Finitely strictly singular operators between James spaces

An operator between Banach spaces is said to be finitely strictly singular if for every ε>0 there exists n such that every subspace E⊆X with dimE?n contains a vector x such that ‖Tx‖<ε‖x‖. We show that, for 1?p<q<∞, the formal inclusion operator from Jp to Jq is finitely strictly singular. As a consequence, we obtain that the strictly singular operator with no invariant subspaces constructed by C. Read is actually finitely strictly singular. These results are deduced from the following fact: if k?n then every k-dimensional subspace of Rn contains a vector x with ‖x?_∞‖=1 such that xmi=i(−1) for some m₁<?<mk.     

On regular 4-coverings and their application for lattice coverings in normed planes

It is well known that the famous covering problem of Hadwiger is completely solved only in the planar case, i.e.: any planar convex body can be covered by four smaller homothetical copies of itself. Lassak derived the smallest possible ratio of four such homothets (having equal size), using the notion of regular 4-covering. We will continue these investigations, mainly (but not only) referring to centrally symmetric convex plates. This allows to interpret and derive our results in terms of Minkowski geometry (i.e., the geometry of finite dimensional real Banach spaces). As a tool we also use the notion of quasi-perfect and perfect parallelograms of normed planes, which do not differ in the Euclidean plane. Further on, we will use Minkowskian bisectors, different orthogonality types, and further notions from the geometry of normed planes, and we will construct lattice coverings of such planes and study related Voronoi regions and gray areas. Discussing relations to the known bundle theorem, we also extend Miquel’s six-circles theorem from the Euclidean plane to all strictly convex normed planes.     

A note on Jordan-von Neumann constant and James constant

Let X be a non-trivial Banach space. L. Maligranda conjectured CNJ(X)?1+J²(X)/4 for James constant J(X) and von Neumann-Jordan constant CNJ(X) of X. Satit Saejung gave a proof of it in 2006. In this note, we show that the last step in Satit Saejung's proof is not valid. Using his proof, the result should be . On the other hand, we give a new proof of CNJ(X)?1+J²(X)/4. As an application, we give a relation between J(X) and J(lp(X)).     

The von Neumann-Jordan constant, weak orthogonality and normal structure in Banach spaces

We give some sufficient conditions for normal structure in terms of the von Neumann-Jordan constant, the James constant and the weak orthogonality coefficient introduced by B. Sims. In the rest of the paper, the von Neumann-Jordan constant and the James constant for the Bynum space are computed, and are used to show that our results are sharp.

     

Another look at Cesàro sequence spaces

We consider the Cesàro sequence space cesp as a closed subspace of the infinite ?p-sum of finite dimensional spaces. We easily obtain many known results, for example, cesp has property (β) of Rolewicz, uniform Opial property, and weak uniform normal structure. We also consider some generalized Cesàro sequence spaces. Finally, we compute the von Neumann-Jordan and James constants of the two-dimensional Cesàro sequence space when 1<p?2.     

The characteristic of convexity of a Banach space and normal structure

We present some sufficient conditions for normal structure of Banach spaces and their dual spaces in terms of the characteristic of convexity, the James constant, and the coefficient of weak orthogonality. Many known results are improved and strengthened. We also show that some of our results are sharp.     

Carl Neumann’s Contributions to Electrodynamics

I examine the publications of Carl Neumann (1832–1925) on electrodynamics, which constitute a major part of his work and which illuminate his approach to mathematical physics. I show how Neumann contributed to physics at an important stage in its development and how his work led to a polemic with Hermann Helmholtz (1821–1894). Neumann advanced and extended the ideas of the Königsberg school of mathematical physics. His investigations were aimed at founding a mathematically exact physical theory of electrodynamics, following the approach of Carl G.J. Jacobi (1804–1851) on the foundation of a physical theory as outlined in Jacobis lectures on analytical mechanics. Neumanns work also shows how he clung to principles that impeded him in appreciating and developing new ideas such as those on field theory that were proposed by Michael Faraday (1791–1867) and James Clerk Maxwell (1831–1879).Karl-Heinz Schlote works as a historian of mathematics in the Arbeitsgruppe für Geschichte der Naturwissenschaften und Mathematik at the Sächsische Akademie der Wissenschaften in Leipzig, Germany.     

K-Theory for the Banach Algebra of Operators on James''s Quasi-reflexive Banach Spaces

We prove that the K-groups of the Banach algebra of bounded, linear operators on the pth James space , where 1 < p < , are given by and . Moreover, for each Banach space and each non-zero, closed ideal contained in the ideal of inessential operators, we show that and . This enables us to calculate the K-groups of for each Banach space which is a direct sum of finitely many James spaces and -spaces.     

On a new geometric constant related to the modulus of smoothness of a Banach space

We shall introduce a new geometric constant A(X) of a Banach space X,which is closely related to the modulus of smoothness ρX(τ),and investigate it in relation with the constant A2(X) by Baronti et al.,the von Neumann–Jordan constant CNJ(X) and the James constant J(X).A sequence of recent results on these constants as well as some other geometric constants will be strengthened and improved.     

On the calculation of the James constant of Lorentz sequence spaces

In [M. Kato, L. Maligranda, On James and Jordan-von Neumann constants of Lorentz sequence spaces, J. Math. Anal. Appl. 258 (2001) 457-465], the James constant of the 2-dimensional Lorentz sequence space d(2)(ω,q) is computed in the case where 2?q<∞. It is an open problem to compute it in the case where 1?q<2. In this paper, we completely determine the James constant of d(2)(ω,q) in the case where 1?q<2.