On Convexity of Composition and Multiplication Operators on Weighted Hardy Spaces

Abstract

A bounded linear operator 𝑇 on a Hilbert space ℋ, satisfying ‖𝑇²ℎ‖²+‖ℎ‖²≥2‖𝑇ℎ‖² for every ℎ∈ℋ, is called a convex operator. In this paper, we give necessary and sufficient conditions under which a convex composition operator on a large class of weighted Hardy spaces is an isometry. Also, we discuss convexity of multiplication operators.

1. Introduction and Preliminaries

We denote by 𝐵(ℋ) the space of all bounded linear operators on a Hilbert space ℋ. An operator 𝑇∈𝐵(ℋ) is said to be convex, if

‖𝑇²ℎ‖²+‖ℎ‖²≥2‖𝑇ℎ‖² (1.1) for each ℎ∈ℋ. Note that if 𝑇 is a convex operator then the sequence (‖𝑇^𝑛ℎ‖²)_𝑛∈𝐍 forms a convex sequence for every ℎ∈ℋ. Taking Δ_𝑇=𝑇^∗𝑇−𝐼, it is easily seen that 𝑇 is a convex operator if and only if 𝑇^∗Δ_𝑇𝑇≥Δ_𝑇.

A weighted Hardy space is a Hilbert space of analytic functions on the open unit disc 𝐃 for which the sequence (𝑧^𝑗)^∞_𝑗=0 forms a complete orthogonal set of nonzero vectors. It is usually assumed that ‖1‖=1. Writing 𝛽(𝑗)=‖𝑧^𝑗‖, this space is denoted by 𝐻²(𝛽) and its norm is given by

‖^∞_𝑗=0𝑎_𝑗𝑧^𝑗‖²=^∞_𝑗=0|𝑎_𝑗|²𝛽(𝑗)². (1.2)

Let 𝜑 be an analytic map of the open unit disc 𝐃 into itself, and define 𝐶_𝜑(𝑓)=𝑓∘𝜑 whenever 𝑓 is analytic on 𝐃. The function 𝜑 is called the symbol of the composition operator. For a positive integer 𝑛, the 𝑛th iterate of 𝜑, denoted by 𝜑_𝑛, is the function obtained by composing 𝜑 with itself 𝑛 times; also 𝜑₀ is defined to be the identity function. Denote the reproducing kernel at 𝑧∈𝐃, for the space 𝐻²(𝛽), by 𝐾_𝑧. Then ⟨𝑓,𝐾_𝑧⟩=𝑓(𝑧) for every 𝑓∈𝐻²(𝛽). It is known that 𝐶^∗_𝜑(𝐾_𝑧)=𝐾_𝜑(𝑧) for all 𝑧 in 𝐃. The generating function for 𝐻²(𝛽) is the function given by

𝑘(𝑧)=^∞_𝑗=0𝑧^𝑗𝛽(𝑗)². (1.3) This function is analytic on 𝐃. Moreover, if 𝑤∈𝐃 then 𝐾_𝑤(𝑧)=𝑘(𝑤𝑧) and ‖𝐾_𝑤‖²=𝑘(|𝑤|²) (see [1]).

Recently, there has been a great interest in studying operator theoretic properties of composition and weighted composition operators, see, for example, monographs [1, 2], papers [3–16], as well as the reference therein.

Isometric operators on weighted Hardy spaces, especially those that are composition operators are discussed by many authors. Isometries of the Hardy space 𝐻² among composition operators are characterized in [17, page 444], [18] and [12, page 66]. Indeed, it is shown that the only composition operators on 𝐻² that are isometries are the ones induced by inner functions vanishing at the origin. Bayart [5] generalized this result and showed that every composition operator on 𝐻² which is similar to an isometry is induced by an inner function with a fixed point in the unit disc. The surjective isometries of 𝐻^𝑝, 1≤𝑝<∞ that are weighted composition operators have been described by Forelli [19]. Carswell and Hammond [6] proved that the isometric composition operators of the weighted Bergman space 𝐴²_𝛼 are the rotations. Cima and Wogen [20] have characterized all surjective isometries of the Bloch space. Furthermore, the identification of all isometric composition operators on the Bloch space is due to Colonna [8]. Some related results can be found also in [3, 4, 6, 21–25].

