It is shown that for every pair of natural numbers $m\geq n\geq 1$, there exists a compact Fréchet space $X_{m,n}$ such that \item {(a)} $\mathop{\rm dim}X_{m,n}=n$, $\mathop{\rm ind}X_{m,n}=\mathop{\rm Ind}X_{m,n}=m$, and \item {(b)} every component of $X_{m,n}$ is homeomorphic to the $n$-dimensional cube $I^n$. \endgraf \noindent This yields new counter-examples to the theorem on dimension-lowering maps in the cases of inductive dimensions.
In this paper, we give some estimates for the essential norm and a new characterization for the boundedness and compactness of weighted composition operators from weighted Bergman spaces and Hardy spaces to the Bloch space., Songxiao Li, Ruishen Qian, Jizhen Zhou., and Obsahuje bibliografické odkazy
We consider a nonnegative superbiharmonic function $w$ satisfying some growth condition near the boundary of the unit disk in the complex plane. We shall find an integral representation formula for $w$ in terms of the biharmonic Green function and a multiple of the Poisson kernel. This generalizes a Riesz-type formula already found by the author for superbihamonic functions $w$ satisfying the condition $0\le w(z)\le C(1-|z|)$ in the unit disk. As an application we shall see that the polynomials are dense in weighted Bergman spaces whose weights are superbiharmonic and satisfy the stated growth condition near the boundary.
We study sub-Bergman Hilbert spaces in the weighted Bergman space $A^2_\alpha $. We generalize the results already obtained by Kehe Zhu for the standard Bergman space $A^2$.