Remarks on hyponormal Toeplitz operators with nonharmonic symbols

Sumin Kim; Jongrak Lee

doi:10.1515/math-2023-0114

Article Open Access

Remarks on hyponormal Toeplitz operators with nonharmonic symbols

Sumin Kim and Jongrak Lee

Published/Copyright: September 2, 2023

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

Abstract

In this article, we present some necessary or sufficient conditions for the hyponormality of Toeplitz operator T φ on the Bergman space A 2 ( D ) . In particular, we consider the Toeplitz operator T φ with nonharmonic symbols φ under certain assumptions.

Keywords: Toeplitz operators; hyponormal operators; nonharmonic symbols; Bergman space

MSC 2010: 47B35; 47B20; 30H20

1 Introduction

Let ℒ ( ℋ ) be the set of bounded linear operators on the separable complex Hilbert space ℋ . An operator T ∈ ℒ ( ℋ ) is normal if its self-commutator [ T ∗ , T ] ≔ T ∗ T − T T ∗ = 0 and hyponormal if [ T ∗ , T ] ≥ 0 . Let d A be the normalized area measure on the open unit disk D in C , and L 2 ( D ) is a Hilbert space of square-integrable measurable functions on D with the inner product

⟨ f , g ⟩ = ∫ D f ( z ) g ( z ) ¯ d A ( z ) .

The Bergman space A 2 ( D ) is the space of analytic functions in L 2 ( D ) . Recall that the power series representation of f ∈ A 2 ( D ) is

f ( z ) = ∑ n = 0 ∞ a n z n where ∑ n = 0 ∞ ∣ a n ∣ 2 n + 1 < ∞ .

The multiplication operator M φ with symbol φ ∈ L ∞ ( D ) is defined by M φ f = φ f for f ∈ L 2 ( D ) . For any φ ∈ L ∞ ( D ) , the Toeplitz operator T φ on A 2 ( D ) is defined by T φ f = P ( φ f ) for f ∈ A 2 ( D ) . Here, P is the orthogonal projection from L 2 ( D ) onto A 2 ( D ) .

The basic properties of the Bergman space and the Hardy space are well known in a few studies [1–4]. In the case of Hardy space, the hyponormality of Toeplitz operators has been studied in [5–8] and references therein for more details. Cowen [5] characterized the hyponormality of Toeplitz operator T φ on the Hardy space by the properties of the symbol φ ∈ L ∞ ( T ) . Cowen’s method is to reconstruct the operator-theoretic problem of hyponormal Toeplitz operator into the problem of finding a solution to equations of functionals. Recently, Hwang [9], Hwang and Lee [10], and Ko and Lee [11] characterized the hyponormality of Toeplitz operators on A 2 ( D ) with harmonic symbols.

The hyponormality of Toeplitz operator is translation invariant, so we can assume that the constant term is zero. We list the well-known properties of Toeplitz operators T φ on A 2 ( D ) . Let f and g be in L ∞ ( D ) and α , β ∈ C , then T α f + β g = α T f + β T g , T f ∗ = T f ¯ , and T f ¯ T g = T f ¯ g if f or g is analytic.

Recently, Simanek [12] deduced the necessary and sufficient conditions for the hyponormality of Toeplitz operators with the nonharmonic symbols φ ( z ) = z n + C ∣ z ∣ s ( s > 0 , C ∈ C ). This is a special case for the symbol φ ( z ) = a m , n z m z ¯ n + a i , j z i z ¯ j such that n = 0 and i = j . Afterward, Fleeman and Liaw [13] studied the sufficient condition for hyponormal Toeplitz operators T φ on A 2 ( D ) with nonharmonic symbols of the form φ ( z ) = a m , n z m z ¯ n + a i , j z i z ¯ j with m > n , i > j , and m − n > i − j . Also, in our previous work [14], we characterized the hyponormal Toeplitz operators T φ with nonharmonic symbols φ ( z ) = a z m z ¯ n + b z s z ¯ t with m − n = s − t acting on A 2 ( D ) . In this article, we consider the necessary or sufficient condition for the hyponormality of Toeplitz operators with nonharmonic symbols on the Bergman spaces.

The purpose of this article is to characterize the hyponormal Toeplitz operators T φ with nonharmonic symbols φ ( z ) = a z m z ¯ n + b z s z ¯ t with m − n ≠ s − t acting on A 2 ( D ) . We first present some lemmas and well-known results for the hyponormal Toeplitz operators. Next, under suitable conditions on the nonharmonic symbols φ , we study some necessary or sufficient conditions for the hyponormality of Toeplitz operator T φ on A 2 ( D ) .

