Toeplitz-Superposition Operators on Analytic Bloch Spaces

The important purpose of this current work is to study a new class of operators, the so-called Toeplitz-superposition operators as an expansion of the weighted known composition operators, induced by such continuous entire functions mapping on bounded specific sets. Minutely, we have deeply discussed the conditions for boundedness of this new type of operators between certain types of some holomorphic Bloch classes with some specific values of the weighted functions.


Introduction
Fundamentals of the needed analytic function spaces as well as the types of concerned operators are briefly introduced. The paper focuses first on the concerned setting of certain classes of function spaces and the new defined operator, which in turn is motivated essentially by some certain classical concepts of known operators such as superposition operators as well as Toeplitz operator. There is an emphasis in the concerned paper on intensive tying together the needed type of analytic function spaces and the concerned operators, to illustrate the roles of the obtained results.
All of the needed information to justify the target of this research is collected in this concerned section. Moreover, here, basic concerned concepts, the Bloch space of analytictype, certain needed concerned lemmas, and superposition and Toeplitz operators are presented.
Let D = fz ∈ ℂ : |z|<1g be the open unit disk in ℂ, and let H ðDÞ denote the class of all analytic functions in D. Let dA ðzÞ = dx dy denote the concerned Lebesgue measures on D: Numerous intensive studies on analytic Bloch-type spaces are researched in literature (see [1][2][3][4][5] and others).
The space B 1 is called the Bloch space and denoted by B (see [3]).
The following interesting needed lemma has been proved in [6].
The following useful integral estimate is well known and can be found in [7]. Lemma 2. Let s > 0 and t > −1 . Then For a > −1 and p ∈ ð0,∞Þ, the weighted Bergman spaces A p a ðDÞ is the space of all functions h ∈ H ðDÞ, for which When a = 0, we simply write A p ðDÞ for A p 0 ðDÞ, and when p = 2, A 2 a ðDÞ is a Hilbert space. It is well known that the Bergman kernel K z ðwÞ of the Hilbert space A 2 a ðDÞ is given by The Bergman projection P a is the orthogonal projection from L 2 ðD, dA a Þ onto Hilbert space A 2 a ðDÞ, which given as: For a > −1 and h ∈ H ðDÞ, the Toeplitz-type operator T a u with symbol u ∈ H ∞ ðDÞ is defined by This paper is organized as follows: during Section 2, we have defined the Toeplitz-superposition operators on the normed (metric) subspaces. Throughout Section 3, we establish the conditions for the Toeplitz-superposition operators to be bounded from a-Bloch space B a into b-Bloch space B b , in the case a ∈ ð0, 1Þ and b > a or b < a. Section 4 is devoted to a study the boundedness of Toeplitzsuperposition operators between weighted Bloch spaces in the case 0 < a ≤ b or a = 0, b > 0.
Remark 3. It is concerned remarkable to say that two concerned quantities N h and N * h , where both depending on the concerned function h ∈ H ðDÞ , the expression N h ≲ N * h , can be satisfied when we have a concerned positive constant C 1 , h can be written to say that there is an equivalence relation between the concerned quantities N h and N * h :