Herein, we are interested in studying the convexity of composition and multiplication operators acting on a weighted Hardy space 𝐻²(𝛽). First, we give some preliminary facts on convex operators. Next, we will offer necessary and sufficient conditions under which a convex composition operator may be isometry on a large class of weighted Hardy spaces containing Hardy, Bergman, and Dirichlet spaces. We also discuss on convexity of the adjoint of a composition operator. Finally, we will obtain similar results for multiplication operators and their adjoints. For a good reference on isometric multiplication operators the reader can see [3].

Throughout this paper, 𝑇 is assumed to be a bounded linear operator on a Hilbert space ℋ. It is easy to see that for every convex operator 𝑇, the sequence (𝑇^∗𝑛Δ_𝑇𝑇^𝑛)_𝑛 forms an increasing sequence. We use this fact to prove the following theorem.

Theorem 1.1. If 𝑇 is a convex operator then so is every nonnegative integer power of 𝑇.

Proof. We argue by using mathematical induction. The convexity of 𝑇 implies that the result holds for 𝑘=1. Suppose that 𝑇^∗𝑛Δ_𝑇^𝑛𝑇^𝑛≥Δ_𝑇^𝑛, then 𝑇^∗𝑛+1Δ_𝑇^𝑛+1𝑇^𝑛+1−Δ_𝑇^𝑛+1=𝑇^∗𝑛+1(𝑇^∗Δ_𝑇^𝑛𝑇+Δ_𝑇)𝑇^𝑛+1−Δ_𝑇^𝑛+1=𝑇^∗2(𝑇^∗𝑛Δ_𝑇^𝑛𝑇^𝑛)𝑇²+𝑇^∗𝑛+1Δ_𝑇𝑇^𝑛+1−Δ_𝑇^𝑛+1≥𝑇^∗2Δ_𝑇^𝑛𝑇²+𝑇^∗𝑛Δ_𝑇𝑇^𝑛−Δ_𝑇^𝑛+1=𝑇^∗2(𝑇^∗𝑛𝑇^𝑛−𝐼)𝑇²+𝑇^∗𝑛Δ_𝑇𝑇^𝑛−𝑇^∗(𝑇^∗𝑛𝑇^𝑛−𝐼)𝑇−Δ_𝑇=𝑇^∗𝑛(𝑇^∗2𝑇²)𝑇^𝑛−𝑇^∗2𝑇²+𝑇^∗𝑛Δ_𝑇𝑇^𝑛−𝑇^∗𝑛(𝑇^∗𝑇)𝑇^𝑛+𝑇^∗𝑇−Δ_𝑇=𝑇^∗𝑛(𝑇^∗2𝑇²−𝐼)𝑇^𝑛−𝑇^∗2𝑇²+𝐼≥2𝑇^∗𝑛Δ_𝑇𝑇^𝑛−𝑇^∗2𝑇²+𝐼≥2𝑇^∗Δ_𝑇𝑇−𝑇^∗2𝑇²+𝐼=𝑇^∗Δ_𝑇𝑇−Δ_𝑇≥0. (1.4) So the result holds for 𝑘=𝑛+1.

Proposition 1.2. If 𝑇 is a convex operator, then for every nonnegative integer 𝑛, 𝑇^∗𝑛𝑇^𝑛≥𝑛Δ_𝑇+𝐼. (1.5)

Proof. We give the assertion by using mathematical induction on 𝑛. The result is clearly true for 𝑛=1. Suppose that 𝑇^∗𝑛𝑇^𝑛≥𝑛Δ_𝑇+𝐼. Thus, 𝑇^∗𝑛+1𝑇^𝑛+1=𝑇^∗(𝑇^∗𝑛𝑇^𝑛)𝑇≥𝑇^∗(𝑛Δ_𝑇+𝐼)𝑇=𝑛𝑇^∗Δ_𝑇𝑇+𝑇^∗𝑇=𝑛(𝑇^∗2𝑇²−2𝑇^∗𝑇+𝐼)+𝑛𝑇^∗𝑇+𝑇^∗𝑇−𝑛𝐼≥(𝑛+1)𝑇^∗𝑇−𝑛𝐼=(𝑛+1)Δ_𝑇+𝐼. (1.6) So the result holds for 𝑘=𝑛+1.

