Hyponormality on a weighted Bergman space of an annulus with a general harmonic symbol

Houcine Sadraoui; Borhen Halouani

doi:10.1515/math-2025-0212

Article Open Access

Hyponormality on a weighted Bergman space of an annulus with a general harmonic symbol

Houcine Sadraoui and Borhen Halouani

Published/Copyright: November 24, 2025

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$Open Mathematics$

From the journal Open Mathematics Volume 23 Issue 1

Abstract

In this work we consider the hyponormality of Toeplitz operators on the Bergman space of the annulus with a logarithmic weight. We give necessary conditions when the symbol is of the form φ + ψ ̄ where both φ and ψ are analytic on the annulus z ∈ C ; 1 / 2 < | z | < 1 and of the form ∑ n ≥ 1 a n z n + b n 1 z n . We also show a sufficient condition for hyponormality on the Bergman space of the unit disk with a square Logarithmic weight.

Keywords: hyponormality; Toeplitz operators; positive matrices; analytic functions; weighted Bergman space; annulus

MSC 2020: 47B35; 47B20; 15B48; 30Bxx

1 Introduction

A bounded operator T on a Hilbert space is hyponormal if T*T − TT* is positive. Hyponormality of Toeplitz operators on the Hardy space was considered by Cowen [1],2]. The first work on hyponormality on the Bergman space can be found in [3]. A general necessary condition, in the case of a harmonic symbol, is shown. An improvement of the necessary condition, which consists of a local version of it, uses function theory and is due to Ahern and Cuckovic [4]. Their method was recently extended to weighted Bergman spaces [5]. An improvement of the necessary condition in a special case is due Curto and Cuckovic [6]. Many partial results can be found in the literature, we cite for example results due to Phukon [7], Hwang [8],9]. Some of the results on hyponormality on the Bergman space of the unit disk, in the case of non-harmonic symbols, are due to Fleeman, and Liaw [10] and Simanek [11]. Sufficient conditions for hyponormality when the analytic part of the symbol is a monomial are given in [3]. Most of the other results on hyponormality on the Bergman space deal mostly with the case of specific symbols and use matrix computations. Some of these results can be found in [9]. Recent results on hyponormality on the Bergman space of an annulus can be found in [12],13]. In this work we generalize the necessary condition for hyponormality of Toeplitz operators B φ + ψ ̄ , on the Bergman space of a annulus with a logarithmic weight [13], to the case where both φ and ψ are of the form ∑ n ≥ 1 a n z n + ∑ n ≥ 1 b n 1 z n . We begin with definitions and notations. Set C 1 / 2 = z ∈ C ; 1 / 2 < | z | < 1 . The space L μ 2 is the space of measurable functions f on C _1/2 such that ∫ C 1 / 2 | f | 2 d μ ( z ) < ∞ where d μ ( z ) = 8 π ( 3 − 2 ⁡ ln ⁡ 2 ) r | log ⁡ r | d r d θ . The subspace of L μ 2 consisting of analytic functions is denoted by A μ 2 . If f is analytic on C _1/2, we have f = ∑ − ∞ − 1 c n z n + ∑ 0 ∞ c n z n and

‖ f ‖ 2 = 1 3 − 2 ⁡ ln ⁡ 2 ∑ 0 ∞ 2 2 n + 2 − 1 − ( 2 n + 2 ) ln ⁡ 2 2 2 n ( n + 1 ) 2 | c n | 2 + 8 ( ln ⁡ 2 ) 2 3 − 2 ⁡ ln ⁡ 2 | c − 1 | 2 + 4 3 − 2 ⁡ ln ⁡ 2 ∑ 2 ∞ 2 2 n − 1 ( n − 1 ) ln ⁡ 2 − 2 2 n − 2 + 1 ( n − 1 ) 2 | c − n | 2 .

The space A μ 2 has the following orthonormal basis:

{ e n ; n ≥ 0 } ∪ { e − 1 } ∪ { e − n ; n ≥ 2 } = 3 − 2 ⁡ ln ⁡ 2 2 n ( n + 1 ) 2 2 n + 2 − 1 − ( 2 n + 2 ) ln ⁡ 2 z n , n ≥ 0 ∪ 3 − 2 ⁡ ln ⁡ 2 2 2 ln ⁡ 2 1 z ∪ 3 − 2 ⁡ ln ⁡ 2 2 n − 1 2 2 n − 1 ( n − 1 ) ln ⁡ 2 − 2 2 n − 2 + 1 1 z n , n ≥ 2 .

For h bounded measurable on C _1/2 we define the Toeplitz operator B _h by B _h(f) = P(hf) where P is the orthogonal projection of L μ 2 onto A μ 2 . We also define Hankel operators by H _h(f) = (I − P)(hf). We start with some properties of Toeplitz operators in general. These properties are easy to prove and the proof is omitted.

2 General properties of Toeplitz and Hankel operators

Lemma 2.1.

