We provide several characterizations of the bounded and the compact weighted composition operators from the Bloch space ℬ and the analytic Besov spaces B_p (with 1 < p < ∞) into the Zygmund space 𝒵. As a special case, we show that the bounded (resp., compact) composition operators from ℬ, B_p, and H^∞ to 𝒵 coincide. In addition, the boundedness and the compactness of the composition operator can be characterized in terms of the boundedness (resp., convergence to 0, under the boundedness assumption of the operator) of the Zygmund norm of the powers of the symbol.

1. Introduction

Let 𝔻 be the open unit disk in the complex plane ℂ and H(𝔻) the space of analytic functions on 𝔻. A function f ∈ H(𝔻) is said to belong to the Bloch space ℬ if

()

Under the norm defined by ∥f∥_ℬ = |f(0)| + β_f, ℬ is a conformally invariant Banach space. The functions in the Bloch space satisfy the following growth condition:

()

An important class of Möbius invariant spaces is given by the analytic Besov spaces B_p, with 1 < p < ∞, consisting of the functions f ∈ H(𝔻) such that

()

where dA denotes the normalized area measure on the unit disk. The quantity b_p is a seminorm and the Besov norm is defined by

()

By Lemma 1.1 of [1], for f ∈ B_p,

()

Thus, the space B_p is continuously embedded in the Bloch space. Moreover, by Theorem 9 in [2], the functions in B_p satisfy the growth condition

()

Let 𝒵 denote the set of all functions

such that

()

where the supremum is taken over all θ ∈ ℝ and h > 0. By Theorem 5.3 of [3] and the Closed Graph theorem, an analytic function f on 𝔻 belongs to 𝒵 if and only if f^′ ∈ ℬ. Furthermore,

()

The quantities in (8) are just seminorms for the space 𝒵, as they do not distinguish between functions differing by a linear function. The norm

()

yields a Banach space structure on 𝒵, which is called the Zygmund space. For more information on the Zygmund space on the unit disk, see, for example, [3].

Let S(𝔻) denote the set of all analytic self-maps of 𝔻. Each φ ∈ S(𝔻) induces the composition operator on H(𝔻) defined by

()

for f ∈ H(𝔻) and z ∈ 𝔻. We refer the interested reader to [4, 5] for the theory of the composition operators.

Let u ∈ H(𝔻). The multiplication operator M_u is defined as

()

for f ∈ H(𝔻) and z ∈ 𝔻. The composition product uC_φ of M_u and C_φ yields a linear operator on H(𝔻) called the weighted composition operator with symbols u and φ.

In recent years, considerable interest has emerged in the study of the weighted composition operators due to the important role they play in the study of the isometries on many Banach spaces of analytic functions, such as the Hardy space H^p (for 1 ≤ p ≤ ∞, p ≠ 2) [6, 7], the weighted Bergman space [8], and the disk algebra [9].

There is a very extensive literature on the composition operators and the weighted composition operators between the Bloch space and other spaces of analytic functions in one and several complex variables. The study on such operators between the Zygmund space and some other spaces of analytic functions is more recent and not yet well developed. The weighted composition operators and related operators from the Zygmund space to Bloch-type spaces have been investigated by the second author and Stević in [10, 11].

Research on the composition operators between various Banach spaces of analytic functions has shown that the compact composition operators acting on different Banach spaces are often the same. For example, in [12], it was shown that the composition operator C_φ mapping into the Bloch space is compact as an operator acting on H^∞ if and only if it is compact as an operator acting on the analytic Besov space B_p (with 1 < p < ∞), on the space BMOA, and on the Bloch space itself. Also, an interesting question that arises in operator theory is whether a countable set of functions on the range of the operator exists whose norm approaching 0 is sufficient to characterize the compactness of the operator. In [13], it was shown that the sequence of powers of the symbol {φⁿ} has this property with regards to the composition operator on the Bloch space as well as the space BMOA. In fact, for X = H^∞, B_p, BMOA, ℬ, the operator C_φ : X → ℬ is compact if and only if as n → ∞. For the weighted composition operator uC_φ, this is not the case, although a few cases are known in which the sequence of norms is sufficient to characterize boundedness and compactness (see [14, 15], see also [16] for analogous questions for the operators mapping into BMOA). In all examples known to us, the space X is small compared to the codomain of the operator. These results prompted us to investigate analogous questions when the operator maps into the Zygmund space.

