Weighted composition operators on Bicomplex bloch space

Introduction

The theory of bicomplex numbers is an area of active research for quite a long time since the innovative work of Segre in search of special algebras. In 1892, Segre proposed the concept of bicomplex numbers which can be thought of as a generalization of complex numbers. Because not every non zero bicomplex number has a multiplicative inverse, the set of bicomplex numbers is a commutative ring with unity that contains the field of complex numbers but does not form a field, the study of zero divisors in bicomplex analysis is introduced. For a long time, bicomplex numbers have been explored. In recent years, a large amount of work has been published. G. B. Price's work [8] is a thorough examination of bicomplex numbers. One can refer to [1,5,8] for a systematic study of basic of bicomplex functional analysis. Composition operators are extensively studied on various holomorphic function spaces of different domains in ℂ and ℂ 𝑛 . There are many papers on the study of composition operators on Bloch space and Bloch type spaces of holomorphic functions, see [2,6] and references therein. The boundedness and the compactness of weighted composition operators on Bloch space was characterised by Ohno, see [7] . My results can be viewed as a generalisations of their results.

Preliminaries

In this section, we summarize some common facts about bicomplex numbers and bicomplex holomorphic functions. The set of bicomplex numbers 𝔹ℂ is defined as.

where 𝑖 and 𝑗 are imaginary units such that 𝑖𝑗 = 𝑗𝑖, 𝑖 2 = 𝑗 2 = -1 and ℂ(𝑖) is the set of complex numbers with the imaginary unit 𝑖. The set 𝔹ℂ of bicomplex numbers form a ring under the usual addition and multiplication of bicomplex numbers. " "Moreover, 𝔹ℂ is a module over itself. The set of positive hyperbolic number is denoted by 𝔻 + which is a subset of 𝑠𝑒𝑡 𝑜𝑓 ℎ𝑦𝑝𝑒𝑟𝑏𝑜𝑙𝑖𝑐 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝔻 is given by

We can discuss three conjugations for bicomplex numbers in the same way we can do for usual complex numbers, because 𝔹ℂ comprises two imaginary units with squares equal to -1 and a hyperbolic units with square equal to 1.

1. 𝑍 ̅ = 𝑤 ̅ 1 + 𝑗𝑤 ̅ 2 (𝑡ℎ𝑒 𝑏𝑎𝑟 -𝑐𝑜𝑛𝑗𝑢𝑔𝑎𝑡𝑖𝑜𝑛);

2. 𝑍 † = 𝑤 1 -𝑗𝑤 2 , (𝑡ℎ𝑒 † -𝑐𝑜𝑛𝑗𝑢𝑔𝑎𝑡𝑖𝑜𝑛);

where 𝑤 ̅ 1 , 𝑤 ̅ 2 denote the usual complex conjugates to 𝑤 1 , 𝑤 2 ∈ ℂ(𝑖).

If 𝑍 = 𝑤 1 + 𝑗𝑤 2 ≠ 0 is such that 𝑍 . 𝑍 † = 𝑤 1 2 + 𝑤 2 2 = 0, then 𝑍 is a zero divisor. The set of zero divisors ℕℂ of 𝔹ℂ is, thus, given by

and is called the null cone and ℕℂ 0 = ℕℂ ∪ {0}.

The hyperbolic numbers 𝑒 and 𝑒 † defined as

are zero divisors, which are linearly independent in the ℂ(𝑖)vector space and satisfy the following properties:

Any bicomplex number 𝑍 = 𝑤 1 + 𝑗𝑤 2 can be uniquely written as " "

where 𝑧 1 = 𝑤 1 -𝑖𝑤 2 and 𝑧 2 = 𝑤 1 + 𝑖𝑤 2 are elements of ℂ(𝑖). Formula (2.1) is called the idempotent representation of a bicomplex number 𝑍. A hyperbolic number 𝛼 = 𝛽 1 + 𝑘𝛽 2 in idempotent representation can be written as

where 𝛼 1 = 𝛽 1 + 𝛽 2 and 𝛼 2 = 𝛽 1 -𝛽 2 are real numbers. We say that 𝛼 is a positive hyperbolic number if 𝛼 1 ≥ 0 and 𝛼 2 ≥ 0. Writing these hyperbolic numbers in their idempotent form 𝛼 = 𝛼 1 𝑒 + 𝛼 2 𝑒 † and 𝛾 = 𝛾 1 𝑒 + 𝛾 2 𝑒 † , with real numbers 𝛼 1 , 𝛼 2 , 𝛾 1 and 𝛾 2 , we have that

