Abstract: In this paper, we study the structural properties of optimal control of spatially distributed systems. Such systems consist of an infinite collection of possibly heterogeneous linear control systems that are spatially interconnected via certain distant-dependent coupling functions over arbitrary graphs. We study the structural properties of optimal control problems with infinite-horizon linear quadratic criteria, by analyzing the spatial structure of the solution to the corresponding operator Lyapunov and Riccati equations. The key idea of the paper is the introduction of a special class of operators called spatially decaying (SD). These operators are a generalization of translation invariant operators used in the study of spatially invariant systems. We prove that given a control system with a state-space representation consisting of SD operators, the solution of operator Lyapunov and Riccati equations are SD. Furthermore, we show that the kernel of the optimal state feedback for each subsystem decays in the spatial domain, with the type of decay (e.g., exponential, polynomial or logarithmic) depending on the type of coupling between subsystems.

Abstract: On the scientific work of M. Sh. Birman in 1998-2007 by M. Solomyak and T. Suslina Continuation of the list of publications of M. Sh. Birman by T. Suslina and D. Yafaev Perturbations of the continuous spectrum of a singular elliptic operator by varying the boundary and the boundary conditions by M. Sh. Birman Asymptotic behavior of the eigenfunctions of many-particle Schrodinger operator. I. One-dimensional particles by V. S. Buslaev and S. B. Levin Spectral asymptotics of the Maxwell operator on Lipschitz manifolds with boundary by M. N. Demchenko and N. D. Filonov Spectral inequalities for Schrodinger operators with surface potentials by R. L. Frank and A. Laptev On the spectrum of the Dirichlet Laplacian in a narrow infinite strip by L. Friedlander and M. Solomyak Hardy inequalities for simply connected planar domains by A. Laptev and A. V. Sobolev Lyapunov functions of periodic matrix-valued Jacobi operators by E. Korotyaev and A. Kutsenko The spectral flow, the Fredholm index, and the spectral shift function by A. Pushnitski On the spectrum of a translationally invariant Pauli operator by G. Raikov Discrete spectrum distribution of the Landau operator perturbed by an expanding electric potential by G. Rozenblum and A. V. Sobolev. On the comparison of the Dirichlet and Neumann counting functions by Y. Safarov Absolutely continuous spectrum of multi-dimensional Schrodinger operators with slowly decaying potentials by O. Safronov On discrete spectrum of the perturbed periodic magnetic Schrodinger operator with degenerate lower edge of the spectrum by R. Shterenberg Homogenization of periodic second order differential operators including first order terms by T. A. Suslina Improved Berezin-Li-Yau inequalities with a remainder term by T. Weidl Spectral and scattering theory of fourth order differential operators by D. R. Yafaev.

Abstract: In 1952 Lee and Yang proposed the program of analyzing phase transitions in terms of zeros of partition functions. Linear operators preserving non-vanishing properties are essential in this program and various contexts in complex analysis, probability theory, combinatorics, and matrix theory. We characterize all linear operators on finite or infinite-dimensional spaces of multivariate polynomials preserving the property of being non-vanishing whenever the variables are in prescribed open circular domains. In particular, this solves the higher dimensional counterpart of a long-standing classification problem originating from classical works of Hermite, Laguerre, Hurwitz and P\'olya-Schur on univariate polynomials with such properties.

Abstract: We solve various master equations to obtain density operators' infinite operator-sum representation via a new approach, i.e., by virtue of the thermo-entangled state representation that has a fictitious mode as a counterpart mode of the system mode. The corresponding Kraus operators from the point of view of quantum channel are derived, whose normalization conditions are proved. Miscellaneous characters possessed by different quantum channels, such as decoherence, phase diffusion, damping, and amplification, can be shown explicitly in the entangled state representation of the density operators. Squeezing transformation is applied to the density operator for decoherence to generate a master equation for describing the phase sensitive process. Partial trace method for deriving new density operators is also introduced. Throughout our discussion, the technique of integration within an ordered product (IWOP) of operators is fully used.

Abstract: We study the boundedness and compactness of the products of composition operators and integral type operators from H ∞ to the Bloch space on the unit disk.

Abstract: We consider differences of composition operators between given weighted Banach spaces or Hv0 of analytic functions with weighted sup-norms and give estimates for the distance of these differences to the space of compact operators. We also study boundedness and compactness of the operators. Some examples illustrate our results.

Abstract: Given a symmetric, semi-bounded, second order elliptic differential operator on a bounded domain with $C^{1,1}$ boundary, we provide a Krein-type formula for the resolvent difference between its Friedrichs extension and an arbitrary self-adjoint one.