Let φ and ψ be bounded measurable on C _1/2. The following holds:

B _φ+ψ = B _φ + B _ψ.
B φ * = B φ ̄ .
B _φ B _ψ = B _φψ if ψ is analytic or φ is conjugate analytic.
B φ * B φ − B φ B φ * = H φ ̄ * H φ if φ is analytic.

As in the case of the Bergman space of the unit disk, the following proposition gives equivalent forms of hyponormality. The Douglas lemma [14] is used to show (c) implies (d).

Proposition 2.2.

For φ ₁ and φ ₂ bounded analytic on C _1/2 the following statements are equivalents:

B φ 1 + φ 2 ̄ is hyponormal.
B φ 2 ̄ B φ 2 − B φ 2 B φ 2 ̄ ≤ B φ 1 ̄ B φ 1 − B φ 1 B φ 1 ̄ .
H φ 2 ̄ * H φ 2 ≤ H φ 1 ̄ * H φ 1 .
H φ 2 ̄ = K H φ 1 ̄ where K is a bounded operator of norm less than or equal 1.

Computations involving the projection are given in the next lemma.

Lemma 2.3.

[15] The orthogonal projection of L μ 2 onto A μ 2 satisfies the following properties:

P ( z m z n ̄ ) = 2 2 m + 2 − 1 − ( 2 m + 2 ) ln ⁡ 2 2 2 m ( m + 1 ) 2 2 2 ( m − n ) ( m − n + 1 ) 2 2 2 ( m − n ) + 2 − 1 − ( 2 ( m − n ) + 2 ) ln ⁡ 2 z m − n , m ≥ n ≥ 0.
P ( z m z n ̄ ) = 1 4 2 2 m + 2 − 1 − ( 2 m + 2 ) ln ⁡ 2 2 2 m ( m + 1 ) 2 ( ( n − m − 1 ) 2 2 2 ( n − m ) − 1 ( ( n − m ) − 1 ) ln ⁡ 2 − 2 2 ( n − m ) − 2 + 1 1 z n − m , n − m ≥ 2, m ≥ 0.
P ( z m z m + 1 ̄ ) = 1 8 ( ln ⁡ 2 ) 2 2 2 m + 2 − 1 − ( 2 m + 2 ) ln ⁡ 2 2 2 m ( m + 1 ) 2 1 z , m ≥ 0.
P ( 1 z m z n ̄ ) = 2 2 m − 1 ( m − 1 ) ln ⁡ 2 − 2 2 m − 2 + 1 ( m − 1 ) 2 ( m + n − 1 ) 2 2 2 ( m + n ) − 1 ( m + n − 1 ) ln ⁡ 2 − 2 2 ( m + n ) − 2 + 1 1 z m + n , m ≥ 2, n ≥ 0.
P ( 1 z z n ̄ ) = 2 ( ln ⁡ 2 ) 2 n 2 2 2 n + 1 n ⁡ ln ⁡ 2 − 2 2 n + 1 1 z n + 1 , n ≥ 1.
P ( 1 z m ̄ z n ) = 2 2 n + 2 − 1 − ( 2 n + 2 ) ln ⁡ 2 2 2 n ( n + 1 ) 2 2 2 ( m + n ) ( m + n + 1 ) 2 2 2 ( m + n ) + 2 − 1 − ( 2 ( m + n ) + 2 ) ln ⁡ 2 z m + n , n ≥ 0, m ≥ 0.
P ( 1 z m ̄ 1 z n ) = 4 2 2 n − 1 ( n − 1 ) ln ⁡ 2 − 2 2 n − 2 + 1 ( n − 1 ) 2 2 2 ( m − n ) ( m − n + 1 ) 2 2 2 ( m − n ) + 2 − 1 − ( 2 ( m − n ) + 2 ) ln ⁡ 2 z m − n , m ≥ n, n ≥ 2.
P ( 1 z m ̄ 1 z ) = 2 ( ln ⁡ 2 ) 2 2 2 m m 2 2 2 m − 1 − 2 m ⁡ ln ⁡ 2 z m − 1 , m ≥ 1.
P ( 1 z m ̄ 1 z n ) = 2 2 n − 1 ( n − 1 ) ln ⁡ 2 − 2 2 n − 2 + 1 ( n − 1 ) 2 ( n − m − 1 ) 2 2 2 ( n − m ) − 1 ( n − m − 1 ) ln ⁡ 2 − 2 2 ( n − m ) − 2 + 1 1 z n − m , n − m ≥ 2, m ≥ 1.
P ( 1 z m ̄ 1 z m + 1 ) ) = 1 2 ( ln ⁡ 2 ) 2 m 2 2 2 m + 1 m ⁡ ln ⁡ 2 − 2 2 m + 1 1 z , m ≥ 1.