2 Hyponormal Toeplitz operators with nonharmonic symbols

First, we present several auxiliary lemmas to prove results.

Lemma 2.1

[9] For any s , t ∈ N ,

P ( z ¯ t z s ) = s − t + 1 s + 1 z s − t , if s ≥ t , 0 , if s < t .

Lemma 2.2

[9] For any nonnegative integer m, we deduce that

z ¯ m ∑ i = 0 ∞ c i z i 2 = ∑ i = 0 ∞ 1 i + m + 1 ∣ c i ∣ 2 ,
P ( z ¯ m ∑ i = 0 ∞ c i z i ) 2 = ∑ i = m ∞ i − m + 1 ( i + 1 ) 2 ∣ c i ∣ 2 .

Hwang [9], Hwang and Lee [10], and Sadraoui [15] characterized the hyponormality of Toeplitz operators T g ¯ + f with bounded and analytic functions f and g by the well-known inequality

(2.1) ‖ ( I − P ) ( g ¯ k ) ‖ ≤ ‖ ( I − P ) ( f ¯ k ) ‖

for every k in A 2 ( D ) . Furthermore, many authors have used the inequality (2.1) to study the hyponormal Toeplitz operators with harmonic symbols. However, in the case of the hyponormality of T φ on A 2 ( D ) with the nonharmonic symbol φ , we cannot apply the inequality (2.1) to φ , since φ cannot be separated by analytic and coanalytic parts. Therefore, we directly calculate the self-commutator [ T φ ∗ , T φ ] of T φ . Recently, Fleeman and Liaw [13] and Kim and Lee [14] considered the hyponormal Toeplitz operators T φ with nonharmonic symbol φ on the Bergman spaces A 2 ( D ) by assuming some assumptions for the symbols φ .

2.1 Necessary condition for hyponormal Toeplitz operators

In this subsection, we consider the necessary condition for hyponormal Toeplitz operators with nonharmonic symbols. In our previous work [14], we studied the necessary condition for the hyponormality of Toeplitz operators T φ with nonharmonic symbols of the form φ ( z ) = a z m z ¯ n + b z s z ¯ t with m ≥ n , t ≥ s , m ≠ t , and m − n = t − s on A 2 ( D ) .

Theorem 2.3

[14] Let φ ( z ) = a z m z ¯ n + b z s z ¯ t with nonnegative integers m , n , s , and t with m ≥ n , t ≥ s , m ≠ t , m − n = t − s , and nonzeros a , b ∈ C . If T φ on A 2 ( D ) is hyponormal, then

∣ a ∣ 2 ≥ max ( 2 m − n ) 2 ( t + m − n ) 2 , Λ ( m , n , t , s ) ∣ b ∣ 2 , if t > m , ∣ a ∣ 2 ≥ max ( m + 1 ) 2 ( t + 1 ) 2 , Λ ( m , n , t , s ) ∣ b ∣ 2 , if t < m ,

where Λ ( m , n , t , s ) = max i ∈ [ m − n , ∞ ) ( t + i − s + 1 ) ( t + i + 1 ) 2 − ( s + i − t + 1 ) ( s + i + 1 ) 2 ( m + i − n + 1 ) ( m + i + 1 ) 2 − ( n + i − m + 1 ) ( n + i + 1 ) 2 .

We will now consider more specific and necessary condition for the hyponormality of Toeplitz operators with nonharmonic symbol φ ( z ) = a z m z ¯ n + b z s z ¯ t without the assumption m − n = t − s .

Theorem 2.4

For any nonnegative integers m , n , s , and t , let φ ( z ) = a z m z ¯ n + b z s z ¯ t with m ≥ n , t ≥ s , and nonzeros a , b ∈ C . Suppose that T φ on A 2 ( D ) is hyponormal. Then, the following statements hold.

If m > t and m − n < t − s , then
∣ a ∣ 2 ≥ ( t − s + 1 ) ( m + 1 ) 2 ( m − n + 1 ) ( t + 1 ) 2 ∣ b ∣ 2 .
If m < t and m − n > t − s , then
∣ a ∣ 2 ≥ ( 2 t − 2 s ) ( m + t − s ) 2 ( m − n + t − s ) ( 2 t − s ) 2 ∣ b ∣ 2 .

