Upper triangular matrix operators with diagonal (T_1,T_2), T_2 k-nilpotent

If \(T=\left(\begin{array}{clcr}T_1&\quad C\\ 0&\quad T_2\end{array}\right) \in B(\mathcal{X }_1\oplus \mathcal{X }_2)\) is a Banach space upper triangular operator matrix with diagonal \((T_1, T_2)\) such that \(T_2\) is \(k\)-nilpotent for some integer \(k\ge 1\), then \(T\) inherits a number of its spectral properties, such as SVEP, Bishop’s property \((\beta )\) and the equality of Browder and Weyl spectrum, from those of \(T_1\). This paper studies such operators. The conclusions are then applied to provide a general framework for results pertaining (for example) to Browder, Weyl type theorems and supercyclicity for classes of Hilbert space operators, such as \(k\)-quasi hyponormal, \(k\)-quasi isometric and \(k\)-quasi paranormal operators, defined by a positivity condition.     

Reverse isotope dilution alpha spectrometry using plutionium-239 spike for determining plutonium concentration in high burn-up fuel samples without purification from americium-241 and a bulk of other impurities

A reverse isotope dilution alpha spectrometric /R-IDAS/ method using²³⁹Pu as a spike is described for the determination of plutonium concentration in high burn-up fuel samples wth²³⁸Pu/(²³⁹Pu+²⁴⁰Pu) alpha activity ratio >0.5, without resorting to any purification from²⁴¹Am and a bulk of other impurities. It involves the addition of a pre-clibrated spike solution to a known aliquot of the plutonium sample solution followed by source preparation using TEG as a spreading agent. The results obtained on a number of plutonium samples containing 20–80% of²⁴¹Am /alpha activity wise/ using this method are compared with those achieved by R-IDAS using purification with TTA, with respect to precision and accuracy. Precision and accuracy of 0.5% are demonstrated. This method eliminates the need of any separation and purification of plutonium from²⁴¹Am and a bulk of other impurities like uranium.     

Polaroid operators and Weyl's theorem

``Polaroid elements" represent an attempt to abstract part of the condition, ``Weyl's theorem holds" for operators.

     

Hereditarily polaroid operators, SVEP and Weyl's theorem

A Banach space operator T∈B(X) is hereditarily polaroid, T∈HP, if every part of T is polaroid. HP operators have SVEP. It is proved that if T∈B(X) has SVEP and R∈B(X) is a Riesz operator which commutes with T, then T+R satisfies generalized a-Browder's theorem. If, in particular, R is a quasi-nilpotent operator Q, then both T+Q and T^∗+Q^∗ satisfy generalized a-Browder's theorem; furthermore, if Q is injective, then also T+Q satisfies Weyl's theorem. If A∈B(X) is an algebraic operator which commutes with the polynomially HP operator T, then T+N is polaroid and has SVEP, f(T+N) satisfies generalized Weyl's theorem for every function f which is analytic on a neighbourhood of σ(T+N), and f^∗(T+N) satisfies generalized a-Weyl's theorem for every function f which is analytic on, and constant on no component of, a neighbourhood of σ(T+N).     

Lightlike submanifolds of semi-Riemannian manifolds

The purpose of this paper is to initiate a study of the differential geometry of lightlike (degenerate) submanifolds of semi-Riemannian manifolds. We construct the transversal vector bundle for an arbitrary lightlike submanifold and obtain results on the geometric structures induced on it.