3 The necessary condition

Set φ = φ ₁ + φ ₂ with φ ₁ = ∑ _n≥1 a _n z ⁿ, φ 2 = ∑ n ≥ 1 b n 1 z n and ψ = ψ ₁ + ψ ₂, with ψ ₁ = ∑ _n≥1 c _n z ⁿ, ψ 2 = ∑ n ≥ 1 d n 1 z n . We have the following identity

(1) B φ ̄ B φ − B φ B φ ̄ = B φ 1 ̄ B φ 1 − B φ 1 B φ 1 ̄ + B φ 2 ̄ B φ 2 − B φ 2 B φ 2 ̄ + B φ 1 ̄ B φ 2 − B φ 2 B φ 1 ̄ + B φ 2 ̄ B φ 1 − B φ 1 B φ 2 ̄

The following lemmas were proven in [13],15].

Lemma 3.1.

Let φ ₁ = ∑ _n≥1 a _n z ⁿ be bounded and analytic on C _1/2. For k and l ≥ 1 we have

⟨ B φ 1 ̄ B φ 1 − B φ 1 B φ 1 ̄ ( e k ) , e l ⟩ = ∑ n + k − l ≥ 1 , n ≥ 1 a n + k − l ̄ a n 2 k ( k + 1 ) 2 l ( l + 1 ) 2 2 k + 2 − 1 − ( 2 k + 2 ) ln ⁡ 2 ( 2 2 ( n + k ) + 2 − 1 − ( 2 ( n + k ) + 2 ) ln ⁡ 2 ) 2 2 l + 2 − 1 − ( 2 l + 2 ) ln ⁡ 2 2 2 ( n + k ) ( n + k + 1 ) 2 − ∑ n + k − l ≥ 1 , l ≥ n ≥ 1 a n + k − l ̄ a n 2 2 k + 2 − 1 − ( 2 k + 2 ) ln ⁡ 2 2 k ( k + 1 ) 2 l ( l + 1 ) 2 2 l + 2 − 1 − ( 2 l + 2 ) ln ⁡ 2 2 2 ( l − n ) ( l − n + 1 ) 2 ( 2 2 ( l − n ) + 2 − 1 − ( 2 ( l − n ) + 2 ) ln ⁡ 2 ) − 1 8 ( ln ⁡ 2 ) 2 a k + 1 ̄ a l + 1 2 2 k + 2 − 1 − ( 2 k + 2 ) ln ⁡ 2 2 k ( k + 1 ) 2 2 l + 2 − 1 − ( 2 l + 2 ) ln ⁡ 2 2 l ( l + 1 ) − 1 4 ∑ n ≥ l + 2 a n + k − l ̄ a n 2 2 k + 2 − 1 − ( 2 k + 2 ) ln ⁡ 2 2 k ( k + 1 ) 2 l ( l + 1 ) 2 2 l + 2 − 1 − ( 2 l + 2 ) ln ⁡ 2 ( n − l − 1 ) 2 ( 2 2 ( n − l ) − 1 ( n − l − 1 ) ln ⁡ 2 − 2 2 ( n − l ) − 2 + 1 ) .

Lemma 3.2.

Let φ 2 = ∑ 1 ∞ b n 1 z n be bounded and analytic on C _1/2. For k and l ≥ 1 we have

⟨ B φ 2 ̄ B φ 2 − B φ 2 B φ 2 ̄ ( e k ) , e l ⟩ = ∑ k ≥ n ≥ 1 , n + l − k ≥ 1 b n + l − k ̄ b n ( 2 k ( k + 1 ) 2 l ( l + 1 ) ( 2 2 ( k − n ) + 2 − 1 − ( 2 ( k − n ) + 2 ) ln ⁡ 2 ) [ 2 2 ( k + 2 ) − 1 − ( 2 k + 2 ) ln ⁡ 2 ] [ 2 2 l + 2 − 1 − ( 2 l + 2 ) ln ⁡ 2 ] 2 2 ( k − n ) ( k − n + 1 ) 2 + 8 ( ln ⁡ 2 ) 2 b l + 1 ̄ b k + 1 2 k ( k + 1 ) 2 2 ( k + 2 ) − 1 − ( 2 k + 2 ) ln ⁡ 2 2 l ( l + 1 ) 2 2 l + 2 − 1 − ( 2 l + 2 ) ln ⁡ 2 + 4 ∑ n ≥ k + 2 b n + l − k ̄ b n 2 k ( k + 1 ) 2 l ( l + 1 ) [ 2 2 ( k − n ) − 1 ( k − n − 1 ) ln ⁡ 2 − 2 2 ( k − n ) − 2 + 1 ] [ 2 2 ( k + 2 ) − 1 − ( 2 k + 2 ) ln ⁡ 2 ] [ 2 2 l + 2 − 1 − ( 2 l + 2 ) ln ⁡ 2 ] ( k − n − 1 ) 2 − ∑ n ≥ 1 , n + l − k ≥ 1 b n + l − k ̄ b n [ 2 2 ( k + 2 ) − 1 − ( 2 k + 2 ) ln ⁡ 2 ] [ 2 2 l + 2 − 1 − ( 2 l + 2 ) ln ⁡ 2 ] 2 2 ( n + l ) ( n + l + 1 ) 2 2 k ( k + 1 ) 2 l ( l + 1 ) [ 2 2 ( n + l ) + 2 − 1 − ( 2 ( n + l ) + 2 ) ln ⁡ 2 ] .