Proof

We know that T φ is hyponormal if and only if for any k i ( z ) = ∑ i = 0 ∞ c i z i ∈ A 2 ( D ) ,

⟨ ( T φ * T φ − T φ T φ * ) k i ( z ) , k i ( z ) ⟩ ≥ 0 .

Using Lemmas 2.1 and 2.2, we have that T φ is hyponormal if and only if

T φ ∑ i = 0 ∞ c i z i 2 − T φ * ∑ i = 0 ∞ c i z i 2 = ∣ a ∣ 2 ∑ i = 0 ∞ m + i − n + 1 ( m + i + 1 ) 2 ∣ c i ∣ 2 + ∣ b ∣ 2 ∑ i = t − s ∞ s + i − t + 1 ( s + i + 1 ) 2 ∣ c i ∣ 2 − ∣ a ∣ 2 ∑ i = m − n ∞ n + i − m + 1 ( n + i + 1 ) 2 ∣ c i ∣ 2 − ∣ b ∣ 2 ∑ i = 0 ∞ t + i − s + 1 ( t + i + 1 ) 2 ∣ c i ∣ 2 + 2 Re a b ¯ ∑ i = m − n ∞ i + 1 ( n + i + 1 ) ( t + i + 1 ) c i − m + n c ¯ t − s + i − 2 Re a b ¯ ∑ i = t − s ∞ i + 1 ( m + i + 1 ) ( s + i + 1 ) c ¯ i + m − n c i − t + s ≥ 0

for any c i ∈ C ( i = 0 , 1 , 2 , … ) . Since c i ’s are arbitrary, set c i = 0 for any i ≥ min { t − s , m − n } . Then, hyponormality of T φ implies that

(2.2) ∣ a ∣ 2 ∑ i = 0 min { t − s , m − n } m + i − n + 1 ( m + i + 1 ) 2 ∣ c i ∣ 2 ≥ ∣ b ∣ 2 ∑ i = 0 min { t − s , m − n } t + i − s + 1 ( t + i + 1 ) 2 ∣ c i ∣ 2 .

For 0 ≤ i < min { t − s , m − n } , define the function

ζ ( i ) = ( t + i − s + 1 ) ( m + i + 1 ) 2 ( m + i − n + 1 ) ( t + i + 1 ) 2 .

If m > t and m − n < t − s , we have that
( m + i + 1 ) 2 ( m + i + 2 ) 2 > ( t + i + 1 ) 2 ( t + i + 2 ) 2 and m − n + i + 2 m − n + i + 1 > t − s + i + 2 t − s + i + 1 .
Therefore,
ζ ( i ) = ( t + i − s + 1 ) ( m + i + 1 ) 2 ( m + i − n + 1 ) ( t + i + 1 ) 2 > ( t + i − s + 2 ) ( m + i + 2 ) 2 ( m + i − n + 2 ) ( t + i + 2 ) 2 = ζ ( i + 1 ) ,
and so ζ ( i ) is a decreasing function for 0 ≤ i < m − n . Hence, if T φ is hyponormal, then
∣ a ∣ 2 ≥ ( t − s + 1 ) ( m + 1 ) 2 ( m − n + 1 ) ( t + 1 ) 2 ∣ b ∣ 2 .
If m < t and m − n > t − s , then by the similar argument as in (i), we obtain that ζ ( i ) is an increasing function for 0 ≤ i < t − s . Hence, if T φ is hyponormal, then
∣ a ∣ 2 ≥ ( 2 t − 2 s ) ( m + t − s ) 2 ( m − n + t − s ) ( 2 t − s ) 2 ∣ b ∣ 2 .
This completes the proof.□

Corollary 2.5

Let φ ( z ) = a z m z ¯ n + b z s z ¯ t with nonnegative integers m , n , s , and t with m ≥ n , t ≥ s , and nonzeros a , b ∈ C . Suppose that T φ on A 2 ( D ) is hyponormal. If m > t and m − n > t − s , or m < t and m − n < t − s , then

∣ a ∣ 2 ≥ max 0 ≤ i < min { m − n , t − s } ( t + i − s + 1 ) ( m + i + 1 ) 2 ( m + i − n + 1 ) ( t + i + 1 ) 2 ∣ b ∣ 2 .

