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Some bounded operators are part of this paper.Through this paper we shall obtain common properties of Some bounded operators in Г-Hilbert space. Also, introduced 2-self-adjoint operators and it’s spectrum in Г-Hilbert Space. Characterizations of these operators are also part of this literature.

We introduce the variable integral and the smooth exponent Besov spaces associated to non-negative self-adjoint operators. Then we give the equivalent norms via the Peetre type maximal functions and atomic decomposition of these spaces.

In this paper, we study the nonlocal Fokker-Planck equations (FPEs) associated with Levy-driven scalar stochastic dynamical systems. We first derive the Fokker-Planck equation for the case of multiplicative symmetric α-stable noises, by the adjoint operator method. Then we construct a finite difference scheme to simulate the nonlocal FPE on either bounded or infinite domain. It is shown that the semi-discrete scheme satisfies the discrete maximum principle and converges. Some experiments are conducted to validate the numerical method. Finally, we extend the results to the asymmetric case and present an application to the nonlinear filtering problem.

We extend the classical boundary values (0.1) g ( a ) = − W ( u a ( λ 0 , ⋅ ) , g ) ( a ) = lim x ↓ a ⁡ g ( x ) u ˆ a ( λ 0 , x ) , g [ 1 ] ( a ) = ( p g ′ ) ( a ) = W ( u ˆ a ( λ 0 , ⋅ ) , g ) ( a ) = lim x ↓ a ⁡ g ( x ) − g ( a ) u ˆ a ( λ 0 , x ) u a ( λ 0 , x ) , for regular Sturm–Liouville operators associated with differential expressions of the type τ = r ( x ) − 1 [ − ( d / d x ) p ( x ) ( d / d x ) + q ( x ) ] for a.e. x ∈ ( a , b ) ⊂ R , to the case where τ is singular on ( a , b ) ⊆ R and the associated minimal operator T m i n is bounded from below. Here u a ( λ 0 , ⋅ ) and u ˆ a ( λ 0 , ⋅ ) denote suitably normalized principal and nonprincipal solutions of τ u = λ 0 u for appropriate λ 0 ∈ R , respectively. Our approach to deriving the analog of (0.1) in the singular context employing principal and nonprincipal solutions of τ u = λ 0 u is closely related to a seminal 1992 paper by Niessen and Zettl [58] . We also recall the well-known fact that the analog of the boundary values in (0.1) characterizes all self-adjoint extensions of T m i n in the singular case in a manner familiar from the regular case. We briefly discuss the singular Weyl–Titchmarsh–Kodaira m-function and finally illustrate the theory in some detail with the examples of the Bessel, Legendre, and Kummer (resp., Laguerre) operators.

This paper presents numerical solution of second order singularly perturbed self-adjoint boundary value problems using weighted residual method of Galerkin type. First, for the given problem, the residue was computed using appropriate approximated basis function which satisfies all the boundary conditions. Then, using the chosen weighting function, integrating the weighted residue over the domain and the given differential equation is transformed to linear systems of algebraic equations. Further, these algebraic equations were solved using Galerkin method. To validate the applicability of the proposed method, two model examples have been considered and solved for different values of perturbation parameter and with different order of basis function. Additionally, convergence of error bounds has been established for the method. As it can be observed from the numerical results, the present method approximates the exact solution very well. Moreover, the present method gives better accuracy when the order of basis function is increased and it also improves the result of the methods existing in the literature. Keywords: Singularly perturbed problems, Self-adjoint problem, Galerkin method, Boundary value problems.

In this paper we study weighted composition operators W g , ψ commuting with self-adjoint weighted composition operators W f , ϕ on the Hardy space H 2 . In particular, we investigate which combina...

In the setting of various complex Banach spaces we consider the questions of when a square of a Hermitian operator is also Hermitian, and when a Hermitian operator has a Hermitian square root. We also describe the structure of Hermitian projections and Hermitian square roots of the identity operator.

Some Choi–Davis–Jensen’s type trace inequalities for convex functions are proved. Also, we generalize these inequalities for any arbitrary operator mean via operator monotone decreasing functions. In particular, we present some new order among $$\mathrm{tr}(\Phi (C)A)$$ and $$\mathrm{tr}(\Phi (C)A^{-1})$$ . New refinements of some power type trace inequalities via reverse and refinement of Young’s inequality are established. Among our results, we obtain new versions of the Holder type trace inequality for any arbitrary operator mean.