Denote by λ k , l the matrix of B φ 1 ̄ B φ 1 − B φ 1 B φ 1 ̄ in the orthonormal basis { e n , n ∈ Z } , and by γ k , l the matrix of B φ 2 ̄ B φ 2 − B φ 2 B φ 2 ̄ . On the Hardy space of the unit disk H ², denote by T _ω the Toeplitz operator on H ² with symbol ω, and denote by θ k , l , χ k , l respectively the matrices of the (possibly unbounded) Toeplitz operators T 2 | φ 1 ′ | 2 and T 2 | φ 2 ̃ ′ | 2 , where φ 2 ̃ = ∑ n ≥ 1 b n ̄ z n . Note some of our results in [13] are in terms of matrices m _l,l+p, while here for our purposes we express convergence results in terms of m _l+p,p. We need the following lemma.

Lemma 3.3.

Assume that φ 1 ′ ∈ H 2 . Then we have lim_l→∞ l ² λ _l+p,l = θ _s+p,s

Proof.

We have

λ l + p , l = ∑ n ≥ p + 1 a n − p ̄ a n 2 l ( l + 1 ) 2 l + p ( l + p + 1 ) 2 2 l + 2 − 1 − ( 2 l + 2 ) ln ⁡ 2 ( 2 2 ( n + l ) + 2 − 1 − ( 2 ( n + l ) + 2 ) ln ⁡ 2 ) 2 2 ( l + p ) + 2 − 1 − ( 2 ( l + p ) + 2 ) ln ⁡ 2 2 2 ( n + l ) ( n + l + 1 ) 2 − ∑ l + p ≥ n ≥ p + 1 a n − p ̄ a n 2 2 l + 2 − 1 − ( 2 l + 2 ) ln ⁡ 2 2 l ( l + 1 ) 2 l + p ( l + p + 1 ) × 2 2 ( l + p ) + 2 − 1 − ( 2 ( l + p ) + 2 ) ln ⁡ 2 2 2 ( l + p − n ) ( l + p − n + 1 ) 2 ( 2 2 ( l + p − n ) + 2 − 1 − ( 2 ( l + p − n ) + 2 ) ln ⁡ 2 ) − 1 8 ( ln ⁡ 2 ) 2 a l + 1 ̄ a l + p + 1 2 2 l + 2 − 1 − ( 2 l + 2 ) ln ⁡ 2 2 l ( l + 1 ) 2 2 ( l + p ) + 2 − 1 − ( 2 ( l + p ) + 2 ) ln ⁡ 2 2 l + p ( l + p + 1 ) − 1 4 ∑ n ≥ l + p + 2 a n − p ̄ a n 2 2 l + 2 − 1 − ( 2 l + 2 ) ln ⁡ 2 2 l ( l + 1 ) 2 l + p ( l + p + 1 ) 2 2 ( l + p ) + 2 − 1 − ( 2 ( l + p ) + 2 ) ln ⁡ 2 ( n − l − p − 1 ) 2 ( 2 2 ( n − l − p ) − 1 ( n − l − p − 1 ) ln ⁡ 2 − 2 2 ( n − l − p ) − 2 + 1 ) .

With obvious notations write

λ l + p , l = ∑ p + 1 ≤ n ≤ l + p a n − p ̄ a n R n , l , p + a l + 1 ̄ a l + p + 1 S p , l + ∑ n ≥ l + p + 2 a n − p ̄ a n T n , p , l .

An involved, but elementary computation, shows that

lim l → ∞ l 2 R n , l , p = 2 n ( n − p ) ,

and that

l 2 | R n , l , p | ≤ C n ( n − p ) , p + 1 ≤ n ≤ l + p , C is a constant .

Writing ∑ p + 1 ≤ n ≤ l + p a n − p ̄ a n l 2 R n , l , p as an integral with respect to the counting measure and applying the dominated convergence theorem we see, since∑ _n≥1 n ²|a _n|² is convergent, that

lim l → ∞ ∑ p + 1 ≤ n ≤ l + p a n − p ̄ a n l 2 R n , l , p = 2 ∑ n ≥ p + 1 n ( n − p ) a n − p ̄ a n .

Clearly

lim l → ∞ l 2 a l + 1 ̄ a l + p + 1 S p , l = 0 .

Similarly we show

lim l → ∞ l 2 ∑ n ≥ l + p + 2 a n − p ̄ a n T n , p , l = 0 .

