Let $(X,d)$ be an infinite compact metric space, let $(B,parallel . parallel)$ be a unital Banach space, and take $alpha in (0,1).$ In this work, at first we define the big and little $alpha$-Lipschitz vector-valued (B-valued) operator algebras, and consider the little $alpha$-lipschitz $B$-valued operator algebra, $lip_{alpha}(X,B)$. Then we characterize its second dual space.

we establish a sharp maximal function estimate for some vector-valued multilinear singular integral operators. as an application, we obtain the $(l^p, l^q)$-norm inequality for vector-valued multilinear operators.

‎let $x,y$ be normed spaces with $l(x,y)$ the space of continuous‎ ‎linear operators from $x$ into $y$‎. ‎if ${t_{j}}$ is a sequence in $l(x,y)$,‎ ‎the (bounded) multiplier space for the series $sum t_{j}$ is defined to be‎ [ ‎m^{infty}(sum t_{j})={{x_{j}}in l^{infty}(x):sum_{j=1}^{infty}%‎ ‎t_{j}x_{j}text{ }converges}‎ ‎]‎ ‎and the summing operator $s:m^{infty}(sum t_{j})rightarrow y$ associat...

we establish a sharp maximal function estimate for some vector-valued multilinear singular integral operators. as an application, we obtain the $(l^p, l^q)$-norm inequality for vector-valued multilinear operators.

‎Let $X,Y$ be normed spaces with $L(X,Y)$ the space of continuous‎ ‎linear operators from $X$ into $Y$‎. ‎If ${T_{j}}$ is a sequence in $L(X,Y)$,‎ ‎the (bounded) multiplier space for the series $sum T_{j}$ is defined to be‎ [ ‎M^{infty}(sum T_{j})={{x_{j}}in l^{infty}(X):sum_{j=1}^{infty}%‎ ‎T_{j}x_{j}text{ }converges}‎ ‎]‎ ‎and the summing operator $S:M^{infty}(sum T_{j})rightarrow Y$ associat...

let $t$ be a bounded operator on the banach space $x$ and $ph$ be an analytic self-map of the unit disk $bbb{d}$‎. ‎we investigate some operator theoretic properties of‎ ‎bilateral composition operator $c_{ph‎, ‎t}‎: ‎f ri t circ f circ ph$ on the vector-valued hardy space $h^p(x)$ for $1 leq p leq‎ ‎+infty$.‎ ‎compactness and weak compactness of $c_{ph‎, ‎t}$ on $h^p(x)$‎ ‎are characterized an...

Let $T$ be a bounded operator on the Banach space $X$ and $ph$ be an analytic self-map of the unit disk $Bbb{D}$‎. ‎We investigate some operator theoretic properties of‎ ‎bilateral composition operator $C_{ph‎, ‎T}‎: ‎f ri T circ f circ ph$ on the vector-valued Hardy space $H^p(X)$ for $1 leq p leq‎ ‎+infty$.‎ ‎Compactness and weak compactness of $C_{ph‎, ‎T}$ on $H^p(X)$‎ ‎are characterized an...