scholarly journals On Self-Adjoint Linear Relations

2021 ◽  
Vol 27_NS1 (1) ◽  
pp. 1-7
Author(s):  
Péter Berkics
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Linear Relation ◽  
Sufficient Condition ◽  
Classical Approach ◽  
Necessary And Sufficient Condition ◽  
Von Neumann ◽  
Necessary And Sufficient ◽  
Densely Defined ◽  
General Setup

A linear operator on a Hilbert space , in the classical approach of von Neumann, must be symmetric to guarantee self-adjointness. However, it can be shown that the symmetry could be omitted by using a criterion for the graph of the operator and the adjoint of the graph. Namely, S is shown to be densely defined and closed if and only if . In a more general setup, we can consider relations instead of operators and we prove that in this situation a similar result holds. We give a necessary and sufficient condition for a linear relation to be densely defined and self-adjoint.

scholarly journals On the solution of Nevanlinna Pick problem with selfadjoint extensions of symmetric linear relations in Hilbert space

1997 ◽  
Vol 20 (3) ◽  
pp. 457-464
Author(s):  
A. A. El-Sabbagh
Keyword(s):  
Hilbert Space ◽  
Linear Relation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Resolvent Matrix ◽  
Linear Relations ◽  
Symmetric Linear Relation ◽  
Necessary And Sufficient ◽  
Selfadjoint Extensions ◽  
Pick Problem

The representation of Nevanlinna Pick Problem is well known, see [7], [8] and [11]. The aim of this paper is to find the necessary and sufficient condition for the solution of Nevanlinna Pick Problem and to show that there is a one-to-one correspondence between the solutions of the Nevanlinna Pick Problem and the minimal selfadjoint extensions of symmetric linear relation in Hilbert space. Finally, we define the resolvent matrix which gives the solutions of the Nevanlinna Pick Problem.

Horodeckis criterion of separability of mixed states in von Neumann and C*-algebras

2014 ◽  
Vol 17 (04) ◽  
pp. 1450028 ◽  
Author(s):  
Marek Miller ◽  
Robert Olkiewicz
Keyword(s):  
Quantum Systems ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Mixed States ◽  
Von Neumann ◽  
C Algebras ◽  
Necessary And Sufficient

The Horodeckis necessary and sufficient condition of separability of mixed states is generalized to arbitrary composite quantum systems.

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 Sums of A Pair of Orthogonal Frames

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 582
Author(s):  
Ghanshyam Bhatt
Keyword(s):  
Linear Operator ◽  
Bounded Linear Operator ◽  
Sufficient Condition ◽  
Simple Construction ◽  
Necessary And Sufficient Condition ◽  
Dual Frames ◽  
Bounded Linear ◽  
Frame Bounds ◽  
Necessary And Sufficient

Frames are more stable as compared to bases under the action of a bounded linear operator. Sums of different frames under the action of a bounded linear operator are studied with the help of analysis, synthesis and frame operators. A simple construction of frames from the existing ones under the action of such an operator is presented here. It is shown that a frame can be added to its alternate dual frames, yielding a new frame. It is also shown that the sum of a pair of orthogonal frames is a frame. This provides an easy construction of a frame where the frame bounds can be computed easily. Moreover, for a pair of orthogonal frames, the necessary and sufficient condition is presented for their alternate dual frames to be orthogonal. This allows for an easy construction of a large number of new frames.

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 On characterization of integrable sesquilinear forms

2012 ◽  
Vol 62 (6) ◽  
Author(s):  
A. Sherstnev ◽  
O. Tikhonov
Keyword(s):  
Normal State ◽  
Von Neumann Algebra ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Sesquilinear Forms ◽  
Von Neumann ◽  
Sesquilinear Form ◽  
Faithful Normal State ◽  
Necessary And Sufficient

AbstractWe give a necessary and sufficient condition for a sesquilinear form to be integrable with respect to a faithful normal state on a von Neumann algebra.

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.

Linear Laws in Physics

1998 ◽  
Vol 12 (16n17) ◽  
pp. 1751-1754
Author(s):  
Liqiu Wang
Keyword(s):  
Linear Relation ◽  
General Linear ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Frame Indifference ◽  
Material Frame Indifference ◽  
Necessary And Sufficient ◽  
Universal Law ◽  
Material Frame

The empirical laws in physics, relating two frame-indifferent spatial vectors linearly, are shown to be both necessary and sufficient condition for a general linear relation between two vectors to satisfy the principle of material frame indifference, a more universal law in physics.

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