scholarly journals On an infinite-dimensional differential equation in vector distribution with discontinuous regular functions in a right hand side

1996 ◽  
Vol 9 (1) ◽  
pp. 1-10 ◽  
Author(s):  
Michael V. Basin
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Unbounded Operator ◽  
Infinite Dimensional ◽  
Regular Functions ◽  
Right Hand ◽  
Equivalent Equation ◽  
Vector Distribution ◽  
Existence Conditions ◽  
Necessary And Sufficient

An infinite-dimensional differential equation in vector distribution in a Hilbert space is studied in case of an unbounded operator and discontinuous regular functions in a right-hand side. A unique solution (vibrosolution) is defined for such an equation, and the necessary and sufficient existence conditions for a vibrosolution are proved. An equivalent equation with a measure, which enables us to directly compute jumps of a vibrosolution at discontinuity points of a distribution function, is also obtained. The application of the obtained results to control theory is discussed in the conclusion.

scholarly journals On the Almost Periodic Solutions of Differential Equations on Hilbert Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Nguyen Thanh Lan
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
Hilbert Spaces ◽  
Sufficient Conditions ◽  
Almost Periodic Solution ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Necessary And Sufficient

For the differential equation , on a Hilbert space , we find the necessary and sufficient conditions that the above-mentioned equation has a unique almost periodic solution. Some applications are also given.

scholarly journals A new proof of Donoghue's interpolation theorem

2004 ◽  
Vol 2 (3) ◽  
pp. 253-265 ◽  
Author(s):  
Yacin Ameur
Keyword(s):  
Hilbert Space ◽  
Radon Measure ◽  
Interpolation Theorem ◽  
Dimensional Case ◽  
Infinite Dimensional ◽  
Positive Radon Measure ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
New Interpretation ◽  
Necessary And Sufficient

We give a new proof and new interpretation of Donoghue's interpolation theorem; for an intermediate Hilbert spaceH∗to be exact interpolation with respect to a regular Hilbert coupleH¯it is necessary and sufficient that the norm inH∗be representable in the form‖f‖∗=(∫[0,∞](1+t−1)K2(t,f;H¯)2dρ(t))1/2with some positive Radon measureρon the compactified half-line[0,∞]. The result was re-proved in [1] in the finite-dimensional case. The purpose of this note is to extend the proof given in [1] to cover the infinite-dimensional case. Moreover, the presentation of the aforementioned proof in [1] was slightly flawed, because we forgot to include a reference to ‘Donoghue's Lemma’, which is implicitly used in the proof. Hence we take this opportunity to correct that flaw.

scholarly journals An inverse problem for a second-order differential equation in a Banach space

2004 ◽  
Vol 2004 (12) ◽  
pp. 997-1005 ◽  
Author(s):  
Y. Eidelman
Keyword(s):  
Differential Equation ◽  
Unbounded Operator ◽  
Operator Function ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Cosine Operator Function ◽  
Second Order Differential Equation ◽  
The Right ◽  
Necessary And Sufficient

We consider the problem of determining the unknown term in the right-hand side of a second-order differential equation with unbounded operator generating a cosine operator function from the overspecified boundary data. We obtain necessary and sufficient conditions of the unique solvability of this problem in terms of location of the spectrum of the unbounded operator and properties of its resolvent.

Unitary equivalence and reductibility or invertibly weighted shifts

1971 ◽  
Vol 5 (2) ◽  
pp. 157-173 ◽  
Author(s):  
Alan Lambert
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Bounded Sequence ◽  
Complex Hilbert Space ◽  
Weighted Shifts ◽  
Infinite Dimensional ◽  
Reducing Subspaces ◽  
Positive Weights ◽  
Completely Reducible ◽  
Necessary And Sufficient

Let H be a complex Hilbert space and let {A1, A2, …} be a uniformly bounded sequence of invertible operators on H. The operator S on l2(H) = H ⊕ H ⊕ … given by S〈x0, x1, …〉 = 〈0, A1x0, A2x1, …〉 is called the invertibly veighted shift on l2(H) with weight sequence {An }. A matricial description of the commutant of S is established and it is shown that S is unitarily equivalent to an invertibly weighted shift with positive weights. After establishing criteria for the reducibility of S the following result is proved: Let {B1, B2, …} be any sequence of operators on an infinite dimensional Hilbert space K. Then there is an operator T on K such that the lattice of reducing subspaces of T is isomorphic to the corresponding lattice of the W* algebra generated by {B1, B2, …}. Necessary and sufficient conditions are given for S to be completely reducible to scalar weighted shifts.

scholarly journals On the equivalence of separability and extendability of quantum states

2017 ◽  
Vol 29 (04) ◽  
pp. 1750012 ◽  
Author(s):  
B. V. Rajarama Bhat ◽  
K. R. Parthasarathy ◽  
Ritabrata Sengupta
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Gaussian State ◽  
Bounded Operators ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Separability Property ◽  
Quantum De Finetti Theorem ◽  
Necessary And Sufficient ◽  
Symmetric States

Motivated by the notions of [Formula: see text]-extendability and complete extendability of the state of a finite level quantum system as described by Doherty et al. [Complete family of separability criteria, Phys. Rev. A 69 (2004) 022308], we introduce parallel definitions in the context of Gaussian states and using only properties of their covariance matrices, derive necessary and sufficient conditions for their complete extendability. It turns out that the complete extendability property is equivalent to the separability property of a bipartite Gaussian state. Following the proof of quantum de Finetti theorem as outlined in Hudson and Moody [Locally normal symmetric states and an analogue of de Finetti’s theorem, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 33(4) (1975/76) 343–351], we show that separability is equivalent to complete extendability for a state in a bipartite Hilbert space where at least one of which is of dimension greater than 2. This, in particular, extends the result of Fannes, Lewis, and Verbeure [Symmetric states of composite systems, Lett. Math. Phys. 15(3) (1988) 255–260] to the case of an infinite dimensional Hilbert space whose C* algebra of all bounded operators is not separable.