We see that 2 ∑ n ≥ p + 1 n ( n − p ) a n − p ̄ a n = θ s + p , s , where θ i , j is the matrix of the Hardy space Toeplitz operator T 2 | φ 1 ′ | 2 . □

The following lemma can be shown using the same method and the proof is omitted.

Lemma 3.4.

Assume φ 2 ̃ ′ ∈ H 2 . Then we have lim_l→∞ l ² γ _l+p,l = χ _s+p,s.

Let us compute the matrix of B φ 1 ̄ B φ 2 − B φ 2 B φ 1 ̄ in the orthonormal basis { e n , n ∈ Z } . Set Q i = 2 i ( i + 1 ) 2 2 i + 2 − 1 − ( 2 i + 2 ) ln ⁡ 2 .

Lemma 3.5.

The matrix of B φ 1 ̄ B φ 2 − B φ 2 B φ 1 ̄ is given by ζ l , k where

ζ l , k = Q k Q l ∑ 1 ≤ n ≤ k − l − 1 1 Q n + l 2 a n ̄ b k − n − l − 1 Q k Q l ∑ 1 ≤ n ≤ k − l − 1 a n ̄ b k − n − l Q k − n 2 .

Proof.

We have for l, k ≥ 1

B φ 1 ̄ B φ 2 e k , e l = Q k Q l ( 3 − 2 ⁡ ln ⁡ 2 ) ∑ m , n ≥ 1 a n ̄ b m z k − m , z n + l = Q k Q l ∑ 1 ≤ n ≤ k − l − 1 1 Q n + l 2 a n ̄ b k − n − l

and B φ 1 ̄ B φ 2 e k , e l = 0 if k − l ≤ 1. Similarly we have

B φ 2 B φ 1 ̄ e k , e l = ∑ m , l ≥ 1 a n ̄ b m P z ̄ n e k , P 1 z ̄ m e l = ∑ m , l ≥ 1 a n ̄ b m ( 3 − 2 ⁡ ln ⁡ 2 ) Q k Q l P ( z ̄ n z k ) , P 1 z ̄ m z l = ∑ 1 ≤ n ≤ k − l − 1 a n ̄ b k − n − l ( 3 − 2 ⁡ ln ⁡ 2 ) Q k Q l Q k − n 2 Q k 2 z k − n , Q k − n 2 Q l 2 z k − n = 1 Q k Q l ∑ 1 ≤ n ≤ k − l − 1 a n ̄ b k − n − l Q k − n 2

and B φ 2 B φ 1 ̄ e k , e l = 0 if k − l ≤ 1. □

Notice that ζ l , k l , k ≥ 2 is upper triangular. Since B φ 2 ̄ B φ 1 − B φ 1 B φ 2 ̄ is the adjoint of B φ 1 ̄ B φ 2 − B φ 2 B φ 1 ̄ we deduce the following lemma.

Lemma 3.6.

The matrix of B φ 1 ̄ B φ 2 − B φ 2 B φ 1 ̄ is given by

η l , k = Q k Q l ∑ 1 ≤ n ≤ l − k − 1 1 Q n + k 2 a n b l − n − k ̄ − 1 Q k Q l ∑ 1 ≤ n ≤ l − k − 1 a n b l − n − k ̄ Q l − n 2 , k , l ≥ 2 .

Let us rewrite the expression of η _l,k.

η l + p , l = Q l Q l + p ∑ 1 ≤ n ≤ p − 1 1 Q n + l 2 a n b p − n ̄ − 1 Q l Q l + p ∑ 1 ≤ n ≤ p − 1 a n b p − n ̄ Q l + p − n 2 = ∑ 1 ≤ n ≤ p − 1 a n b p − n ̄ Q l Q l + p Q n + l 2 − Q l + p − n 2 Q l Q l + p .

We get an asymptotic expression of η _l+p,l.

Lemma 3.7.

lim l → ∞ l 2 η l + p , l = − 2 ∑ 1 ≤ n ≤ p − 1 n ( p − n ) a n b p − n ̄ .

The matrix σ i , j given by σ i + p , i = − 2 ∑ 1 ≤ n ≤ p − 1 n ( p − n ) a n b p − n ̄ and σ i , i + p = σ i + p , i ̄ is the matrix of the Hardy space Toeplitz operator with symbol − 4 R e φ 1 ′ φ 2 ̃ ′ − ( a 1 b 1 ̄ + 2 ( a 2 b 1 ̄ + a 1 b 2 ̄ ) z ) , where φ 2 ̃ ( z ) = ∑ n ≥ 1 b n ̄ z n . Let Φ i , j denote the matrix of B φ ̄ B φ − B φ B φ ̄ in the basis { e n , n ∈ Z } . From (1) using Lemmas 3.3, 3.4, and 3.7 we deduce the following

Lemma 3.8.

