Some relations between eigenvalues and matrix elements of linear operators

1966 ◽  
pp. 201-216 ◽  
Author(s):  
I. C. Gohberg ◽  
A. S. Markus
Keyword(s):  
Linear Operators ◽  
Matrix Elements

LOCALIZATION AT WEAK DISORDER: SOME ELEMENTARY BOUNDS

1994 ◽  
Vol 06 (05a) ◽  
pp. 1163-1182 ◽  
Author(s):  
MICHAEL AIZENMAN
Keyword(s):  
Time Evolution ◽  
Elementary Proof ◽  
Random Potential ◽  
Linear Operators ◽  
Matrix Elements ◽  
Weak Disorder ◽  
Evolution Operators ◽  
Translation Invariant ◽  
The Matrix ◽  
Relevant Range

An elementary proof is given of localization for linear operators H = Ho + λV, with Ho translation invariant, or periodic, and V (·) a random potential, in energy regimes which for weak disorder (λ → 0) are close to the unperturbed spectrum σ (Ho). The analysis is within the approach introduced in the recent study of localization at high disorder by Aizenman and Molchanov [4]; the localization regimes discussed in the two works being supplementary. Included also are some general auxiliary results enhancing the method, which now yields uniform exponential decay for the matrix elements <0|P[a,b] exp (−itH)|x> of the spectrally filtered unitary time evolution operators, with [a, b] in the relevant range.

Approximation in statistical sense to B -continuous functions by positive linear operators

2010 ◽  
Vol 47 (3) ◽  
pp. 289-298 ◽  
Author(s):  
Fadime Dirik ◽  
Oktay Duman ◽  
Kamil Demirci
Keyword(s):  
Compact Subset ◽  
Real Line ◽  
Statistical Convergence ◽  
Approximation Theorem ◽  
Continuous Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Statistical Approximation ◽  
Classical Version ◽  
Statistical Sense

In the present work, using the concept of A -statistical convergence for double real sequences, we obtain a statistical approximation theorem for sequences of positive linear operators defined on the space of all real valued B -continuous functions on a compact subset of the real line. Furthermore, we display an application which shows that our new result is stronger than its classical version.

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