In this work, besides giving characterizations of the bounded and the compact operator uC_φ from the Besov and the Bloch spaces into 𝒵 in terms of function theoretic conditions on the symbols u and φ, we provide boundedness and the compactness criteria for the operator uC_φ, in terms of the norms in 𝒵 of functions in the range of the operator belonging to certain one-parameter families. For both boundedness and compactness, one of these characterizations involves the sequence .

Finally, we show that there are no nontrivial bounded multiplication operators from the Bloch space to the Zygmund space, and that a composition operator from ℬ or B_p into 𝒵 is bounded (resp., compact) if and only if it is bounded (resp., compact) as an operator acting on the space H^∞ of bounded analytic functions on 𝔻 under the supremum norm. One of the characterizing conditions for boundedness (resp., compactness, under the boundedness assumption of the operator) of such operators is that (resp., ) as k → ∞. We found this phenomenon quite surprising due to the size of the spaces B_p and ℬ in comparison to that of the Zygmund space.

2. Boundedness

We begin the section by characterizing the bounded weighted composition operators from ℬ to the Zygmund space. We note that since the constant function 1 belongs to all spaces under consideration, the boundedness of uC_φ requires that u = uC_φ1 ∈ 𝒵. Thus, we will assume throughout that u ∈ 𝒵.

For a fixed a ∈ 𝔻 and for z ∈ 𝔻, set

()

For an integer k ≥ 0 and z ∈ 𝔻, let p_k(z) = z^k.

Theorem 1. Let u ∈ 𝒵 and φ ∈ S(𝔻). Then the following conditions are equivalent.

(a)
The operator uC_φ : ℬ → 𝒵 is bounded.
(b)
, and .
(c)
N₁∶ = sup _z∈𝔻(1−|z|²) | u(z)φ^′(z) ²|, N₂∶ = sup _z∈𝔻(1−|z|²) | 2u^′(z)φ^′(z) + u(z)φ^′′(z)|,
()
and H are finite.
(d)
The quantities M₁∶ = sup _z∈𝔻(1 − |z|²)|u^′′(z)|log (e/(1 − |φ(z)|²)),
()
are finite.

To prove Theorem 1, we will use the following result which follows from the characterization of the bounded weighted composition operators from H^∞ to the Zygmund space given in Theorem 1 of [17].

Lemma 2. Let u ∈ 𝒵 and φ ∈ S(𝔻). Then the following conditions are equivalent.

(a)
.
(b)
The quantities N₁, N₂, A, and B are finite.
(c)
The quantities M₂ and M₃ are finite.

Proof. (a) ⇒ (b) Since the sequence {p_k} is bounded in ℬ with supremum norms no greater than 1, if uC_φ : ℬ → 𝒵 is bounded, then for each integer k ≥ 0, we have

()

Therefore, the supremum of

over all integers k ≥ 0 is finite. Likewise,

. Therefore,

()

(b) ⇒ (c) follows at once from Lemma 2.

(c) ⇒ (d) Suppose that H, N₁, N₂, A, and B are finite. By Lemma 2 we see that M₂ and M₃ are finite. For each nonnegative integer n and each a ∈ 𝔻, a direct calculation shows that

()

Setting

()

for w ∈ 𝔻, from (17) we obtain

()

Subtracting (19) from (20), we get

()

Hence, from (21) and (22), we deduce

()

Therefore, taking the modulus and multiplying by (1−|w|²), we obtain

()

Taking the supremum over all w ∈ 𝔻, we see that M₁ is finite.

(d) ⇒ (a) Suppose that (d) holds. As a consequence of Theorem 5.1.5 of [18], for each f ∈ ℬ,

()

Thus, using (2) and (25), for an arbitrary z in 𝔻, we have

()

In addition, again by (2), we have

()

The boundedness of uC_φ : ℬ → 𝒵 follows from (27) and (26) after taking the supremum over all z ∈ 𝔻.

We now analyze the case of the weighted composition operator acting on the analytic Besov spaces B_p for 1 < p < ∞. The case p = 1 has been established by the second author in [19]. For 1 < p < ∞ and a ∈ 𝔻, define

()

Then, as shown in [12], h_p,a ∈ B_p and

is bounded by a constant L_p independent of a.

