Topological structure of the space of composition operators onH ?

Differences of Composition Operators Between Weighted Banach Spaces of Holomorphic Functions

“…However, it is not compact, since w(r ) v(φ(r )) ρ(φ(r ), ψ(r )) = 8 (1 − r ) 2 t (r − 1) 3 For examples of compact and noncompact differences of composition operators C φ − C ψ : H ∞ → H ∞ , see [19,Example 1]. The change of the behaviour of the operator C φ − C ψ depending on the weights v and w is emphasized in our last example.…”

“…The case of operators defined on weighted Banach spaces of the type defined above was treated, for example, in [4,5] and [6]. Differences of composition operators have been investigated more recently; see [10,13,14,19] and [20]. In this article we are mainly interested in finding an expression for the essential norm C φ − C ψ e , that is, the distance of C φ − C ψ to the space of compact operators, when C φ − C ψ is a bounded operator from H ∞ v into H ∞ w ; compare with [10] and [12] for the case of H ∞ .…”

ESSENTIAL NORMS OF LINEAR COMBINATIONS OF COMPOSITION OPERATORS ON h∞

“…Hence C ϕ1 − C ϕ2 e = 0. By [12,Theorem 3], lim z→1 ρ(ϕ 1 (z), ϕ 2 (z)) = 0. By Theorem 2.5, C ϕ1 − C ϕ2 < 2.…”

“…Recently, the norm, the essential norm and the topological structure of composition operators on H ∞ have been studied (see [5,6,8,9,10,12]). But the exact value of the essential norm of the difference of composition operators C ϕ −C ψ e on H ∞ is not yet known.…”

Weighted composition operators between weighted Banach spaces of holomorphic functions and weighted Bloch type space

“…Composition operators and weighted composition operators acting between various spaces of analytic functions have been investigated by several authors (see e.g. [13], [7], [11], [2], [4], [3], [8], [12]). In [13] and [12] weighted composition operators between weighted Bloch type spaces resp.…”