The Essential Norm of the Generalized Hankel Operators on the Bergman Space of the Unit Ball in Cn

and Applied Analysis 3 to define a kind of generalized Hankel operator:


Introduction
Let B be the open unit ball in C n , m the Lebesgue measure on C n normalized so that m B 1, H B denotes the class of all holomorphic functions on B. The Bergman space A 2 B is the Banach space of all holomorphic functions f on B such that B |f z | 2 dm z < ∞.It is easy to show that A 2 B is a closed subspace of L 2 B, dm .
There is an orthogonal projection of L 2 B, dm onto A 2 B , denoted by P and where I is the identity operator.
Since the Hankel operator H f is connected with the Toeplitz operator, the commutator, the Bloch space, and the Besov space, it has been extensively studied.Important papers in this context are 1, 2 for the case n 1 and 3-5 for the case n > 1.It is known that H f is bounded on A 2 B if and only if f ∈ β B and H f is compact A 2 B if and only if f ∈ β 0 B , where

1.3
Rf is the radial derivative of f defined by β B is called the Bloch space, and β 0 B is called the little Bloch space. For and dλ z 1 − |z| 2 −2 dm z is the invariant volume measure on D, B p D is called the Besov space on D. This theorem expresses that there is a cutoff of H f at p 1.
For n >

1.7
Obviously, c n depends on the dimension n of the unit ball.In 1993, Peloso 3 replaced f z − f w with to define a kind of generalized Hankel operator: Δ j f z, w K z, w g w dm w . 1.9 Here, Clearly, if j 1, H f,1 and H f,1 are just the classical Hankel operator H f .He proved that H f,j has the same boundedness and compactness properties as and only if f is a polynomial of degree at most j − 1.So the value of "cutoff" of H f,j is 2n/j; this means that the cutoff constant c n depends not only on the dimension but also on the degree of the polynomial and we are able to lower the cutoff constant by increasing j.
The cutoff phenomenon expressed that the generalized Hankel operator H f,j defined by Peloso and the classical Hankel operator H f are different.
In the present paper, we will consider the generalized Hankel operators H f,j defined by Peloso on the Bergman space A p B which is the Banach space of all holomorphic functions f on B such that B |f z | p dm z < ∞, for p > 1.
For f z ∈ H B , j is a positive integer, and we define the generalized Hankel operators H f,j and H f,j of order j with symbol f by

1.12
Luo and Ji-Huai 6 studied the boundedness, compactness, and the Schatten class property of the generalized Hankel operator H f,j on the Bergman space A p B p > 1 , which extended the known results.
We will study the essential norm of this kind of generalized Hankel operators H f,j and H f,j .We recall that the essential norm of a bounded linear operator T is the distance from T to the compact operators; that is, The essential norm of a bounded linear operator T is connected with the compactness of the operator T and the spectrum of the operator T.
We know that T ess 0 if and only if T is compact, so that estimates on T ess lead to conditions for T to be compact.Thus, we will obtain a different proof of the compactness of the generalized Hankel operators H f,j and H f,j .
Throughout the paper, C denotes a positive constant, whose value may change from one occurrence to the next one.

Preliminaries
For any fixed point a ∈ B − {0}, z ∈ B, define the Möbius transformation ϕ a by where s a 1 − |a| 2 and P a is the orthogonal projection from C n onto the one-dimensional subspace a generated by a, Q a is the orthogonal projection from C n onto C n ! a .It is clear that

2.2
Lemma 2.1.For every a ∈ B, ϕ a has the following properties: Proof.The proofs can be found in 7 .

2.3
Then, Here, the notation a z ∼ b z means that the ratio a z /b z has a positive finite limit as |z| → 1 − .Proof.This is in 7, Theorem 1.12 .
, then k ξ z has the following properties: Proof.It is obvious.
Lemma 2.4.Let K ξ z K z, ξ .Then, for any positive integer j, Proof.The proof is obtained by the definition of H f,j and H f,j and the reproducing property of K z, ξ , through the direct computation to get them.
Lemma 2.5.Let j be any positive integer, f ∈ H B , and 0 < q < ∞, then there is a constant C independent of f, such that where R j f is the jth order radial derivative of f, Proof.This is in 3, Proposition 3.2 .
Lemma 2.6.Let j be any positive integer, f ∈ H B , and 0 < ρ < 1, p > 1, then Proof. 1 Write F w, z for Δ j f w, z .Using the change of variables w ϕ z ξ , we obtain and set q q / q − 1 .Then, applying H ölder's inequality to * , we obtain Because of our choice of q , it follows that −ρq > −1 and is bounded by a constant.Therefore, applying 3, Theorem 3.4 , we get 2 The proof of 2 is similar to that of 1 .Δ j f z, w K z, w g w dm w .

3.1
Suppose that H f,j and H f,j are bounded on A p B , then the following quantities are equivalent: 1 H f,j ess and H f,j ess , Particularly, H f,j and H f,j are compact on Proof.First, we will prove that By the definition of k ξ z of Lemmas 2.3 and 2.4, we have dm τ .

3.8
We first show that T 1 is compact.Let {g l } be a sequence weakly converging to 0 and p p/ p − 1 , by Hölder's inequality, then we have

3.9
By Lemma 2.6, we get

3.11
So, T 1 is compact.For g ∈ A p and p p/ p − 1 , by Hölder's inequality, dm w dm z .

3.15
By the same argument of 3, Theorem 3.4 , we know that B χ B 0;ρ,1 z F ϕ z ξ , z p q dm ξ 1/q ≤ C sup 3.17 We complete the proof of Theorem 3.1.

1 / 1
− z, w n 1 is the Bergman kernel on B. For a function f ∈ H B , define the Hankel operator H f : A 2 B → A 2 B ⊥ with symbol f by H f g I − P fg B f z − f w K z, w g w dm w , 1.2 |Δ j f ξ, z | p |K z, ξ | p dm z .Use the change of variables z ϕ ξ τ in the integral I, and recall that dm z 1 − |ξ| 2 |1 − τ, ξ | 2 n 1 z K z, w g w dm w , T 2 g z χ B 0;ρ,1 z B F w, z K z, w g w dm w .
Then, the cutoff phenomenon of H f appears at p 2n.If c n denotes the value of "cutoff," then 2 − n 1 dm z is the invariant volume measure on B. B p B is called the Besov space on B.
Let f ∈ H B , j any positive integer, p > 1, and the generalized Hankel operators H f,j , H f,j defined on A p B by Theorem 3.1.