In the first part of this note we give a rather short proof of a generalization of Stenger’s lemma about the compression $$A_0$$ to $${{\mathfrak {H}}}_0$$ of a self-adjoint operator A in some Hilbert space $${{\mathfrak {H}}}={{\mathfrak {H}}}_0\oplus {{\mathfrak {H}}}_1$$ . In this situation, $$S:=A\cap A_0$$ is a symmetry in $${{\mathfrak {H}}}_0$$ with the canonical self-adjoint extension $$A_0$$ and the self-adjoint extension A with exit into $${{\mathfrak {H}}}$$ . In the second part we consider relations between the resolvents of A and $$A_0$$ like M.G. Krein’s resolvent formula, and corresponding operator models.

We provide quantitative estimates on the location of eigenvalues of one-dimensional discrete Dirac operators with complex $$\ell ^p$$ -potentials for $$1\le p \le \infty $$ . As a corollary, subsets of the essential spectrum free of embedded eigenvalues are determined for small $$\ell ^1$$ -potential. Further possible improvements and sharpness of the obtained spectral bounds are also discussed.

Let H be a separable complex Hilbert space with dim H ≥ 3, Bs(H) be the Lie algebra of all bounded self-adjoint operators on H, and let F:iBs(H)→[d,∞] with d≥ 0 be a radial unitary similarity invar...

We examine an example of a self-adjoint ordinary differential operator going back to Naimark. This operator is remarkable because its point and continuous spectra intersect. We find the spectrum and eigenfunctions of this operator “following Landau and Lifshitz,” i.e., following the rules stated in their book Quantum Mechanics and based on plausible heuristic physical arguments and analogies with linear algebra, which, to our knowledge, have not been rigorously mathematically justified so far. Then we adduce arguments in support of reasonableness of the results obtained by this method, which is conventional for physicists. The arguments are based on the analysis of the independently calculated Green function of the operator.

We establish conditions for the absolute instability of the solutions of linear and nonlinear difference equations with self-adjoint operator coefficients.

This paper deals with the multi-term generalisation of the time-fractional diffusion-wave equation for general operators with discrete spectrum, as well as for positive hypoelliptic operators, with homogeneous multi-point time-nonlocal conditions. Several examples of the settings where our nonlocal problems are applicable are given. The results for the discrete spectrum are also applied to treat the case of general homogeneous hypoelliptic left-invariant differential operators on general graded Lie groups, by using the representation theory of the group. For all these problems, we show the existence, uniqueness, and the explicit representation formulae for the solutions., 32 pages

Motivated by a problem in local differential geometry of Cauchy--Riemann (CR) structures of hypersurface type, we find a canonical form for pairs consisting of a nondegenerate Hermitian form and a self-adjoint antilinear operator, or, equivalently, consisting of a nondegenerate Hermitian form and a symmetric bilinear form. This generalizes the only previously known results on simultaneous normalization of such pairs, namely, the results of Benedetti and Cragnolini (1984) on simultaneous diagonalization of these pairs in the case where the Hermitian form is positive definite and of Hong and Horn (1986), where a criterion for simultaneous diagonalization is given., 22 pages

In this paper we show that every non-cycle finite transitive directed graph has a Cuntz-Krieger family whose WOT-closed algebra is $B(\mathcal{H})$. This is accomplished through a new construction that reduces this problem to in-degree $2$-regular graphs, which is then treated by applying the periodic Road Coloring Theorem of B\'eal and Perrin. As a consequence we show that finite disjoint unions of finite transitive directed graphs are exactly those finite graphs which admit self-adjoint free semigroupoid algebras., Comment: Added missing reference. 16 pages 2 figures