If 𝛾 -𝛼 ∈ 𝔻 + \{0}, we write 𝛾 ≻ 𝛼. This imples that 𝑧 ∈ 𝔻 + is equivalent to 𝑧 ⪰ 0 and that 𝑧 ∈ 𝔻 + \{0} is equivalent to 𝑧 ≻ 0.

A module defined over the ring of bicomplex numbers 𝔹ℂ (or ring of hyperbolic numbers 𝔻) is called a 𝔹ℂ-module (or 𝔻module). Consider the set 𝔛 1 = 𝔛𝑒 and 𝔛 2 = 𝔛𝑒 † .Then 𝔛 1 ∩ 𝔛 2 = {0}. Thus, we can write

where 𝔛 1 = 𝔛𝑒 and 𝔛 2 = 𝔛𝑒 † are ℂ(𝑖)-vector (or ℝ-vector) spaces. Equation (2.2) is called the idempotent decomposition of 𝔛. The 𝔹ℂ module can be endowed canonically with the hyperbolic, or 𝔻-valued norm denoted by ‖⋅‖ 𝔻 as follows:

Definition 2.2 Let Ω ⊂ 𝔹ℂ → 𝔹ℂ be a function defined on the domain Ω. The derivative f′(z 0 )of the function f at a point z 0 ∈ Ω is the limit, if it exists,

such that 𝑧 -𝑧 0 is an invertible bicomplex number. If 𝑓 is derivable for all 𝑧 ∈ Ω, we say that it is a bicomplex holomorphic function in Ω.

Theorem 2.3 A bicomplex valued function f: Ω → 𝔹ℂ defined as

is a bicomplex holomorphic function on Ω if and only if f 1 and f 2 are complex holomorphic functions on Ω 1 and Ω 2 respectively. It derivatives is given by

The unit disk in ℂ(𝑖) will be denoted by 𝑈. The bidisk 𝕌 in 𝔹ℂ is the product-type domain:

Note that such bidisk should be seen as the cartesian product 𝑈 × 𝑈 in the idempotent ℂ 2 (𝑖).

3 Bio-complex Bloch spaces Definition 3.1 [9] A bicomplex holomorphic function f in 𝕌 is said to be bicomplex Bloch space ℬ if

The little bicomplex Bloch space ℬ 0 is the closed subspace of ℬ consisting of functions 𝑓 with :

The hyperbolic norm in the bicomplex Bloch space is defined as

With this hyperbolic norm, the bicomplex Bloch space is complete. By idempotent decomposition it is immediate that:

where ℬ 𝑙 and ℬ 0,𝑙 are the complex Bloch spaces, where 𝑙 = 1,2.

Lemma 3.2 [9] Let a ∈ 𝕌 and define the bicomplex Mobius transformation T: 𝕌 → 𝕌 as:

In particular ∥ 𝑇(𝑧) ∥ 𝔻 = 1 if and only if ∥ 𝑧 ∥ 𝔻 = 1, that is, if 𝑧 belongs to the distinguished boundary of 𝕌.

Lemma 3.3 [9] Let f: 𝕌 → 𝕌 be a bicomplex holomorphic function. Then

The equality occurs if and only if 𝑓 is a Mobius transformation of the bidisk. Let 𝑓: 𝕌 → 𝔹ℂ be a bicomplex holomorphic function and 𝜙: 𝕌 → 𝕌 be a bicomplex holomorphic self map of 𝕌. Thus, with 𝑧 = 𝑧 1 𝑒 + 𝑧 2 𝑒 † ∈ 𝕌, we can write 𝑓(𝑧) = 𝑓 1 (𝑧 1 )𝑒 + 𝑓 2 (𝑧 2 )𝑒 † , where 𝑓 1 and 𝑓 2 are complex valued holomorphic functions on the unit disk 𝑈 and 𝜙(𝑧) = 𝜙 1 (𝑧 1 )𝑒 + 𝜙 2 (𝑧 2 )𝑒 † , where 𝜙 1 and 𝜙 2 are holomorphic self maps of the unit disk 𝑈. Definition 3.4 [4] Let ϕ: 𝕌 → 𝕌 be a bicomplex holomorphic self map of 𝕌. We define the composition operator C ϕ induced from ϕ on ℬ by

