Weyl type theorems and hypercyclic operators II

In this note, the relation between hypercyclic operator matrices (or supercyclic operator matrices) and the operator matrices which satisfy Weyl type theorems is discussed. Also, using a variant of the essential approximate point spectrum, we give the necessary and sufficient conditions for for which a-Browder's theorem or a-Weyl's theorem holds.

     

A note on the colorful fractional Helly theorem

Helly’s theorem is a classical result concerning the intersection patterns of convex sets in ${\mathbb{R}}^d$. Two important generalizations are the colorful version and the fractional version. Recently, Bárány et al. combined the two, obtaining a colorful fractional Helly theorem. In this paper, we give an improved version of their result.     

An Intersection Theorem on an Unbounded Set and Its Application to the Fair Allocation Problem

We prove the following theorem. Let m and n be any positive integers with mn, and let be a subset of the n-dimensional Euclidean space ⁿ . For each i=1, . . . , m, there is a class of subsets M _i ^j of ^Tn . Assume that for each i=1, . . . , m, that M _i ^j is nonempty and closed for all i, j, and that there exists a real number B(i, j) such that and its jth component _{xjB(i, j)} imply . Then, there exists a partition of {1, . . . , n} such that for all i and We prove this theorem based upon a generalization of a well-known theorem of Birkhoff and von Neumann. Moreover, we apply this theorem to the fair allocation problem of indivisible objects with money and obtain an existence theorem.     

An Elementary Proof of the Fundamental Theorem of Projective Geometry (Dedicated to Alfred Frölicher)

The following version of the fundamental theorem is proved: Let V, W be vector spaces and g: P(V)\E P(W) a morphism between the associated projective spaces. If the image of g is not contained in a line, then there exists a semilinear map f: V W which induces g. The difficulty lies in the fact that the homomorphism of division rings associated to the map f can be nonsurjective. As an application, a short proof of Wigner's theorem is also proposed.     

Essential point spectra of operator matrices trough local spectral theory

For A∈B(X), B∈B(Y) and C∈B(Y,X), let MC be the operator defined on X⊕Y by . In this paper, we study defect set (Σ(A)∪Σ(B))?Σ(MC), where Σ is the Browder spectrum, the essential approximate point spectrum and Browder essential approximate point spectrum. We then give application for Weyl's and Browder's theorems.     

Essential approximate point spectra and Weyl's theorem for operator matrices

When A∈B(H) and B∈B(K) are given, we denote by MC the operator acting on the infinite dimensional separable Hilbert space H⊕K of the form . In this paper, it is shown that a 2×2 operator matrix MC is upper semi-Fredholm and ind(MC)?0 for some C∈B(K,H) if and only if A is upper semi-Fredholm and

     

-Weyl's theorem for operator matrices

If is a upper triangular matrix on the Hilbert space , then -Weyl's theorem for and need not imply -Weyl's theorem for , even when . In this note we explore how -Weyl's theorem and -Browder's theorem survive for operator matrices on the Hilbert space.

     

A perturbation result for the asymptotic behavior of matrix powers

We prove a perturbation result for the asymptotic behavior of the sequence (A ⁿ c) _nN , whereAG|(d), the space of invertibled×d matrices, andc ^d .