Boundedness and Essential Norm of an Operator between Weighted-Type Spaces of Holomorphic Functions on a Unit Ball

: The boundedness of a sum-type operator between weighted-type spaces is characterized and its essential norm is estimated


Introduction
By N k , where k ∈ Z, we denote the set {n ∈ Z : n ≥ k}.Let B(a, r) = {z ∈ C n : |z − a| < r}, where a ∈ C n , r ≥ 0, |z| = z, z and z, w = ∑ n j=1 z j w j , z, w ∈ C n .Further, let B = B(0, 1), S = ∂B, dV(z) be the n-dimensional Lebesgue measure on B, H(B) be the space of holomorphic functions on B and S(B) be the family of holomorphic self-maps of B. For some basics on the functions in H(B), consult, e.g., [1].For some other presentations of the theory, see also [2,3].If f ∈ C(B) and f (z) ≥ 0, z ∈ B, then we call it a weight function and write f ∈ W(B).µ ∈ W(B) is radial if µ(z) = µ(|z|), z ∈ B. If µ ∈ W(B) is radial and non-increasing in |z|, and lim |z|→1 µ(z) = 0, then it is typical.If X is a normed space, then B X = {x : x X ≤ 1}.
Let X and Y be two normed spaces.A linear operator T : X → Y is bounded if there is C ≥ 0 such that T f Y ≤ C f X , f ∈ X, and we write T ∈ L(X, Y).The operator is compact if it maps bounded sets into relatively compact ones ( [4][5][6][7]), and we write T ∈ K(X, Y).The essential norm of T ∈ L(X, Y) is T e,X→Y = inf{ T + K X→Y : K ∈ K(X, Y)}.
If µ ∈ W(B), then the space of f ∈ H(B), such that is the weighted-type space H ∞ µ (B) = H ∞ µ .The little weighted-type space H ∞ µ,0 (B) = H ∞ µ,0 contains f ∈ H(B) such that lim |z|→1 µ(z)| f (z)| = 0.For some information on these function spaces see, e.g., [8][9][10][11][12][13][14].For several technical and theoretical reasons, these spaces are suitable choices for studying concrete linear operators from or to them.Each ϕ ∈ S(B) induces the composition operator C ϕ f (z) = f (ϕ(z)).Each u ∈ H(B) induces the multiplication operator M u f (z) = u(z) f (z).The radial derivative of f ∈ H(B) is f (z) = ∑ n j=1 z j D j f (z), where D j f (z where by D k f we denote the differentiation operator of the kth order f (k) (for k = 0, the identity operator is obtained).There has been some interest in these operators, integral-type operators (for some of them see, e.g., [14][15][16]), and their products.Besides the products of C ϕ and M u , there have been some investigations into the products of D and C ϕ .One of the first papers on these products was [17], where D • C ϕ between Bergman and Hardy spaces was studied.Ohno in [18] studied the products between Hardy spaces.S. Li and S. Stević then studied the operators between various spaces (see, e.g., [19], where we studied the products from H ∞ and the Bloch space to nth weighted-type spaces, and the related references therein).For some later investigations of the operators see, e.g., [20][21][22].The operator D • M u on Bloch-type spaces was studied in [23].
Motivated by the above-mentioned product-type operators, researchers started investigating some more complex operators.The operator D m ϕ,u := M u C ϕ D m is a natural generalization of the product C ϕ D and has been investigated in depth.One of the first studies of the operator was conducted in [24].Zhu studied the operator from Bergman-type spaces to some weighted-type spaces.The research was continued in [25], where the operator from Bloch-type spaces to weighted Bergman spaces was studied, and in [26], where the operator on weighted Bergman spaces was studied.In several papers, we have studied the operator between various spaces of holomorphic functions (see [27], where we studied the operator from the mixed-norm space to the nth weighted-type space, and the related references therein).For some later studies of the operator, see, e.g., [28][29][30][31][32][33].The operator ϕ,u , was introduced in [34] (see also [35]).
Here, we continue our research in [27, 34,36,37,44,45] by studying the boundedness and compactness and estimating the essential norm of the operators S m u,ϕ acting between weighted-type spaces of holomorphic functions.
By C we denote some positive constants.If we write a b (respectively, a b), then there is C > 0 such that a ≤ Cb (respectively, a ≥ Cb).If a b and b a, then we write a b.
Proof.For any fixed r ∈ (0, 1), the Cauchy-Schwarz and Cauchy inequalities imply for z ∈ B and f ∈ H(B).From (2), we have By the above two inequalities, we have for every f ∈ H ∞ µ (B) and z ∈ B. If we replace f by k−1 f in (4), then we obtain Since it holds that 1 µ(w) from which (3) holds for each m ∈ N.
where (a Lemma 4. Assume µ ∈ W(B) satisfies condition (6), where α > 0, m ∈ N, w ∈ B, f α w,t is defined in (7), and (a (18) and (19).Then, for each l ∈ {1, . . ., m}, there is where c hold.Moreover, we have sup w∈B h k in (20) by c k .Then, from (17), we get Lemma 2.5 in [19] shows that the determinant of the system, is not equal to zero.This implies that there is a unique solution (24).For these c k values, ( 20) satisfies ( 21) and (22).Finally, Lemma 2 implies sup w∈B h The following lemma is well known as a characterization of the compactness of a closed set in the little weighted-type space.Its proof is a slight modification of the proof of Lemma 1 in [54].Thus, we omit the proof.
Lemma 6.Let Y be a Banach space of holomorphic functions on B and µ be a typical weight function on B. Then, T : H ∞ µ,0 (B) → Y is compact if and only if it is weakly compact.