Instabilité de vecteurs propres d’opérateurs linéaires

1999 ◽  
Vol 42 (1) ◽  
pp. 104-117 ◽  
Author(s):  
Ludmila Nikolskaia
Keyword(s):  
Hilbert Space ◽  
Bounded Operator ◽  
Linear Operators ◽  
Suitable Choice ◽  
Geometric Properties ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Linear Perturbations ◽  
Necessary And Sufficient

AbstractWe consider some geometric properties of eigenvectors of linear operators on infinite dimensional Hilbert space. It is proved that the property of a family of vectors (xn) to be eigenvectors of a bounded operator T (admissibility property) is very instable with respect to additive and linear perturbations. For instance, (1) for the sequence to be admissible for every admissible (xn) and for a suitable choice of small numbers it is necessary and sufficient that the perturbation sequence be eventually scalar: there exist such that (Theorem 2); (2) for a bounded operator A to transform admissible families (xn) into admissible families (Axn) it is necessary and sufficient that A be left invertible (Theorem 4).

On nonlinear axisymmetric equilibria in the magnetohydrodynamic approximation

1977 ◽  
Vol 18 (3) ◽  
pp. 537-550
Author(s):  
M. L. Woolley
Keyword(s):  
Differential Equation ◽  
Spatial Variation ◽  
Sufficient Conditions ◽  
Hydrodynamic Pressure ◽  
Analytic Solutions ◽  
Closed Curves ◽  
Exact Analytic Solutions ◽  
Equivalent Equation ◽  
Necessary And Sufficient ◽  
Insight Into

The second order elliptic differential equation which describes the static equilibrium of an axisymmetric ideally conducting plasma toroid, in the magnetohydrodynamic approximation, may be transformed into an equivalent equation which admits at least one class of exact analytic solutions for which the original equation is essentially nonlinear. Necessary and sufficient conditions are given for the existence of systems in this category having closed toroidal flux surfaces and a result is proved which gives some insight into the mathematics by which the spatial variation of the hydrodynamic pressure can be represented by families of nested closed curves.

TO A TENSOR ANALYSIS OF THE SOLVABILITY OF THE PROBLEM OF REALIZATION OF A SECOND-ORDER BILINEAR SYSTEM WITH DELAY

2021 ◽  
Vol 2 ◽  
pp. 79-92
Author(s):  
Anatoly Lakeyev ◽  
◽  
Vyacheslav Rusanov ◽  
Andrey Banshchikov ◽  
◽  
...  
Keyword(s):  
Hilbert Space ◽  
Evolution Equations ◽  
Sufficient Conditions ◽  
Bilinear System ◽  
Second Order ◽  
Infinite Dimensional ◽  
Type Property ◽  
Machine Interface ◽  
Linear Differential ◽  
Necessary And Sufficient

The analytical conditions (necessary and sufficient) are defined for the solvability of the problem of differential realization of a continuous beam of controlled trajectory curves in the class of bilinear nonautonomous ordinary differential equations (with delay and without it) of the second order in a real separable Hilbert space. The problem under consideration belongs to the type of inverse problems for an additive combination of nonstationary linear and bilinear operators of evolution equations in an infinite-dimensional Hilbert space. The meta-language of this theory is the constructions of tensor products of Hilbert spaces, the structures of lattices with ortho-complementation, and the functional apparatus of the nonlinear Rayleigh-Ritz operator. It is shown that in the case of a finite bundle of trajectories, the presence of a sublinearity-type property of this operator allows us to obtain sufficient conditions for the existence of such realizations. Along the way, the topological-metric conditions for the continuity of the projectivization of the nonlinear Rayleigh-Ritz functional operator with the calculation of the fundamental group of its image are justified. The results obtained provide the motivation for the development of a qualitative theory of nonlinear structural identification of higher-order multi-linear differential models (e.g. for processes, induced by the «brain–machine» interface-platform of the type of Neuralink).

scholarly journals Random motion of strings and related stochastic evolution equations

1983 ◽  
Vol 89 ◽  
pp. 129-193 ◽  
Author(s):  
Tadahisa Funaki
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Basic Equation ◽  
Evolution Equations ◽  
Random Motion ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Infinite Dimensional ◽  
The Third ◽  
Elastic String

In this paper, we shall investigate the random motion of an elastic string by using the theory of infinite dimensional stochastic differential equations. The paper consists of three main parts and appendices. In the first part (§2), we shall derive a basic equation which describes the random motion of a string. Several properties of this equation will be investigated in § 3, 4 and 5. In the third part (§ 6), we shall deal with a stochastic differential equation on a Hilbert space as a generalization of the equation of the string.

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