Under the assumptions φ 1 ′ , φ 2 ̃ ′ ∈ H 2 , we have

lim l → ∞ l 2 Φ l + p , l = θ s + p , s + χ s + p , s + σ s + p , s .

We have our first main result where we denote by Ψ i , j the matrix of B ψ ̄ B ψ − B ψ B ψ ̄ .

Theorem 3.9.

Assume φ ₁, φ ₂ are analytic and bounded, with φ 1 ′ , φ 2 ̃ ′ ∈ H 2 and that B φ + ψ ̄ is hyponormal. Then ψ 1 ′ , ψ 2 ̃ ′ ∈ H 2 and we have

| ψ 1 ′ | 2 + | ψ 2 ̃ ′ | 2 − 2 R e ψ 1 ′ ψ 2 ̃ ′ − ( c 1 d 1 ̄ + 2 ( c 2 d 1 ̄ + c 1 d 2 ̄ ) z ) ≤ | φ 1 ′ | 2 + | φ 2 ̃ ′ | 2 − 2 R e φ 1 ′ φ 2 ̃ ′ − ( a 1 b 1 ̄ + 2 ( a 2 b 1 ̄ + a 1 b 2 ̄ ) z )

a.e on the unit circle.

Proof.

Hyponormality implies

i 2 Ψ i , i ≤ i 2 Φ i , i = i 2 ( λ i , i + γ i , i ) .

Thus

lim inf i → ∞ i 2 Ψ i , i ≤ lim inf i → ∞ i 2 Φ i , i .

From Lemmas 3.3 and 3.4 we have

lim inf i → ∞ i 2 Φ i , i = ∑ 1 ∞ 2 n 2 | a n | 2 + | b n | 2 < ∞

since φ 1 ′ , φ 2 ̃ ′ ∈ H 2 . Write Ψ_i,i = C _i,i + D _i,i, where C i , j and D i , j , respectively, denote the matrices of B ψ 1 ̄ B ψ 1 − B ψ 1 B ψ 1 ̄ and B ψ 2 ̄ B ψ 2 − B ψ 2 B ψ 2 ̄ . Using Lemma 3.2 we have with obvious notations

C i , i = ∑ 1 ≤ n ≤ i E i , n | c n | 2 + F i | c i + 1 | 2 + ∑ n ≥ i + 2 G i , n | c n | 2

where, for example, we have

E i , n = 2 2 i ( i + 1 ) 2 2 2 i + 2 − 1 − ( 2 i + 2 ) ln ⁡ 2 ( 2 2 ( n + i ) + 2 − 1 − ( 2 ( n + i ) + 2 ) ln ⁡ 2 ) 2 2 ( n + i ) ( n + i + 1 ) 2 − 2 2 i + 2 − 1 − ( 2 i + 2 ) ln ⁡ 2 2 2 i ( i + 1 ) 2 2 2 ( i − n ) ( i − n + 1 ) 2 ( 2 2 ( i − n ) + 2 − 1 − ( 2 ( i − n ) + 2 ) ln ⁡ 2 ) .

We see that E _i,n is positive, since it is the ith diagonal term of B z ̄ n B z n − B z n B z ̄ n . Clearly F _i and G _i,n are positive. A computation shows that

lim i → ∞ i 2 E i , n = 2 n 2 .

Writing ∑ _1≤n≤i i ² E _i,n|c _n|² as an integral with respect to the counting measure and applying Fatou’s lemma we obtain

lim inf i → ∞ i 2 C i , i = ∑ n ≥ 1 2 n 2 | c n | 2 ≤ ∑ 1 ∞ 2 n 2 | a n | 2 + | b n | 2 < ∞ .

We can deduce that i ² ∑ _n≥i+2 G _i,n|c _n|² → 0 as i → ∞ in a similar way, and it is clear that lim_i→∞ i ² F _i|c _i+1|² = 0. Thus ψ 1 ′ ∈ H 2 . We similarly verify that ψ 2 ̃ ′ ∈ H 2 . We get that lim_i→∞ i ²Ψ_i+p,i = ϱ _s+p,s where (ϱ _i,j) is the matrix of the Hardy space Toeplitz operator with symbol 2 | ψ 1 ′ | 2 + 2 | ψ 2 ̃ ′ | 2 − 4 R e ψ 1 ′ ψ 2 ̃ ′ − ( c 1 d 1 ̄ + 2 ( c 2 d 1 ̄ + c 1 d 2 ̄ ) z ) . Finally, by a property of Toeplitz forms [16], Theorem (c), p. 19], the inequality H ψ ̄ * H ψ ≤ H φ ̄ * H φ implies

| ψ 1 ′ | 2 + | ψ 2 ̃ ′ | 2 − 2 R e ψ 1 ′ ψ 2 ̃ ′ − ( c 1 d 1 ̄ + 2 ( c 2 d 1 ̄ + c 1 d 2 ̄ ) z ≤ | φ 1 ′ | 2 + | φ 2 ̃ ′ | 2 − 2 R e φ 1 ′ φ 2 ̃ ′ − ( a 1 b 1 ̄ + 2 ( a 2 b 1 ̄ + a 1 b 2 ̄ ) z .