Proposition 1.3. Let 𝑇∈ℬ(ℋ) be a convex operator and let ℎ∈ℋ be such that sup_𝑘≥0‖𝑇^𝑘ℎ‖<∞. If Δ_𝑇≥0, then ‖𝑇ℎ‖=‖ℎ‖.

Proof. By applying Proposition 1.2, we observe that for every nonnegative integer 𝑛, 𝑛⟨Δ_𝑇ℎ,ℎ⟩+‖ℎ‖²≤‖𝑇^𝑛ℎ‖²≤sup_𝑘≥0‖𝑇^𝑘ℎ‖²<∞. (1.7) Letting 𝑛→∞, the positivity of Δ_𝑇 implies that Δ_𝑇ℎ=0; hence, ‖𝑇ℎ‖=‖ℎ‖.

Proposition 1.4. Let {𝑒_𝑛}^∞_𝑛=0 be an orthonormal basis for ℋ and let 𝑇∈ℬ(ℋ) be a convex operator satisfying Δ_𝑇≥0. Suppose that there is a nonnegative integer 𝑖 and a scalar 𝛼_𝑖 with 0<|𝛼_𝑖|≤1 so that 𝑇𝑒_𝑖=𝛼_𝑖𝑒_𝑖, then ℳ=∨_𝑛≠𝑖{𝑒_𝑛} is an invariant subspace for 𝑇.

Proof. Using Proposition 1.2, we see that ‖𝑒_𝑖‖²≥‖𝛼^𝑛_𝑖𝑒_𝑖‖²=‖𝑇^𝑛𝑒_𝑖‖²=⟨𝑇^∗𝑛𝑇^𝑛𝑒_𝑖,𝑒_𝑖⟩≥𝑛⟨Δ_𝑇𝑒_𝑖,𝑒_𝑖⟩+‖𝑒_𝑖‖² (1.8) for every 𝑛≥0. Let 𝑛→∞. Since Δ_𝑇 is a positive operator, we conclude that Δ_𝑇𝑒_𝑖=0. Consequently, 𝑇^∗𝑒_𝑖=(1/𝛼_𝑖)𝑇^∗𝑇𝑒_𝑖=(1/𝛼_𝑖)𝑒_𝑖. Now, if 𝑓∈ℳ then ⟨𝑇𝑓,𝑒_𝑖⟩=0; hence, 𝑇𝑓∈ℳ.

2. Composition Operators

Our purpose in this section is to discuss on convex composition operators on a weighted Hardy space. Recall that an operator 𝑇 in 𝐵(ℋ) is an isometry, if Δ_𝑇=0. At first, we give an example of a nonisometric composition operator 𝑇 on a weighted Hardy space such that 𝑇^∗Δ_𝑇𝑇≥Δ_𝑇≥0. For simplicity of notation, Δ_𝐶𝜑 is denoted by Δ_𝜑.

Example 2.1. Consider the weighted Hardy space 𝐻²(𝛽) with weight sequence (𝛽(𝑛))_𝑛 given by 𝛽(𝑛)=𝑛+1. Define 𝜑∶𝐃→𝐃 by 𝜑(𝑧)=𝑧². It is easily seen that 𝐶_𝜑(𝐻²(𝛽))⊆𝐻²(𝛽), and an application of the closed graph theorem shows that 𝐶_𝜑 is bounded. Now, a simple calculation shows that ⟨(𝐶^∗_𝜑Δ_𝜑𝐶_𝜑−Δ_𝜑)(𝑧^𝑘),𝑧^𝑘⟩=‖𝐶_𝜑2𝑧^𝑘‖²−2‖𝐶_𝜑𝑧^𝑘‖²+‖𝑧^𝑘‖²>0 (2.1) for all 𝑘≥0; besides ⟨Δ_𝜑𝑧^𝑘,𝑧^𝑘⟩=‖𝐶_𝜑𝑧^𝑘‖²−‖𝑧^𝑘‖² (2.2) which is positive for all 𝑘≥1, and zero whenever 𝑘=0. It follows that 𝐶^∗_𝜑Δ_𝜑𝐶_𝜑≥Δ_𝜑≥0, but 𝐶_𝜑 is not an isometry.

