On Berezin norm and Berezin number inequalities for sum of operators

Najla Altwaijry; Kais Feki; Nicusor Minculete

doi:10.1515/dema-2023-0159

Article Open Access

On Berezin norm and Berezin number inequalities for sum of operators

Najla Altwaijry , Kais Feki and Nicusor Minculete

Published/Copyright: May 8, 2024

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From the journal Demonstratio Mathematica Volume 57 Issue 1

Abstract

The aim of this study is to obtain several inequalities involving the Berezin number and the Berezin norm for various combinations of operators acting on a reproducing kernel Hilbert space. First, we present some bounds regarding the Berezin number associated with W * Q + W * Q ′ , where W , Q , and Q ′ are three bounded linear operators. Next, several Berezin norm and Berezin number inequalities for the sum of n operators are established.

Keywords: Berezin number; Berezin norm; sum; inequality

MSC 2010: 47A30; 15A60; 47A12

1 Introduction

Many articles related to numerical radius of an operator or Berezin number of an operator on a reproducing kernel Hilbert space (RKHS) have been studied for their many applications in engineering, quantum computing, quantum mechanics, numerical analysis, and differential equations [1–9]. There are important results for the upper and lower bounds of Berezin number in the literature (see, e.g., [10–19]).

To characterize the Berezin number and the Berezin norm, we first present some concepts and properties of the bounded linear operators on a Hilbert space.

Let ℰ be a complex Hilbert space, endowed with the inner product ⟨ ⋅ , ⋅ ⟩ and associated norm ‖ ⋅ ‖ . Let L ( ℰ ) denote the C ∗ -algebra of all bounded linear operators on ℰ , with identity I . An operator Q ∈ L ( ℰ ) is called positive if ⟨ Q x , x ⟩ ≥ 0 for all x ∈ ℋ , and then, we write Q ≥ 0 . If a bounded linear operator Q on ℰ is positive, then there exists a unique positive bounded linear operator denoted by Q 1 2 such that Q = ( Q 1 2 ) 2 . An important operator used is the absolute value of Q , denoted by ∣ Q ∣ , which is defined by ∣ Q ∣ = ( Q * Q ) 1 2 . It is easy to see that ∣ Q ∣ ≥ 0 . For Q ∈ L ( ℰ ) , we have the following numerical values: the operator norm ‖ Q ‖ given by ‖ Q ‖ = sup x ∈ ℰ , ‖ x ‖ = 1 ‖ Q x ‖ and the numerical radius of the operator Q defined by ω ( Q ) = sup x ∈ ℰ , ‖ x ‖ = 1 ∣ ⟨ Q x , x ⟩ ∣ . We remark that ω ( Q ) ≤ ‖ Q ‖ . When Q * Q = Q Q * , we say that Q is a normal operator, which, in fact, shows us that ∣ Q * ∣ = ∣ Q ∣ and ω ( Q ) = ‖ Q ‖ . It is important to mention that the operator norm and the numerical radius norm are equivalent, because we have the inequalities: ‖ Q ‖ 2 ≤ ω ( Q ) ≤ ‖ Q ‖ for every Q ∈ L ( ℰ ) . We also have ω ( Q ) ≤ ω ( ∣ Q ∣ ) . For some properties of the numerical radius, see, for example, [20–22] and references therein.