□

Corollary 3.10.

If φ ₁, φ ₂, ψ ₁, ψ ₂ are analytic and bounded, with φ 1 ′ , φ 2 ̃ ′ in H ² and satisfy c 1 d 1 ̄ = c 2 d 1 ̄ + c 1 d 2 ̄ = a 1 b 1 ̄ = a 2 b 1 ̄ + a 1 b 2 ̄ = 0 . Then if B φ + ψ ̄ is hyponormal we have ψ 1 ′ , ψ 2 ̃ ′ are in H ² and | ψ 1 ′ − ψ 2 ̃ ′ | ≤ | φ 1 ′ − φ 2 ̃ ′ | a.e on the unit circle.

4 The sufficient condition

In the second part of this work we show a sufficient condition for hyponormality of Toeplitz operators on the Bergman space of the unit disk, with a square logarithmic weight, in the case the symbol is of the form z m + g ̄ , where g is a polynomial of degree m. We start with some definitions and notations. We consider the Hilbert space of measurable functions on the unit disk D such that ∫ _D|f(z)|² dν(z) < ∞, where d ν ( z ) = 2 π ( log | z | ) 2 d A ( z ) , and dA(z) is the Lebesgue measure. When f is analytic on D, we have f = ∑a _n z ⁿ and ‖ f ‖ 2 = ∫ D | f ( z ) | 2 d μ ( z ) = ∑ 1 ( n + 1 ) ( n + 2 ) 2 | a n | 2 . Denote by L a , ν 2 the closed subspace of such functions. Its orthonormal basis is given by { n + 1 ( n + 2 ) z n , n ≥ 0 } . Toeplitz operators are defined by B _φ(f) = P(fφ), where φ is a bounded and measurable function on the disk, f is in L a , ν 2 , and P is the orthogonal projection of L ²(D, dν) on L a , ν 2 . Hankel operators are defined by H _φ(f) = (I − P)(fφ). Basic properties of Toeplitz operators and Hankel operators on L a , ν 2 are identical to the ones on A μ 2 listed in the first part, and to the case of the unweighted Bergman space of the unit disk [17],18]. We need the following computational lemma, where m ≥ 1.

Lemma 4.1.

The matrix of H z m ̄ * H z m ̄ in the orthonormal basis i + 1 ( i + 2 ) z i , i ≥ 0 is diagonal and is given by:

d i = ( i + 1 ) ( i + 2 ) 2 ( i + 1 + m ) ( i + 2 + m ) 2 , i < m , ( i + 1 ) ( i + 2 ) 2 ( i + 1 + m ) ( i + 2 + m ) 2 − ( i + 1 − m ) ( i + 2 − m ) 2 ( i + 1 ) ( i + 2 ) 2 , i ≥ m .

This leads to the following proposition.

Proposition 4.2.

Let m, n be two integers with n < m. The operator B z m + λ z ̄ n is hyponormal on L a , ν 2 if and only if | λ | ≤ n + 1 m + 1 n + 2 m + 2 .

Proof.

Hyponormality is equivalent to three inequalities

(2) | λ | 2 ( i + 1 ) ( i + 2 ) 2 ( i + 1 + n ) ( i + 2 + n ) 2 ≤ ( i + 1 ) ( i + 2 ) 2 ( i + 1 + m ) ( i + 2 + m ) 2 , i ≤ n − 1 ,

(3) | λ | 2 ( i + 1 ) ( i + 2 ) 2 ( i + 1 + n ) ( i + 2 + n ) 2 − ( i + 1 − n ) ( i + 2 − n ) 2 ( 1 + 1 ) ( i + 2 ) 2 ≤ ( i + 1 ) ( i + 2 ) 2 ( i + 1 + m ) ( i + 2 + m ) 2 , n ≤ i ≤ m − 1 ,

(4) | λ | 2 ( i + 1 ) ( i + 2 ) 2 ( i + 1 + n ) ( i + 2 + n ) 2 − ( i + 1 − n ) ( i + 2 − n ) 2 ( 1 + 1 ) ( i + 2 ) 2 ≤ ( i + 1 ) ( i + 2 ) 2 ( i + 1 + m ) ( i + 2 + m ) 2 − ( i + 1 − m ) ( i + 2 − m ) 2 ( 1 + 1 ) ( i + 2 ) 2 , i ≥ m .

Inequality (2) is equivalent to

| λ | ≤ min i + 1 + n i + 1 + m i + 2 + n i + 2 + m , i ≤ n − 1 .

Clearly i + 1 + n i + 1 + m i + 2 + n i + 2 + m increases with i. Thus (2) is equivalent to

| λ | ≤ n + 1 m + 1 n + 2 m + 2 .

