arXiv subjects

Anton R. Schep

Publications and source records attributed to Anton R. Schep.

3 recordsLinked to original sources

arXiv2022

The essential spectrum, norm, and spectral radius of abstract multiplication operators

Let $E$ be a complex Banach lattice and $T$ is an operator in the centrum $Z(E)=\{T: |T|\le λI \mbox{ for some } λ\}$ of $E$. Then the essential norm $\|T\|_{e}$ of $T$ equals the essential spectral radius $r_{e}(T)$ of $T$. We also prove $r_{e}(T)=\max\{\|T_{A^{d}}\|, r_{e}(T_{A})\}$, where $T_{A}$ is the atomic part of $T$ and $T_{A^{d}}$ is the non-atomic part of $T$. Moreover $r_{e}(T_{A})=\limsup_{\mathcal F}λ_{a}$, where $\mathcal F$ is the Fréchet filter on the set $A$ of all positive atoms in $E$ of norm one and $λ_{a}$ is given by $T_{A}a=λ_{a}a$ for all $a\in A$.

math.FA↗

arXiv2016

Two-sided multiplication operators on the space of regular operators

Let $W$, $X$, $Y$ and $Z$ be Dedekind complete Riesz spaces. For $A\in L^{r}(Y, Z)$ and $B\in L^{r}(W, X)$ let $M_{A,\,B}$ be the two-sided multiplication operator from $L^{r}(X, Y)$ into $L^r(W,\,Z)$ defined by $M_{A,\,B}(T)=ATB$. We show that for every $0\leq A_0\in L^{r}_{n}(Y, Z)$, $|M_{A_0, B}|(T)=M_{A_0, |B|}(T)$ holds for all $B\in L^{r}(W, X)$ and all $T\in L^{r}_{n}(X, Y)$. Furthermore, if $W$, $X$, $Y$ and $Z$ are Dedekind complete Banach lattices such that $X$ and $Y$ have order continuous norms, then $|M_{A,\, B}|=M_{|A|, \,|B|}$ for all $ A\in L^{r}(Y, Z)$ and all $B\in L^{r}(W, X)$. Our results generalize the related results of Synnatzschke and Wickstead, respectively.

math.FA↗

arXiv2008

Products and Factors of Banach function spaces

Given two Banach function spaces we study the pointwise product space E.F, especially for the case that the pointwise product of their unit balls is again convex. We then give conditions on when the pointwise product E . M(E, F)=F, where M(E,F) denotes the space of multiplication operators from E into F.

math.FA↗