scholarly journals Extensions of dissipative operators with closable imaginary part

2021 ◽  
Vol 41 (3) ◽  
pp. 381-393
Author(s):  
Christoph Fischbacher
Keyword(s):  
Hilbert Space ◽  
Complex Hilbert Space ◽  
Dissipative Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Order Operator ◽  
Dissipative Operators ◽  
First Order ◽  
Necessary And Sufficient ◽  
Symmetric Operators

Given a dissipative operator \(A\) on a complex Hilbert space \(\mathcal{H}\) such that the quadratic form \(f \mapsto \text{Im}\langle f, Af \rangle\) is closable, we give a necessary and sufficient condition for an extension of \(A\) to still be dissipative. As applications, we describe all maximally accretive extensions of strictly positive symmetric operators and all maximally dissipative extensions of a highly singular first-order operator on the interval.

Geometry on the Unit Ball of a Complex Hilbert Space

1978 ◽  
Vol 30 (01) ◽  
pp. 22-31 ◽  
Author(s):  
Kyong T. Hahn
Keyword(s):  
Hilbert Space ◽  
Unit Ball ◽  
Hyperbolic Geometry ◽  
Complex Hilbert Space ◽  
Open Unit ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Open Unit Ball ◽  
Similar Role ◽  
Necessary And Sufficient

Furnishing the open unit ball of a complex Hilbert space with the Carathéodory-differential metric, we construct a model which plays a similar role as that of the Poincaré model for the hyperbolic geometry. In this note we study the question whether or not through a point in the model not lying on a given line there exists a unique perpendicular, and give a necessary and sufficient condition for the existence of a unique perpendicular. This enables us to divide a triangle into two right triangles. Many trigonometric identities in a general triangle are easy consequences of various identities which hold on a right triangle.

scholarly journals Existence of nonoscillatory solutions of first order nonlinear neutral equations

1990 ◽  
Vol 32 (2) ◽  
pp. 180-192 ◽  
Author(s):  
Lu Wudu
Keyword(s):  
Nonoscillatory Solution ◽  
Sufficient Condition ◽  
Neutral Equation ◽  
Necessary And Sufficient Condition ◽  
Nonoscillatory Solutions ◽  
Neutral Equations ◽  
First Order ◽  
Image Position ◽  
Necessary And Sufficient

AbstractConsider the nonlinear neutral equationwhere pi(t), hi(t), gj(t), Q(t) Є C[t0, ∞), limt→∞hi(t) = ∞, limt→∞gj(t) = ∞ i Є Im = {1, 2, …, m}, j Є In = {1, 2, …, n}. We obtain a necessary and sufficient condition (2) for this equation to have a nonoscillatory solution x(t) with limt→∞ inf|x(t)| > 0 (Theorems 5 and 6) or to have a bounded nonoscillatory solution x(t) with limt→∞ inf|x(t)| > 0 (Theorem 7).

scholarly journals On B- Covariant Derivative of First Order for Some Tensors in different Spaces

2021 ◽  
Vol 2 (2) ◽  
pp. 30-37
Author(s):  
Alaa A. Abdallah ◽  
A. A. Navlekar ◽  
Kirtiwant P. Ghadle
Keyword(s):  
Curvature Tensor ◽  
Covariant Derivative ◽  
Torsion Tensor ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
First Order ◽  
Recurrence Property ◽  
Necessary And Sufficient ◽  
The Relationship

In this paper, we study the relationship between Cartan's second curvature tensor $P_{jkh}^{i}$ and $(h) hv-$torsion tensor $C_{jk}^{i}$ in sense of Berwald. Moreover, we discuss the necessary and sufficient condition for some tensors which satisfy a recurrence property in $BC$-$RF_{n}$, $P2$-Like-$BC$-$RF_{n}$, $P^{\ast }$-$BC$-$RF_{n}$ and $P$-reducible-$BC-RF_{n}$.

On inequalities for powers of linear operators and for quadratic forms

1981 ◽  
Vol 89 (1-2) ◽  
pp. 25-50 ◽  
Author(s):  
Vũ Qúôc Phóng
Keyword(s):  
Quadratic Forms ◽  
Linear Operators ◽  
Dissipative Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Dense Domain ◽  
The Best Possible Constant ◽  
Bilinear Functional ◽  
Image Position ◽  
Necessary And Sufficient

SynopsisLetHbe a Hilbert space in which a symmetric operatorSwith a dense domainDsis given and letShave a finite deficiency index (r, s). This paper contains a necessary and sufficient condition for validity of the following inequalities of Kolmogorov typeand a method for calculating the best possible constantsCn,m(S).Moreover, let φ be a symmetric bilinear functional with a dense domainDφsuch thatDs⊂Dφand φ(f, g) = (Sf, g) for allf∈Ds,g∈Dφ. A necessary and sufficient condition for validity of the inequalityas well as a method for calculating the best possible constantKare obtained. Then an analogous approach is worked out in order to obtain the best possible additive inequalities of the formThe paper is concluded by establishing the best possible constants in the inequalitieswhereTis an arbitrary dissipative operator. The theorems are extensions of the results of Ju. I. Ljubič, W. N. Everitt, and T. Kato.

