GEOMETRIC AND FIXED POINT PROPERTIES IN PRODUCTS OF NORMED SPACES

Given two (real) normed (linear) spaces $X$ and $Y$, let $X\otimes _{1}Y=(X\otimes Y,\Vert \cdot \Vert )$, where $\Vert (x,y)\Vert =\Vert x\Vert +\Vert y\Vert$. It is known that $X\otimes _{1}Y$ is $2$-UR if and only if both $X$ and $Y$ are UR (where we use UR as an abbreviation for uniformly rotund). We prove that if $X$ is $m$-dimensional and $Y$ is $k$-UR, then $X\otimes _{1}Y$ is $(m+k)$-UR. In the other direction, we observe that if $X\otimes _{1}Y$ is $k$-UR, then both $X$ and $Y$ are $(k-1)$-UR. Given a monotone norm $\Vert \cdot \Vert _{E}$ on $\mathbb{R}^{2}$, we let $X\otimes _{E}Y=(X\otimes Y,\Vert \cdot \Vert )$ where $\Vert (x,y)\Vert =\Vert (\Vert x\Vert _{X},\Vert y\Vert _{Y})\Vert _{E}$. It is known that if $X$ is uniformly rotund in every direction, $Y$ has the weak fixed point property for nonexpansive maps (WFPP) and $\Vert \cdot \Vert _{E}$ is strictly monotone, then $X\otimes _{E}Y$ has WFPP. Using the notion of $k$-uniform rotundity relative to every $k$-dimensional subspace we show that this result holds with a weaker condition on $X$.

COMPACT FACTORIZATION OF OPERATORS WITH λ-COMPACT ADJOINTS

Let λ be a symmetric, normal sequence space equipped with a k-symmetric, monotone norm ‖.‖λ. Also, assume that (λ, ‖.‖λ) is AK-BK. Corresponding to this sequence space λ, we study compactness of the operator ideal Kλ. We proved compactness, completeness and injectivity of the dual operator ideal Kλd. We also investigate the factorization of these operators.

THE FIXED POINT PROPERTY IN DIRECT SUMS AND MODULUS

We show that the direct sum $\mathop{({X}_{1} \oplus \cdots \oplus {X}_{r} )}\nolimits_{\psi } $ with a strictly monotone norm has the weak fixed point property for nonexpansive mappings whenever $M({X}_{i} )\gt 1$ for each $i= 1, \ldots , r$. In particular, $\mathop{({X}_{1} \oplus \cdots \oplus {X}_{r} )}\nolimits_{\psi } $ enjoys the fixed point property if Banach spaces ${X}_{i} $ are uniformly nonsquare. This combined with the earlier results gives a definitive answer for $r= 2$: a direct sum ${X}_{1} {\mathop{\oplus }\nolimits}_{\psi } {X}_{2} $ of uniformly nonsquare spaces with any monotone norm has the fixed point property. Our results are extended to asymptotically nonexpansive mappings in the intermediate sense.