August 2014 - Banach - mathdept.okstate.edu

Abstract of a paper by Julio Becerra Guerrero, Gines Lopez-Perez and Abraham Rueda Zoca
by alspach＠math.okstate.edu 08 Aug '14

08 Aug '14

This is an announcement for the paper "Octahedral norms in spaces of operators" by Julio Becerra Guerrero, Gines Lopez-Perez and Abraham Rueda Zoca. Abstract: We study octahedral norms in the space of bounded linear operators between Banach spaces. In fact, we prove that $L(X,Y)$ has octahedral norm whenever $X^*$ and $Y$ have octahedral norm. As a consequence the space of operators $L(\ell_1 ,X)$ has octahedral norm if, and only if, $X$ has octahedral norm. These results also allows us to get the stability of strong diameter 2 property for projective tensor products of Banach spaces, which is an improvement of the known results about the size of nonempty relatively weakly open subsets in the unit ball of the projective tensor product of Banach spaces. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B22 Remarks: 16 pages Submitted from: glopezp(a)ugr.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1407.6038 or http://arXiv.org/abs/1407.6038

1 0

Abstract of a paper by Ben Wallis
by alspach＠math.okstate.edu 08 Aug '14

08 Aug '14

This is an announcement for the paper "Constructing Banach ideals using upper $\ell_p$-estimates" by Ben Wallis. Abstract: Using upper $\ell_p$-estimates for normalized weakly null sequence images, we describe a new family of operator ideals $\mathcal{WD}_{\ell_p}^{(\infty,\xi)}$ with parameters $1\leq p\leq\infty$ and $1\leq\xi\leq\omega_1$. These classes contain the completely continuous operators, and are distinct for all choices $1\leq p\leq\infty$ and, when $p\neq 1$, for all choices $\xi\neq\omega_1$. For the case $\xi=1$, there exists an ideal norm $\|\cdot\|_{(p,1)}$ on the class $\mathcal{WD}_{\ell_p}^{(\infty,1)}$ under which it forms a Banach ideal. Archive classification: math.FA Mathematics Subject Classification: 47L20, 46B45, 46A45, 46B25 Submitted from: wallis(a)math.niu.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1407.5948 or http://arXiv.org/abs/1407.5948

1 0

Abstract of a paper by Iian B. Smythe
by alspach＠math.okstate.edu 08 Aug '14

08 Aug '14

This is an announcement for the paper "Borel equivalence relations in the space of bounded operators" by Iian B. Smythe. Abstract: We consider various notions of equivalence in the space of bounded operators on a Hilbert space, including modulo finite rank operators, modulo Schatten $p$-classes, and modulo compact operators. Using Hjorth's theory of turbulence, the latter two are shown to be not classifiable by countable structures, while the first cannot be reduced to the orbit equivalence relation of any Polish group action. The results for modulo finite rank and modulo compact operators are also shown for the restrictions of these equivalence relations to the space of projection operators. Families of non-classifiable equivalence relations on sequence spaces are described and utilized in these results. Archive classification: math.LO math.OA Mathematics Subject Classification: Primary 03E15, 47B10, Secondary 47C15, 46A45 Remarks: 36 pages Submitted from: ibs24(a)cornell.edu The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1407.5325 or http://arXiv.org/abs/1407.5325

1 0