Simply connected topological spaces of weighted composition operators

1 Introduction

Let H ( B N ) be the space of analytic functions on the open unit ball

and H ∞ ( B N ) the space of bounded analytic functions on B N with the supremum norm ∥ ⋅ ∥ ∞ . When N = 1 , the unit open ball reduces to the unit open disk D in the complex plane ℂ . Let S ( B N ) be the set of analytic self-maps of B N . Every φ = ( φ 1 , … , φ N ) ∈ S ( B N ) induces a composition operator C φ defined by C φ f = f ∘ φ for f ∈ H B N . If u ∈ H ( B N ) , the multiplication operator M u : H ( B N ) → H ( B N ) is defined by M u ( f ) = u ⋅ f for any f ∈ H ( B N ) . If u ∈ H ( B N ) and φ ∈ S ( B N ) , we define the weighted composition operator W u , φ to be the product M u C φ .

Much effort has been expended on characterizing those analytic maps which induce bounded or compact composition operators between classical spaces of analytic functions. Readers interested in this topic can refer to [1,2,3,4], all of which are excellent sources for the development of the theory of composition operators and function spaces.

An active topic is the topological structure of the space of composition operators acting on a given function space. When X is a Banach space of analytic functions, we write C ( X ) for the topological space of composition operators on X under the operator norm topology. The investigation of the topological structure of C ( H 2 ( D ) ) was initiated by Berkson [5] in 1981. The central problem focuses on the relations between the structure of C ( H 2 ( D ) ) and the compactness of its members.

In 1990, Shapiro and Sundberg [6] gave results on compact difference and isolation. They posed the following fundamental question:

(*) Do the compact composition operators form a connected component of the set C ( H 2 ( D ) ) ?

In [7], Gallardo-Gutiérrez et al. answered the question by demonstrating that, while the compact operators form a connected subset, there are non-compact operators belonging to the same component. In [8], Bourdon determined when two linear-fractional composition operators on the Hardy space H 2 ( D ) belong to the same component in C ( H 2 ( D ) ) . He showed that two linear-fractional composition operators in the same component may fail to have compact difference. In [9], Hammond and MacCluer completely described the component structure of C ( H p , H q ) when 0 < q < p ≤ ∞ . Moorhouse and Toews [10] used Carleson measure techniques to give a sufficient condition for the difference of two composition operators on A α 2 ( D ) to be compact. In 2005, Moorhouse [11] answered the question of compact difference of composition operators acting on A λ 2 ( D ) , λ > − 1 , and gave a partial answer to the component structure of C ( A λ 2 ( D ) ) . Recently, Choe, Koo and Park [12,13] extended Moorhouse’s characterization to the unit polydisk and unit ball in ℂ N . In 2015, Hosokawa, Izuchi and Ohno [14] investigated the topological space of weighted composition operators acting between some Hilbert spaces on the unit disk in general, and they also considered the Hilbert-Schmidt norm topology. Inspired by the aforementioned studies, we are interested in studying the topological spaces of weighted composition operators further and giving a more detailed characterization for those topological spaces. We are concerned of not only the path connectedness but also the fundamental group of those topological spaces. We now state our main result.

The paper is organized as follows: some basic definitions and lemmas are introduced in Section 2. Our main results are proved in Section 3; that is, we prove the topological spaces of nonzero weighted composition operators acting between certain Hilbert spaces of analytic functions on B N are simply connected. We also prove that Hilbert-Schmidt topological spaces of nonzero weighted composition operators are simply connected in Section 4. Finally, some applications are listed in Section 5.

2 Preliminaries

Recall that a topological space is said to be simply connected if its fundamental group is {0} , or equivalently every closed continuous path can be homotopic to a point path. Inspired by the discussions in [14], we continue to investigate the simply connected topological spaces in higher dimensional case. One of the interesting problems is to characterize the topological spaces of weighted composition operators acting between Dirichlet spaces and Hardy spaces or weighted Bergman spaces.

