Essential norm of some extensions of the generalized composition operators between kth weighted-type spaces

We calculate the essential norm of some extensions of the generalized composition operators between kth weighted-type spaces on the unit disk in the complex plane, considerably extending some results in the literature.


Introduction
Let D be the open unit disk in the complex plane C, H(D) the class of all holomorphic functions on D, and S(D) the class of all holomorphic self-maps of D.
Let μ(z) be a positive continuous function on D (weight) and k ∈ N  . The kth weightedtype space denoted by W (k) μ (D) = W (k) μ is defined as follows: The space was introduced in [] where the composition operators from the weighted Bergman space to the space were studied. Some other concrete operators on the space were later studied in [-]. If k = , then b W () μ (·) is a norm on space W () μ , the so-called weighted-type space ([, ]). If k ∈ N, then it is easy to see that b W (k) μ (·) is a semi-norm on W (k) μ . It is not a norm on the space since from b W (k) μ (f ) =  it follows that f (k) (z) = , z ∈ D, and consequently f (z) = p k- (z), where p k- is a polynomial of degree at most k -. However, it is a norm on the quotient space W (k) μ /P k- , where P k- is the space of all polynomials of degree less than or equal k -. Indeed, let f + P k- ∈ W (k) μ /P k- , and, based on the definition of a norm on a quotient space, let Then, if f + P k- W (k) μ /P k- = , by using () and (), we have from which it follows that f ∈ P k- , that is, f + P k- = P k- =  W (k) On the other hand, there are some natural algebraic isomorphisms between some quotient spaces and some spaces of holomorphic functions. Namely, we have and Indeed, for each class g z j , and the map L(g + P k- ) := f g is a linear bijection from H(D)/P k- onto H k (D), as well as from W (k) μ /P k- onto W (k) μ,k (D). Hence, we can identify the quotient spaces with the corresponding subspaces of holomorphic functions satisfying the conditions f (j) () = , j = , k -.
From () and () it follows that this fact along with the above mentioned algebraic isomorphism shows that the spaces ( ) are isometrically isomorphic, that is, . So, they can be identified, and we can regard it to be the same if where μ is a weight and k ∈ N  (for k =  we use the standard convention l- j=l a j = , l ∈ Z). Then it is easy to see that () defines a norm on space W (k) μ , and that (W (k) μ , · W (k) μ ) is a Banach space. The normed space is a natural generalization of the weighted-type, Bloch-type and Zygmund-type spaces (see, e.g., [-]).
Let L : X → Y be a linear bounded operator, that is, it maps bounded sets of X into bounded sets of Y . By L X→Y , we denote the operator norm of L : X → Y , that is, Essential norm of a bounded operator L : X → Y is defined by that is, as the distance of operator L to the set of compact operators K(X, Y ). Let be the standard differentiation operator on H(D). By D k we will denote the composition of (exactly) k differentiation operators, that is, if f ∈ H(D), then Let where k ∈ N and f ∈ H(D). It is clear that D k I k f = f for every f ∈ H(D), that is, where Id X denotes the identity operator on space X. It is also easy to see that where we regard that D  is the identity operator. Beside this, by using the Newton-Leibnitz-type formula for holomorphic functions k times, we have where k ∈ N and f ∈ H(D), from which it follows that for every f ∈ H(D)/P k- , that is, I k D k is the identity operator on H(D)/P k- , and consequently on its subspaces, such as are Let ϕ ∈ S(D). Then by C ϕ we denote the composition operator on H(D), which is defined by C ϕ (f )(z) = f (ϕ(z)).
Let u ∈ H(D). Then by M g is denoted the multiplication operator on H(D), which is The product of operators C ϕ and M g , that is, is called the weighted composition operator and is denoted by gC ϕ . These three operators have been considerably studied on various spaces of holomorphic functions (see, for example, [, , , , ] and the references therein).
Let ϕ ∈ S(D), g ∈ H(D) and k ∈ N. We define an operator on H(D) as follows: for f ∈ H(D). For k =  is obtained the generalized composition operator in [], which was later studied or generalized, for example, in [, -]. For some related operators; see, also [-] and the references therein. Note that from () it immediately follows that Motivated by [, , ] here we calculate the essential norm of operator () between two kth weighted-type spaces. For some related results see also [, ].

