in order to calculate a chosen spectral series, it is sufficient to assume that the integral over only one of the cycles takes integer values, and different cycles determine different parts of the spectrum.

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16

Böttcher, Albrecht, and Peter Otte. "The first Szegö limit theorem for non-selfadjoint operators in the Følner algebra." MATHEMATICA SCANDINAVICA 97, no. 1 (September 1, 2005): 115. http://dx.doi.org/10.7146/math.scand.a-14967.

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Анотація:

We determine the first order asymptotics of the trace of $f(P_nUP_n)$ and the determinant $\det P_nUP_n$ for operators $U$ belonging to the Følner algebra associated with the sequence $\{P_n\}$ and satisfying an "index zero" condition. We present three different proofs of the main result in the case where $U$ is a normal operator.

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17

OLKHOVSKY, V. S., and E. RECAMI. "NEW DEVELOPMENTS IN THE STUDY OFTIMEAS A QUANTUM OBSERVABLE." International Journal of Modern Physics B 22, no. 12 (May 10, 2008): 1877–97. http://dx.doi.org/10.1142/s0217979208039162.

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Анотація:

Some results are briefly reviewed and developments are presented on the study of Time in quantum mechanics as an observable, canonically conjugate to energy. Operators for the observable Time are investigated in particle and photon quantum theory. In particular, this paper deals with the hermitian (more precisely, maximal hermitian, but non-selfadjoint) operator for Time which appears: (i) for particles, in ordinary non-relativistic quantum mechanics; and (ii) for photons (i.e., in first-quantization quantum electrodynamics).

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18

Gerhold, Malte, and Michael Skeide. "Interacting Fock spaces and subproduct systems." Infinite Dimensional Analysis, Quantum Probability and Related Topics 23, no. 03 (September 2020): 2050017. http://dx.doi.org/10.1142/s0219025720500174.

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Анотація:

We present a new more flexible definition of interacting Fock space that allows to resolve in full generality the problem of embeddability. We show that the same is not possible for regularity. We apply embeddability to classify interacting Fock spaces by squeezings. We give necessary and sufficient criteria for when an interacting Fock space has only bounded creators, giving thus rise to new classes of non-selfadjoint and selfadjoint operator algebras.

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19

Seri, Marcello, Andreas Knauf, Mirko Degli Esposti, and Thierry Jecko. "Resonances in the two-center Coulomb systems." Reviews in Mathematical Physics 28, no. 07 (August 2016): 1650016. http://dx.doi.org/10.1142/s0129055x16500161.

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Анотація:

We investigate the existence of resonances for two-center Coulomb systems with arbitrary charges in two dimensions, defining them in terms of generalized complex eigenvalues of a non-selfadjoint deformation of the two-center Schrödinger operator. We construct the resolvent kernels of the operators and prove that they can be extended analytically to the second Riemann sheet. The resonances are then analyzed by means of perturbation theory and numerical methods.

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20

Aniello, Paolo, Stefano Mancini, and Vincenzo Parisi. "A p-Adic Model of Quantum States and the p-Adic Qubit." Entropy 25, no. 1 (December 31, 2022): 86. http://dx.doi.org/10.3390/e25010086.

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Анотація:

We propose a model of a quantum N-dimensional system (quNit) based on a quadratic extension of the non-Archimedean field of p-adic numbers. As in the standard complex setting, states and observables of a p-adic quantum system are implemented by suitable linear operators in a p-adic Hilbert space. In particular, owing to the distinguishing features of p-adic probability theory, the states of an N-dimensional p-adic quantum system are implemented by p-adic statistical operators, i.e., trace-one selfadjoint operators in the carrier Hilbert space. Accordingly, we introduce the notion of selfadjoint-operator-valued measure (SOVM)—a suitable p-adic counterpart of a POVM in a complex Hilbert space—as a convenient mathematical tool describing the physical observables of a p-adic quantum system. Eventually, we focus on the special case where N=2, thus providing a description of p-adic qubit states and 2-dimensional SOVMs. The analogies—but also the non-trivial differences—with respect to the qubit states of standard quantum mechanics are then analyzed.

