Réflexivité d’une extension d’un opérateur normal par un opérateur nilpotent

Let T be an extension of a one to one normal operator A by a nilpotent operator N. In this paper we calculate the defect of reflexivity of T. We give a necessary and sufficient condition to insure the reflexivity of such extensions. In particular, it is shown that T is reflexive when N is reflexive.     

Bragg diffraction of finite beams by thick gratings: Two rival theories

Two rival techniques used in the analysis of Bragg diffraction of two-dimensional light beams by thick unslanted phase gratings, namely Fourier plane wave decomposition and 2-D coupled-wave theory, are compared. The advantages and regions of applicability of each are discussed, and the conditions found under which they yield identical results. In particular, the results of plane-wave decomposition are used to provide quantitative conditions for validity of 2-D coupled-wave theory. These conditions are not easily derived using any other technique, and set quantitative limits to how narrow or fast-varying the incident distribution may be before 2-D coupled-wave theory fails. It is also shown that 2-D coupled-wave theory is inadequate when the parameterv ₀, due to Kogelnik (1969), which is proportional to the product of coupling rate and grating thickness, is very large.     

Une remarque sur l'orthogonalité de l'image au noyau d'une dérivation généralisée

In this paper we introduce the notion of the pair of operators having the Fuglede-Putnam property in a two-sided ideal of . The characterization of this class allows us to generalize the recent result of F. Kittaneh. We also give some applications of this result.

     

Bragg diffraction of gaussian beams by thick gratings: Numerical evaluations by plane-wave decomposition

The validity of 2-D coupled-wave theory is investigated for the case of Bragg diffraction of a Gaussian beam by a thick unslanted phase grating in transmission mode. Fourier plane wave decomposition theory, as presented in a companion article, is used to test 2-D coupled-wave theory under circumstances when its validity is in question, namely when the incident distribution is very narrow or fast-varying, or when the parameterv _o, due to Kogelnik (1969), which is proportional to the product of coupling rate and grating thickness, is very large. Numerical evaluations of the field patterns at output from the grating, obtained using plane-wave decomposition, are used to explore the effect of the gradual violation of the conditions (from the previous article) under which 2-D coupled-wave theory is valid.     

Réflexivité d'une extension d'un opérateur sous-normal par un opérateur algébrique

A bounded linear operator on a Hilbert space is said to be reflexive if the operators which leave invariant the invariant subspaces of T are wot-limits of polynomials in T. In this paper we give a necessary and sufficient condition for an extension of a subnormal operator by an algebraic one to be reflexive.We also give a formula for the reflexivity defect of such extensions.