Proposition 2.2. Suppose that 𝑇∶𝐻²(𝛽)→𝐻²(𝛽) is a convex operator satisfying 𝑇1=1 and Δ_𝑇≥0, then 𝑀={𝑓∈𝐻²(𝛽)∶𝑓(0)=0} (2.3) is a nontrivial invariant subspace of 𝑇.

Proof. Clearly 𝑀 is a nontrivial closed subspace of 𝑇. To show that 𝑀 is invariant for 𝑇, apply Proposition 1.4 for the Hilbert space ℋ=𝐻²(𝛽), the orthonormal basis {𝑒_𝑛}_𝑛 given by 𝑒_𝑛=𝑧^𝑛/𝛽(𝑛),𝑖=0 and 𝛼₀=1.

Example 2.3. Consider the Bergman space 𝐴²(𝐃) consisting of all analytic functions 𝑓 on the open unit disc 𝐃, for which ‖𝑓‖²=_𝐃|𝑓(𝑧)|²𝑑𝐴(𝑧)<∞, (2.4) where 𝑑𝐴(𝑧) is the normalized Lebesgue area measure on 𝐃. If 𝑓∈𝐴²(𝐃) is represented by 𝑓(𝑧)=∑^∞_𝑛=0𝑎_𝑛𝑧^𝑛, then ‖𝑓‖²=^∞_𝑛=0|𝑎_𝑛|²𝑛+1. (2.5) Also, {𝑧^𝑘}_𝑘 forms an orthogonal basis for 𝐴²(𝐃). Fix nonnegative integers 𝑘 and 𝑛, and observe that ‖𝐶^𝑛_𝜑𝑧^𝑘‖²=‖𝜑^𝑛_𝑘‖²=_𝐃|𝜑^𝑛_𝑘(𝑧)|²𝑑𝐴(𝑧)≤_𝐃𝑑𝐴(𝑧)=1. (2.6) Thus, Proposition 1.3 implies that 𝐶^∗_𝜑Δ_𝜑𝐶_𝜑≥Δ_𝜑≥0 if and only if 𝐶_𝜑 is an isometry. In this case, taking 𝑇=𝐶_𝜑 and 𝑓(𝑧)=𝑧 in Proposition 2.2, we conclude that 𝜑(0)=0; thus, the Schwarz lemma implies that |𝜑(𝑧)|≤|𝑧| for all 𝑧∈𝐃. On the other hand, if 𝑓(𝑧)=𝑧 then _𝐃|𝜑(𝑧)|²𝑑𝐴(𝑧)=‖𝐶_𝜑𝑓‖²=‖𝑓‖²=_𝐃|𝑧|²𝑑𝐴(𝑧), (2.7) and so |𝜑(𝑧)|=|𝑧| almost everywhere with respect to the area measure. Hence, 𝜑(𝑧)=𝑒^𝑖𝜃𝑧 for some 𝜃∈[0,2𝜋).

Example 2.4. Consider the Hardy space 𝐻²(𝐃). If 𝜑 is an analytic self-map of the unit disc, then 𝜑 induces a bounded composition operator, and ‖𝐶^𝑛_𝜑𝑧^𝑘‖≤1 for all nonnegative integers 𝑛 and 𝑘. Thus, by Proposition 1.3, 𝐶^∗_𝜑Δ_𝜑𝐶_𝜑≥Δ_𝜑≥0 if and only if 𝐶_𝜑 is an isometry.