Let A be a symmetric linear relation in the Hilbert space ℌ with unequal deficiency indices n−A < n+(A). A self-adjoint linear relation $$ \tilde{A}\supset A $$ in some Hilbert space $$ \tilde{\mathrm{\mathfrak{H}}}\supset \mathrm{\mathfrak{H}} $$ is called an (exit space) extension of A. We study the compressions $$ C\left(\tilde{A}\right)={P}_{\mathrm{\mathfrak{H}}}\tilde{A}\upharpoonright \mathrm{\mathfrak{H}} $$ of extensions $$ \tilde{A}=\tilde{A^{\ast }}. $$ Our main result is a description of compressions $$ C\left(\tilde{A}\right) $$ by means of abstract boundary conditions, which are given in terms of a limit value of the Nevanlinna parameter τ(λ) from the Krein formula for generalized resolvents. We describe also all extensions $$ \tilde{A}=\tilde{A^{\ast }}. $$ of A with the maximal symmetric compression $$ C\left(\tilde{A}\right) $$ and all extensions $$ \tilde{A}=\tilde{A^{\ast }}. $$ of the second kind in the sense of M.A. Naimark. These results generalize the recent results by A. Dijksma, H. Langer and the author obtained for symmetric operators A with equal deficiency indices n+(A) = n−(A).

Let ℋ be a Krein space with fundamental symmetry J. Starting with a canonical block-operator matrix representation of J, we study the regular subspaces of ℋ. We also present block-operator matrix representations of the J-self-adjoint projections for the regular subspaces of ℋ, as well as for the regular complements of the isotropic part in a pseudo-regular subspace of ℋ.

The paper deals with quasi linear maps on two by two matrices over Banach and $$C^{\ast}$$ -algebras. Let $$\varphi:A\to X$$ be a homogeneous map between Banach algebra $$A$$ and a linear space $$X$$ . Let us take its amplification $$\psi=\varphi^{(2)}$$ to two by two matrix structure $$M_{2}(A)$$ over $$A$$ . If $$\psi(x+x^{2})=\psi(x)+\psi(x^{2})$$ for all $$x$$ , then $$\varphi$$ is linear. Ramifications for self adjoint parts of Banach $$\ast$$ -algebras and $$C^{\ast}$$ -algebras as well applications to Mackey–Gleason problem are given.

In this paper, we present an initial value technique for solving self-adjoint singularly perturbed linear boundary value problems. The original problem is reduced to its normal form and the reduced problem is converted to first order initial value problems. This replacement is significant from the computational point of view. The classical fourth order Runge-Kutta method is used to solve these initial value problems. This approach to solve singularly perturbed boundary-value problems is numerically very appealing. To demonstrate the applicability of this method, we have applied it on several linear examples with left-end boundary layer and right-end layer. From the numerical results, the method seems accurate and solutions to problems with extremely thin boundary layers are obtained.

In this paper, we prove that any $W^{2,1}$ strong solution to second-order non-divergence form elliptic equations is locally $W^{2,\infty}$ and piecewise $C^{2}$ when the leading coefficients and data are of piecewise Dini mean oscillation and the lower-order terms are bounded. Somewhat surprisingly here the interfacial boundaries are only required to be $C^{1,\text{Dini}}$. We also derive global weak-type $(1,1)$ estimates with respect to $A_{1}$ Muckenhoupt weights. The corresponding results for the adjoint operator are established. Our estimates are independent of the distance between these surfaces of discontinuity of the coefficients., 41 pages; submitted

In this paper, we construct finite-difference approximations of optimal-control problems involving non-self-adjoint convection-diffusion elliptic equations with discontinuous coefficients and states and examine the convergence of these approximations. Control functions in these problems are the coefficients of the convective-transfer operator in the equation of state and its right-hand side. We study the well-posedness of problems considered. For finite-difference approximations, we obtain estimates of the exactness by the state and the convergence rate by the functional and prove the weak convergence by the control. In addition, we regularize approximations in the Tikhonov sense.

Wave equation–based migration velocity analysis techniques aim to construct a kinematically accurate velocity model for imaging or as an initial model for full waveform inversion applications. The most popular wave equation–based migration velocity analysis method is differential semblance optimization, where the velocity model is iteratively updated by minimizing the unfocused energy in an extended image volume. However, differential semblance optimization suffers from artefacts, courtesy of the adjoint operator used in imaging, leading to poor convergence. Recent findings show that true amplitude imaging plays a significant role in enhancing the differential semblance optimization's gradient and reducing the artefacts. Here, we focus on a pseudo‐inverse operator to the horizontally extended Born as a true amplitude imaging operator. For laterally inhomogeneous models, the operator required a derivative with respect to a vertical shift. Extending the image vertically to evaluate such a derivative is costly and impractical. The inverse operator can be simplified in laterally homogeneous models. We derive an extension of the approach to apply the full inverse formula and evaluate the derivative efficiently. We simplified the implementation by applying the derivative to the imaging condition and utilize the relationship between the source and receiver wavefields and the vertical shift. Specifically, we verify the effectiveness of the approach using the Marmousi model and show that the term required for the lateral inhomogeneity treatment has a relatively small impact on the results for many cases. We then apply the operator in differential semblance optimization and invert for an accurate macro‐velocity model, which can serve as an initial velocity model for full waveform inversion.