Let Θ be a non-empty set and ℱ ( Θ , C ) be the set of all functions from Θ to C , where C is the field of the complex numbers. A set ℰ Θ included in ℱ ( Θ , C ) is called a reproducing kernel Hilbert space (RKHS) on Θ if ℰ Θ is a Hilbert space (with identity I Θ ), and for every λ ∈ Θ , the linear evaluation functional E λ : ℰ Θ → C given by E λ ( f ) = f ( λ ) is bounded. Using the Riesz representation theorem, we show that for each λ ∈ Θ , there exists a unique vector k λ ∈ ℰ Θ such that f ( λ ) = E λ ( f ) = ⟨ f , k λ ⟩ for all f ∈ ℰ Θ . Here, the function k λ is called the reproducing kernel for the element λ , and the set { k λ ; λ ∈ Θ } is called the reproducing kernel of ℰ Θ . When k λ ≠ 0 , we denote k ^ λ = k λ ‖ k λ ‖ for λ ∈ Θ as the normalized reproducing kernel of ℰ Θ and we note that the set { k ^ λ : λ ∈ Θ } is a total set in ℰ Θ . For Q ∈ L ( ℰ Θ ) , the Berezin symbol (or Berezin transform) of Q , which has been first introduced by Berezin [23,24], is the bounded function Q ˜ : Θ → C defined by Q ˜ ( λ ) = ⟨ Q k λ ^ , k λ ^ ⟩ . If the operator Q is self-adjoint ( Q = Q * ), then Q ˜ ( λ ) ∈ R , and if the operator Q is positive, then Q ˜ ( λ ) ≥ 0 .

The Berezin symbol has been investigated in detail for the Toeplitz and Hankel operators on the Hardy and Bergman spaces; it is widely applied in the various questions of analysis and uniquely determines the operator (i.e., for all λ ∈ Θ , Q ˜ ( λ ) = W ˜ ( λ ) implies Q = W ). For further information about the Berezin symbol, we refer previously published articles [25–28] and references therein.

The Berezin set and the Berezin number of an operator Q are, respectively, defined by

Ber ( Q ) ≔ { Q ˜ ( λ ) ; λ ∈ Θ } and ber ( Q ) ≔ sup λ ∈ Θ ∣ Q ˜ ( λ ) ∣ = sup λ ∈ Θ ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ .

By some simple calculations, we obtain 0 ≤ ber ( Q ) ≤ ω ( Q ) ≤ ‖ Q ‖ for all Q ∈ L ( ℰ Θ ) . Karaev [1] showed that ‖ Q ‖ 2 ≤ ber ( Q ) does not hold for every Q ∈ L ( ℰ Θ ) .

It is easy to see that ber ( I Θ ) = 1 and ∣ ⟨ Q k λ , k λ ⟩ ∣ ≤ ber ( Q ) ‖ k λ ‖ 2 , for all reproducing kernels k λ .

For every Q , W ∈ L ( ℰ Θ ) , we have the following properties:

ber ( α Q ) = ∣ α ∣ ber ( Q ) , for all α ∈ C ;
ber ( Q + W ) ≤ ber ( Q ) + ber ( W ) .

From the aforementioned considerations regarding the Berezin number, it follows that ber ( ⋅ ) is a norm on L ( ℰ Θ ) . Similarly, for ber ( Q ) , the following concept can be found in [14,29]: c ( Q ) ≔ inf λ ∈ Θ { ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ } .

For Q ∈ L ( ℰ Θ ) , the Berezin norm of Q is given by

‖ Q ‖ ber ≔ sup { ∣ ⟨ Q k ^ λ , k ^ μ ⟩ ∣ ; λ , μ ∈ Θ } ,

where k ^ λ , k ^ μ are the two normalized reproducing kernels of the space ℰ Θ (see [30,31]). It is easy to see that ∣ ⟨ Q k ^ λ , k ^ μ ⟩ ∣ ≤ ‖ Q ‖ ber for all k ^ λ , k ^ μ . It is important to note that ‖ ⋅ ‖ ber does not verify, in general, the submultiplicativity property (see [32]). It is important to mention that

(1.1) ber ( Q ) ≤ ‖ Q ‖ ber ≤ ‖ Q ‖ , ∀ Q ∈ L ( ℰ Θ ) .

It should be mentioned here that Inequalities (1.1) are in general strict. However, Bhunia et al. proved in [33] that if Q ∈ L ( ℰ Θ ) is a positive operator, then

(1.2) ber ( Q ) = ‖ Q ‖ ber .

Remark 1.1

It is very crucial to mention that Relation (1.2) may not be true, in general, for self-adjoint operators (see [33]).