Proof

In the proof of Theorem 2.4, if T φ is hyponormal, then equation (2.2) holds. If, m > t and m − n > t − s (or m < t and m − n < t − s ), then ζ ( i ) is not a monotone function for i , therefore

∣ a ∣ 2 ≥ max 0 ≤ i < min { m − n , t − s } ( t + i − s + 1 ) ( m + i + 1 ) 2 ( m + i − n + 1 ) ( t + i + 1 ) 2 ∣ b ∣ 2 . □

Corollary 2.6

Let φ ( z ) = a z m z ¯ + b z z ¯ t with nonnegative integers m , t , and nonzeros a , b ∈ C . Suppose that T φ on A 2 ( D ) is hyponormal. If m < t , then

∣ a ∣ 2 ≥ ( 2 t − 2 ) ( m + t − 1 ) 2 ( m + t − 2 ) ( 2 t − 1 ) 2 ∣ b ∣ 2 ,

and if m > t , then

∣ a ∣ 2 ≥ t ( m + 1 ) 2 m ( t + 1 ) 2 ∣ b ∣ 2 .

Proof

Let f ( x ) = ( t + x ) ( m + x + 1 ) 2 ( m + x ) ( t + x + 1 ) 2 . Then, by a direct calculation,

f ′ ( x ) = ( m + x + 1 ) ( t + x + 1 ) ( m − t ) { − x 2 + ( 2 − m − t ) x + m + t + 1 − m t } ( m + x ) 2 ( t + x + 1 ) 4

and the last term in the numerator has a maximum at x = 2 − m − t 2 < 0 , and so if m > t , then f ( x ) is a decreasing function for x ≥ 0 . Similarly, if m < t , then f ( x ) is an increasing function for x ≥ 0 . By Corollary 2.5, if m < t , then for 0 ≤ i < t − 1 , ( t + i ) ( m + i + 1 ) 2 ( m + i ) ( t + i + 1 ) 2 is increasing, and so

max 0 ≤ i < t − 1 ( t + i ) ( m + i + 1 ) 2 ( m + i ) ( t + i + 1 ) 2 = ( 2 t − 2 ) ( m + t − 1 ) 2 ( m + t − 2 ) ( 2 t − 1 ) 2 .

Hence,

∣ a ∣ 2 ≥ ( 2 t − 2 ) ( m + t − 1 ) 2 ( m + t − 2 ) ( 2 t − 1 ) 2 ∣ b ∣ 2 .

If instead m > t , then for 0 ≤ i < m − 1 , ( t + i ) ( m + i + 1 ) 2 ( m + i ) ( t + i + 1 ) 2 is decreasing, and so

max 0 ≤ i < m − 1 ( t + i ) ( m + i + 1 ) 2 ( m + i ) ( t + i + 1 ) 2 = t ( m + 1 ) 2 m ( t + 1 ) 2 .

Hence,

∣ a ∣ 2 ≥ t ( m + 1 ) 2 m ( t + 1 ) 2 ∣ b ∣ 2 .

□

2.2 Sufficient condition for hyponormal Toeplitz operators

In this subsection, we consider the sufficient condition for hyponormal Toeplitz operators with nonharmonic symbols. We record here some results on the sufficient condition for the hyponormality of Toeplitz operators on the Bergman spaces with nonharmonic symbols, which have been recently developed in the study by Fleeman and Liaw [13].

Theorem 2.7

[13] Suppose f ( z ) = a m , n z m z ¯ n and g ( z ) = a i , j z i z ¯ j with m > n , i > j , and m − n > i − j . Then, T f + g is hyponormal if, for each k ≥ 0 , the term

(2.3) a m , n a i , j m − n + k + 1 ( m + k + 1 ) 2 + a i , j a m , n i − j + k + 1 ( i + k + 1 ) 2

is sufficiently large.

Here, sufficiently large refers to the condition where the term (2.3) is larger than the one defined as C k + D k in Remark 2, as mentioned in [13]. Furthermore, they proved the sufficient conditions for the hyponormality of Toeplitz with nonharmonic polynomials of fixed relative degree.

Theorem 2.8

[13] Let φ ( z ) = a 1 z m 1 z ¯ n 1 + ⋯ + a k z m k z ¯ n k with m 1 − n 1 = ⋯ = m k − n k = δ ≥ 0 , and a i all lying in the same quarter-plane 1 ≤ i ≤ k (i.e., we have max 1 ≤ i , j ≤ k ∣ arg ( a i ) − arg ( a j ) ∣ ≤ π 2 ), then T φ is hyponormal.

