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 Ben Wallis Mathematics , 2015, Abstract: Suppose $X$ is a real or complexified Banach space containing a complemented copy of $\ell_p$, $p\in(1,2)$, and a copy (not necessarily complemented) of either $\ell_q$, $q\in(p,\infty)$, or $c_0$. Then $\mathcal{L}(X)$ and $\mathcal{L}(X^*)$ each admit continuum many closed ideals. If in addition $q\geq p'$, $\frac{1}{p}+\frac{1}{p'}=1$, then the closed ideals of $\mathcal{L}(X)$ and $\mathcal{L}(X^*)$ each fail to be linearly ordered. We obtain additional results in the special cases of $\mathcal{L}(\ell_1\oplus\ell_q)$ and $\mathcal{L}(\ell_p\oplus c_0)$, $1  Ryan Causey Mathematics , 2015, Abstract: We construct a class of minimal trees and use these trees to establish a number of coloring theorems on general trees. Among the applications of these trees and coloring theorems are quantification of the Bourgain$\ell_p$and$c_0$indices, dualization of the Bourgain$c_0$index, establishing sharp positive and negative results for constant reduction, and estimating the Bourgain$\ell_p$index of an arbitrary Banach space$X$in terms of a subspace$Y$and the quotient$X/Y$.  Anna Pelczar-Barwacz Mathematics , 2011, Abstract: We present condition on higher order asymptotic behaviour of basic sequences in a Banach space ensuring the existence of bounded non-compact strictly singular operator on a subspace. We apply it in asymptotic$\ell_p$spaces,$1\leq p<\infty$, in particular in convexified mixed Tsirelson spaces and related asymptotic$\ell_p$HI spaces.  Anil Kumar Karn Mathematics , 2012, Abstract: We introduce a notion of p-orthogonality in a general Banach space$1 \le p \le \infty$. We use this concept to characterize$\ell_p$-spaces among Banach spaces and also among complete order smooth p-normed spaces. We further introduce a notion of$p$-orthogonal decomposition in order smooth p-normed spaces. We prove that if the$\infty$-orthogonal decomposition holds in an order smooth$\infty$-normed space, then the 1-orthogonal decomposition holds in the dual space. We also give an example to show that the above said decomposition may not be unique.  Mathematics , 2012, Abstract: A natural class of ideals, almost isometric ideals, of Banach spaces is defined and studied. The motivation for working with this class of subspaces is our observation that they inherit diameter 2 properties and the Daugavet property. Lindenstrauss spaces are known to be the class of Banach spaces that are ideals in every superspace; we show that being an almost isometric ideal in every superspace characterizes the class of Gurariy spaces.  Mathematics , 2014, Abstract: We prove that in the reflexive range$1
 Mathematics , 2015, Abstract: Let $\lambda$ be an infinite cardinal number and let $\ell_\infty^c(\lambda)$ denote the subspace of $\ell_\infty(\lambda)$ consisting of all functions which assume at most countably many non zero values. We classify all infinite dimensional complemented subspaces of $\ell_\infty^c(\lambda)$, proving that they are isomorphic to $\ell_\infty^c(\kappa)$ for some cardinal number $\kappa$. Then we show that the Banach algebra of all bounded linear operators on $\ell_\infty^c(\lambda)$ or $\ell_\infty(\lambda)$ has the unique maximal ideal consisting of operators through which the identity operator does not factor. Using similar techniques, we obtain an alternative to Daws' approach description of the lattice of all closed ideals of $\mathscr{B}(X)$, where $X = c_0(\lambda)$ or $X=\ell_p(\lambda)$ for some $p\in [1,\infty)$, and we classify the closed ideals of $\mathscr{B}(\ell_\infty^c(\lambda))$ that contain the ideal of weakly compact operators.