An implicit-explicit multistep method based on the backward difference formulae (BDF) is proposed for time discretization of parabolic equations with a non-self-adjoint operator. Implicit and expli...

A boundary value problem for a second-order parabolic equation with a non-self-adjoint operator is considered. Such problems are mathematicalmodels for a number of problems, describing convective-diffusion processes of matter transfer, breakdown mechanisms of laser activity in plasma, etc. While studying the physics of breakdown, one should take into account the avalanche-like increase in the number of free electrons due to multiphoton ionization processes under the influence of optical pulses. This requires the inclusion of related boundary conditions in the problem formulation. An important circumstance that must be taken into account when developing a method for solving the problem is fulfillment of a certain conservation law for its solution. To solve the boundary value problem an approach based on the finite difference method is proposed. The approximation of the equation and boundary conditions is constructed so that the difference scheme is completely conservative. It approximates the original problem with the second order in the spatial variable and in time, and it has the second order of convergence. To effectively solve a system of linear algebraic equations at each time layer, the sweep method for complex systems in combination with the non-monotonic sweep method for systems with a tridiagonal matrix is used. Software based on computer mathematics MATLAB is developed to perform numerical calculations. It is obtained an approximate solution of an applied problem for different instants of time, as well as values of an absorption coefficient, the change in sign of which determines the transition of the plasma in a laser-active state.

A priori, a posteriori, and mixed type upper bounds for the absolute change in Ritz values of self-adjoint matrices in terms of submajorization relations are obtained. Some of our results prove recent conjectures by Knyazev, Argentati, and Zhu, which extend several known results for one dimensional subspaces to arbitrary subspaces. In addition, we improve Nakatsukasa's version of the $\tan \Theta$ theorem of Davis and Kahan. As a consequence, we obtain new quadratic a posteriori bounds for the absolute change in Ritz values., Comment: 20 pages. This is the version of the paper which appears in SIMAX, with minor changes (one of them in the title of the paper)

The main purpose of this paper is to discuss operator Jensen inequality for convex functions, without appealing to operator convexity. Several variants of this inequality will be presented, and some applications will be shown too.

Define $ A $ a unbounded self-adjoint operator on Hilbert space $ X $. Let $ \{ A_n \} $ be its resolvent approximation sequence with closed range $ \mathcal{R}(A_n) (n \in \mathrm{N}) $, that is, $ A_n (n \in \mathrm{N}) $ are all self-adjoint on Hilbert space $ X $ and \begin{equation*} \hbox{ \raise-2mm\hbox{$\textstyle s-\lim \atop \scriptstyle {n \to \infty}$}} R_��(A_n) = R_��(A)\quad (��\in \mathrm{C} \setminus \mathrm{R}), \ \textrm{where} \ R_ ��(A) := (��I-A)^{-1}. \end{equation*} The Moore-Penrose inverse $ A^\dagger_n \in \mathcal{B}(X) $ is a natural approximation to the Moore-Penrose inverse $ A^\dagger $. This paper shows that: $ A^\dagger $ is continuous and strongly converged by $ \{ A^\dagger_n \} $ if and only if $ \sup\limits_n \Vert A^\dagger_n \Vert < +\infty $. On the other hand, this result tells that arbitrary bounded computational scheme $ \{ A^\dagger_n \} $ induced by resolvent approximation $ \{ A_n \} $ is naturally instable (that is, $ \sup_n \Vert A^\dagger_n \Vert = \infty $) for any self-adjoint operator equation with non-closed range, for example, free Schr��dinger operator, Schr��dinger operator with Coulumb potential and Schr��dinger operator in model of many particles. This implies the infeasibility to globally and approximately solve non-closed range self-ajoint operator equation by resolvent approximation., Ninth Version