The uniform exponential stability of a class of linear differential–difference equations in a Hilbert space

1981 ◽  
Vol 89 (3-4) ◽  
pp. 201-215 ◽  
Author(s):  
Richard Datko
Keyword(s):  
Hilbert Space ◽  
Phase Space ◽  
Exponential Stability ◽  
Difference Equations ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Uniform Exponential Stability ◽  
Linear Differential ◽  
Necessary And Sufficient

SynopsisA necessary and sufficient condition is given for the uniform exponential stability of certain autonomous differential–difference equations whose phase space is a Hilbert space. It is shown that this property is preserved when the delays depend homogeneously on a nonnegative parameter.

scholarly journals Oscillation for nonlinear delay difference equations

2001 ◽  
Vol 32 (4) ◽  
pp. 275-280 ◽  
Author(s):  
X. H. Tang
Keyword(s):  
Oscillatory Behavior ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Delay Difference Equation ◽  
First Order ◽  
Condition Of Oscillation ◽  
Necessary And Sufficient ◽  
Delay Difference Equations ◽  
Delay Difference ◽  
Nonlinear Delay

The oscillatory behavior of the first order nonlinear delay difference equation of the form $$ x_{n+1} - x_n + p_n x_{n-k}^{\alpha} = 0, ~~~ n = 0, 1, 2, \ldots ~~~~~~~ \eqno{(*)} $$ is investigated. A necessary and sufficient condition of oscillation for sublinear equation (*) ($ 0 < \alpha < 1 $) and an almost sharp sufficient condition of oscillation for superlinear equation (*) ($ \alpha > 1 $) are obtained.

scholarly journals Existence of minimizers for one-dimensional vectorial non-semicontinuous functionals with second order lagrangian

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Sandro Zagatti
Keyword(s):  
Existence Result ◽  
Variational Problems ◽  
Convex Envelope ◽  
Minimum Problem ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
One Dimensional ◽  
First Order ◽  
Existence Of Minimizers ◽  
Necessary And Sufficient

<p style='text-indent:20px;'>We study the minimum problem for functionals of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \mathcal{F}(u) = \int_{I} f(x, u(x), u^ \prime(x), u^ {\prime\prime}(x))\,dx, \end{equation} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where the integrand <inline-formula><tex-math id="M1">\begin{document}$ f:I\times \mathbb{R}^m\times \mathbb{R}^m\times \mathbb{R}^m \to \mathbb{R} $\end{document}</tex-math></inline-formula> is not convex in the last variable. We provide an existence result assuming that the lower convex envelope <inline-formula><tex-math id="M2">\begin{document}$ \overline{f} = \overline{f}(x,p,q,\xi) $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M3">\begin{document}$ f $\end{document}</tex-math></inline-formula> with respect to <inline-formula><tex-math id="M4">\begin{document}$ \xi $\end{document}</tex-math></inline-formula> is regular and enjoys a special dependence with respect to the i-th single components <inline-formula><tex-math id="M5">\begin{document}$ p_i, q_i, \xi_i $\end{document}</tex-math></inline-formula> of the vector variables <inline-formula><tex-math id="M6">\begin{document}$ p,q,\xi $\end{document}</tex-math></inline-formula>. More precisely, we assume that it is monotone in <inline-formula><tex-math id="M7">\begin{document}$ p_i $\end{document}</tex-math></inline-formula> and that it satisfies suitable affinity properties with respect to <inline-formula><tex-math id="M8">\begin{document}$ \xi_i $\end{document}</tex-math></inline-formula> on the set <inline-formula><tex-math id="M9">\begin{document}$ \{f&gt; \overline{f}\} $\end{document}</tex-math></inline-formula> and with respect to <inline-formula><tex-math id="M10">\begin{document}$ q_i $\end{document}</tex-math></inline-formula> on the whole domain. We adopt refined versions of the integro-extremality method, extending analogous results already obtained for functionals with first order lagrangians. In addition we show that our hypotheses are nearly optimal, providing in such a way an almost necessary and sufficient condition for the solvability of this class of variational problems.</p>

A two-parameter eigenvalue problem involving complex potentials

1990 ◽  
Vol 116 (1-2) ◽  
pp. 177-191
Author(s):  
M. Faierman
Keyword(s):  
Hilbert Space ◽  
Eigenvalue Problem ◽  
Differential Equations ◽  
Complex Potentials ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Parameter System ◽  
Complete Set ◽  
Necessary And Sufficient ◽  
Two Parameter

SynopsisWe consider a two-parameter system of ordinary differential equations of the second order involving complex potentials and show that, unlike the case of real potentials, the eigenfunctions of the system do not necessarily form a complete set in the usual Hilbert space associated with the problem. We also give a necessary and sufficient condition for the eigenfunctions to be complete. Finally, we establish some results concerning the eigenvalues of the system.

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