Theorem 3. Let u ∈ 𝒵, φ ∈ S(𝔻), and 1 < p < ∞. Then the following conditions are equivalent.

(a)
The operator uC_φ : B_p → 𝒵 is bounded.
(b)
, and .
(c)
The quantities N₁, N₂, A, and B in Theorem 1 and H_p are finite.
(d)
The quantities M₂ and M₃ in Theorem 1 and
()
are finite.

Proof. Note that, unlike the case of the Bloch space, the sequence {p_k} is unbounded in B_p. Therefore, first condition in (b) is not an immediate consequence of the boundedness of the operator.

(a) ⇒ (d) Suppose uC_φ : B_p → 𝒵 is bounded. Since , it follows that

()

Moreover, for a ∈ 𝔻, f_a and g_a are in B_p and their Besov norms are bounded by constants independent of a. Indeed, noting that

, with

, and using the Möbius invariance of the Besov seminorm, we have

()

Likewise, since

, and observing that b_p(p₂) ≤ 2b_p(p₁), again by the Möbius invariance of b_p, we obtain

()

Therefore, by the boundedness of uC_φ, the quantities A and B are finite.

Next observe that since p₁ and p₂ are in B_p and uC_φ is bounded, uC_φ(p₁) and uC_φ(p₂) are in 𝒵, so arguing as in the proof of Theorem 1 in [17], we see that N₁ and N₂ are finite as well. By Lemma 2, it follows that M₂ and M₃ are finite. To prove that M₄ is finite, let w ∈ 𝔻 and note that

()

Recalling the notation v(w) = 2u^′(w)φ^′(w) + u(w)φ^′′(w), a straightforward calculation shows that

()

Thus, by the boundedness of uC_φ, we have

()

Taking the supremum over all w ∈ 𝔻, it follows that M₄ is finite, as desired.

(d) ⇒ (a) Assume (d) holds. As a consequence of Theorem 5.1.5 of [18] and (5), for each f ∈ B_p,

()

Thus, using (6) and (36), for any z ∈ 𝔻, we have

()

Moreover, again by (6) and (5), we have

()

Taking the supremum over all z ∈ 𝔻 and using (38) and (37), we conclude that uC_φ : B_p → 𝒵 is a bounded operator.

The equivalence of (c) and (d) follows from (34) and Lemma 2. The equivalence of (b) and (c) is an immediate consequence of Lemma 2.

3. Compactness

We begin the section with a useful compactness criterion which will be used to characterize the compact weighted composition operators from the Bloch space and the Besov spaces to the Zygmund space. Its proof is based on standard arguments similar to those outlined in Proposition 3.11 of [4].

Lemma 4. Let u ∈ H(𝔻), φ ∈ S(𝔻), and 1 < p < ∞. Then, the operator uC_φ from ℬ (resp., B_p) into 𝒵 is compact if and only if it uC_φ : ℬ → 𝒵 (resp., uC_φ : B_p → 𝒵) is bounded and , as k → ∞, whenever {f_k} is a bounded sequence in ℬ (resp., B_p) converging to zero uniformly on compact subsets of 𝔻.

We will also make use the following result that follows from the characterization of the compact weighted composition operators from H^∞ to 𝒵 given in Theorem 2 of [17].

Lemma 5. Let u ∈ 𝒵 and φ ∈ S(𝔻) be such that any of the equivalent conditions in Lemma 2 hold. Then the following conditions are equivalent.

(a)
.
(b)
.
(c)
lim _|φ(z)|→1(1−|z|²) | u^′′(z)| = lim _|φ(z)|→1((1−|z|²) | u(z)φ^′(z) ²|/(1−|φ(z)|²) ²) = 0, and
()

In the next theorem we focus on the weighted composition operators acting on the Bloch space. We will use the one-parameter family {h_a : a ∈ 𝔻} introduced in (12).

Theorem 6. Let u ∈ 𝒵, φ ∈ S(𝔻) and suppose that the operator uC_φ : ℬ → 𝒵 is bounded. Then the following conditions are equivalent.

