arXiv subjects
Publications and source records attributed to Jan-David Hardtke.
It has been proved in [J.-D. Hardtke, J. Math. Phys. Anal. Geom. 16, no.2, 119--137 (2020)] that a Köthe-Bochner space $E(X)$ is locally octahedral/locally almost square if $X$ has the respective property and the simple functions are dense in $E(X)$. Here we show that the result still holds true without the density assumption. The proof makes use of the Kuratowski-Ryll-Nardzewski Theorem on measurable selections.
We give a proof of the well-known fact that the category of nearness spaces is bireflective in the category of merotopic spaces which uses Zorn's Lemma instead of the usual construction by transfinite induction.
Roughly speaking, Buckingham's $Π$-Theorem provides a method to "guess" the structure of physical formulas simply by studying the dimensions (the physical units) of the involved quantities. Here we will prove a quantitative version of Buckingham's Theorem, which is "purely mathematical" in the sense that it does make any explicit reference to physical units.
In 2017, O. Deiser and C. Lasser obtained an explicit formula for the $n$-th derivative of the inverse tangent function. We calculate this derivative by a different method based on Faà di Bruno's formula. Comparing the two results leads to the following identity for binomial coefficients: $$\sum_{i=m}^{[n/2]}\frac{(-1)^i}{4^i}\binom{i}{m}\binom{n-i}{i}=\frac{(-1)^m}{2^n}\binom{n+1}{2m+1},$$ where $n,m\in \mathbb{N}_0$ and $m\leq [n/2]$. As was pointed out to the author by C. Krattenthaler, this formula is a special case of Gauß's formula for the hypergeometric function $_2F_1$.
We consider a certain type of geometric properties of Banach spaces, which includes for instance octahedrality, almost squareness, lushness and the Daugavet property. For this type of properties, we obtain a general reduction theorem, which, roughly speaking, states the following: if the property in question is stable under certain finite absolute sums (for example finite $\ell^p$-sums), then it is also stable under the formation of corresponding Köthe-Bochner spaces (for example $L^p$-Bochner spaces). From this general theorem, we obtain as corollaries a number of new results as well as some alternative proofs of already known results concerning octahedral and almost square spaces and their relatives, diameter-two-properties, lush spaces and other classes.
We study the question whether properties like local/weak almost squareness and local octahedrality pass down from an absolute sum $X\oplus_F Y$ to the summands $X$ and $Y$.
The notion of direct integrals introduced by Haydon, Levy and Raynaud in 1991 is a generalisation of the well-known concept of Köthe-Bochner spaces of vector-valued functions (using a family of target spaces instead of just one space). Here we will discuss some classical geometric properties like strict convexity, local uniform convexity and uniform convexity in direct integrals. We will also consider strongly convex and very convex Köthe-Bochner spaces.
We prove some integral inequalities related to Feng Qi's inequality (2000) and obtain a few corollaries.
Given a Banach space $X$, we consider Cesàro spaces $\text{Ces}_p(X)$ of $X$-valued functions over the interval $[0,1]$, where $1\leq p<\infty$. We prove that if $X$ has the Opial/uniform Opial property, then certain analogous properties also hold for $\text{Ces}_p(X)$. We also prove a result on the Opial/uniform Opial property of Cesàro spaces of vector-valued sequences.
A Banach space $X$ is said to have the ball generated property (BGP) if every closed, bounded, convex subset of $X$ can be written as an intersection of finite unions of closed balls. In 2002 S. Basu proved that the BGP is stable under (infinite) $c_0$- and $\ell^p$-sums for $1<p<\infty$. We will show here that for any absolute, normalised norm $\|\cdot\|_E$ on $\mathbb{R}^2$ satisfying a certain smoothness condition the direct sum $X\oplus_E Y$ of two Banach spaces $X$ and $Y$ with respect to $\|\cdot\|_E$ enjoys the BGP whenever $X$ and $Y$ have the BGP.
We prove some results concerning the WORTH property and the García-Falset coefficient of absolute sums of infinitely many Banach spaces. The Opial property/uniform Opial property of infinite $\ell^p$-sums is also studied and some properties analogous to the Opial property/uniform Opial property for Lebesgue-Bochner spaces $L^p(μ,X)$ are discussed.
X. Huang et al. recently introduced the notion of generalised lush (GL) spaces, which, at least for separable spaces, is a generalisation of the concept of lushness introduced by K. Boyko et al. in 2007. The main result of Huang et al. is that every GL-space has the so called Mazur-Ulam property (MUP). In this note, we will prove some properties of GL-spaces (further than those already established by Huang et al.), for example, every $M$-ideal in a GL-space is again a GL-space, ultraproducts of GL-spaces are again GL-spaces, and if the bidual $X^{**}$ of a Banach space $X$ is GL, then $X$ itself still has the MUP.
In 2000 Kadets et al. introduced the notions of acs, luacs and uacs spaces, which form common generalisations of well-known rotundity and smoothness properties of Banach spaces. In a recent preprint the author introduced some further related notions and investigated the behaviour of these geometric properties under the formation of absolute sums. This paper is in a sense a continuation of the previous work. Here we will study the behaviour of said properties under the formation of Köthe-Bochner spaces, thereby generalising some results of Sirotkin on the acs, luacs and uacs properties of $L^p$-Bochner spaces.
We study the notions of acs, luacs and uacs Banach spaces which were introduced by V. Kadets et al. in 2000 and form common generalisations of the usual rotundity and smoothness properties of Banach spaces. In particular, we are interested in (mainly infinite) absolute sums of such spaces. We also introduce some new classes of spaces that lie inbetween those of acs and uacs spaces and study their behaviour under the formation of absolute sums as well.
Recently P. Das, S. Dutta and E. Savas introduced and studied the notions of strong $A^I$-summability with respect to an Orlicz function $F$ and $A^I$-statistical convergence, where $A$ is a non-negative regular matrix and $I$ is an ideal on the set of natural numbers. In this note, we will generalise these notions by replacing $A$ with a family of matrices and $F$ with a family of Orlicz functions or moduli and study the thus obtained convergence methods. We will also give an application in Banach space theory, presenting a generalisation of Simons' $\sup$-$\limsup$-theorem to the newly introduced convergence methods (for the case that the filter generated by the ideal $I$ has a countable base), continuing the author's previous work.
Recently Alan D. Sokal gave a very short and completely elementary proof of the uniform boundedness principle. The aim of this note is to point out that by using a similiar technique one can give a considerably short and simple proof of a stronger statement, namely a principle of condensation of singularities for certain double-sequences of non-linear operators on quasi-Banach spaces, which is a bit more general than a result of I.\,S. Gál.
We extend the well-known Rainwater-Simons convergence theorem to various generalized convergence methods such as strong matrix summability, statistical convergence and almost convergence. In fact we prove these theorems not only for boundaries but for the more general notion of (I)-generating sets introduced by Fonf and Lindenstrauss.
Let $X$ be a real Banach space. A subset $B$ of the dual unit sphere of $X$ is said to be a boundary for $X$, if every element of $X$ attains its norm on some functional in $B$. The well-known Boundary Problem originally posed by Godefroy asks whether a bounded subset of $X$ which is compact in the topology of pointwise convergence on $B$ is already weakly compact. This problem was recently solved by H.Pfitzner in the positive. In this note we collect some stronger versions of the solution to the Boundary Problem, most of which are restricted to special types of Banach spaces. We shall use the results and techniques of Pfitzner, Cascales et al., Moors and others.