The objective of this study is to obtain several characterizations of Berezin numbers for various combinations of operators. To this end, this article is organized in the following manner: in Section 2, we collect a few results that are required to state and prove the results in the subsequent section. Section 3 contains our main results and presents several Berezin number inequalities regarding the Berezin number associated with the W * Q + W * Q ′ operator. Next, we study the Berezin norm and Berezin number inequalities for sum of operators.

2 Useful lemmas

The aim of this section is to collect some well-known lemmas that will be useful in proving our results. Throughout this article, ( ℰ , ⟨ ⋅ , ⋅ ⟩ ) denotes a complex Hilbert space. The norm associated with ⟨ ⋅ , ⋅ ⟩ will be denoted by ‖ ⋅ ‖ .

Our starting lemma is proved in [34] and provides an improvement of the well-known Cauchy-Schwarz inequality.

Lemma 2.1

Let x , y ∈ ℰ and ν ∈ [ 0 , 1 ] . Then,

∣ ⟨ x , y ⟩ ∣ 2 ≤ ν ( ‖ x ‖ 2 ‖ y ‖ 2 − ∣ ⟨ x , y ⟩ ∣ 2 ) + ∣ ⟨ x , y ⟩ ∣ 2 ν ‖ x ‖ 2 − 2 ν ‖ y ‖ 2 − 2 ν ≤ ‖ x ‖ 2 ‖ y ‖ 2 ,

for any x , y ∈ ℰ .

Another result of the aforementioned type is given by Alomari [35], namely:

Lemma 2.2

Let x , y ∈ ℰ and ν ∈ [ 0 , 1 ] . Then,

(2.1) ∣ ⟨ x , y ⟩ ∣ 2 ≤ ν ‖ x ‖ 2 ‖ y ‖ 2 + ( 1 − ν ) ∣ ⟨ x , y ⟩ ∣ ‖ x ‖ ‖ y ‖ ≤ ‖ x ‖ 2 ‖ y ‖ 2 ,

for any x , y ∈ ℰ .

For ν = 1 2 in Inequality (2.1), we obtain, after rearranging the terms, the following inequality:

∣ ⟨ x , y ⟩ ∣ 2 ≤ ∣ ⟨ x , y ⟩ ∣ ∥ x ∥ ∥ y ∥ + 1 2 ( ∥ x ∥ 2 ∥ y ∥ 2 − ∣ ⟨ x , y ⟩ ∣ 2 ) ≤ ∥ x ∥ 2 ∥ y ∥ 2 ,

for all x , y ∈ ℰ . This inequality is proved by Kittaneh and Moradi in [21], which is another refinement of the Cauchy-Schwarz inequality.

Remark 2.1

It follows from Inequality (2.1) that

∣ ⟨ x , y ⟩ ∣ 2 ≤ 1 3 ‖ x ‖ 2 ‖ y ‖ 2 + 2 3 ∣ ⟨ x , y ⟩ ∣ ‖ x ‖ ‖ y ‖ ,

for all x , y ∈ ℰ .

The classical Schwarz inequality for a positive operator Q ∈ L ( ℰ ) is given as:

(2.2) ∣ ⟨ Q x , y ⟩ ∣ 2 ≤ ⟨ Q x , x ⟩ ⟨ Q y , y ⟩ ,

for any vectors x , y ∈ ℰ . Kato [36] established a companion of the Schwarz Inequality (2.2), which asserts

∣ ⟨ Q x , y ⟩ ∣ 2 ≤ ⟨ ∣ Q ∣ 2 θ x , x ⟩ ⟨ ∣ Q * ∣ 2 ( 1 − θ ) y , y ⟩ ,

for every operator Q ∈ L ( ℰ ) , for any vectors x , y ∈ ℰ , and θ ∈ [ 0 , 1 ] . For θ = 1 2 we obtain a result attributed to Halmos [37, pp. 75–76]:

(2.3) ∣ ⟨ Q x , y ⟩ ∣ ≤ ⟨ ∣ Q ∣ x , y ⟩ ⟨ ∣ Q * ∣ y , y ⟩ ,

for every Q ∈ L ( ℰ ) and for all x , y ∈ ℰ .

The inequality in the following lemma deals with positive operators and is known as the McCarthy inequality.