(a)
The operator uC_φ : ℬ → 𝒵 is compact.
(b)
.
(c)
.
(d)
lim _|φ(z)|→1(1−|z|²) | u^′′(z) | log (e/(1−|φ(z)|²)) = lim _|φ(z)|→1((1−|z|²) | u(z)φ^′(z) ²|/(1−|φ(z)|²) ²) = 0, and
()

Proof. (a) ⇒ (b) Suppose uC_φ : ℬ → 𝒵 is a compact operator. Since the sequence {p_k} is bounded in ℬ and converges to 0 uniformly on compact subsets of 𝔻, then by Lemma 4,

()

On the other hand, if {w_n} is a sequence in 𝔻 such that |φ(w_n)| → 1 as n → ∞, then

is a bounded sequence in ℬ converging to 0 uniformly on compact subsets of 𝔻, so again by Lemma 4, it follows that

()

proving (b).

(b) ⇒ (c) follows from Lemma 4.

()

as |φ(w)| → 1. The desired result now follows from the last formula and Lemma 4.

(d) ⇒ (a) Assume (d) holds. Then, in particular,

()

Thus, recalling the notation v = 2u^′φ^′ + uφ^′′, for any ε > 0, there is a constant r ∈ (0,1), such that

()

whenever r<|φ(z)| < 1.

Let {f_k} be a sequence in ℬ with and converging to 0 uniformly on compact subsets of 𝔻. In light of Lemma 4, it suffices to show that as k → ∞. Using (45), for |φ(w)| > r, and recalling (2) and (25), we have

()

By Cauchy′s estimate, if {f_k} is a sequence which converges to zero on compact subset of 𝔻, then so do the sequences

and

. Hence, for |φ(w)| ≤ r, we have

()

Since |(uC_φf_k)(0)| = |u(0)f_k(φ(0))| → 0 and

()

as k → ∞, from (46) and (47) we deduce that , as k → ∞. This completes the proof.

We now turn our attention to the weighted composition operators mapping B_p into the Zygmund space. We will use the one-parameter family {h_p,a : a ∈ 𝔻} defined in (28).

Theorem 7. Let u ∈ 𝒵, φ ∈ S(𝔻), 1 < p < ∞ and assume the operator uC_φ : B_p → 𝒵 is bounded. Then the following conditions are equivalent.

(a)
The operator uC_φ : B_p → 𝒵 is compact;
(b)
;
(c)
;
(d)
= lim _|φ(z)|→1((1−|z|²) | u(z)φ^′(z) ²|/(1−|φ(z)|²) ²) = 0, and
()

Proof. Assume uC_φ : B_p → 𝒵 is compact. Since {h_p,φ(w)} is bounded in B_p and convergent to 0 uniformly on compact subsets of 𝔻, if {w_n} is a sequence in 𝔻 such that |φ(w_n)| → 1 as n → ∞, by Lemma 4 it follows that as n → ∞. Similarly, since the sequences and are bounded in B_p and converge to 0 uniformly on compact subsets of 𝔻, it follows that and converge to 0 as n → ∞, proving (c). Therefore, by Lemma 5, it follows that as k → ∞, so (b) holds as well.

On the other hand, by (34) and Lemma 5, for all w ∈ 𝔻, we have

()

which proves that (d) holds as well. Thus, to prove the equivalence of (a)–(d), it suffices to show that (d) implies (a). Since the proof is similar to the proof of (d) implies (a) in Theorem 6, we omit the details.

Observe that conditions (d) in Theorems 1 and 6 coincide with the corresponding conditions (d) in Theorems 3 and 7 by taking p = ∞. We could indeed have adopted the notation B_∞ in place of ℬ, as the Bloch space is widely viewed as a limit of B_p as p → ∞, which could have allowed us to unify parts of Theorems 1 and 3 (resp., Theorems 6 and 7) into a single statement taking 1 < p ≤ ∞. However, we did not do so because the family of functions {h_∞,a} is not suitable to characterize the bounded and the compact weighted composition operators from the Bloch space into the Zygmund space.

4. Multiplication and the Composition Operators

In this section we highlight the results concerning the boundedness and the compactness of the multiplication operator and the composition operator, which to the best of our knowledge have not appeared in the literature.

For the case of the multiplication operator, the finiteness of M₃ in part (d) of Theorems 1 and 3, combined with the continuous inclusions of H^∞ and of B_p into ℬ and Theorems 1 and 2 in [19], yields the following result.

Corollary 8. For u ∈ 𝒵 and 1 ≤ p < ∞, the following statements are equivalent.