To overcome the assumption max 1 ≤ i , j ≤ k ∣ arg ( a i ) − arg ( a j ) ∣ ≤ π 2 in Theorem 2.8, we consider the hyponormality of Toeplitz operators T φ to nonharmonic symbols of the form φ ( z ) = a z m z ¯ m − 1 + b z m + 1 z ¯ m + c z m + 2 z ¯ m + 1 ( m ≥ 1 ) under certain assumptions acting on A 2 ( D ) .

Theorem 2.9

Suppose that φ ( z ) = a z m z ¯ m − 1 + b z m + 1 z ¯ m + c z m + 2 z ¯ m + 1 with m ≥ 1 and arg a = arg b + π 2 = arg c + π or arg a = arg b − π 2 = arg c + π . If ∣ a ∣ > 2 m + 3 2 m − 1 ∣ c ∣ , then T φ on A 2 ( D ) is hyponormal.

Proof

For φ ( z ) = a z m z ¯ m − 1 + b z m + 1 z ¯ m + c z m + 2 z ¯ m + 1 , we have

T φ ∑ i = 0 ∞ c i z i 2 − T φ * ∑ i = 0 ∞ c i z i 2 = P a z ¯ m − 1 ∑ i = 0 ∞ c i z i + m + P b z ¯ m ∑ i = 0 ∞ c i z i + m + 1 + P c z ¯ m + 2 ∑ i = 0 ∞ c i z i + m + 2 2 − P a ¯ z ¯ m ∑ i = 0 ∞ c i z i + m − 1 + P b ¯ z ¯ m + 1 ∑ i = 0 ∞ c i z i + m + P c ¯ z ¯ m + 2 ∑ i = 0 ∞ c i z i + m + 1 2 .

By using the Lemmas 2.1 and 2.2, we deduce that

T φ ∑ i = 0 ∞ c i z i 2 − T φ * ∑ i = 0 ∞ c i z i 2 = ∣ a ∣ 2 ∑ i = 0 ∞ i + 2 ( m + i + 1 ) 2 ∣ c i ∣ 2 + ∣ b ∣ 2 ∑ i = 0 ∞ i + 2 ( m + i + 2 ) 2 ∣ c i ∣ 2 + ∣ c ∣ 2 ∑ i = 0 ∞ i + 2 ( m + i + 3 ) 2 ∣ c i ∣ 2 + 2 ∑ i = 0 ∞ ( i + 2 ) Re ( a b ¯ ) ( m + i + 1 ) ( m + i + 2 ) + ( i + 2 ) Re ( a c ¯ ) ( m + i + 1 ) ( m + i + 3 ) + ( i + 2 ) Re ( b c ¯ ) ( m + i + 2 ) ( m + i + 3 ) ∣ c i ∣ 2 − ∣ a ∣ 2 ∑ i = 1 ∞ i ( m + i ) 2 ∣ c i ∣ 2 − ∣ b ∣ 2 ∑ i = 1 ∞ i ( m + i + 1 ) 2 ∣ c i ∣ 2 − ∣ c ∣ 2 ∑ i = 1 ∞ i ( m + i + 2 ) 2 ∣ c i ∣ 2 − 2 ∑ i = 1 ∞ i Re ( a ¯ b ) ( m + i ) ( m + i + 1 ) + i Re ( a ¯ c ) ( m + i ) ( m + i + 2 ) + i Re ( b ¯ c ) ( m + i + 1 ) ( m + i + 3 ) ∣ c i ∣ 2 .

Since arg a = arg b + π 2 = arg c + π or arg a = arg b − π 2 = arg c + π , we have that Re ( a b ¯ ) = Re ( b c ¯ ) = 0 and Re ( a c ¯ ) = − ∣ a ∣ ∣ c ∣ , and hence T φ is hyponormal if and only if

