Rainis Haller

University of Tartu · Institute of Mathematics

We prove that the Lipschitz-free space over a metric space M is locally almost square whenever M is a length space. Consequently, the Lipschitz-free space is locally almost square if and only if it has the Daugavet property. We also show that a Lipschitz-free space is never almost square.

We prove that Banach spaces ℓ1 ⊕2 R and X ⊕∞ Y , with strictly convex X and Y , have plastic unit balls (we call a metric space plastic if every non-expansive bijection from this space onto itself is an isometry).

We solve some open problems regarding diameter two properties within the class of Banach spaces of real-valued Lipschitz functions by using the de Leeuw transform. Namely, we show that: the diameter two property, the strong diameter two property, and the symmetric strong diameter two property are all different for these spaces of Lipschitz function...

We prove that the Lipschitz-free space over a metric space M is locally almost square whenever M is a length space. Consequently, the Lipschitz-free space is locally almost square if and only if it has the Daugavet property. We also show that a Lipschitz-free space is never almost square.

We prove that Banach spaces $\ell_1\oplus_2\mathbb{R}$ and $X\oplus_\infty Y$, with strictly convex $X$ and $Y$, have plastic unit balls (we call a metric space plastic if every non-expansive bijection from this space onto itself is an isometry).

Inspired by R. Whitley's thickness index the last named author recently introduced the Daugavet index of thickness of Banach spaces. We continue the investigation of the behavior of this index and also consider two new versions of the Daugavet index of thickness, which helps us solve an open problem which connect the Daugavet indices with the Dauga...

A Δ-point x of a Banach space is a norm-one element that is arbitrarily close to convex combinations of elements in the unit ball that are almost at distance 2 from x . If, in addition, every point in the unit ball is arbitrarily close to such convex combinations, x is a Daugavet point. A Banach space X has the Daugavet property if and only if ever...

A Daugavet-point (resp.~$\Delta$-point) of a Banach space is a norm one element $x$ for which every point in the unit ball (resp.~element $x$ itself) is in the closed convex hull of unit ball elements that are almost at distance 2 from $x$. A Banach space has the well-known Daugavet property (resp.~diametral local diameter 2 property) if and only i...

We study Banach spaces with the property that, given a finite number of slices of the unit ball, there exists a direction such that all these slices contain a line segment of length almost 2 in this direction. This property was recently named the symmetric strong diameter two property by Abrahamsen, Nygaard, and P\~oldvere. The symmetric strong dia...

A $\Delta$-point $x$ of a Banach space is a norm one element that is arbitrarily close to convex combinations of elements in the unit ball that are almost at distance $2$ from $x$. If, in addition, every point in the unit ball is arbitrarily close to such convex combinations, $x$ is a Daugavet-point. A Banach space $X$ has the Daugavet property if...

We introduce and study Banach spaces which have property CWO, i.e., every finite convex combination of relatively weakly open subsets of their unit ball is open in the relative weak topology of the unit ball. Stability results of such spaces are established, and we introduce and discuss a geometric condition---property (co)---on a Banach space. Pro...

We study Banach spaces with the property that, given a finite number of slices of the unit ball, there exists a direction such that all these slices contain a line segment of length almost 2 in this direction. This property was recently named the symmetric strong diameter two property by Abrahamsen, Nygaard, and P\~oldvere. The symmetric strong dia...

We prove that the following three properties for a Banach space are all different from each other: every finite convex combination of slices of the unit ball is (1) relatively weakly open, (2) has nonempty interior in relative weak topology of the unit ball, and (3) intersects the unit sphere. In particular, the $1$-sum of two Banach spaces does no...

We prove that, if Banach spaces $X$ and $Y$ are $\delta$-average rough, then their direct sum with respect to an absolute norm $N$ is $\delta/N(1,1)$-average rough. In particular, for octahedral $X$ and $Y$ and for $p$ in $(1,\infty)$ the space $X\oplus_p Y$ is $2^{1-1/p}$-average rough, which is in general optimal. Another consequence is that for...

We prove that, if Banach spaces $X$ and $Y$ are $\delta$-average rough, then their direct sum with respect to an absolute norm $N$ is $\delta/N(1,1)$-average rough. In particular, for octahedral $X$ and $Y$ and for $p$ in $(1,\infty)$ the space $X\oplus_p Y$ is $2^{1-1/p}$-average rough, which is in general optimal. Another consequence is that for...

