### $(r,p)$-absolutely summing operators on the space $C(T,X)$ and applications.

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We prove that, for a compact metric space X not reduced to a point, the existence of a bilinear mapping ⋄: C(X) × C(X) → C(X) satisfying ||f⋄g|| = ||f|| ||g|| for all f,g ∈ C(X) is equivalent to the uncountability of X. This is derived from a bilinear version of Holsztyński's theorem [3] on isometries of C(X)-spaces, which is also proved in the paper.

We present a characterization of continuous surjections, between compact metric spaces, admitting a regular averaging operator. Among its consequences, concrete continuous surjections from the Cantor set 𝓒 to [0,1] admitting regular averaging operators are exhibited. Moreover we show that the set of this type of continuous surjections from 𝓒 to [0,1] is dense in the supremum norm in the set of all continuous surjections. The non-metrizable case is also investigated. As a consequence, we obtain...

We construct a totally disconnected compact Hausdorff space K₊ which has clopen subsets K₊” ⊆ K₊’ ⊆ K₊ such that K₊” is homeomorphic to K₊ and hence C(K₊”) is isometric as a Banach space to C(K₊) but C(K₊’) is not isomorphic to C(K₊). This gives two nonisomorphic Banach spaces (necessarily nonseparable) of the form C(K) which are isomorphic to complemented subspaces of each other (even in the above strong isometric sense), providing a solution to the Schroeder-Bernstein problem for Banach spaces...

For every countable ordinal α, we construct an ${l}_{1}$-predual ${X}_{\alpha}$ which is isometric to a subspace of $C\left({\omega}^{{\omega}^{{\omega}^{\alpha}+2}}\right)$ and isomorphic to a quotient of $C\left({\omega}^{\omega}\right)$. However, ${X}_{\alpha}$ is not isomorphic to a subspace of $C\left({\omega}^{{\omega}^{\alpha}}\right)$.

For a ${C}^{1}$-function $f$ on the unit ball $\mathbb{B}\subset {\u2102}^{n}$ we define the Bloch norm by ${\parallel f\parallel}_{\U0001d505}=sup\parallel \tilde{d}f\parallel ,$ where $\tilde{d}f$ is the invariant derivative of $f,$ and then show that $${\parallel f\parallel}_{\U0001d505}=\underset{\genfrac{}{}{0pt}{}{z,w\in \mathbb{B}}{z\ne w}}{sup}{(1-|z|}^{2}{)}^{1/2}{(1-|w|}^{2}{)}^{1/2}\frac{\left|f\right(z)-f(w\left)\right|}{|w-{P}_{w}z-{s}_{w}{Q}_{w}z|}.$$