This paper gives a note on weighted composition operators on the weighted Bergman space, which shows that for a fixed composition symbol, the weighted composition operators are bounded on the weighted Bergman space only with bounded weighted symbols if and only if the composition symbol is a finite Blaschke product.
We use the Weierstrass a-function associated with a lattice in the complex plane to construct finite dimensional zero-based subspaces and quasi-invariant subspaces of given index in the Bargmann-Fock space.
The authors introduce a kind of slowly increasing cohomology HS＊（X） for a discrete metric space X with polynomial growth, and construct a character map from the slowly increasing cohomology HS＊ （X） into HC＊ont （S（X））, the continuous cyclic cohomol- ogy of the smooth subalgebra S（X） of the uniform Roe algebra B＊ （X）. As an application, it is shown that the fundamental cocycle, associated with a uniformly contractible complete Riemannian manifold M with polynomial volume growth and polynomial contractibility radius growth, is slowly increasing.
