and Applied Analysis 3 The following lemma is the crucial criterion for the compactness of Cφ, whose proof is an easy modification of the proof of Proposition 3.11 in 1 . Lemma 2.4. Assume that φ is a holomorphic self-map of D. Then Cφ : Bp → Bq is compact if and only if Cφ is bounded and for any bounded sequence {fm}m∈N in Bp which converges to zero uniformly on compact subsets of D, we have ∥...

In [V. Mathai, K-theory of twisted group C∗-algebras and positive scalar curvature, Contemp. Math. 231 (1999) 203–225], we established a natural connection between the Baum-Connes conjecture and noncommutative Bloch theory, viz., the spectral theory of projectively periodic elliptic operators on covering spaces. We elaborate on this connection here and provide significant evidence for a fundame...

Let φ and ψ be analytic self-maps of the open unit disk D . Using pseudo-hyperbolic distance ρ(φ ,ψ) , we characterize the boundedness and compactness of the differences of generalized composition operators (C φ −Ch ψ ) f (z) = ∫ z 0 [ f ′(φ(ξ ))g(ξ )− f ′(ψ(ξ ))h(ξ )]dξ , z ∈ D between two Bloch-type spaces on D . The results generalize the corresponding results on the single generalized compo...

Let φ(z) = (φ 1 (z),. .. , φ n (z)) be a holomorphic self-map of U n and ψ(z) a holomorphic function on U n , where U n is the unit polydisk of C n. Let p ≥ 0, q ≥ 0, this paper gives some necessary and sufficient conditions for the weighted composition operator W ψ,φ induced by ψ and φ to be bounded and compact between p-Bloch space B p (U n) and q-Bloch space B q (U n).