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21

KHOCHMAN, ABDALLAH. "RESONANCES AND SPECTRAL SHIFT FUNCTION FOR THE SEMI-CLASSICAL DIRAC OPERATOR." Reviews in Mathematical Physics 19, no. 10 (November 2007): 1071–115. http://dx.doi.org/10.1142/s0129055x0700319x.

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Анотація:

We consider the selfadjoint operator H = H0+ V, where H0is the free semi-classical Dirac operator on ℝ3. We suppose that the smooth matrix-valued potential V = O(〈x〉-δ), δ > 0, has an analytic continuation in a complex sector outside a compact. We define the resonances as the eigenvalues of the non-selfadjoint operator obtained from the Dirac operator H by complex distortions of ℝ3. We establish an upper bound O(h-3) for the number of resonances in any compact domain. For δ > 3, a representation of the derivative of the spectral shift function ξ(λ,h) related to the semi-classical resonances of H and a local trace formula are obtained. In particular, if V is an electro-magnetic potential, we deduce a Weyl-type asymptotics of the spectral shift function. As a by-product, we obtain an upper bound O(h-2) for the number of resonances close to non-critical energy levels in domains of width h and a Breit–Wigner approximation formula for the derivative of the spectral shift function.

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22

Marcoux, L. W., and A. R. Sourour. "Relative annihilators and relative commutants in non-selfadjoint operator algebras." Journal of the London Mathematical Society 85, no. 2 (January 31, 2012): 549–70. http://dx.doi.org/10.1112/jlms/jdr065.

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23

Tunca, Gülen Başcanbaz, and Elgiz Bairamov. "Discrete spectrum and principal functions of non-selfadjoint differential operator." Czechoslovak Mathematical Journal 49, no. 4 (December 1999): 689–700. http://dx.doi.org/10.1023/a:1022488631049.

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24

Davidson, K. R., R. H. Levene, L. W. Marcoux, and H. Radjavi. "On the topological stable rank of non-selfadjoint operator algebras." Mathematische Annalen 341, no. 2 (November 8, 2007): 239–53. http://dx.doi.org/10.1007/s00208-007-0180-5.

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25

Davidson, K. R., R. H. Levene, L. W. Marcoux, and H. Radjavi. "On the topological stable rank of non-selfadjoint operator algebras." Mathematische Annalen 341, no. 4 (April 1, 2008): 963–64. http://dx.doi.org/10.1007/s00208-008-0229-0.

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26

WORONOWICZ, S. L. "QUANTUM EXPONENTIAL FUNCTION." Reviews in Mathematical Physics 12, no. 06 (June 2000): 873–920. http://dx.doi.org/10.1142/s0129055x00000344.

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Анотація:

A special function playing an essential role in the construction of quantum "ax+b"-group is introduced and investigated. The function is denoted by Fℏ(r,ϱ), where ℏ is a constant such that the deformation parameter q2=e-iℏ. The first variable r runs over non-zero real numbers; the range of the second one depends on the sign of r: ϱ=0 for r>0 and ϱ=±1 for r<0. After the holomorphic continuation the function satisfies the functional equation [Formula: see text] The name "exponential function" is justified by the formula: [Formula: see text] where R, S are selfadjoint operators satisfying certain commutation relations and [R+S] is a selfadjoint extension of the sum R+S determined by operators ρ and σ appearing in the formula. This formula will be used in a forthcoming paper to construct a unitary operator W satisfying the pentagonal equation of Baaj and Skandalis.

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27

Kukushkin, Maksim V. "Natural Lacunae Method and Schatten–Von Neumann Classes of the Convergence Exponent." Mathematics 10, no. 13 (June 26, 2022): 2237. http://dx.doi.org/10.3390/math10132237.

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Анотація:

Our first aim is to clarify the results obtained by Lidskii devoted to the decomposition on the root vector system of the non-selfadjoint operator. We use a technique of the entire function theory and introduce a so-called Schatten–von Neumann class of the convergence exponent. Considering strictly accretive operators satisfying special conditions formulated in terms of the norm, we construct a sequence of contours of the power type that contrasts the results by Lidskii, where a sequence of contours of the exponential type was used.

