Bounded Operator Coefficients Research Articles

Second regularized trace of a differential operator of Sturm–Liouville type

We consider a Sturm–Liouville‐type differential operator defined on a separable Hilbert space with bounded operator coefficient and periodic boundary conditions given on a finite interval. We obtain an explicit formula which we call the second regularized trace formula of the mentioned operator.

О комплексных операторных функциях комплексного операторного переменного

We consider a family of complex operator functions whose domain and range of values are included in the real Banach algebra of bounded linear complex operators acting in the Banach space of complex vectors over the field of real numbers. It is shown that the study of a function from this family can be reduced to the study of a pair of real operator functions of two real operator variables. The main elementary functions of this family are considered: power function; exponent; trigonometric functions of sine, cosine, tangent, cotangent, secant, cosecant; hyperbolic sine, cosine, tangent, cotangent, secant, cosecant; the main property of the exponent is proved. A complex Euler operator formula is obtained. Relations that express sine and cosine in terms of the exponent are found. For the trigonometric functions of sine and cosine, addition formulas are justified. The periodicity of the exponent and trigonometric functions of sine, cosine, tangent, cotangent is proved; reduction formulas for these functions are provided. The main complex operator trigonometric identity is obtained. Equalities connecting trigonometric and hyperbolic functions are found. The main complex operator hyperbolic identity is established. For the hyperbolic functions of sine and cosine, addition formulas are indicated. As an example of an elementary function from the family of complex operator functions under consideration, a rational function is considered, a special case of which is the characteristic operator polynomial of a linear homogeneous differential equation of n-th order with constant bounded operator coefficients in a real Banach space.

Assignment of the upper Bohl exponent for linear periodic control systems in Hilbert spaces

A linear continuous-time control system with time-varying linear bounded operator coefficients in a Hilbert space is considered. The controller in the system has the form of linear state feedback with a time-varying linear bounded gain operator function. We study the problem of arbitrary assignment of the upper Bohl exponent by state feedback control. The property of exact controllability for the open-loop system is studied, some necessary and sufficient conditions are obtained for exact controllability. For periodic systems, it is proved that if the open-loop system is exactly controllable then one can shift the upper Bohl exponent of the closed-loop system by any pregiven number with respect to the upper Bohl exponent of the free system by means of linear state feedback with a periodic gain operator function. This implies arbitrary assignability of the upper Bohl exponent by linear state feedback. Finally, an illustrative example is presented.

Stationary solutions of a second-order differential equation with operator coefficients

Necessary and sufficient conditions are given for the existence of a unique stationary solution to the second-order linear differential equation with bounded operator coefficients, perturbed by a stationary process. In the case when the corresponding “algebraic” operator equation has separated roots, the new representation of the stationary solution of the considered differential equation is obtained.

On the regularized trace of a differential operator of Sturm-Liouville type

In this work, we study a spectral problem for the abstract Sturm-Liouville operator with a bounded operator coefficient V(t) and with periodic boundary conditions on the interval [0, ?], and we present a regularized trace formula for this operator.

A Second Regularized Trace Formula for a Fourth Order Differential Operator

In applications, many states given for a system can be expressed by orthonormal elements, called “state elements”, taken in a separable Hilbert space (called “state space”). The exact nature of the Hilbert space depends on the system; for example, the state space for position and momentum states is the space of square-integrable functions. The symmetries of a quantum system can be represented by a class of unitary operators that act in the Hilbert space. The operators called ladder operators have the effect of lowering or raising the energy of the state. In this paper, we study the spectral properties of a self-adjoint, fourth-order differential operator with a bounded operator coefficient and establish a second regularized trace formula for this operator.

On Higher Regularized Traces of a Differential Operator with Bounded Operator Coefficient Given in a Finite Interval

In this work, we find a higher regularized trace formula for a regular Sturm–Liouville differential operator with operator coefficient.

On Assignment of the Upper Bohl Exponent for Linear Time-Invariant Control Systems in a Hilbert Space by State Feedback

We consider a linear continuous-time control system with time-invariant linear bounded operator coefficients in a Hilbert space. The controller in the system has the form of linear state feedback with a time-varying linear bounded gain operator function. We study the problem of arbitrary assignment for the upper Bohl exponent by state feedback control. We prove that if the open-loop system is exactly controllable then one can shift the upper Bohl exponent of the closed-loop system by any pregiven number with respect to the upper Bohl exponent of the free system. This implies arbitrary assignability of the upper Bohl exponent by linear state feedback. Finally, an illustrative example is presented.