Recall that the Dirichlet space 𝒟 is the set of all functions analytic on 𝐃 whose derivatives lie in the Bergman space 𝐴²(𝐃). The Dirichlet norm is defined by

‖𝑓‖²_𝒟=|𝑓(0)|²+_𝐃|𝑓^′(𝑧)|²𝑑𝐴(𝑧). (2.8) If 𝜑 is a univalent self-map of 𝐃, then 𝐶_𝜑 is bounded on 𝒟 [2, page 18]. Also, the area formula [1, page 36], shows that

‖𝐶_𝜑𝑓‖²_𝒟=|(𝑓𝑜𝜑)(0)|²+_𝐃|𝑓^′(𝑧)|²𝑛_𝜑(𝑧)𝑑𝐴(𝑧), (2.9) where 𝑛_𝜑(𝑧) is, as usual, the counting function defined as the cardinality of the set {𝑤∈𝐃∶𝜑(𝑤)=𝑧}.

In the next theorem, we characterize all convex composition operators 𝐶_𝜑 on 𝒟 satisfying Δ_𝜑≥0. Note that we cannot use Proposition 1.3 for the Dirichlet space, thanks to the fact that in general the positive powers of 𝐶_𝜑 are not uniformly bounded on the 𝑧^𝑖's.

Theorem 2.5. If 𝐶_𝜑 is convex on the Dirichlet space 𝒟, then Δ_𝜑≥0 if and only if 𝐶_𝜑 is an isometry.

Proof. One implication is clear. Suppose that Δ_𝜑 is a positive operator, and take 𝑇=𝐶_𝜑 in Proposition 2.2. Since the identity function is in the subspace 𝑀={𝑓∈𝒟∶𝑓(0)=0}, we conclude that 𝜑(0)=0. Thus, in light of (2.9), to show that 𝐶_𝜑 is an isometry it is sufficient to prove that _𝐃|𝑓^′(𝑧)|(1−𝑛_𝜑)(𝑧)𝑑𝐴(𝑧)=0,∀𝑓∈𝒟. (2.10) Let 𝑓 be any function in the Dirichlet space 𝒟. Then 0≤⟨(𝐶^∗_𝜑Δ_𝜑𝐶_𝜑−Δ_𝜑)(𝑓),𝑓⟩=_𝐃|𝑓^′(𝑧)|²(𝑛_𝜑2−2𝑛_𝜑+1)(𝑧)𝑑𝐴(𝑧). (2.11) Furthermore, 0≤⟨Δ_𝜑𝑓,𝑓⟩=_𝐃|𝑓^′(𝑧)|²(𝑛_𝜑−1)(𝑧)𝑑𝐴(𝑧). (2.12) By summing up these two relations we get _𝐃|𝑓^′(𝑧)|²(𝑛_𝜑2−𝑛_𝜑)(𝑧)𝑑𝐴(𝑧)≥0. (2.13) But 𝑛_𝜑2(𝑧)≤𝑛_𝜑(𝑧), and so _𝐃|𝑓^′(𝑧)|²(𝑛_𝜑2−𝑛_𝜑)(𝑧)𝑑𝐴(𝑧)=0,∀𝑓∈𝒟. (2.14) This, in turn, implies that 𝑛_𝜑2(𝑧)=𝑛_𝜑(𝑧) almost everywhere. Substituting this in (2.11), and then considering (2.12) the assertion will be completed.

Observe that if 𝜑(0)=0, 𝑛_𝜑2−2𝑛_𝜑+1≥0 almost everywhere, and 𝐶_𝜑 is bounded on 𝒟 then it is convex. Indeed,

⟨(𝐶^∗_𝜑Δ_𝜑𝐶_𝜑−Δ_𝜑)𝑓,𝑓⟩=_𝐃|𝑓^′(𝑧)|²(𝑛_𝜑2−2𝑛_𝜑+1)(𝑧)𝑑𝐴(𝑧)≥0. (2.15)

In the next theorem, we turn to the adjoint of a composition operator and give necessary and sufficient conditions under which a convex operator 𝐶^∗_𝜑 is an isometry.

Theorem 2.6. Let 𝜑 be an analytic self-map of 𝐃 with 𝜑(0)=0. If 𝐶^∗_𝜑 is a convex operator on 𝐻²(𝛽), then it is an isometry if and only if Δ_𝐶^∗_𝜑≥0.

