We study the structure of Lipschitz and Hölder-type spaces and their preduals on general metric spaces, and give applications to the uniform structure of Banach spaces. In particular we resolve a problem of Weaver who asks wether if M is a compact metric space and 0 < α < 1, it is always true the space of Hölder continuous functions of class α is isomorphic to l∞. We show that, on the contrary, if M is a compact convex subset of a Hilbert space this isomorphism holds if and only if...

Let $E$ be a complex Banach space, with the unit ball $B_E$. We study the spectrum of a bounded weighted composition operator $u{C}_{\varphi }$ on $H^{\infty }\left(B_E\right)$ determined by an analytic symbol $\varphi$ with a fixed point in $B_E$ such that $\varphi \left(B_E\right)$ is a relatively compact subset of $E$, where $u$ is an analytic function on $B_E$.

Let $\varphi$ be an entire self-map of ${ℂ}^N$, $u_0$ be an entire function on ${ℂ}^N$ and $𝐮=(u_1,\cdots ,u_N)$ be a vector-valued entire function on ${ℂ}^N$. We extend the Stević-Sharma type operator to the classcial Fock spaces, by defining an operator $T_{u_0,𝐮,\varphi }$ as follows: $$-.4pt{T}_{u_0,𝐮,\varphi }f=u_0·f\circ \varphi +\sum _{i=1}^N{u}_i·\frac{\partial f}{\partial z_i}\circ \varphi .$$ We investigate the boundedness and compactness of $T_{u_0,𝐮,\varphi }$ on Fock spaces. The complex symmetry and self-adjointness of $T_{u_0,𝐮,\varphi }$ are also characterized.

Let ${\left\{\beta \left(n\right)\right\}}_{n=0}^{\infty }$ be a sequence of positive numbers and $1\le p<\infty$. We consider the space $H^p\left(\beta \right)$ of all power series $f\left(z\right)={\sum }_{n=0}^{\infty }\widehat{f}\left(n\right)z^n$ such that ${\sum }_{n=0}^{\infty }{\left|\widehat{f}\left(n\right)\right|}^p\beta {\left(n\right)}^p<\infty$. We investigate strict cyclicity of $H_p^{\infty }\left(\beta \right)$, the weakly closed algebra generated by the operator of multiplication by $z$ acting on $H^p\left(\beta \right)$, and determine the maximal ideal space, the dual space and the reflexivity of the algebra $H_p^{\infty }\left(\beta \right)$. We also give a necessary condition for a composition operator to be bounded on $H^p\left(\beta \right)$ when $H_p^{\infty }\left(\beta \right)$ is strictly cyclic.

This paper gives a characterization of surjective isometries on spaces of continuously differentiable functions with values in a finite-dimensional real Hilbert space.