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28

MARCOUX, L. W., and A. R. SOUROUR. "CONJUGATION-INVARIANT SUBSPACES AND LIE IDEALS IN NON-SELFADJOINT OPERATOR ALGEBRAS." Journal of the London Mathematical Society 65, no. 02 (April 2002): 493–512. http://dx.doi.org/10.1112/s002461070100299x.

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29

Shakhmurov, Veli B. "Maximal regular boundary value problems in Banach-valued function spaces and applications." International Journal of Mathematics and Mathematical Sciences 2006 (2006): 1–26. http://dx.doi.org/10.1155/ijmms/2006/92134.

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Анотація:

The nonlocal boundary value problems for differential operator equations of second order with dependent coefficients are studied. The principal parts of the differential operators generated by these problems are non-selfadjoint. Several conditions for the maximal regularity and the Fredholmness in Banach-valuedLp-spaces of these problems are given. By using these results, the maximal regularity of parabolic nonlocal initial boundary value problems is shown. In applications, the nonlocal boundary value problems for quasi elliptic partial differential equations, nonlocal initial boundary value problems for parabolic equations, and their systems on cylindrical domain are studied.

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30

Gil’, Michael. "Norm estimates for functions of non-selfadjoint operators nonregular on the convex hull of the spectrum." Demonstratio Mathematica 50, no. 1 (October 26, 2017): 267–77. http://dx.doi.org/10.1515/dema-2017-0026.

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Анотація:

Abstract We consider a bounded linear operator A in a Hilbert space with a Hilbert-Schmidt Hermitian component (A − A*)/2i. A sharp norm estimate is established for functions of A nonregular on the convex hull of the spectrum. The logarithm, fractional powers and meromorphic functions of operators are examples of such functions. Our results are based on the existence of a sequence An (n = 1, 2, ...) of finite dimensional operators strongly converging to A, whose spectra belongs to the spectrum of A. Besides, it is shown that the resolvents and holomorphic functions of An strongly converge to the resolvent and corresponding function of A.

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31

MUTLU, Gökhan, and Esra KIR ARPAT. "Spectral Analysis of Non-selfadjoint Second Order Difference Equation with Operator Coefficient." Sakarya University Journal of Science 24, no. 3 (June 1, 2020): 501–7. http://dx.doi.org/10.16984/saufenbilder.627496.

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32

Lomonosov, Victor. "The Bishop–Phelps Theorem Fails for Uniform Non-selfadjoint Dual Operator Algebras." Journal of Functional Analysis 185, no. 1 (September 2001): 214–19. http://dx.doi.org/10.1006/jfan.2001.3754.

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33

Dragomir, Silvestru Sever. "Some integral inequalities for operator monotonic functions on Hilbert spaces." Special Matrices 8, no. 1 (July 22, 2020): 172–80. http://dx.doi.org/10.1515/spma-2020-0108.

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Анотація:

AbstractLet f be an operator monotonic function on I and A, B∈I (H), the class of all selfadjoint operators with spectra in I. Assume that p : [0.1], →ℝ is non-decreasing on [0, 1]. In this paper we obtained, among others, that for A ≤ B and f an operator monotonic function on I,\matrix{0 \hfill & { \le \int\limits_0^1 {p\left( t \right)f\left( {\left( {1 - t} \right)A + tB} \right)dt - \int\limits_0^1 {p\left( t \right)dt\int\limits_0^1 {f\left( {\left( {1 - t} \right)A + tB} \right)dt} } } } \hfill \cr {} \hfill & { \le {1 \over 4}\left[ {p\left( 1 \right) - p\left( 0 \right)} \right]\left[ {f\left( B \right) - f\left( A \right)} \right]} \hfill \cr }in the operator order.Several other similar inequalities for either p or f is differentiable, are also provided. Applications for power function and logarithm are given as well.

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34

Yokuş, Nihal. "Principal Functions of Non-Selfadjoint Sturm-Liouville Problems with Eigenvalue-Dependent Boundary Conditions." Abstract and Applied Analysis 2011 (2011): 1–12. http://dx.doi.org/10.1155/2011/358912.

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Анотація:

We consider the operator generated in by the differential expression , and the boundary condition , where is a complex-valued function and , with . In this paper we obtain the properties of the principal functions corresponding to the spectral singularities of .