Let d v denote the volume measure on B N , normalized so that v ( B N ) = 1 . The surface measure on S N will be denoted by d σ . Once again, we have normalized σ so that σ ( S N ) = 1 . The normalizing constants are the volume of B N and the surface area of S N . Let H 2 ( B N ) be the classical Hardy-Hilbert space on B N , namely, the space of holomorphic functions f on B N such that

It is well known that

where α = ( α 1 , … , α N ) and α ! = α 1 ! ⋯ α N ! . The inner product of H 2 ( B N ) is given by

where f , g ∈ H 2 and f ⁎ , g ⁎ are radial functions of f , g . The reproducing kernel of H 2 ( B N ) is given by

For λ > − 1 , the weighted Bergman space A λ 2 ( B N ) on B N is the space of f ∈ H ( B N ) satisfying

where d v λ = Γ ( N + λ + 1 ) N ! Γ ( λ + 1 ) ( 1 − ∥ z ∥ 2 ) λ d v ( z ) . The inner product of A λ 2 ( B N ) is given by

where f , g ∈ A λ 2 . The reproducing kernel of A λ 2 ( B N ) is given by

When λ = 0 , A 0 2 ( B N ) is the classical Bergman space. It is well known that

where | α | = α 1 + α 2 + ⋯ + α N . We denote by D ( B N ) the Dirichlet space; that is, the space of holomorphic functions f on B N such that all the functions

belong to L 2 B N , d v ( z ) ( 1 − ∥ z ∥ 2 ) N + 1 for any positive integer n = | α | > N / 2 . It is known that holomorphic f ( z ) = ∑ α c α z α belongs to D ( B N ) if and only if ∑ α | α | α ! | α | ! | c α | 2 < ∞ , and

The inner product of D ( B N ) is given by

where both f ( z ) = ∑ α a α z α and g ( z ) = ∑ α b α z α are in D ( B N ) . The reproducing kernel of D ( B N ) is given by

We first study the topological spaces of weighted composition operators acting between general Hilbert spaces of analytic functions on B N . As applications, we give several simply connected topological spaces of weighted composition operators. Let ℋ denote a Hilbert space of analytic functions on B N , with norm ∥ ⋅ ∥ ℋ , that satisfies the following conditions:

(c1) The point evaluation ε w : f ↦ f ( w ) is a bounded linear functional on ℋ for any w ∈ B N , and sup ∥ w ∥ ≤ r ∥ ε w ∥ ℋ < ∞ for every 0 < r < 1 .

(c2) H ∞ ( B N ) ⋅ ℋ ⊂ ℋ and ∥ f g ∥ ℋ ≤ ∥ f ∥ ∞ ∥ g ∥ ℋ for every f ∈ H ∞ ( B N ) and g ∈ ℋ .

(c3) ∥ 1 ∥ ℋ = 1 and { z α / ∥ z α ∥ ℋ : α = ( α 1 , … , α N ) with α i ≥ 0 } is an orthonormal basis in ℋ .

(c4) For every f ∈ ℋ and 0 ≤ r ≤ 1 , we have f r ( z ) ≔ f ( r z ) ∈ ℋ .

By (c3), ℋ contains all polynomials of z = ( z 1 , … , z N ) . By (c2), we have ∥ f ∥ ℋ = ∥ f ⋅ 1 ∥ ℋ ≤ ∥ f ∥ ∞ ∥ 1 ∥ ℋ = ∥ f ∥ ∞ for f ∈ H ∞ . We note that the aforementioned conditions are very common for many classical Hilbert spaces of analytic functions on B N . Let C w ( ℋ ) denote the space of nonzero bounded weighted composition operators on ℋ with the operator norm topology; that is,

Let ℋ 1 denote another Hilbert space of analytic functions on B N , with ℋ 1 ⊂ ℋ , satisfying conditions (c1), (c3) and (c4). Hence, ℋ 1 contains all polynomials of z. Furthermore, we assume that ℋ 1 satisfies:

(c5) ∥ f ∥ ℋ ≤ ∥ f ∥ ℋ 1 for every f ∈ ℋ 1 .

