# Essential Norm of Difference of Composition Operators from Weighted Bergman Spaces to Bloch-Type Spaces.

1. Introduction and Preliminaries

Let D be the open unit disk in the complex plane C, H(D) the space of all holomorphic functions on D, and S(D) the set of all holomorphic self-maps of D. For [phi] [member of] S(D), the composition operator [C.sub.[phi]] is a linear operator defined by

[C.sub.[phi]]f = f [omicron] [phi], f [member of] H(D). (1)

For a general reference on composition operator we refer to . To understand the topological structures of spaces of composition operators, many people have studied difference of composition operators on different spaces of holomorphic functions; see, for example, [2-26] and references therein. Recently, Zhu and Yang  characterized boundedness and compactness of the differences of two composition operators from weighted Bergman spaces to Bloch spaces. Motivated by these results, in this paper we compute essential norm of difference of composition operators from weighted Bergman spaces to Bloch-type spaces.

Throughout this paper, constants are denoted by C; they are positive and not necessary the same in each occurrence. The notation A [??] B means that A is less than or equal to a constant times B and D [??] E means that D is greater than or equal to a constant times E. When A [??] B and A [??] B, then we write A [??] B.

For 0 < [alpha] < [infinity] and 0 < p < [infinity], the weighted Bergman space [A.sup.p.sub.[alpha]] is the space of all functions f [member of] H(D) such that

[mathematical expression not reproducible] (2)

where dA(z) is the normalized area measure on D. For any z [member of] D, the following point-evaluation estimate holds:

[mathematical expression not reproducible]. (3)

The following equivalent norm in the weighted Bergman spaces is well-known. Let 0 < p < [infinity], n [member of] N, and f [member of] H(D). Then f [member of] [A.sup.p.sub.[alpha]] if and only if, for all n [member of] N, [f.sup.(n)](z)[(1 - [[absolute value of z].sup.2]).sup.n] [member of] [L.sup.P]([(1 - [[absolute value of z].sup.2]).sup.[alpha]] d[A.sub.[alpha]]), and

[mathematical expression not reproducible]. (4)

Thus, if [mathematical expression not reproducible]. Therefore, by (3) applied to the function [f.sup.(n)], for every z in D, we have

[mathematical expression not reproducible]. (5)

Moreover, by Lemma 2 in , for any z, w [member of] D, there is a constant C > 0 such that

[mathematical expression not reproducible] (6)

for all f [member of] [A.sup.p.sub.[alpha]]. For more about weighted Bergman spaces, we refer the reader to [27, 28].

Let [mu] be a positive continuous function on D, which we call a weight. A weight [mu] is called typical if it is radial; that is, [mu](z) = [mu]([absolute value of z]), for each z [member of] D and []([absolute value of z]), decreasingly converges to 0 as [absolute value of z] [right arrow] 1. Throughout this paper by a weight, we shall mean a typical weight. For a weight [mu], the weighted Bloch-type space [B.sub.[mu]] on D is a Banach space of all analytic functions f on D such that

[mathematical expression not reproducible]. (7)

The space [B.sub.[mu]] is a Banach space with the norm

[mathematical expression not reproducible]. (8)

In particular, the space [mathematical expression not reproducible] is the classical Bloch space and we denote it by B.

2. Main Results

In this section, we compute upper and lower bounds for the essential norm of difference of composition operators mapping the weighted Bergman space into Bloch-type spaces.

Given Banach spaces X and Y, we recall that the essential norm of a bounded linear map T : X [right arrow] Y is defined as

[mathematical expression not reproducible]; (9)

that is, the essential norm of an operator T : X [right arrow] Y is the distance of T from the set of all compact operators from X to Y. Clearly, if T : X [right arrow] Y is compact, then [[parallel]T[parallel].sub.eX[right arrow]Y] = 0.