Main results
In this section we prove the main results in this paper.
Theorem  Assume that μ and ν are weights, k, m ∈ N  , and that the operator L : is also a compact operator, from which along with the definition of the essential norm of an operator, it easily follows that (   ) ν be a compact operator and f ∈ W (k) μ . Then by where a j = f (j) ()/j!, j = , k -, is defined an extension of operator K on the whole space ν , which is obviously a compact operator. Denote the set of such obtained operators K by K. Let μ be a bounded operator, then the operator We have From () and (), equality () follows.
Theorem  Assume that μ and ν are weights, k, l, m ∈ N, m ≥ k, and that the operator C g ϕ,k : Proof First we prove the following inequality: We show that which by Theorem  is equivalent to () (recall that when m = k and l = , we naturally and it follows that where the strict inequality can occur here.
Hence, we see that the operator , and from (), () and the boundedness of the operator C g ϕ,k : W (m+l-) μ /P m+l- → W (m) ν , it follows that the operator gC ϕ : is also bounded.

Due to () and (), we have
for every f ∈ W (m+l-) μ /P m+l- . Indeed, since m ≥ k and l ∈ N we have m + l - ≥ k, so by () we have I k D k f = f , and further from () we get C g ϕ,k f ∈ W (m) μ /P k- . By another application of () is obtained (). Hence, by using ()-() and some simple estimates, we have By taking the supremum in () over the unit ball in W (m+l-) μ /P m+l- , and then taking the infimum in such obtained inequality over the set of all compact operators K : , we get (). Now we prove the following inequality: (   ) To do this first note that since () holds for every f ∈ W (m+l-k-) μ , the operator I k : is bounded by the assumption, and D k : is also bounded due to the inequality in (), we see that the operator ν be a compact operator. Then the operator Using this facts and (), it follows that By taking the supremum in () over the unit ball in W (m+l-k-) μ , and then taking the infimum in such obtained inequality over the set of all compact operators K : ν , the inequality () is obtained. From () and () equality () follows.
Before we formulate our next results, we want to say that their proofs are related to the one of Theorem , but we will give all the differences for the completeness.
Theorem  Assume that μ and ν are weights, k, l, m ∈ N, m ≥ k, and that the operator C g ϕ,k : We show that which by Theorem  is equivalent to (). and it follows that I k f ∈ W (m) μ /P m- , that is, operator I k maps space W (m-k) is bounded, and since for every h ∈ W (m+l-) Hence, we see that the operator () maps W (m-k) μ /P m-k- to W (m+l-k-) , and from (), () and the boundedness of C g ϕ,k : it follows that the operator gC ϕ : W (m-k) μ /P m-k- → W (m+l-k-) is also bounded. Beside this, since m ≥ k, we see that () holds for every f ∈ W (m) μ /P m- .
ν be a compact operator. Then the operator From this, () and (), we have By taking the supremum in () over the unit ball in W (m) μ /P m- , and then taking the infimum in such obtained inequality over the set of all compact operators K : , we see that the operator I k : is bounded by the assumption, and D k : ν is also bounded due to the inequality in (), we see that ν be a compact operator. Then the operator Using this fact along with () and (), we have By taking the supremum in () over the unit ball in W (m-k) μ , and then taking the infimum in such obtained inequality over the set of all compact operators K : , we get (). From () and () is directly obtained (), as desired.
Theorem  Assume that μ and ν are weights, k, l, m ∈ N, m ≥ k, and that the operator C g ϕ,k : W (m+l-) Proof First we prove the following inequality: Assume that f ∈ W (m+l-k-) μ /P m+l-k- . Then, since () and () hold, we see that the operator I k maps the space W (m+l-k-) ν /P m- is bounded, and since for every h ∈ W (m) ν /P m- , () holds and we have D k : Moreover, we see that () holds. Hence, we see that the operator () maps the space W (m+l-k-) μ /P m+l-k- into W (m-k) ν /P m-k- , and from (), () and the boundedness of the operator C g ϕ,k :