Atkinson Theorem Research Articles

General Form of Generalized Invertible Operators in Banach Spaces

We prove theorems on the general form of generalized invertible operators in Banach spaces in the case where the operators are topologically Noetherian or topologically Fredholm. These theorems generalize the well-known Nikol’skii theorem on the general form of Fredholm operators and the Atkinson theorem on the general form of Noetherian operators in function spaces.

Welfare Analysis in an Extended Harris-Todaro Model: An Application of the Atkinson Theorem

I analyze the welfare effects of a policy of modern sector enlargement (MSENL), and a policy of increasing the efficiency of on-the-job search from the urban informal sector (IEOS) in a generalized Harris-Todaro model. I show that MSENL causes a Lorenz worsening of the income distribution and IEOS causes a Lorenz improvement. In a rare direct application of the Atkinson theorem, I conclude that MSENL decreases social welfare and IEOS increases social welfare for all anonymous, increasing and Schur-concave social welfare functions.

Welfare Analysis in an Extended Harris-Todaro Model: An Application of the Atkinson Theorem

AbstractI analyze the welfare effects of a policy of modern sector enlargement (MSENL), and a policy of increasing the efficiency of on-the-job search from the urban informal sector (IEOS) in a generalized Harris-Todaro model. I show that MSENL causes a Lorenz worsening of the income distribution and IEOS causes a Lorenz improvement. In a rare direct application of the Atkinson theorem, I conclude that MSENL decreases social welfare and IEOS increases social welfare for all anonymous, increasing and Schur-concave social welfare functions.

Local linear dependence seen through duality II

A vector space S of linear operators between finite-dimensional vector spaces U and V is called locally linearly dependent (in abbreviated form: LLD) when every vector x∈U is annihilated by a non-zero operator in S. By a duality argument, one sees that studying LLD operator spaces amounts to studying vector spaces of matrices with rank less than the number of columns, or, alternatively, vector spaces of non-injective operators.In this article, this insight is used to obtain classification results for LLD spaces of small dimension or large essential range (the essential range being the sum of all the ranges of the operators in S). We show that such classification theorems can be obtained by translating into the context of LLD spaces Atkinson's classification of primitive spaces of bounded rank matrices; we also obtain a new classification theorem for such spaces that covers a range of dimensions for the essential range that is roughly twice as large as that in Atkinson's theorem. In particular, we obtain a classification of all 4-dimensional LLD operator spaces for fields with more than 3 elements (beforehand, such a classification was known only for algebraically closed fields and in the context of primitive spaces of matrices of bounded rank).These results are applied to obtain improved upper bounds for the maximal rank in a minimal LLD operator space.

The Atkinson Theorem in HilbertC*-Modules overC*-Algebras of Compact Operators

The concept of unbounded Fredholm operators on HilbertC*-modules over an arbitraryC*-algebra is discussed and the Atkinson theorem is generalized for bounded and unbounded Feredholm operators on HilbertC*-modules overC*-algebras of compact operators. In the framework of HilbertC*-modules overC*-algebras of compact operators, the index of an unbounded Fredholm operator and the index of its bounded transform are the same.

Integrable Renormalization II: The General Case

We extend the results we obtained in an earlier work. The cocommutative case of rooted ladder trees is generalized to a full Hopf algebra of (decorated) rooted trees. For Hopf algebra characters with target space of Rota-Baxter type, the Birkhoff decomposition of renormalization theory is derived by using the Rota-Baxter double construction, respectively Atkinson's theorem. We also outline the extension to the Hopf algebra of Feynman graphs via decorated rooted trees.

A note on pro-poor growth

This paper proposes a ‘poverty growth curve’ that measures whether economic growth is pro-poor or not pro-poor. Our methodology is developed based on Atkinson's theorem linking the generalized Lorenz curve and changes in poverty. This methodology is applied to both Thailand's unit record household surveys and international cross-country data. Empirical study illustrates that our approach to measuring pro-poor growth provides conclusive results in the majority of cases.

Typical Recurrence for Lifts of mean Rotation Zero Annulus Homeomorphisms

This paper is about typical (uniform topology dense Gg) properties of homeomorphisms of the torus or annulus which preserve a fixed measure and have mean rotation zero. We first show that ergodicity is typical (Theorem 1). We then show that the lift (to the universal covering space) of such a homeomorphism of the annulus is the skew product of the annulus homeomorphism with respect to a skewing function of mean zero. Hence Atkinson's Theorem on skew products, together with Theorem 1, implies that it is typical for an annulus homeomorphism of mean rotation zero to have a recurrent lift (Theorem 3). Standard arguments then give the Poincare-Birkhoff Fixed Point Theorem as a corollary.

Gelfand theory of certainC *-algebras on nonasymptotically flat manifolds

Let Ω be a Riemannian manifold with finitely many conical ends. Under certain conditions which do not require Ω to be asymptotically flat, we study aC*-algebra containing pseudodifferential operators on Ω. Results on compact commutators and the Gelfand space are presented. Criteria for differential operators or systems within reach to be Fredholm are just simple consequences of Atkinson's theorem and our results.

On the Measurement of Tax Progressivity: An Implication of the Atkinson Theorem

On the Measurement of Tax Progressivity: An Implication of the Atkinson Theorem

Marketable Pollution Permits and Acid Rain Externalities: A Comment and Some Further Evidence

In a recent paper, Scott Atkinson (1983) has provided some important insights into the design of systems of marketable permits for the control of air pollution. Atkinson reaches two major conclusions: 1. A system of marketable permits that minimizes the control costs for the attainment of local ambient air-quality standards for So2 is likely to increase significantly the extent of long-range sulfate depositions (acid rain) as compared to a traditional command-and-control (CAC) strategy. 2. Solely from the perspective of local air quality, the cost-minimizing system of marketable permits must imply higher levels of local pollution relative to an alternative system of emissions permits or to a prototype CAC system. We have no quibbles with the first point. Moreover, Atkinson's simulation results suggest that the trade-off between local air pollution and the long-range transport of sulfur is a serious issue. This is largely a matter of stack height. Higher chimneys allow sulfur emissions to escape the local environment only to result in increased sulfate pollution at more distant locations. This suggests that stack height must be treated as a critical variable in the design of systems to control jointly local and 'global' pollution. Our concern in this note is with Atkinson's second point. We shall show, first, that the Atkinson theorem is formally incorrect: the cost-minimizing permit system need not result in increased local pollution. However, as a practical matter, Atkinson may well be right. We shall supplement his simulation results with some findings from another set of simulations for a different air pollutant, particulate matter, in the Baltimore Air Quality Control