Proof. Suppose that Δ_𝐶^∗_𝜑≥0, and assume that 𝜑 is not the identity or an elliptic automorphism. By the Denjoy-Wolff theorem 𝜑_𝑛 converges uniformly to zero on compact subsets of 𝐃 [1], and so for every 𝑧∈𝐃, lim_𝑛→∞‖𝐾_{𝜑𝑛(𝑧)}‖=‖𝐾₀‖. (2.16) Proposition 1.2 coupled with the fact that 𝐶^∗𝑛_𝜑𝐾_𝑧=𝐾_{𝜑𝑛(𝑧)} implies that for all 𝑧∈𝐃 and all nonnegative integers 𝑛, ‖𝐾_{𝜑𝑛(𝑧)}‖²≥𝑛(‖𝐾_𝜑(𝑧)‖²−‖𝐾_𝑧‖²)+‖𝐾_𝑧‖². (2.17) Furthermore, the positivity of Δ_𝑇 shows that ‖𝐾_𝜑(𝑧)‖≥‖𝐾_𝑧‖. Thus, in light of (2.16) and (2.17) we conclude that ‖𝐾_𝑧‖=‖𝐾_𝜑(𝑧)‖ for all 𝑧∈𝐃, and so ‖𝐾_𝑧‖=‖𝐾_{𝜑𝑛(𝑧)}‖ for every positive integer 𝑛. Consequently, ‖𝐾_𝑧‖=‖𝐾₀‖ for all 𝑧∈𝐃. It follows that 1=‖𝐾₀‖²=‖𝐾_𝑧‖²=𝑘(|𝑧|²)=1+^∞_𝑗=1(|𝑧|²)^𝑗𝛽(𝑗)²,for𝑧∈𝐃. (2.18) This contradiction shows that 𝜑 is the identity or an elliptic automorphism. Thus, there is a 𝜃∈[0,2𝜋) so that 𝜑(𝑧)=𝑒^𝑖𝜃𝑧 for all 𝑧∈𝐃. Now, if 𝜔∈𝐃 then 𝐶^∗_𝜑𝐾_𝜔(𝑧)=𝐾_𝜑(𝜔)(𝑧)=𝑘(𝜑(𝜔)𝑧)=𝐾_𝜔(𝑒^−𝑖𝜃𝑧)=𝐾_𝜔(𝜑⁻¹(𝑧))=𝐶_𝜑−1𝐾_𝜔(𝑧). (2.19)
It follows that 𝐶^∗_𝜑=𝐶_𝜑−1. But it is easily seen that ‖𝐶_𝜑−1𝑓‖=‖𝑓‖ for every 𝑓∈𝐻²(𝛽). Hence, 𝐶^∗_𝜑 is an isometry. The converse is obvious.

3. Multiplication Operators

This section deals with convex multiplication operators on a weighted Hardy space. Recall that a multiplier of 𝐻²(𝛽) is an analytic function 𝜑 on 𝐃 such that 𝜑𝐻²(𝛽)⊆𝐻²(𝛽). The set of all multipliers of 𝐻²(𝛽) is denoted by 𝑀(𝐻²(𝛽)). It is known that 𝑀(𝐻²(𝛽))⊆𝐻^∞. In fact, if 𝜑∈𝑀(𝐻²(𝛽)) and 𝑓 is the constant function 1 then for every positive integer 𝑛 and for every 𝑧∈𝐃 we have

|𝜑(𝑧)|=|⟨𝑀^𝑛_𝜑𝑓,𝐾_𝑧⟩|^1/𝑛≤‖𝑀^𝑛_𝜑𝑓‖^1/𝑛‖𝐾_𝑧‖^1/𝑛≤‖𝑀_𝜑‖‖𝐾_𝑧‖^1/𝑛. (3.1) Now, letting 𝑛→∞, we conclude that 𝜑 is bounded. This coupled with the fact that 𝜑∈𝐻²(𝛽) implies that 𝜑∈𝐻^∞. If 𝜑 is a multiplier, then the multiplication operator 𝑀_𝜑, defined by 𝑀_𝜑𝑓=𝜑𝑓, is bounded on 𝐻²(𝛽). Also note that for each 𝜆∈𝐃, 𝑀^∗_𝜑𝐾_𝜆=𝜑(𝜆)𝐾_𝜆.