Proof.
Let

Boundedness
First, we consider the operator S m u,ϕ : To analyze S m u,ϕ , the growth condition for | m f | in Lemma 1 and the functions f α w,k and h (l) w defined in Lemmas 2 and 4, respectively, play an important role in our argument.The class of all typical weights satisfying conditions ( 2) and ( 6) is denoted by W 1 (B).
Furthermore, if it is bounded, then we have Proof.By Lemma 1, we have . By this inequality, we see that condition (25) implies k u,ϕ : for some polynomial P k whose coefficients are all non-negative.Since |ϕ(w , and thus we obtain from which, together with |ϕ(w)| ∑ n j=1 |ϕ j (w)|, we have for any w ∈ B with |ϕ(w)| ≤ 1/2.Combining ( 26) and (27), we get for each k ∈ N. Thus, we accomplish the proof.
Corollary 1.Under the assumptions of Theorem 1, the followings statements are equivalent: is bounded if and only if u and ϕ satisfy (25).Hence, Theorem 1 implies the desired claim.
) is also bounded.As in the proof of Theorem 1, condition (28) can be verified by the functions To prove the other direction, we assume that S m u,ϕ : is bounded and ( 28) is true for j = 1, m.By Theorem 1, it is enough to prove for j = 1, m.
If |ϕ(w)| > 0, by Lemma 4, then there is h and sup w∈B h By considering the boundedness of S m u,ϕ , we have Hence, it follows that sup By (28), we have sup and so J m < ∞.
Next, we assume that (29) holds for j = s + 1, m, for s ∈ {1, 2, . . ., m − 1}.For h (s) ϕ(w) as in Lemma 4, we see that sup w∈B h From (30), it follows that , so that we get sup On the other hand, by (28), we have sup Hence, (29) holds for j = s and thus for j = 1, m.
For the same reasons as in Corollary 1, we get the following corollary.
Corollary 2. Under the assumptions of Theorem 2, the followings statements are equivalent: (a) All the operators j u j ,ϕ : Proof.First, suppose that k u,ϕ : H ∞ µ,0 (B) → H ∞ ν (B) is bounded and (31) holds.Since and the norms on the spaces H ∞ ν,0 (B) and H ∞ ν (B) are the same, it immediately follows that the boundedness of k u,ϕ : In order to derive the condition (31), we consider the functions f j (z) = z j for j = 1, n.Since µ is typical, we see from which (31) easily follows.
if and only if j u j ,ϕ : Proof.Suppose that S m u,ϕ : H ∞ µ,0 (B) → H ∞ ν,0 (B) is bounded and (32) holds.Theorem 3 shows that it is enough to prove that j u j ,ϕ : H ∞ µ,0 (B) → H ∞ ν (B) are bounded for j = 1, m.For this purpose, it is sufficient to show the boundedness of for j = 1, m.Now, looking back at the proof of Theorem 2, by Lemma 4, there exists a function h and sup w∈B h . Hence, as in the proof of Theorem 2, we obtain sup On the other hand, the assumption (32) indicates and so we obtain Thus, (33) holds for j = m.We can also prove that (33) holds for all j = 1, m by exactly the same argument as in the proof of Theorem 2. Hence, Theorem 1 implies are bounded, and so The other direction is trivial from Theorem 3.

