Hyponormality on a Weighted Bergman Space

Denote by Aα the space of analytic functions on DR such that k f k2 <∞. It is known that Aα is a Hilbert space [1], and an orthonormal basis is given by enðzÞ = ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Γðn + α + 2Þ p /Rn ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n!Γðα + 1Þ p Þzn. The Toeplitz operator with symbol f on Aα is defined by T f ðkÞ = Pð f kÞ where f is bounded and measurable on DR, k is in A 2 α, and P is the orthogonal projection of LðDR, dμαðzÞÞ onto Aα. Hankel operators are defined by Hf ðkÞ = ðI − PÞð f kÞ, f and k as before. Recall that a bounded operator T on a Hilbert space is hyponormal if T∗T − TT∗ is a positive operator. Hyponormality on the Hardy space is studied by Cowen in [2, 3]. Unweighted Bergman spaces on the unit disk ðα = 0, R = 1Þ and Toeplitz operators on these spaces were considered in [4, 5] and [6]. Hyponormality was first considered by Sadraoui in [7]. An improvement of the necessary condition is due to Ahern and Cuckovic [8]. A new necessary condition in a special case is found by Cuckovic and Curto in [9]. Many results on hyponormality on weighted Bergman spaces treat special cases. We cite for example [1, 10]. In this work for simplicity’s sake, we consider the case α = 1 and R = 1. Under a smoothness assumption, we give a fairly general necessary condition for the hyponormality of Toeplitz operators with a symbol of the form f + g where f and g are bounded and analytic. We also give sufficient conditions when f is a monomial and g a polynomial without assuming R = 1. We begin by recalling some general properties of Toeplitz operators.


Introduction
Let D R denote the disk of radius R in the complex plane, d μ α ðzÞ = ð1/πR 2 Þð1 − jz/Rj 2 Þ α dAðzÞ where dAðzÞ is the Lebesgue measure on D R and α > −1. L 2 ðD R , dμðzÞÞ denotes the Hilbert space of complex-valued functions on D R that are square integrable with respect to μ. We write kf k 2 = Ð D R jf ðzÞj 2 dμðzÞ. When f is analytic on D R , we have f z ð Þ = 〠 ∞ n=0 a n z n , f k k 2 = 〠 ∞ n=0 R 2n n!Γ α + 1 ð Þ Γ n + α + 2 ð Þ a n j j 2 : ð1Þ Denote by A 2 α the space of analytic functions on D R such that k f k 2 < ∞. It is known that A 2 α is a Hilbert space [1], and an orthonormal basis is given by e n ðzÞ = ð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Γðn + α + 2Þ p /R n ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi n!Γðα + 1Þ p Þz n . The Toeplitz operator with symbol f on A 2 α is defined by T f ðkÞ = Pðf kÞ where f is bounded and measurable on D R , k is in A 2 α , and P is the orthogonal projection of L 2 ðD R , dμ α ðzÞÞ onto A 2 α . Hankel operators are defined by H f ðkÞ = ðI − PÞðf kÞ, f and k as before. Recall that a bounded operator T on a Hilbert space is hyponormal if T * T − TT * is a positive operator. Hyponormality on the Hardy space is studied by Cowen in [2,3]. Unweighted Bergman spaces on the unit disk ðα = 0, R = 1Þ and Toeplitz operators on these spaces were considered in [4,5] and [6]. Hyponormality was first considered by Sadraoui in [7]. An improvement of the necessary condition is due to Ahern and Cuckovic [8].
A new necessary condition in a special case is found by Cuckovic and Curto in [9]. Many results on hyponormality on weighted Bergman spaces treat special cases. We cite for example [1,10]. In this work for simplicity's sake, we consider the case α = 1 and R = 1. Under a smoothness assumption, we give a fairly general necessary condition for the hyponormality of Toeplitz operators with a symbol of the form f + g where f and g are bounded and analytic. We also give sufficient conditions when f is a monomial and g a polynomial without assuming R = 1.
We begin by recalling some general properties of Toeplitz operators.

Some General Properties
We assume f , g are in L ∞ ðD R Þ. Then, we have: The use of these properties leads to describing hyponormality in more than one form. These are easy to prove by using Douglas lemma [11].
Proposition 1. Let f , g be bounded and analytic on D R . Then, the following are equivalent: [label=)] where C is of norm less than or equal to one We also need the following lemmas.

