Path connected components in weighted composition operators on $h^\infty $ and $H^\infty $ with the operator norm

Abstract

We consider the component problem on the sets of weighted composition operators on the spaces of bounded harmonic and analytic functions on the open unit disk with the operator norms, respectively. Especially, we shall determine path connected components in the sets of noncompact weighted composition operators. u(z) = ∂D u(e iθ)P z (e iθ) dm(e iθ) for z ∈ D, where P z is the Poisson kernel for the point z ∈ D and m is the normalized Lebesgue measure on ∂D. Then u ∈ h ∞. Besides, for f ∈ h ∞ , f * = f on D. Let S(D) be the set of analytic self-maps of D. For each ϕ ∈ S(D), we may define the composition operator C ϕ by C ϕ f = f •ϕ, where f is analytic on D. Composition operators on various analytic function spaces have been studied extensively during the past few decades. See [2, 14] for an overview of these results. Presently, some of the open questions in this field are related to the topological structure of the set of composition operators. For even f ∈ h ∞ , C ϕ f = f • ϕ is also harmonic on D. So the question of properties of C ϕ on h ∞ naturally arises. In [1], Choa and the first and third authors first considered properties of composition operators on h ∞ and, although h ∞ is not algebra, introduced the concept of weighted composition operators M u C ϕ on h ∞ defined by (M u C ϕ f)(z) = ∂D u(e iθ)(f • ϕ) * (e iθ)P z (e iθ) dm(e iθ)

Weighted Composition Operators Between Different Fock Spaces

“…Weighted composition operators on these spaces appeared in some works (see, for instance, [6,7,8,12]) with different applications. There is a great number of topics on operators of such a type: boundedness and compactness [5,10], compact differences [18], topological structure [3,14,20,21], dynamical and ergodic properties [1,2,27]. On many spaces, these topics are difficult and not yet solved completely.…”

Boundary vs. interior conditions associated with weighted composition operators

“…We expect that the technique discussed here would be valuable in other problems concerning composition operators; e.g. using the argument developed here, the authors have determined path connected components in the spaces of noncompact weighted composition operators on ∞ and H ∞ with the operator and the essential operator norms in [11,12], respectively.…”

Ограниченные Композиционные Операторы На Весовых Функциональных Пространствах На Единичном Круге

“…Many studies have been done on composition operators on various different spaces of holomorphic functions on the unit disk or the unit ball, such as Hardy spaces, Bergman spaces, Dirichlet spaces, spaces of bounded holomorphic function and weighted Banach spaces with sup-norm (see, e.g., [4][5][6][7][8][9]).…”