Toeplitz-Superposition Operators
Let EðℂÞ denote the set of all entire functions on the complex plane ℂ. For a function ϕ ∈ EðℂÞ, the superposition operator S ϕ : H ðDÞ → H ðDÞ is defined by S ϕ ðhÞ = ðϕ ∘ hÞ. Moreover, if u ∈ H ðDÞ and ϕ ∈ EðℂÞ, the weighted superposition oper-ator S ϕ,u : H ðDÞ → H ðDÞ is defined by S ϕ,u ðhÞðzÞ = uðzÞϕð hðzÞÞ, for all h ∈ H ðDÞ and z ∈ D. Note that, if uðzÞ = 1, then S ϕ,u = S ϕ , for any z ∈ D.
For any normed subspace X ⊂ H ðDÞ, we will consider the set KðXÞ, defined by Now, we define the Toeplitz-superposition operators acting on H ðDÞ. Definition 4. Let two functions ϕ ∈ EðℂÞ and u ∈ H ðDÞ . Then, the Toeplitz-superposition operators T u S ϕ on the normed (metric) subspace X are given by Let α, β be the scalers if ϕ is a fixed entire function and u, v ∈ H ðDÞ. Then, from the definition of Toeplitzsuperposition operators, we have which holds for all h ∈ KðXÞ, and hence, the Toeplitz-Superposition operators are linear on the normed subspace X. It can be seen that whenever u ∈ H ðDÞ, then, the operator T u S ϕ becomes the operator S ϕ,u . So, Toeplitzsuperposition operators can be taken as an extension of weighted superposition operators. The present paper is interested in answering the following interesting questions.
(i) Can we transform one holomorphic function space into another by what kinds of entire functions?
(ii) What are the holomorphic spaces that can be transformed one into another by certain weighted classes of entire functions such as specific analytic polynomials of a certain degree and certain entire-type functions of given type and order?
(iii) When does the holomorphic function φ induces a Toeplitz-superposition operators to form one holomorphic function space into another?
As a concerned result, the obtained results will introduce answers of the above mentioned questions by using the class of Toeplitz-superposition operators that are acting between different classes of Bloch functions.
Also, the answers for some of these concerned questions have been introduced by several authors; the following citations can be stated for interesting and intensive studies [8][9][10][11][12][13][14][15][16][17][18][19][20]. 2 Journal of Function Spaces 3. Boundedness in the case a ∈ ð0, 1Þ and b > a or b < a Several important discussions on boundedness property of the new operator acting on the analytic Bloch spaces are presented in this concerned section. Furthermore, some essential equivalent characterizations for its boundedness are established too. Now, we will introduce the main results of boundedness.
Theorem 5. For a ∈ ð0, 1Þ and b > a . Suppose that u ∈ H ∞ ðDÞ and let ϕ ∈ EðℂÞ , with ϕ ≠ 0 . Then, the Toeplitzsuperposition operator T u S ϕ : we have Now, let the constant R > 0 where h ∈ B a such that ∥h ∥ B a ≤ R, by Lemma 1, we have |hðzÞ | ≲R. Set R 1 = max |z|=R | ϕ ðzÞ | , then |ϕðhðzÞÞ | ≤R 1 . Since b > a, we have the fact that B a ⊂ B b , and since a ∈ ð0, 1Þ, we have that B a ⊂ H ∞ ðDÞ: Thus, where R, R 1 depended only on a, b, and ϕ. This shows that T u S ϕ : B a → B b is bounded. Theorem 6. For 0 < b < a < 1 , let u ∈ L 1 ðDÞ be harmonic and let ϕ ∈ EðℂÞ . Then, the Toeplitz-superposition operator T u S ϕ : B a → B b is bounded if and only if u ∈ H ∞ ðDÞ and ϕ is a constant entire function.
Proof. It is trivial that if u ∈ H ∞ ðDÞ and ϕ is constant, then T u S ϕ : B a → B b is bounded. If ϕ is constant, not identically 0, and T u S ϕ maps B a into B b then it is clear that u ∈ H ∞ ðDÞ. Assume now that u ≠ 0 and ϕ is not constant, and set T u S ϕ maps B a into B b . Let h be the constant function defined by hðζÞ = λ, for all ζ ∈ D, such that ϕðλÞ ≠ 0. Since h ∈ B a , it follows that T u S ϕ hðζÞ = T u ϕðλÞ ∈ B b . This implies that u ∈ B b ⊂ H ∞ ðDÞ, since 0 < b < 1. Finally, since ϕ is not constant, then there is a disk |w − w 0 | <ε and δ > 0, on which |ϕðwÞ | >δ|w|. Set the test function h 0 ðwÞ = w 0 + rð1 − wÞ 1−a ∈ B a . Then, for all w ∈ D, we have But, along with the positive radius, we get juðwÞj/ð j1 − wj a j1 − wzj 2 Þ → ∞, as w → 1. This shows that T u S ϕ : B a → B b is not bounded.

