scholarly journals Translatable radii of an operator in the direction of another operator II

2010 ◽  
Vol 60 (1) ◽  
Author(s):  
Kallol Paul
Keyword(s):  
Eigenvalue Problem ◽  
Unit Vector ◽  
Generalized Eigenvalue Problem ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Generalized Eigenvalue ◽  
Necessary And Sufficient ◽  
Stationary Vector

AbstractOne of the couple of translatable radii of an operator in the direction of another operator introduced in earlier work [PAUL, K.: Translatable radii of an operator in the direction of another operator, Scientae Mathematicae 2 (1999), 119–122] is studied in details. A necessary and sufficient condition for a unit vector f to be a stationary vector of the generalized eigenvalue problem Tf = λAf is obtained. Finally a theorem of Williams ([WILLIAMS, J. P.: Finite operators, Proc. Amer. Math. Soc. 26 (1970), 129–136]) is generalized to obtain a translatable radius of an operator in the direction of another operator.

The Inverse Eigenvalue Problem for Symmetric Doubly Arrow Matrices

2013 ◽  
Vol 444-445 ◽  
pp. 625-627
Author(s):  
Kan Ming Wang ◽  
Zhi Bing Liu ◽  
Xu Yun Fei
Keyword(s):  
Eigenvalue Problem ◽  
Special Kind ◽  
Inverse Eigenvalue Problem ◽  
Symmetric Matrices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Inverse Eigenvalue ◽  
Necessary And Sufficient

In this paper we present a special kind of real symmetric matrices: the real symmetric doubly arrow matrices. That is, matrices which look like two arrow matrices, forward and backward, with heads against each other at the station, . We study a kind of inverse eigenvalue problem and give a necessary and sufficient condition for the existence of such matrices.

Stochastic monotonicity of birth–death processes

1980 ◽  
Vol 12 (01) ◽  
pp. 59-80 ◽  
Author(s):  
Erik A. Van Doorn
Keyword(s):  
Initial Distribution ◽  
Unit Vector ◽  
Death Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Stochastic Monotonicity ◽  
Initial State ◽  
Distribution Vector ◽  
Necessary And Sufficient ◽  
Birth Death

A birth–death process {x(t): t ≥ 0} with state space the set of non-negative integers is said to be stochastically increasing (decreasing) on the interval (t 1, t 2) if Pr {x(t) > i} is increasing (decreasing) with t on (t 1, t 2) for all i = 0, 1, 2, ···. We study the problem of finding a necessary and sufficient condition for a birth–death process with general initial state probabilities to be stochastically monotone on an interval. Concrete results are obtained when the initial distribution vector of the process is a unit vector. Fundamental in the analysis, and of independent interest, is the concept of dual birth–death processes.

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.

scholarly journals Structure of the antieigenvectors of a strictly accretive operator

1998 ◽  
Vol 21 (4) ◽  
pp. 761-766 ◽  
Author(s):  
K. C. Das ◽  
M. Das Gupta ◽  
K. Paul
Keyword(s):  
Accretive Operator ◽  
Unit Vector ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Selfadjoint Operators ◽  
Kantorovich Inequality ◽  
Normal Operators ◽  
Necessary And Sufficient

A necessary and sufficient condition that a vectorfis an antieigenvector of a strictly accretive operatorAis obtained. The structure of antieigenvectors of selfadjoint and certain class of normal operators is also found in terms of eigenvectors. The Kantorovich inequality for selfadjoint operators and the Davis's inequality for normal operators are then easily deduced. A sort of uniqueness is also established for the values ofRe(Af,f)and‖Af‖if the first antieigenvalue, which is equal to minRe(Af,f)/(‖Af‖‖f‖)is attained at the unit vectorf.

scholarly journals On Properties of ClassA(n)andn-Paranormal Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Xiaochun Li ◽  
Fugen Gao
Keyword(s):  
Positive Integer ◽  
Tensor Products ◽  
Unit Vector ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Direct Sum ◽  
Paranormal Operator ◽  
Necessary And Sufficient

Letnbe a positive integer, and an operatorT∈B(ℋ)is called a classA(n)operator ifT1+n2/1+n≥|T|2andn-paranormal operator ifT1+nx1/1+n≥||Tx||for every unit vectorx∈ℋ, which are common generalizations of classAand paranormal, respectively. In this paper, firstly we consider the tensor products for classA(n)operators, giving a necessary and sufficient condition forT⊗Sto be a classA(n)operator whenTandSare both non-zero operators; secondly we consider the properties forn-paranormal operators, showing that an-paranormal contraction is the direct sum of a unitary and aC.0completely non-unitary contraction.

scholarly journals Necessary and Sufficient Condition for the Existence of Solutions to a Discrete Second-Order Boundary Value Problem

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Chenghua Gao
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Value Problem ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Second Order ◽  
Sufficient Condition ◽  
First Eigenvalue ◽  
Necessary And Sufficient Condition ◽  
The First Eigenvalue ◽  
Necessary And Sufficient

This paper is concerned with the existence of solutions for the discrete second-order boundary value problemΔ2u(t-1)+λ1u(t)+g(Δu(t))=f(t),t∈{1,2,…,T},u(0)=u(T+1)=0, whereT>1is an integer,f:{1,…,T}→R,g:R→Ris bounded and continuous, andλ1is the first eigenvalue of the eigenvalue problemΔ2u(t-1)+λu(t)=0,t∈T,u(0)=u(T+1)=0.

Stochastic monotonicity of birth–death processes

1980 ◽  
Vol 12 (1) ◽  
pp. 59-80 ◽  
Author(s):  
Erik A. Van Doorn
Keyword(s):  
Initial Distribution ◽  
Unit Vector ◽  
Death Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Stochastic Monotonicity ◽  
Initial State ◽  
Distribution Vector ◽  
Necessary And Sufficient ◽  
Birth Death

A birth–death process {x(t): t ≥ 0} with state space the set of non-negative integers is said to be stochastically increasing (decreasing) on the interval (t1, t2) if Pr {x(t) > i} is increasing (decreasing) with t on (t1, t2) for all i = 0, 1, 2, ···. We study the problem of finding a necessary and sufficient condition for a birth–death process with general initial state probabilities to be stochastically monotone on an interval. Concrete results are obtained when the initial distribution vector of the process is a unit vector. Fundamental in the analysis, and of independent interest, is the concept of dual birth–death processes.

scholarly journals Extremal Inverse Eigenvalue Problem for a Special Kind of Matrices

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Zhibing Liu ◽  
Yeying Xu ◽  
Kanmin Wang ◽  
Chengfeng Xu
Keyword(s):  
Eigenvalue Problem ◽  
Special Kind ◽  
Inverse Eigenvalue Problem ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Inverse Eigenvalue ◽  
Maximal Eigenvalues ◽  
Principal Submatrices ◽  
Necessary And Sufficient ◽  
Arrow Matrix

We consider the following inverse eigenvalue problem: to construct a special kind of matrix (real symmetric doubly arrow matrix) from the minimal and maximal eigenvalues of all its leading principal submatrices. The necessary and sufficient condition for the solvability of the problem is derived. Our results are constructive and they generate algorithmic procedures to construct such matrices.

Sign in / Sign up

Export Citation Format

Share Document