For [zeta] [member of] D, let [[eta].sub.[zeta]] be the conformal automorphism of D that interchanges 0 and [zeta]:

[[eta].sub.[zeta]](z) = [zeta] - z/1 - [bar.[zeta]]z, z [member of] D. (10)

The pseudo-hyperbolic distance between [zeta] and z is given by

[rho]([zeta], z) = [absolute value of [[eta].sub.[zeta]](z)] = [absolute value of [zeta] - z]/1 - [bar.[zeta]]z]. (11)

For [mu], a typical weight, and [phi], [psi] [member of] S(D), let [D.sup.[phi].sub.[mu]] and [D.sup.[psi].sub.[mu]] be two functions defined as

[mathematical expression not reproducible]. (12)

Theorem 1. Let [alpha] [member of] (1, [infinity]), p [member of] (0, [infinity]), [phi], [psi] [member of] S(D) such that [parallel][phi][parallel].sub.[infinity]] = [parallel][psi][parallel].sub.[infinity]] = 1 and let [mu] be a weight function such that [C.sub.[phi]], [C.sub.[psi]] : [A.sup.p.sub.[alpha]] [right arrow] [B.sub.[mu]] is bounded. Then

[mathematical expression not reproducible]. (13)

Proof. Let [([z.sub.j]).sub.n[member of]N] be a sequence in D such that [absolute value of [phi]([z.sub.j])] [right arrow] 1 as j and

[mathematical expression not reproducible]. (14)

For each k [member of] N, let [f.sub.j,k,[phi]] be defined as

[f.sub.j,k,[phi]](z) = 1 - [[absolute value of [phi]([z.sub.j])].sup.2]/ [(1 - [bar.[phi]([z.sub.j])]z).sup.k+(2+[alpha])/p]. (15)

Now consider the following functions:

[mathematical expression not reproducible]. (16)

Then it is easy to see that [g.sub.j,[phi]] and [h.sub.j,[phi]] are norm bounded sequences in [A.sup.p.sub.[alpha]]. Moreover, both sequences ([g.sub.j,[phi]]) and ([h.sub.j,[phi]]) converge to 0 uniformly on compact subsets of D. Let K : [A.sup.p.sub.[alpha]] [right arrow] [B.sub.[mu]] be any compact operator. Then

[mathematical expression not reproducible]. (17)

Again

[mathematical expression not reproducible]. (18)

Multiplying (18) by [rho]([phi]([z.sub.j]), [psi]([z.sub.j])) and then adding it to (19), we get

[mathematical expression not reproducible]. (20)

Similarly, by considering a sequence [([z.sub.j]).sub.n[member of]N] in D such that [absolute value of [psi]f([z.sub.j])] [right arrow] 1 as j [right arrow] [infinity] and

[mathematical expression not reproducible], (21)

we can prove that

[mathematical expression not reproducible], (22)

Combining (17) and (20) and using the fact that [mathematical expression not reproducible], we have

[mathematical expression not reproducible]. (23)

Combining (17) and (22) and using the fact that [mathematical expression not reproducible], we have

Again, let [([z.sub.j]).sub.n[member of]N] be a sequence in D such that [absolute value of [phi]([z.sub.j])] [conjunction] [absolute value of [psi]([z.sub.j])] [right arrow] 1 as j [infinity] [infinity] and

[mathematical expression not reproducible]. (25)

Then from (18) we have

[mathematical expression not reproducible]. (26)

Using (20) and (26), we have

[mathematical expression not reproducible]. (27)

Combining (17) and (27) and using the fact that [mathematical expression not reproducible], we have

[mathematical expression not reproducible]. (28)

Combining (23), (24), and (28), we get the lower bound as

[mathematical expression not reproducible]. (29)

Let [[tau].sub.k](z) = (k/(k + 1)) z. Then [[parallel][[tau].sub.k][parallel].sub.[infinity]] < 1. Let

[mathematical expression not reproducible]. (30)