∑ i = 0 ∞ ( i + 2 ) ∣ a ∣ 2 ( m + i + 1 ) 2 + ( i + 2 ) ∣ b ∣ 2 ( m + i + 2 ) 2 + ( i + 2 ) ∣ c ∣ 2 ( m + i + 3 ) 2 − 2 ( i + 2 ) ∣ a ∣ ∣ c ∣ ( m + i + 1 ) ( m + i + 3 ) ∣ c i ∣ 2 − ∑ i = 1 ∞ i ∣ a ∣ 2 ( m + i ) 2 + i ∣ b ∣ 2 ( m + i + 1 ) 2 + i ∣ c ∣ 2 ( m + i + 2 ) 2 − 2 i ∣ a ∣ ∣ c ∣ ( m + i ) ( m + i + 2 ) ∣ c i ∣ 2 ≥ 0 ,

or equivalently,

(2.4) 2 ( m + 1 ) 2 ∣ a ∣ 2 + 2 ( m + 2 ) 2 ∣ b ∣ 2 + 2 ( m + 3 ) 2 ∣ c ∣ 2 − 4 ( m + 1 ) ( m + 3 ) ∣ a ∣ ∣ c ∣ ∣ c 0 ∣ 2 + ∑ i = 1 ∞ 2 m 2 + 2 m i − i ( m + i + 1 ) 2 ( m + i ) 2 ∣ a ∣ 2 + 2 m 2 + 2 m i + 4 m + i + 2 ( m + i + 1 ) 2 ( m + i + 2 ) 2 ∣ b ∣ 2 + 2 m 2 + 2 m i + 8 m + 3 i + 8 ( m + i + 2 ) 2 ( m + i + 3 ) 2 ∣ c ∣ 2 − 2 ( 2 m 2 + 2 m i + 4 m + i ) ( m + i ) ( m + i + 1 ) ( m + i + 2 ) ( m + i + 3 ) ∣ a ∣ ∣ c ∣ ∣ c i ∣ 2 ≥ 0 .

By the simple calculations,

2 ( m + 1 ) 2 ∣ a ∣ 2 + 2 ( m + 2 ) 2 ∣ b ∣ 2 + 2 ( m + 3 ) 2 ∣ c ∣ 2 − 4 ( m + 1 ) ( m + 3 ) ∣ a ∣ ∣ c ∣ ≥ 0 ,

and

2 m 2 + 2 m i + 4 m + i + 2 ( m + i + 1 ) 2 ( m + i + 2 ) 2 ∣ b ∣ 2 ≥ 0

for any i ≥ 1 . Set

f ( x ) ≔ ( 2 m 2 + 2 m i − i ) x 2 ( m + i + 1 ) 2 ( m + i ) 2 − 2 ( 2 m 2 + 2 m i + 4 m + i ) x ( m + i ) ( m + i + 1 ) ( m + i + 2 ) ( m + i + 3 ) + 2 m 2 + 2 m i + 8 m + 3 i + 8 ( m + i + 2 ) 2 ( m + i + 3 ) 2

for any i ≥ 1 . Since f is concave upward and the discriminant of f is positive, the larger root of f is:

x = ( m + i ) ( m + i + 1 ) { ( 2 m 2 + 2 m i + 4 m + i ) ± 4 i ( i + 2 ) } ( m + i + 2 ) ( m + i + 3 ) ( 2 m 2 + 2 m i − i ) ,

and this root is increasing for i ≥ 1 and

lim i → ∞ ( m + i ) ( m + i + 1 ) { ( 2 m 2 + 2 m i + 4 m + i ) + 4 i ( i + 2 ) } ( m + i + 2 ) ( m + i + 3 ) ( 2 m 2 + 2 m i − i ) = 2 m + 3 2 m − 1 .

Thus, if ∣ a ∣ > 2 m + 3 2 m − 1 ∣ c ∣ , then

∑ i = 1 ∞ 2 m 2 + 2 m i − i ( m + i + 1 ) 2 ( m + i ) 2 ∣ a ∣ 2 + 2 m 2 + 2 m i + 8 m + 3 i + 8 ( m + i + 2 ) 2 ( m + i + 3 ) 2 ∣ c ∣ 2 − 2 2 m 2 + 2 m i + 4 m + i ( m + i ) ( m + i + 1 ) ( m + i + 2 ) ( m + i + 3 ) ∣ a ∣ ∣ c ∣ ∣ c i ∣ 2 ≥ 0 .

Therefore, the inequality (2.4) holds for any c i ( i ≥ 0 ) , and hence T φ is hyponormal. This completes the proof.□

Corollary 2.10

Suppose that φ ( z ) = a z m z ¯ m − 1 + b z m + 1 z ¯ m + c z m + 2 z ¯ m + 1 with m ≥ 1 , and arg b = arg a + π 2 = arg c + π or arg b = arg a − π 2 = arg c + π . If ∣ b ∣ > 2 m + 3 2 m + 1 ∣ c ∣ , then T φ on A 2 ( D ) is hyponormal.