Essential Norm and Compactness
Here, we investigate the essential norm and the compactness of k u,ϕ and S m u,ϕ .To characterize the compactness of T, it is well known that it is sufficient to evaluate T e .To estimate the essential norm of k u,ϕ or S m u,ϕ , we need the properties of the test functions f α w,k and h w in Lemmas 2 and 4, respectively, plus the fact that f α w,k and h w converge weakly to 0 as |w| → 1 − 0. Since this weak convergence is verified by the condition (15) on µ ∈ W(B), we continue to assume that µ ∈ W 1 (B) and add further condition (15).The class of such weights we denote by W α,k (B).
By letting ρ → 1, we obtain the upper estimate .
To prove the lower estimate for k u,ϕ e , we take a sequence (z j , where f α w,1 are as in Lemma 2.Then, sup j≥1 G j H ∞ µ < ∞.As we pointed out in Remark 1, the assumption (15) on µ implies that G j → 0 uniformly on compact subsets of B as j → ∞.
That is, the lower estimate holds.The proof is accomplished.Corollary 3.Under the assumptions of Theorem 5, the followings statements are equivalent: Proof.By Theorem 5, it is enough to prove the equivalence (b) ⇔ (c).To do this, we estimate the essential norm of the bounded operator k u,ϕ : The upper estimate for this operator is obtained by the arguments in the proof of Theorem 5. On the other hand, we use the weak convergence of the sequence (G j ) j∈N to 0 in H ∞ µ (B) for the lower estimate.In fact, an application of the Hahn-Banach extension theorem implies that G j → 0 weakly in H ∞ µ,0 (B) as j → ∞.Thus, we also see that the essential norm of .
This indicates that (b) ⇔ (c) is true. .
Proof.The case ϕ ∞ < 1 is treated as in Theorem 5. Now, assume ϕ ∞ = 1.For a fixed r ∈ (0, 1), the operator C r is compact on for each z ∈ B, from Lemma 1, we have sup .
By noting (28), the same argument which derives (36) and (37) implies w are as in Lemma 4.Then, we see that sup k≥1 h 21) and ( 22), we have that j h and j h (s) hold for s = 1, m.Hence, it follows from ( 40) and ( 41) that , and so Now, we assume that for s ∈ {1, . . ., m − 1}, holds for j = s + 1, m. Equations ( 40) and ( 41) imply lim sup S m u,ϕ e , from which we easily get lim sup This indicates that (43) holds for j = s, and therefore holds for any j ∈ {1, . . ., m}.Hence, we obtain the lower estimate .
We complete the proof.
The following result is proved exactly by the previous arguments.
Corollary 4.Under the assumptions of Theorem 6, the followings statements are equivalent: ,ϕ e = 0 holds.On the other hand, we obtain for each z ∈ B. We consider the function f j (z) = z j for j = 1, n.Since µ is typical, we see Thus, (45) and (46) give that lim sup Now, we assume that ϕ ∞ = 1.In view of Theorem 5, it is sufficient to prove lim sup Take a sequence (z l ) l∈N ⊂ B such that lim sup If sup l∈N |ϕ(z l )| < 1, then (46) shows that the second limit in (48) is zero.Since the following inequality obviously holds lim sup we see that (47) holds as the upper limit of both sides is zero.
, which proves that (47) really holds.In exactly the same way as in Theorem 9, we also obtain the following result.= 0, j = 1, m.

Conclusions
We studied the boundedness of a recently introduced operator between weighted-type spaces of holomorphic functions and estimated its essential norm.To do this, we gave some methods, ideas and tricks which may be useful in investigations of related concrete linear operators, which will be the focus of our further investigations.