If the weighted composition operator W u , φ : ℋ 1 → ℋ is bounded, then u ∈ ℋ by the fact that 1 ∈ ℋ 1 . Let C w ( ℋ 1 , ℋ ) denote the collection of bounded weighted composition operators W u , φ : ℋ 1 → ℋ such that W u , φ ∈ C w ( ℋ ) . By the closed graph theorem, it is well known that any bounded operator W u , φ : ℋ → ℋ can be restricted to a bounded operator W u , φ : ℋ 1 → ℋ . We can see C w ( ℋ 1 , ℋ ) = C w ( ℋ ) as sets. Denote the operator norm of a bounded linear operator T : ℋ 1 → ℋ by ∥ T ∥ ℋ 1 , ℋ . We equip the set C w ( ℋ 1 , ℋ ) with the topology induced by the norm ∥ ⋅ ∥ ℋ 1 , ℋ . In the following context, we discuss the topological connectedness of the space C w ( ℋ 1 , ℋ ) .

On the other hand, if W u , φ ∈ C w ( ℋ ) , we can use (c5) to obtain that

for every f ∈ ℋ 1 . From this point of view, W u , φ : ℋ 1 → ℋ is bounded and

which implies that the topology of C w ( ℋ ) is stronger than the one of C w ( ℋ 1 , ℋ ) .

We need the next lemma in Section 3 to prove the simple connectedness of C w ( ℋ 1 , ℋ ) .

3 Operator norm topological spaces

Let ℋ and ℋ 1 be spaces satisfying the conditions given in Section 2. In this section, we study the simple connectedness of the topological space C w ( ℋ 1 , ℋ ) with operator norm topology.

Note that the composition operator induced by the self-mapping

is a bounded composition operator C 0 in C w ( ℋ 1 , ℋ ) . If we denote by

it follows from the closed graph theorem that ℳ 0 ( ℋ 1 , ℋ ) ⊂ C w ( ℋ 1 , ℋ ) .

A deformation retraction of a space X onto a subspace A is a family of maps T t : X → X , t ∈ [ 0 , 1 ] such that

1. T 0 = id ,

2. T 1 ( X ) = A ,

3. T t | A = id A for all t,

We now obtain the following theorem, which is the main result of this paper.

4 Hilbert-Schmidt norm topological spaces

We now consider the topological space of Hilbert-Schmidt-weighted composition operators. If X is a separable Hilbert space with orthonormal base { e m } and X ′ is another Hilbert space, recall that a linear operator T : X → X ′ is said to be Hilbert-Schmidt if the Hilbert-Schmidt norm of T

is finite.

By condition (c3), W u , φ ∈ C w ( ℋ ) is Hilbert-Schmidt if and only if

We denote by C w , H S ( ℋ ) the space of Hilbert-Schmidt operators W u , φ in C w ( ℋ ) with the Hilbert-Schmidt norm topology.

We have that W u , φ ∈ C w ( ℋ 1 , ℋ ) is Hilbert-Schmidt if and only if

We denote by C w , H S ( ℋ 1 , ℋ ) the space of all Hilbert-Schmidt-weighted composition operators W u , φ ∈ C w ( ℋ 1 , ℋ ) . We consider C w , H S ( ℋ 1 , ℋ ) with the Hilbert-Schmidt norm topology, which is stronger than the operator norm topology. Hence, a path connected set in C w , H S ( ℋ 1 , ℋ ) is also a path connected set in C w ( ℋ 1 , ℋ ) . Since

by (c5), we have C w , H S ( ℋ ) ⊂ C w , H S ( ℋ 1 , ℋ ) . The next lemma plays an important role.

We are going to study the topological space C w , H S ( ℋ 1 , ℋ ) with the Hilbert-Schmidt norm topology. The strategy is analogous to that in Section 3. We need to consider the topological space

in which open sets are induced by the Hilbert-Schmidt norm.