Boundedness in the case
For a > 0 , let u ∈ L 1 ðDÞ be harmonic and let ϕ ∈ EðℂÞ . Then, T u S ϕ is bounded on B a if and only if u ∈ H ∞ ðDÞ and ϕ is an affine function (linear function plus a translation).
Proof. First, suppose that u ∈ H ∞ ðDÞ and ϕ is an affine function. It is easy to explain T u S ϕ is bounded from B a into itself.
On the other hand, assume that u ∈ H ∞ ðDÞ and ϕ ∈ EðℂÞ does not linear function. Then, by using the Cauchy estimates for ϕ ∈ EðℂÞ, we can find a sequence fw n g ⊂ ℂ, for each n ∈ ℕ such that |w n | → ∞ as n → ∞ and jϕðw n Þj = max jwj=n jϕðwÞj ≳ jw n j 2 . Also, since the weight ð1 − jζj 2 Þ a is typical, we can find a sequence of points fz n g ⊂ D such that|z n | → 1 − , with ð0:5<|z n |<1Þ and such that ð1−|z n | Þ | w n | = 1, for all n ∈ ℕ. Now consider the sequence of functions fh n g contained in B a satisfies ∥h n ∥ B a ≤ 1 and|h n ðz n Þ | = | w n | . Furthermore, we can suppose that h n ðz n Þ = w n . Hence, Because |w n | → ∞ as n → ∞. This shows that T u S ϕ : B a → B a cannot be bounded if ϕ ∈ EðℂÞ is not a linear function.

Journal of Function Spaces
Theorem 8. For 0 < a ≤ b , let u ∈ L 1 ðDÞ be harmonic and let ϕ ∈ EðℂÞ be an increasing and continuous function. Then, T u S ϕ : B a → B b is bounded if and only if u ∈ H ∞ ðDÞ , and for each λ ∈ ð0, 1Þ , there is a positive constant η whenever |w | >η , such that Proof. First, suppose that u ∈ H ∞ ðDÞ and (15) is true. Now, consider R 1 > 0 and let h ∈ B a satisfy ∥h∥ B a ≤ R 1 and select λ ∈ ð0, 1Þ such that λR 1 < 1. Then, there is η > 0, such that |ϕðwÞ | ≲ϕðλ | w | Þ, whenever |w | >η. Thus, since D R = fw ∈ ℂ : |w|≤Rg is a compact set and ϕ ∈ EðℂÞ is a continuous function, we can assume that |ϕðwÞ | ≤1, for all w ∈ D R . Hence, Using that the function ϕ is increasing and the fact that λ < 1, we have This shows that T u S ϕ : B a → B b is bounded. On the other hand, assume that u ∈ H ∞ ðDÞ and ϕ ∈ Eð ℂÞ does not satisfy (15). Then, we can find λ 1 ∈ ð0, 1Þ and a sequence fw n g ⊂ ℂ such that |w n | → ∞ as n → ∞ and |ϕð w n Þ | ≥ϕðλ 1 | w n | Þ, for all n ∈ ℕ. Since the weight ð1 − jzj 2 Þ a is typical, we can find a sequence of points fz n g ⊂ D such that |z n | → 1 − as n → ∞. Thus, we can consider a sequence of functions fh n g contained in B a satisfies ∥h n ∥ B a ≤ 1 and |h n ðz n Þ | ≲|w n |. Now, let z ∈ D and set the function f n ðzÞ = w n h n ðzÞ/h n ðz n Þ for all n ∈ ℕ. Then, we have f n ðz n Þ = w n and ∥f n ∥ B a ≲ 1. For large enough n ∈ ℕ, we obtain