Proof

In a similar way to the proof of Theorem 2.9, T φ is hyponormal if and only if

(2.5) ∑ i = 0 ∞ ( i + 2 ) ∣ a ∣ 2 ( m + i + 1 ) 2 + ( i + 2 ) ∣ b ∣ 2 ( m + i + 2 ) 2 + ( i + 2 ) ∣ c ∣ 2 ( m + i + 3 ) 2 − 2 ( i + 2 ) ∣ b ∣ ∣ c ∣ ( m + i + 2 ) ( m + i + 3 ) ∣ c i ∣ 2 − ∑ i = 1 ∞ i ∣ a ∣ 2 ( m + i ) 2 + i ∣ b ∣ 2 ( m + i + 1 ) 2 + i ∣ c ∣ 2 ( m + i + 2 ) 2 − 2 i ∣ b ∣ ∣ c ∣ ( m + i + 1 ) ( m + i + 3 ) ∣ c i ∣ 2 ≥ 0 .

For i = 0 , the inequality (2.5) holds. For any i ≥ 1 , set

g ( x ) ≔ 2 m 2 + 2 m i + 4 m + i + 2 ( m + i + 1 ) 2 ( m + i + 2 ) 2 x 2 − 4 ( m + 1 ) x ( m + i + 1 ) ( m + i + 2 ) ( m + i + 3 ) + 2 m 2 + 2 m i + 8 m + 3 i + 8 ( m + i + 2 ) 2 ( m + i + 3 ) 2

for any i ≥ 1 . Since f is concave upward and the discriminant of g is positive, the larger root of g is:

x = ( m + i + 1 ) { ( m + i + 3 ) ( 2 m + 2 ) + i 2 + 2 i } ( 2 m 2 + 2 m i + 4 m + i + 2 ) ( m + i + 3 )

and this root is increasing for i ≥ 1 and

lim i → ∞ ( m + i + 1 ) { ( m + i + 3 ) ( 2 m + 2 ) + i 2 + 2 i } ( 2 m 2 + 2 m i + 4 m + i + 2 ) ( m + i + 3 ) = 2 m + 3 2 m + 1 .

Thus, if ∣ b ∣ > 2 m + 3 2 m + 1 ∣ c ∣ , then the inequality (2.5) holds for any c i ( i ≥ 0 ) , and hence T φ is hyponormal. This completes the proof.□

Corollary 2.11

Suppose that φ ( z ) = a z m z ¯ m − 1 + b z m + 1 z ¯ m + c z m + 2 z ¯ m + 1 with m ≥ 1 , and arg a = arg c + π 2 = arg b + π or arg a = arg c − π 2 = arg b + π . If ∣ a ∣ > 2 m + 1 2 m − 1 ∣ b ∣ , then T φ on A 2 ( D ) is hyponormal.

Proof

It is similar to the proof of Theorem 2.9.□

Example 2.12

Let φ ( z ) = a z 2 z ¯ + b z 3 z ¯ 2 + c z 4 z ¯ 3 .

If arg a = arg b + π 2 = arg c + π and ∣ a ∣ > 7 3 ∣ c ∣ , then T φ is hyponormal.
If arg b = arg a + π 2 = arg c + π and ∣ b ∣ > 7 5 ∣ c ∣ , then T φ is hyponormal.
If arg a = arg c + π 2 = arg b + π and ∣ a ∣ > 5 3 ∣ b ∣ , then T φ is hyponormal.

Corollary 2.13

Suppose that φ ( z ) = a z m z ¯ m − 1 + b z m + 1 z ¯ m with m ≥ 1 , and arg a = arg b + π . If ∣ a ∣ > 2 m + 1 2 m − 1 ∣ b ∣ , then T φ on A 2 ( D ) is hyponormal.

Corollary 2.14

Suppose that φ ( z ) = a z m z ¯ m − 1 + b z m + 2 z ¯ m + 1 with m ≥ 1 , and arg a = arg b + π . If ∣ a ∣ > 2 m + 3 2 m − 1 ∣ b ∣ , then T φ on A 2 ( D ) is hyponormal.

Acknowledgment

The authors would like to thank the referee for some helpful suggestions.