(a)
M_u : ℬ → 𝒵 is bounded.
(b)
M_u : ℬ → 𝒵 is compact.
(c)
M_u : B_p → 𝒵 is bounded.
(d)
M_u : B_p → 𝒵 is compact.
(e)
M_u : H^∞ → 𝒵 is bounded.
(f)
M_u : H^∞ → 𝒵 is compact.
(g)
u is identically 0.

For the case of the composition operator, noting that u = 1 implies that u^′ = u^′′ = 0, from (23), we see that for any w ∈ 𝔻,

is a linear combination of

and

. Furthermore, since for a ∈ 𝔻, the function

maps 𝔻 into the right half-plane, the image of the function

is contained in the horizontal strip {w ∈ ℂ:|Im w | < π/2}. Thus,

()

which does not depend on w. Thus, the boundedness of

and

implies the boundedness of

. Moreover, from (34), it follows that for w ∈ 𝔻,

()

Therefore, if

()

then, arguing as above, we see that |h_p,φ(w)(φ(0))| is bounded by a constant independent of w, so

()

From these remarks, Corollary 2 of [17], and Theorem 1 of [19], Theorems 1 and 3 yield the following result.

Corollary 9. For φ ∈ S(𝔻) and 1 ≤ p < ∞, the following statements are equivalent.

(a)
C_φ : ℬ → 𝒵 is bounded.
(b)
.
(c)
φ, φ² ∈ 𝒵, and .
(d)
sup _z∈𝔻((1−z|²)|φ^′′(z)|/(1−|φ(z)|²)) < ∞ and sup _z∈𝔻((1−|z|²) | φ^′(z)|²/(1−|φ(z)|²) ²) < ∞.
(e)
C_φ : B_p → 𝒵 is bounded.
(f)
C_φ : H^∞ → 𝒵 is bounded.

Next note that, by the above remarks, as |φ(w)| → 1,

()

Thus, the convergence to 0 of

and

as |φ(w)| → 1 implies that

and

converge to 0 as well. From Theorems 6 and 7, and from Corollary 3 of [17] and Theorem 2 of [19], we deduce the following result.

Corollary 10. For φ ∈ S(𝔻) such that C_φ : ℬ → 𝒵 is bounded and for 1 ≤ p < ∞, the following statements are equivalent.

(a)
C_φ : ℬ → 𝒵 is compact.
(b)
.
(c)
.
(d)
lim _|φ(z)|→1((1−z|²)|φ^′′(z)|/(1−|φ(z)|²)) = lim _φ|z|→1((1−|z|²) | φ^′(z)|²/(1−|φ(z)|²) ²) = 0.
(e)
C_φ : B_p → 𝒵 is compact.
(f)
C_φ : H^∞ → 𝒵 is compact.

4.1. Concluding Remarks

In this section, we established that the boundedness (resp., compactness) of the multiplication and the composition operator from H^∞ to 𝒵 is equivalent to the boundedness (resp., compactness) of the multiplication and composition operator from B_p or ℬ to 𝒵. It is evident that the same can be said for the weighted composition operator mapping between these spaces for any choice of composition symbol when the multiplicative symbol is linear, or for an arbitrary multiplication symbol in the Zygmund space if the range of the composition symbol is relatively compact. This leads to the question of whether this equivalence holds also for completely general weighted composition operators. As a consequence of [17, 19], the answer is affirmative for the case of the spaces H^∞ and B₁. We suspect this is false for the other Besov spaces or ℬ but have not been able to construct examples to support this claim.

Acknowledgments

The second author is supported by the Foundation for Scientific and Technological Innovation in Higher Education of Guangdong (no. 2012KJCX0096), National Natural Science Foundation of China (no. 11001107), and the Foundation for Distinguished Young Talents in Higher Education of Guangdong (no. LYM11117).

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Weighted Composition Operators from the Bloch Space and the Analytic Besov Spaces into the Zygmund Space

Abstract

1. Introduction

2. Boundedness

3. Compactness

4. Multiplication and the Composition Operators

4.1. Concluding Remarks

Acknowledgments

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Weighted Composition Operators from the Bloch Space and the Analytic Besov Spaces into the Zygmund Space

Abstract

1. Introduction

2. Boundedness

3. Compactness

4. Multiplication and the Composition Operators

4.1. Concluding Remarks

Acknowledgments

References

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