Об основном свойстве комплексной операторной экспоненциальной функции комплексного операторного аргумента

Operator functions e^A, sin B, cos B of the operator argument from the Banach algebra of bounded linear operators acting from E to E are considered in the Banach space E . For trigonometric operator functions sin B, cos B, formulas for the sine and cosine of the sum of the arguments are derived that are similar to the scalar case. In the proof of these formulas, the composition of ranges with operator terms in the form of Cauchy is used. The basic operator trigonometric identity is given. For a complex operator exponential function e^Z of an operator argument Z from the Banach algebra of complex operators, using the formulas for the cosine and sine of the sum, the main property of the exponential function is proved. Operator functions e^At , sin Bt, cos Bt, e^Zt of a real argument t∈(-∞;∞) are considered. The facts stated for the operator functions of the operator argument are transferred to these functions. In particular, the group property of the operator exponent e^Zt is given. The rule of differentiation of the function e^Zt is indicated. It is noted that the operator functions of the real argument t listed above are used in constructing a general solution of a linear n th order differential equation with constant bounded operator coefficients in a Banach space.

Об общем решении линейного однородного дифференциального уравнения в банаховом пространстве в случае комплексных характеристических операторов

A linear inhomogeneous differential equation (LIDE) of the n th order with constant bounded operator coefficients is studied in Banach space. Finding a general solution of LIDE is reduced to the construction of a general solution to the corresponding linear homogeneous differential equation (LHDE). Characteristic operator equation for LHDE is considered in the Banach algebra of complex operators. In the general case, when both real and complex operator roots are among the roots of the characteristic operator equation, the n -parametric family of solutions to LHDE is indicated. Operator functions eAt ; sinBt ; cosBt of real argument t ∈ [0;∞) are used when building this family. The conditions under which this family of solutions form a general solution to LHDE are clarified. In the case when the characteristic operator equation has simple real operator roots and simple pure imaginary operator roots, a specific form of such conditions is indicated. In particular, these roots must commute with LHDE operator coefficients. In addition, they must commute with each other. In proving the corresponding assertion, the Cramer operator-vector rule for solving systems of linear vector equations in a Banach space is applied

О БАНАХОВОЙ АЛГЕБРЕ КОМПЛЕКСНЫХ ОПЕРАТОРОВ

The Banach algebra of complex operators that are used in the study of linear differential equations with constant bounded operator coefficients in a Banach space is consiuder.

PTH Power Integrable Solutions of an Operator-Differential Equation

We study the property of boundedness (or of the pth power integrability) of the solutions of a differential equation with bounded operator coefficient in a Banach space for a special class of “input” functions.

The regularized trace formula for a fourth order differential operator given in a finite interval

In this work, a regularized trace formula for a differential operator of fourth order with bounded operator coefficient is found.

The second regularized trace of a self adjoint differential operator given in a finite interval with bounded operator coefficient

In this work a formula for the second regularized trace of a self adjoint differential operator which is given in a finite interval and with bounded operator coefficient was found.

The third regularized trace formula for the Sturm–Liouville operator with the bounded operator coefficient

In this work, we obtain a regularized trace formula for the Sturm–Liouville operator with a bounded operator coefficient given in the finite interval.

“On Invertible Contractions of Quotients Generated by a Differential Expression and by a Nonnegative Operator Function”

In the present paper, we describe invertible contractions of the maximal quotient generated by a differential expression with bounded operator coefficients and by a nonnegative operator function. We show that the operators inverse to such contractions are integral operators and prove a criterion for such operators to be holomorphic. Using the results obtained, we describe the generalized resolvents of symmetric quotients.

On the General Solution of a Linear nth-Order Differential Equation with Constant Bounded Operator Coefficients in a Banach Space

On the General Solution of a Linear nth-Order Differential Equation with Constant Bounded Operator Coefficients in a Banach Space

Investigation of One Linear Differential Equation by Using Generalized Functions with Values in a Banach Space

We present a generalization of some facts of the theory of generalized functions of slow growth to the case of operator-valued test functions. We propose a construction of regular generalized functions with values in a Banach space. The results obtained are used for the description of slowly increasing solutions of linear homogeneous differential equations with shifted arguments and bounded operator coefficients in a Banach space.

Small stabilizing perturbations of a vector second-order Euler equation with bounded operator coefficients

Small stabilizing perturbations of a vector second-order Euler equation with bounded operator coefficients

A Method for Solving Initial Value Problems for Linear Differential Equations in Hilbert Space Based on The Cayley Transform

A first-order differential equation with a bounded operator coefficient in Hilbert space is considered. In case the coefficient is constant we establish a one-to-one correspondence between the continuous initial value problem and some discrete initial value problem. Using the Cayley transform we give explicit formulas for their solutions as well as formulas connecting corresponding continuous and discrete semigroups. On the basis of these formulas we propose a numerical algorithm for solving initial value problems with a bounded constant operator coefficient which has an exponential rate of convergence. If the operator coefficient is variable we use its piecewise constant approximation and the previously cited algorithm. Error estimates are given.