Lemma 2. For s and t integers, we have
Proof. We have We see that hPð z t z s Þ, z m i = 0 if s < t. For s ≥ t, hPð z t z s Þ, z m i ≠ 0 if and only if m = s − t. In this case, we get On the other hand, if Pð z t z s Þ = λz s−t , then so λ = hPð z t z s Þ, z s−t i/kz s−t k 2 . A computation shows so λ = R 2t ðs!Γðs − t + α + 2Þ/ðs − tÞ!Γðs + α + 2ÞÞ, and the result follows.
Lemma 3. If f = ∑ ∞ n=0 a n z n is bounded analytic on the disk D R , then the ði, jÞ th entry of the matrix of H * f H f with respect to the orthonormal basis fð ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Γðn + α + 2Þ p /R n ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n!Γðα + 1Þ p Þz n g n≥0 is given by: Proof.
We have: Thus, by Lemma 2, Journal of Function Spaces We also have by Lemma 2 Then, a m a n P z n z j À Á , P z m z i À Á : The above inner product is nonzero if i − m = j − n ≥ 0. We get by Lemma 2 When R = 1, we set D 1 = D. We obtain in this case the following corollary.
Corollary 4. If f = ∑ +∞ n=0 a n z n is bounded and analytic on the unit disk D, and if α = 1. Then, the matrix of H * f H f with respect to the orthonormal basis f ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn + 2Þðn + 1Þ p z n g n≥0 is given by Recall that the Hardy space H 2 is defined by a n z n analytic on the unit disk such that 〠 +∞ n=0 a n j j 2 < +∞ In the following theorem, the Toeplitz operator T j f ′j 2 on H 2 is not necessarily bounded. However, its matrix ðΓ i,j Þ i,j in the usual orthonormal basis of H 2 is defined.
The following theorem is an extension of Lemma 2.4.2 [7] to the weighted Bergman space.
Theorem 5. Assume f bounded analytic on the unit disk D and f ′ ∈ H 2 . Let ðΓ i,j Þ be the matrix of the Toeplitz operator on H 2 , T j f ′j 2 . Then, n 2 Δ i+n,j+n ⟶ 2Γ i,j as n ⟶ ∞.
Proof. The symmetric matrix of H * f H f with respect to the orthonormal basis f ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn + 2Þðn + 1Þ p z n g n≥0 is given by (set j = i + p in Corollary 4 with p ≥ 0.) To simplify, set C n,i,p = ði + n + 1Þði + n + 2Þði + n + p + 1Þ ði + n + p + 2Þ. Then, 3 Journal of Function Spaces for any integer n. Set m = p + l in the first sum, and m = l in the second. With some obvious notations, we get A computation shows that It is not difficult to see that and that the following inequality holds: for l ≤ i + n. We have that ∑ l 2 ja l j 2 + ðl + pÞ 2 ja l+p j 2 < ∞ since f ′ ∈ H 2 . Writing the sum ∑ l≤i+n n 2 A n,l,p,i a l a p+l as an integral, where the measure is the counting measure, and using the dominated convergence theorem, we get that Clearly, ffiffiffiffiffiffiffiffiffi ffi C n,i,p p ≤ ði + n + p + 1Þði + n + p + 2Þ. So we have for l ≥ i + n + 1, n 2 B n,l,p,i a l a l+p = n 2 ffiffiffiffiffiffiffiffiffi ffi C n,i,p p for l ≥ i + n + 1. Thus, This shows that 〠 l≥i+n+1 n 2 B n,l,p,i a l a p+l ! n→∞ 0: We deduce that

The Results
Using the previous theorem leads us to our main result which extends Theorem 5.4.3 [7].
Theorem 6. Let f and g be bounded and analytic on the unit disk D, and such that f ′ ∈ H 2 . If T f + g is hyponormal on A 2 1 then g ′ ∈ H 2 , and jg ′ j ≤ jf ′ j a.e. on the unit circle.
Proof. Set g = ∑ n b n z n . All diagonal terms of the matrix of H * f H f − H * g H g are positive; thus, using the same notation as in the previous proof, we have A n,l,p,i a l a p+l + 〠 l≥i+n+1 B n,l,p,i a l a p+l : We have seen that the left hand side tends to 2∑ l 2 ja l j 2 , a finite sum since f ′ ∈ H 2 . Writing ∑ l≤i+n n 2 A n,l,0,i jb l j 2 as an integral with respect to the counting measure, noticing that χ f0,::i+ng ðlÞn 2 A n,l,0,i jb l j 2 ! n→∞ 2l 2 jb l j 2 , and using Fatou's lemma, we deduce that 2∑ l 2 jb l j 2 ≤ 2∑ l 2 ja l j 2 and g ′ ∈ H 2 . If g H g is positive by hypothesis, it follows that the matrix of T j f ′ j 2 −jg ′ j 2 is also positive. A Toeplitz matrix is positive if its symbol is positive [12]. It follows that j f ′ j ≥ jg ′ ja:e on the unit circle.
To obtain sufficient conditions when f is a monomial, we need the following lemma.

Lemma 7.
Let q be a positive integer, the matrix of H * z q H z q is diagonal and given by: In what follows, we will take α = 1. It follows from Proposition 1 that, if p > q, the hyponormality of T z q +λz p is equivalent to the following inequalities: Using logarithmic differentiation, we can see that the right hand side of the last inequality is a decreasing function of i. Thus, the minimum is assumed at i = p − 1, and this proves the lemma.
A similar argument shows the following. Definition 13. For f ∈ A 2 1 , we denote by G f the set We see, from the density of H ∞ in A 2 1 and Proposition 1, that when g and f are in H ∞ , g ∈ G f is equivalent to T f + g is hyponormal. The following proposition lists some properties of G f [7]. We provide the proof for the sake of completeness.
Proposition 14. For f ∈ A 2 1 the following holds: [label=()] (a) G f is convex and balanced (b) If g ∈ G f , then g + c is in G f for any complex number c