Compactness and nuclearity of weighted composition operators on uniform spaces (Q1273289)

Let \((X,\rho_X)\), \((Y,\rho_Y)\) be metric spaces with metrics \(\rho_X\), \(\rho_Y\), as indicated, and let \(\phi:X\to Y\) be a continuous mapping. The space of continuous functions on \(X\) is denoted by \(C(X)\), and if \(A\) is a subspace of \(C(X)\) and \(u\in C(X)\), then the weighted composition operator \(T:A\to C(X)\) is defined by \(T(f)(x)= u(x) f(\phi(x))\), \(x\in A\). If \(\|\cdot\|\) is the norm in \(C(X)\), then a closed subset \(E\subseteq X\) is said to be a peak set with respect to \(A\) if there is a sequence \(\{f_j, j= 1,2,\dots\}\) in \(A\) such that \(\| f_n\|= f_n(x)= 1\) for \(x\in E\), \(n= 1,2,3,\dots\). In the first main result of this paper, the authors state that if the weighted composition operator \(T\) is compact and \(Y\) is a completely connected component of \(\{x\in X:u(x)\neq 0\}\), then for any \(E\) which is a peak set with respect to \(A\), either \(\phi(Y)\subseteq E\) or \(\phi(Y)\cap E=\emptyset\). Among further results derived in the paper are statements relating to the compactness of the operator \(T\) for cases in which (i) \(A= A(B^n)\) is the algebra of analytic functions on the unit ball \(B^n= \{z= (z_1,z_2,\dots, z_n)\in \mathbb{C}^n:(| z_1|^2+| z_2|^2+\cdots+| z_n|^2)< 1\}\); (ii) \(A= A(D^n)\) is the algebra of analytic functions on \(D^n= \{z=(z_1,z_2,\dots, z_n)\in\mathbb{C}^n:| z_k|< 1, 1\leq k\leq n\}\) which are continuous on the closure of \(D^n\). The results relating to the algebras of analytic functions are related to earlier special cases by \textit{H. Kamowitz} [Pac. J. Math. 80, 205-211 (1979; Zbl 0414.47016)] and by \textit{R. K. Singh} and \textit{W. H. Summers} [Proc. Am. Math. Soc. 99, No. 4, 667-670 (1987; Zbl 0635.47032)].