Then, we have the fact that [L.sub.k] : [A.sup.p.sub.[alpha]] [right arrow] [A.sup.p.sub.[alpha]] is compact. Since [C.sub.[phi]] - [C.sub.[psi]] : [A.sup.p.sub.[alpha]] [right arrow] [B.sub.[mu]] is bounded, so ([C.sub.[phi]] - [C.sub.[psi]]) [L.sub.k] : [A.sup.p.sub.[alpha]] [right arrow] [B.sub.[mu]] is compact. Thus

[mathematical expression not reproducible], (31)

where I is the identity operator on [mathematical expression not reproducible]. For any r [member of] (0, 1), we can write

[mathematical expression not reproducible]. (32)

If [absolute value of [phi](z)] [conjunction] [absolute value of [psi](z)] [less than or equal to] r, then we have

[mathematical expression not reproducible]. (33)

Let [absolute value of [phi](z)] [conjunction] [absolute value of [psi](z)] [less than or equal to] r, and then [absolute value of [phi](z)] [less than or equal to] r and [absolute value of [psi](z)] < r. Let w [member of] D be such that [absolute value of w] < r. Denote the straight line segment from kw/(k + 1) to w by [kw/(k + 1), w]. Then the segment

[kw/(k + 1), w] [subset] D(0, r), where D(0, r) = {z : [absolute value of z] [less than or equal to] r}. Thus by (5), we have

[mathematical expression not reproducible]. (34)

Similarly, we can show that

[mathematical expression not reproducible]. (35)

Since [C.sub.[eta]] : [A.sup.p.sub.[alpha]] [right arrow] [B.sub.[mu]] is bounded, where [eta] = [phi] or [psi], so

[mathematical expression not reproducible]. (36)

By taking f(z) = z in (36), we have

[mathematical expression not reproducible]. (37)

Combining (33), (34), and (37), we have

[mathematical expression not reproducible]. (38)

Using (35) with w = [phi](0) and w = [psi](0), we have

[mathematical expression not reproducible]. (39)

Combining (38) and (39), we have

[mathematical expression not reproducible]. (40)

as k [right arrow] x. Finally, we have

[mathematical expression not reproducible]. (41)

For every f [member of] H(D) and z [member of] D, we have

[mathematical expression not reproducible]. (42)

and by Banach-Steinhaus theorem; it converges to zero uniformly on compact subsets of D, so we have

[mathematical expression not reproducible]. (43)

Also, for the boundedness of [C.sub.[phi]], [C.sub.[psi]] : [A.sup.p.sub.[alpha]] [right arrow] [B.sub.[mu]], we have the facts that [sup.sub.z[member of]D] [mu](z)[absolute value of [phi]'(z)] < [infinity] and [sup.sub.z[member of]D] [mu](z)[absolute value of [phi]'(z)] < [infinity]. Using these facts and (41), we see that for each r [member of] (0,1) and [absolute value of [psi](z)] [less than or equal to] r the right hand side of (41) is dominated by a constant multiple of

[mathematical expression not reproducible]. (44)

If [absolute value of [phi](z)] > r, then we see that the right hand side of (41) is dominated by a constant multiple of

[mathematical expression not reproducible]. (45)

Thus

[mathematical expression not reproducible]. (46)

Similarly, we can show that

[mathematical expression not reproducible]. (47)

Combining (31), (32), (40), (46), and (47), we have the fact that

[mathematical expression not reproducible]. (48)

Combining (29) and (48), we get the desired result.

https://doi.org/10.1155/2018/4670904

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The first author acknowledges DST (India) for Inspire fellowship (DST/Inspire fellowship/2013/281). The third author acknowledges NBHM (DAE) (India) for the Project Grant no. 02011/30/2017/R&D II/12068.

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Ram Krishan, (1) Mehak Sharma, (1) and Ajay K. Sharma (iD) (2)

(1) School of Mathematics, Shri Mata Vaishno Devi University, Kakryal, Katra 182320, India

(2) Department of Mathematics, Central University of Jammu, Bagla, Rahya-Suchani, Samba 181143, India

Correspondence should be addressed to Ajay K. Sharma; aksjm76@yahoo.com

Received 10 November 2017; Accepted 31 January 2018; Published 13 March 2018