In what follows, the operator 𝑀_𝜑 is assumed to be convex. First, we present an example of a nonisometric convex multiplication operator 𝑇 with Δ_𝑇≥0.

Example 3.1. Consider the weighted Hardy space 𝐻²(𝛽) with weight sequence (𝛽(𝑛))_𝑛 given by 𝛽(𝑛)=𝑛+1. Define the mapping 𝜑 on 𝐃 by 𝜑(𝑧)=𝑧². Obviously, 𝑀_𝜑 is bounded. Furthermore, it is easy to see that for every nonnegative integer 𝑘, ‖𝑀²_𝜑𝑧^𝑘‖²−2‖𝑀_𝜑𝑧^𝑘‖²+‖𝑧^𝑘‖²>0,‖𝑀_𝜑𝑧^𝑘‖>‖𝑧^𝑘‖. (3.2) Consequently, 𝑀_𝜑 is convex but not an isometry. Besides, Δ_𝑀𝜑 is a positive operator.

Theorem 3.2. Let 𝐻^∞ consist of all multipliers of 𝐻²(𝛽), and let 𝜑∈𝐻^∞ be such that ‖𝜑‖_∞≤1. If 𝑇=𝑀_𝜑 or 𝑇=𝑀^∗_𝜑 then 𝑇^∗Δ_𝑇𝑇≥Δ_𝑇≥0 if and only if 𝑇 is an isometry.

Proof. Suppose that 𝑇 is 𝑀_𝜑 or 𝑀^∗_𝜑 and 𝑇^∗Δ_𝑇𝑇≥Δ_𝑇≥0. Define the linear mapping 𝑆∶𝐻^∞→ℬ(𝐻²(𝛽)) by 𝑆(𝜓)=𝑀_𝜓. An application of the closed graph theorem implies that 𝑆 is bounded. Therefore, there is 𝑐>0 such that for all 𝜓∈𝐻^∞, ‖𝑀_𝜓‖≤𝑐‖𝜓‖_∞. (3.3) It follows that for every 𝑓∈𝐻²(𝛽) and every nonnegative integer 𝑛, ‖𝑀^𝑛_𝜑𝑓‖≤𝑐‖𝜑^𝑛‖_∞‖𝑓‖≤𝑐‖𝑓‖. (3.4) Thus, sup_𝑛≥0‖𝑀^𝑛_𝜑𝑓‖<∞ for every 𝑓∈𝐻²(𝛽). Since ‖𝑀^∗_𝜓‖=‖𝑀_𝜓‖ for all 𝜓∈𝐻^∞, by a similar method one can show that sup_𝑛≥0‖𝑀^∗𝑛_𝜑𝑓‖<∞ for all 𝑓∈𝐻²(𝛽). Therefore, the result follows from Proposition 1.3.

Example 3.3. Let ℋ be the Bergman space or the Hardy space and let 𝑇 be 𝑀_𝜑 or its adjoint on ℋ. It is well known that 𝑀(ℋ)=𝐻^∞. So if 𝜑 is a multiplier with ‖𝜑‖_∞≤1, then by applying the preceding theorem, we observe that 𝑇^∗Δ_𝑇𝑇≥Δ_𝑇≥0 if and only if 𝑇 is an isometry.

We remark herein that if 𝜑(𝑧)=𝑧 and 𝑇=𝑀_𝜑 on the Dirichlet space 𝒟, then it is easily seen that 𝑇^∗Δ_𝑇𝑇≥Δ_𝑇≥0 but 𝑇 is not an isometry.

Acknowledgments

The authors would like to thank Dr. Faghih Ahmadi for her assistance and the referee for a number of helpful comments and suggestions. This research was in part supported by a grant no. (88-GR-SC-27) from Shiraz University Research Council.