Abstract: We study differences of weighted composition operators between weighted Banach spaces H ν ∞ of analytic functions with weighted sup-norms and give an expression for the essential norm of these differences. We apply our result to estimate the essential norm of differences of composition operators acting on Bloch-type spaces.

Abstract: The authors establish λ-central BMO estimates for commutators of singular integral operators with rough kernels on central Morrey spaces Moreover, the boundedness of a class of multisublinear operators on the product of central Morrey spaces is discussed As its special cases, the corresponding results of multilinear Calderon-Zygmund operators and multilinear fractional integral operators can be deduced, respectively

Abstract: The paper is devoted to the study of a multi-dimensional semi-linear elliptic system of equations in an unbounded cylinder with a linear dependence of the components of the non-linearity vector. Problems of this type describe reaction-diffusion waves with the Lewis number different from 1. Due to this property of non-linearity, the corresponding operator does not possess the Fredholm property. Therefore the usual solvability conditions and the conventional methods of nonlinear analysis cannot be directly applied. We reduce the elliptic problem to an integro-differential system of equations and show how to apply the implicit function theorem to it. It allows us to prove existence of waves for the Lewis number different from 1 and sufficiently close to it. Next we prove the Fredholm property of integro-differential operators, their properness, and construct the topological degree. The latter is used to study bifurcations of solutions.

Abstract: The higher Sugawara operators acting on the Verma modules over the affine Kac-Moody algebra at the critical level are related to the higher Hamiltonians of the Gaudin model due to work of Feigin, Frenkel and Reshetikhin. An explicit construction of the higher Hamiltonians in the case of gl_n was given recently by Talalaev. We propose a new approach to these results from the viewpoint of the vertex algebra theory by proving directly the formulas for the higher order Sugawara operators. The eigenvalues of the operators in the Wakimoto modules of critical level are also calculated.

Abstract: Operators on function spaces acting by composition to the right with a fixed selfmap φ of some set are called composition operators of symbol φ. A weighted composition operator is an operator equal to a composition operator followed by a multiplication operator. We summarize the basic properties of bounded and compact weighted composition operators on the Hilbert Hardy space on the open unit disk and use them to study composition operators on Hardy–Smirnov spaces.

Abstract: A construction of Triebel-Lizorkin type spaces associated with flexible decompositions of the frequency space ℝ d is considered. The class of admissible frequency decompositions is generated by a one parameter group of (anisotropic) dilations on ℝ d and a suitable decomposition function. The decomposition function governs the structure of the decomposition of the frequency space, and for a very particular choice of decomposition function the spaces are reduced to classical (anisotropic) Triebel-Lizorkin spaces. An explicit atomic decomposition of the Triebel-Lizorkin type spaces is provided, and their interpolation properties are studied. As the main application, we consider Hormander type classes of pseudo-differential operators adapted to the anisotropy and boundedness of such operators between corresponding Triebel-Lizorkin type spaces is proved.

Abstract: Here we give some approximation theorems concerning pointwise convergence and rate of pointwise convergence for non-convolution type linear operators of the form: with kernels satisfying some general homogeneity assumptions. Here Λ is a non-empty set of indices with a topology and λ0 an accumulation point of Λ in this topology.

Abstract: Some consequences of promoting the object of noncommutativity ${\ensuremath{\theta}}^{ij}$ to an operator in Hilbert space are explored. Its canonical conjugate momentum is also introduced. Consequently, a consistent algebra involving the enlarged set of canonical operators is obtained, which permits us to construct theories that are dynamically invariant under the action of the rotation group. In this framework it is also possible to give dynamics to the noncommutativity operator sector, resulting in new features.

Abstract: In this paper we present two new generalizations of Kantorovich operators based on q-calculus. With the help of Bohman-Korovkin type theorem we obtain some statistical approximation properties for these operators. Also, by using the modulus of continuity, the statistical rate of convergence is established.

Abstract: We consider the approximation of trigonometric operator functions that arise in the numerical solution of wave equations by trigonometric integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behavior if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we propose and analyze a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators. In contrast to standard Krylov methods, the convergence will be independent of the norm of the operator and thus of its spatial discretization. We will discuss efficient implementations for finite element discretizations and illustrate our analysis with numerical experiments.

Abstract: The survey is devoted to spectral stability problems for uniformly elliptic differential operators under the variation of the domain and to the accompanying estimates for the difference of the eigenvalues. Two approaches to the problem are discussed in detail. In the first one it is assumed that the domain is transformed by means of a transformation of a certain class, and the spectral stability with respect to this transformation is investigated. The second approach is based on the notion of a transition operator and allows direct comparison of the eigenvalues on domains which are close in that or another sense.