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35

Gavrilyuk, Ivan P., and Volodymyr L. Makarov. "Stability and Regularization of Difference Schemes in Banach and Hilbert Spaces." Computational Methods in Applied Mathematics 13, no. 2 (April 1, 2013): 139–60. http://dx.doi.org/10.1515/cmam-2013-0005.

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Abstract. The necessary and sufficient conditions for stability of abstract difference schemes in Hilbert and Banach spaces are formulated. Contrary to known stability results we give stability conditions for schemes with non-self-adjoint operator coefficients in a Hilbert space and with strongly positive operator coefficients in a Banach space. It is shown that the parameters of the sectorial spectral domain play the crucial role. As an application we consider the Richardson iteration scheme for an operator equation in a Banach space, in particulary the Richardson iteration with precondition for a finite element scheme for a non-selfadjoint operator. The theoretical results are also the basis when using the regularization principle to construct stable difference schemes. For this aim we start from some simple scheme (even unstable) and derive stable schemes by perturbing the initial operator coefficients and by taking into account the stability conditions. Our approach is also valid for schemes with unbounded operator coefficients.

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36

Torshage, Axel. "Enclosure of the numerical range and resolvent estimates of non-selfadjoint operator functions." Journal of Spectral Theory 10, no. 2 (February 27, 2020): 379–413. http://dx.doi.org/10.4171/jst/297.

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37

Olgun, Murat, Turhan Koprubasi, and Yelda Aygar. "Principal Functions of Non-Selfadjoint Difference Operator with Spectral Parameter in Boundary Conditions." Abstract and Applied Analysis 2011 (2011): 1–10. http://dx.doi.org/10.1155/2011/608329.

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We investigate the principal functions corresponding to the eigenvalues and the spectral singularities of the boundary value problem (BVP) , and , where and are complex sequences, is an eigenparameter, and , for , 1.

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38

Drozdov, Aleksey. "Explicit stability conditions for stochastic integro-differential equations with non-selfadjoint operator coefficients." Stochastic Analysis and Applications 17, no. 1 (January 1999): 23–41. http://dx.doi.org/10.1080/07362999908809586.

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39

Bui, The Anh, and Piero D’Ancona. "Generalized Hardy operators." Nonlinearity 36, no. 1 (November 29, 2022): 171–98. http://dx.doi.org/10.1088/1361-6544/ac9c81.

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Abstract Consider the operator on L 2 ( R d ) L a = ( − Δ ) α / 2 + a | x | − α with 0 < α < m i n { 2 , d } . Under the condition a ⩾ − 2 α Γ ( ( d + α ) / 4 ) 2 Γ ( ( d − α ) / 4 ) 2 the operator is non negative and selfadjoint. We prove that fractional powers L a s / 2 for s ∈ (0, 2] satisfy the estimates L a s / 2 f L p ≲ ( − Δ ) α s / 4 f L p , ( − Δ ) s / 2 f L p ≲ L a α s / 4 f L p for suitable ranges of p. Our result fills the remaining gap in earlier results from Killip et al (2018 Math. Z. 288 1273–98); Merz (2021 Math. Z. 299 101–21); Frank et al (Int. Math. Res. Not. 2021 2284–303). The method of proof is based on square function estimates for operators whose heat kernel has a weak decay.

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40

Król, M., M. Kutniv, and O. Pazdriy. "Exact difference scheme for system nonlinear ODEs of second order on semi-infinite intervals." Mathematical Modeling and Computing 1, no. 1 (2014): 31–44. http://dx.doi.org/10.23939/mmc2014.01.031.

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We constructed and substantiated the exact three-point differential scheme for the numerical solution of boundary value problems on a semi-infinite interval for systems of second order nonlinear ordinary differential equations with non-selfadjoint operator. The existence and uniqueness of the solution of the exact three-point difference scheme and the convergence of the method of successive approximations for its findings are proved under the conditions of existence and uniqueness of the solution of the boundary value problem.

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41

Brown, Malcolm, Marco Marletta, Serguei Naboko, and Ian Wood. "Boundary triplets and M -functions for non-selfadjoint operators, with applications to elliptic PDEs and block operator matrices." Journal of the London Mathematical Society 77, no. 3 (March 20, 2008): 700–718. http://dx.doi.org/10.1112/jlms/jdn006.