Inequality (3) is equivalent to

(5) | λ | 2 ≤ min i + 1 + n i + 1 + m ( i + 2 + n ) 2 ( i + 2 + m ) 2 ( i + 1 ) 2 ( i + 2 ) 4 ( i + 1 ) 2 ( i + 2 ) 4 − ( ( i + 1 ) 2 − n 2 ) ( ( i + 2 ) 2 − n 2 ) 2 , n ≤ i ≤ m − 1 .

It is easy to see that i + 1 + n i + 1 + m ( i + 2 + n ) 2 ( i + 2 + m ) 2 increases with i. The expression 1 − n 2 ( i + 1 ) 2 increases with i, and so does ( 1 − n 2 ( i + 2 ) 2 ) 2 . Thus

1 − 1 − n 2 ( i + 1 ) 2 1 − n 2 ( i + 2 ) 2 2

is decreasing and its inverse ( i + 1 ) 2 ( i + 2 ) 4 ( i + 1 ) 2 ( i + 2 ) 4 − ( ( i + 1 ) 2 − n 2 ) ( ( i + 2 ) 2 − n 2 ) 2 is increasing and the minimum in (5) is assumed at i = n. We conclude that inequality (3) is equivalent to

| λ | ≤ 2 n + 1 n + m + 1 ( 2 n + 2 ) ( n + m + 2 ) ( n + 1 ) ( n + 2 ) 2 ( n + 1 ) 2 ( n + 2 ) 4 − ( ( n + 1 ) 2 − n 2 ) ( ( n + 2 ) 2 − n 2 ) 2 .

Clearly the right hand side of the last inequality is greater than n + 1 m + 1 n + 2 m + 2 . So if inequality (2) is satisfied so is (3). Inequality (4) is equivalent to

| λ | 2 ≤ inf i + 1 + n i + 1 + m ( i + 2 + n ) 2 ( i + 2 + m ) 2 ( i + 1 ) 2 ( i + 2 ) 4 − ( ( i + 1 ) 2 − m 2 ) ( ( i + 2 ) 2 − m 2 ) 2 ( i + 1 ) 2 ( i + 2 ) 4 − ( ( i + 1 ) 2 − n 2 ) ( ( i + 2 ) 2 − n 2 ) 2 , i ≥ m .

Since m > n we have

( i + 1 ) 2 ( i + 2 ) 4 − ( ( i + 1 ) 2 − m 2 ) ( ( i + 2 ) 2 − m 2 ) 2 ( i + 1 ) 2 ( i + 2 ) 4 − ( ( i + 1 ) 2 − n 2 ) ( ( i + 2 ) 2 − n 2 ) 2 > 1 .

As mentioned before i + 1 + n i + 1 + m ( i + 2 + n ) 2 ( i + 2 + m ) 2 increases with i, and thus if inequality (2) is satisfied, so is (4). Thus hyponormality of B z m + λ z ̄ n is equivalent to | λ | ≤ n + 1 m + 1 n + 2 m + 2 . □

Notice the result holds also if m = n. It is not difficult to see that if B z m + g 1 ̄ and B z m + g 2 ̄ are hyponormal on L a , ν 2 , and a and b are two complex numbers such that |a| + |b| ≤ 1, then B z m + a g 1 ̄ + b g 2 ̄ is hyponormal. We state our second main result.

Theorem 4.3.

Assume λ ₁,…,λ _m are complex numbers such that ∑ _1≤n≤m|λ _n| ≤ 1, and set g = ∑ 1 ≤ n ≤ m λ n n + 1 m + 1 n + 2 m + 2 z n . Then we have B g ̄ B g − B g B g ̄ ≤ B z ̄ m B z m − B z m B z ̄ m .

Corresponding author: Borhen Halouani, Department of Mathematics, College of Sciences, King Saud University, P. O Box 2455 Riyadh 11451, Saudi Arabia, E-mail: halouani@ksu.edu.sa

Acknowledgments

The authors would like to thank the referees for useful suggestions and comments which considerably improved the content and the form of this work.

Research ethics: Not applicable.
Informed consent: Not applicable.
Author contributions: Both authors contributed equally and significantly to the study conception and design, material preparation, data collection and analysis. The first draft of the manuscript was written by HS and both authors commented on previous versions of the manuscript. Both authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results and approved the final version of the manuscript.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
Conflict of interest: Authors state no conflicts of interest.
Research funding: The authors would like to extend their sincere appreciation to Ongoing Research Funding program (ORF-2025-1112), King Saud University, Riyadh, Saudi Arabia.
Data availability: Not applicable.

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Received: 2024-05-08

Accepted: 2025-10-04

Published Online: 2025-11-24

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https://doi.org/10.1515/math-2025-0212

Keywords for this article

hyponormality; Toeplitz operators; positive matrices; analytic functions; weighted Bergman space; annulus

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