Funding information: The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2019R1A6A1A10073079). The second author was supported by the NRF funded by the Korea government (No. 2021R1C1C1008713).
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors declare that they have no conflicts of interest.

References

[1] S. Axler, Bergman spaces and their operators, in: J. B. Conway and B. B. Morrell (Eds.), Surveys of Some Recent Results in Operator Theory, Vol. I, Pitman Research Notes in Mathematics, Vol. 171, John Wiley & Sons, New York, 1988, pp. 1–50.Search in Google Scholar

[2] R. G. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, New York, 1972. Search in Google Scholar

[3] P. L. Duren, Theory of Hp Spaces, Academic Press, New York, 1970. Search in Google Scholar

[4] H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman spaces, Springer-Verlag, New York, 2000. 10.1007/978-1-4612-0497-8Search in Google Scholar

[5] C. C. Cowen, Hyponormality of Toeplitz operators, Proc. Amer. Math. Soc. 103 (1988), 809–812, DOI: https://doi.org/10.2307/2046858. 10.1090/S0002-9939-1988-0947663-4Search in Google Scholar

[6] C. Cowen, Hyponormal and subnormal Toeplitz operators, in: J. B. Conway and B. B. Morrell (Eds.), Surveys of Some Recent Results in Operator Theory, Vol. I, Pitman Research Notes in Mathematics, Vol. 171, Longman, 1988, pp. 155–167.Search in Google Scholar

[7] I. S. Hwang and W. Y. Lee, Hyponormality of trigonometric Toeplitz operators, Trans. Amer. Math. Soc. 354 (2002), 2461–2474, DOI: https://doi.org/10.1090/S0002-9947-02-02970-7. 10.1090/S0002-9947-02-02970-7Search in Google Scholar

[8] T. Nakazi and K. Takahashi, Hyponormal Toeplitz operators and extremal problems of Hardy spaces, Trans. Amer. Math. Soc. 338 (1993), 759–769, DOI: https://doi.org/10.1090/S0002-9947-1993-1162103-7. 10.1090/S0002-9947-1993-1162103-7Search in Google Scholar

[9] I. S. Hwang, Hyponormal Toeplitz operators on the Bergman spaces, J. Korean Math. Soc. 42 (2005), 387–403, DOI: https://doi.org/10.4134/JKMS.2005.42.2.387. 10.4134/JKMS.2005.42.2.387Search in Google Scholar

[10] I. S. Hwang and J. Lee, Hyponormal Toeplitz operators on the Bergman spaces, II, Bull. Korean Math. Soc. 3 (2007), no. 3, 517–522,10.4134/BKMS.2007.44.3.517Search in Google Scholar

[11] E. Ko and J. Lee, Remarks on hyponormal Toeplitz operators on the weighted Bergman spaces, Complex Anal. Operator Theory 14 (2020), 83, DOI: https://doi.org/10.1007/s11785-020-01046-7. 10.1007/s11785-020-01046-7Search in Google Scholar

[12] B. Simanek, Hyponormal Toeplitz operators with non-harmonic algebraic symbol, Anal. Math. Phys. 9 (2019), 1613–1626, DOI: https://doi.org/10.1007/s13324-018-00279-2. 10.1007/s13324-018-00279-2Search in Google Scholar

[13] M. Fleeman and C. Liaw, Hyponormal Toeplitz operators with non-harmonic symbol acting on the Bergman space, Oper. Matrices 13 (2019), 61–83, DOI: https://doi.org/10.7153/oam-2019-13-04. 10.7153/oam-2019-13-04Search in Google Scholar

[14] S. Kim and J. Lee, Hyponormality of Toeplitz operators with non-harmonic symbols on the Bergman spaces, J. Inequal. Appl. 67 (2021), 1–13, DOI: https://doi.org/10.1186/s13660-021-02602-1. 10.1186/s13660-021-02602-1Search in Google Scholar

[15] H. Sadraoui, Hyponormality of Toeplitz Operators and Composition Operators, PhD thesis, Purdue University, ProQuest Dissertations, 1992, https://www.proquest.com/openview/7e19b86c64552b36cdcddaa34165a993/1. Search in Google Scholar

Received: 2022-11-01

Revised: 2023-08-13

Accepted: 2023-08-15

Published Online: 2023-09-02

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2023-0114

Keywords for this article

Toeplitz operators; hyponormal operators; nonharmonic symbols; Bergman space

Creative Commons

BY 4.0