Dear all,
The next Banach spaces webinar is on Friday December 4 at 9AM CDT (e.g., Dallas, TX time). Please join us at
https://unt.zoom.us/j/512907580
Speaker: Thomas Schlumprecht (Texas A&M) Title: Banach Spaces which admit lots of closed Operator Ideals
Abstract. We present general conditions which imply that for a Banach space $X$, which has an unconditional basis, the space of bounded linear operators $ L(X)$ has $2^{\frak c}$ ``small'' closed ideals (ideals which are generated by finitely strictly singular operators). The class of spaces which satisfy these conditions include:\ $\ell_p\oplus \ell_q$, $1<p<q<\infty$,\ $\ell_1\oplus \ell_p$, $c_0\oplus \ell_p$, $\ell_\infty\oplus \ell_p$, $1<p<\infty$,\ $T^p_\xi\oplus T^q_\xi$, $1< p<q<\infty$, $T^p_\xi$ being the $p$-convexification of the Tsireson space of order $\xi<\omega_1$,\ $S^p_\xi$, $1\le p<\infty$, $S^p_\xi$ being the $p$-convexification of the Schreier space of order $\xi<\omega_1$,
Using arguments by Beanland, Kania, Laustsen, as well as Gasparis and Leung we show that $\mathcal L(S^p_\xi)$, and $\mathcal L(T^p_\xi)$, for $\xi<\omega$, has $2^{\frak c}$ ``large'' closed ideals (ideals generated by projections on subspaces which are spanned by subsequences of the basis). Moreover, using an unpublished argument by Johnson, and showing a combinatorial result on higher order Schreier families, we also deduce that $\mathcal L(T^p_\xi)$, for $\xi<\omega_1$, has $2^{\frak c}$ large closed ideals.
Part of this talk is on joint work with Dan Freeman and 'Andras Zsak.
For more information about the past and future talks, and videos please visit the webinar website http://www.math.unt.edu/~bunyamin/banach
Best regards, Bunyamin
Bunyamin Sari