We investigate sufficient and necessary conditions for the space of bounded linear operators between two Banach spaces to be rough or average rough. Our main result is that $\mathcal L(X,Y)$ is $\delta$-average rough whenever $X^\ast$ is $\delta$-average rough and $Y$ is alternatively octahedral. This allows us to give a unified improvement of two...

We investigate sufficient and necessary conditions for the space of bounded linear operators between two Banach spaces to be rough or average rough. Our main result is that $\mathcal L(X,Y)$ is $\delta$-average rough whenever $X^\ast$ is $\delta$-average rough and $Y$ is alternatively octahedral. This allows us to give a unified improvement of two...

We prove that the diametral strong diameter 2 property of a Banach space (meaning that, in convex combinations of relatively weakly open subsets of its unit ball, every point has an "almost diametral" point) is stable under 1-sums, i.e., the direct sum of two spaces with the diametral strong diameter 2 property equipped with the 1-norm has again th...

We prove that the diametral strong diameter 2 property of a Banach space (meaning that, in convex combinations of relatively weakly open subsets of its unit ball, every point has an "almost diametral" point) is stable under 1-sums, i.e., the direct sum of two spaces with the diametral strong diameter 2 property equipped with the 1-norm has again th...

A Banach space is said to have the diameter 2 property if the diameter of every nonempty relatively weakly open subset of its unit ball equals 2. In a paper by Abrahamsen, Lima, and Nygaard (Remarks on diameter 2 properties. J. Conv. Anal., 2013, 20, 439–452), the strong diameter 2 property is introduced and studied. This is the property that the d...

We discuss the geometry of Banach spaces whose norm is octahedral or, more generally, locally or weakly octahedral. Our main results characterize these spaces in terms of covering of the unit ball.

It is known that a Banach space has the strong diameter 2 property (i.e. every convex combination of slices of the unit ball has diameter 2) if and only if the norm on its dual space is octahedral (a notion introduced by Godefroy and Maurey). We introduce two more versions of octahedrality, which turn out to be dual properties to the diameter 2 pro...

We study the position of compact operators in the space of all continuous linear operators and its subspaces in terms of ideals. One of our main results states that for Banach spaces X and Y the subspace of all compact operators K (X, Y) is an M(r 1r 2, s 1s 2)-ideal in the space of all continuous linear operators L(X, Y) whenever K (X,X) and K (Y,...

We study Banach spaces X and Y for which the subspace of all compact operators K(X,Y) forms an ideal satisfying the M(r,s)-inequality in the space of all continuous linear operators L(X,Y). We prove that K(X,Y) is an M(r12r2,s12s2)- and an M(r1r22,s1s22)-ideal in L(X,Y) whenever K(X) and L(Y) are M(r1,s1)- and M(r2,s2)-ideals in span(K(X)∪IX) and s...

The aim of the Kaczmarz algorithm is to reconstruct an element in a Hilbert space from data given by inner products of this element with a given sequence of vectors. The main result characterizes sequences of vectors leading to reconstruction of any element in the space. This generalizes some results of Kwapie n and Mycielski. 1. Introduction. Letf...

In this note we study the transitivity of M(r,s)-inequalities and geometry of higher duals of Banach spaces. Our main theorem shows that if X and Y are closed subspaces of a Banach space Z such that X is an ideal satisfying the M(r,s)-inequality in Y and Y is an ideal satisfying the M(r,s)-inequality in Z, then X is an ideal satisfying the M(r/(2−r...

It is well–known that a Banach space X is an M–ideal in its bidual whenever the space K(X, X) of compact operators on X is an M–ideal in the space ℒ(X, X) of bounded operators. The same conclusion holds whenever the space (1, X) of compact operators from 1 to X is an M–ideal in ℒ(1, X). In the present paper, these results are extended by means of a...

Let r h 1 and s be positive numbers. We prove that a Banach space X satisfies the M (r, s)-inequality (i.e. || x***|| ³ r||px***||+s||x***-px***|| "x*** Î X*** \| x^{\ast \ast \ast }\|\geq r\|\pi x^{\ast \ast \ast }\|+s\|x^{\ast \ast \ast }-\pi x^{\ast \ast \ast }\|\qquad \forall x^{\ast \ast \ast }\in X^{\ast \ast \ast }where p \pi is the canonica...

Thesis (doctoral)--Universitas Tartuensis, 2002.