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42

Kir Arpat, Esra. "An eigenfunction expansion of the non-selfadjoint Sturm–Liouville operator with a singular potential." Journal of Mathematical Chemistry 51, no. 8 (June 19, 2013): 2196–213. http://dx.doi.org/10.1007/s10910-013-0208-x.

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43

Buterin, S. A. "On inverse spectral problem for non-selfadjoint Sturm–Liouville operator on a finite interval." Journal of Mathematical Analysis and Applications 335, no. 1 (November 2007): 739–49. http://dx.doi.org/10.1016/j.jmaa.2007.02.012.

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44

Engström, Christian, and Axel Torshage. "Enclosure of the Numerical Range of a Class of Non-selfadjoint Rational Operator Functions." Integral Equations and Operator Theory 88, no. 2 (May 23, 2017): 151–84. http://dx.doi.org/10.1007/s00020-017-2378-6.

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45

Intissar, Abdelkader. "Diagonalization of Non-selfadjoint Analytic Semigroups and Application to the Shape Memory Alloys Operator." Journal of Mathematical Analysis and Applications 257, no. 1 (May 2001): 1–20. http://dx.doi.org/10.1006/jmaa.2000.7148.

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46

Koprubasi, Turhan, and Ram Mohapatra. "Spectral analysis of discrete dirac equation with generalized eigenparameter in boundary condition." Filomat 33, no. 18 (2019): 6039–54. http://dx.doi.org/10.2298/fil1918039k.

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Let L denote the discrete Dirac operator generated in ?2 (N,C2) by the non-selfadjoint difference operators of first order (an+1y(2)n+1 + bny(2)n + pny(1)n = ?y(1)n, an-1y(1)n-1 + bny(1)n + qny(2)n = ?y(2)n, n ? N, (0.1) with boundary condition Xp k=0 (y(2)1?k + y(1)0 ?k)?k=0, (0.2) where (an), (bn), (pn) and (qn), n ? N are complex sequences, ?i; ?i ? C, i = 0, 1, 2,..., p and ? is a eigenparameter. We discuss the spectral properties of L and we investigate the properties of the spectrum and the principal vectors corresponding to the spectral singularities of L, if ?? n=1 |n|(|1-an| + |1+bn| + |pn| + |qn|) < ? holds.

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47

NAKAZAWA, Hideo. "The Principle of Limiting Absorption for the Non-Selfadjoint Schrödinger Operator with Energy Dependent Potential." Tokyo Journal of Mathematics 23, no. 2 (December 2000): 519–36. http://dx.doi.org/10.3836/tjm/1255958686.

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48

Ciaglia, Florio M., Fabio Di Cosmo, Alberto Ibort, Giuseppe Marmo, Luca Schiavone, and Alessandro Zampini. "Causality in Schwinger’s Picture of Quantum Mechanics." Entropy 24, no. 1 (January 1, 2022): 75. http://dx.doi.org/10.3390/e24010075.

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This paper begins the study of the relation between causality and quantum mechanics, taking advantage of the groupoidal description of quantum mechanical systems inspired by Schwinger’s picture of quantum mechanics. After identifying causal structures on groupoids with a particular class of subcategories, called causal categories accordingly, it will be shown that causal structures can be recovered from a particular class of non-selfadjoint class of algebras, known as triangular operator algebras, contained in the von Neumann algebra of the groupoid of the quantum system. As a consequence of this, Sorkin’s incidence theorem will be proved and some illustrative examples will be discussed.

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49

Liu, Yixuan, Guoliang Shi, and Jun Yan. "An inverse problem for non-selfadjoint Sturm–Liouville operator with discontinuity conditions inside a finite interval." Inverse Problems in Science and Engineering 27, no. 3 (May 16, 2018): 407–21. http://dx.doi.org/10.1080/17415977.2018.1470624.

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50

Faierman, M. "Eigenvalue Asymptotics for the Non-Selfadjoint Operator Induced by a Parameter-Elliptic Multi-Order Boundary Problem." Integral Equations and Operator Theory 74, no. 1 (June 26, 2012): 25–42. http://dx.doi.org/10.1007/s00020-012-1977-5.

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