Further, from lemma [see [4] ], we can write

where 𝐶 𝜙 𝑙 , for 𝑙 = 1,2 is a composition operator on complex Bloch spaces. Proof. For any 𝑓 ∈ ℬ, there exist two holomorphic functions 𝑓 1 , 𝑓 2: 𝑈 → ℂ(𝑖) such that 𝑓(𝑧) = 𝑓 1 (𝑧 1 )𝑒 + 𝑓 2 (𝑧 2 )𝑒 † . Since 𝑓 ∈ ℬ, it follows that 𝑓 1 and 𝑓 2 lie in the complex Bloch spaces. So by using [7] , it follows that ∥ 𝑓(𝑧) -𝑓(𝑤) ∥ 𝔻 =∥ (𝑓 1 (𝑧 1 ) -𝑓 1 (𝑤

Proof. For each 𝑓 ∈ ℬ and 𝑧 ∈ 𝕌, we can write 𝑓(𝑧) as 𝑓(𝑧) = 𝑓 1 (𝑧 1 )𝑒 + 𝑓 2 (𝑧 2 )𝑒 † , so by using [7] , it follows that

Proposition 3.8 Let ℬ be a bicomplex Bloch space and let 𝑓: 𝕌 → 𝔹ℂ be a bicomplex holomorphic function. If 𝑓 ∈ ℬ, then 𝑓𝑜 𝜙 ∈ ℬ.

Proof. If 𝑓 ∈ ℬ, then (1-∥ 𝑧 ∥ 𝔻 2 ) ∥ 𝑓′(𝑧) ∥ 𝔻 ≺ ∞. Now.

∥ (𝑓𝑜 𝜙) ′ (𝑧) ∥ 𝔻 =∥ 𝑓′(𝜙(𝑧)). 𝜙′(𝑧) ∥ 𝔻 =∥ 𝑓′(𝜙(𝑧)) ∥ 𝔻 . ∥ 𝜙′(𝑧) ∥ 𝔻

(1-∥ 𝑧 ∥ 𝔻 2 ) ∥ (𝑓𝑜𝜙)′(𝑧) ∥ 𝔻 = (1-∥ 𝑧 ∥ 𝔻 2 ) ∥ 𝑓′(𝜙(𝑧)) ∥ 𝔻 ∥ 𝜙′(𝑧) ∥ 𝔻 .

Since, 𝜙(𝑧) is bicomplex mobius transformation on 𝕌, so

Conditions for boundedness and compactness

Theorem 4.1 Let u be a bicomplex holomorphic function on 𝕌 and 𝜙: 𝕌 → 𝕌 be a bicomplex holomorphic self map of 𝕌. Then u. C ϕ is bounded on a bicomplex Bloch space ℬ if and only if the following are satisfied:

Proof. Suppose 𝑢. 𝐶 𝜙 is bounded on a bicomplex Bloch space ℬ. By above theorem, 𝑢. 𝐶 𝜙 is bounded on bicomplex Bloch space if and only if 𝑢 1 . 𝐶 𝜙 1 and 𝑢 2 . 𝐶 𝜙 2 are bounded on complex Bloch spaces. We can decompose the L.H.S of (i) and (ii) part as:

By [ [7] , theorem 1], sup 𝑧 𝑙 ∈𝑈 (1-∥ 𝑧 𝑙 ∥ 𝑙 2 ) ∥ 𝑢 𝑙 ′(𝑧 𝑙 ) ∥ 𝑙 log Thus, it follows that

Conversely, suppose that (i) and (ii) conditions are hold. We have to show that 𝑢. 𝐶 𝜙 is bounded on bicomplex Bloch space ℬ. Further,

where we use (3.1) Therefore, from [7] , we have lim

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