Lemma 2.3

[38, Theorem 1.4] Let Q ∈ L ( ℰ ) be a positive operator and x ∈ ℰ be such that ‖ x ‖ = 1 . Then, for all r ≥ 1 , we have

⟨ Q x , x ⟩ r ≤ ⟨ Q r x , x ⟩ .

If 0 ≤ r ≤ 1 , then the aforementioned inequality is reversed.

The interesting inequality in the following lemma is proved by Buzano in [39].

Lemma 2.4

Let x , y , e ∈ ℰ be such that ∥ e ∥ = 1 . Then,

∣ ⟨ x , e ⟩ ⟨ e , y ⟩ ∣ ≤ 1 2 ( ∥ x ∥ ∥ y ∥ + ∣ ⟨ x , y ⟩ ∣ ) .

In the next lemma, we recall a known inequality due to Bohr (see e.g. [40]).

Lemma 2.5

Let a k be the positive real numbers for all i ∈ { 1 , 2 , … , d } . Then, for all r ≥ 1 , we have

∑ k = 1 d a k r ≤ d r − 1 ∑ k = 1 d a k r .

Our final lemma is given by Dragomir in [41].

Lemma 2.6

For any x , y , z ∈ ℰ , we have

∣ ⟨ x , y ⟩ ∣ 2 + ∣ ⟨ x , z ⟩ ∣ 2 ≤ ‖ x ‖ 2 ( max { ‖ y ‖ 2 , ‖ z ‖ 2 } + ∣ ⟨ y , z ⟩ ∣ ) .

3 Main results

In this section, we denote by ( ℰ Θ , ⟨ ⋅ , ⋅ ⟩ ) a RKHS on a set Θ with associated norm ‖ ⋅ ‖ .

Our first result in this article reads as follows.

Theorem 3.1

Let Q , Q ′ , W ∈ L ( ℰ Θ ) , and ν ∈ ( 0 , 1 ) . Then, we have

ber 2 ( W * Q + W * Q ′ ) ≤ 2 ν ( 1 − ν ) 1 + ν − ν 2 ( ber ( W * Q ) ∥ ∣ Q ∣ 2 + ∣ W ∣ 2 ∥ ber + ber ( W * Q ′ ) ∥ ∣ Q ′ ∣ 2 + ∣ W ∣ 2 ∥ ber ) + 1 − ν + ν 2 1 + ν − ν 2 ∥ ∣ Q ∣ 4 + ∣ Q ′ ∣ 4 + 2 ∣ W ∣ 4 ∥ ber .

Proof

We take the first inequality from Lemma 2.1:

(3.1) ∣ ⟨ x , y ⟩ ∣ 2 ≤ ν 1 + ν ‖ x ‖ 2 ‖ y ‖ 2 + 1 1 + ν ∣ ⟨ x , y ⟩ ∣ 2 ν ‖ x ‖ 2 − 2 ν ‖ y ‖ 2 − 2 ν ,

for any x , y ∈ ℰ and ν ∈ [ 0 , 1 ] . If we replace x and y by Q k ^ λ and W k ^ λ , respectively, in (3.1), and we note that ‖ k ^ λ ‖ = 1 , then we have

∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ 2 ≤ ν 1 + ν ‖ Q k ^ λ ‖ 2 ‖ W k ^ λ ‖ 2 + 1 1 + ν ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ 2 ν ‖ Q k ^ λ ‖ 2 − 2 ν ‖ W k ^ λ ‖ 2 − 2 ν .

Using the same idea as in [34], for the aforementioned inequality, we deduce

∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ 2 ≤ ν 1 + ν ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ ⟨ ∣ W ∣ 2 k ^ λ , k ^ λ ⟩ + 1 1 + ν ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ 2 ν ( ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ ⟨ ∣ W ∣ 2 k ^ λ , k ^ λ ⟩ ) 1 − ν = ν 1 + ν ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ ⟨ ∣ W ∣ 2 k ^ λ , k ^ λ ⟩ + 1 1 + ν ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ ν ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ 1 − ν ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ ν ⟨ ∣ W ∣ 2 k ^ λ , k ^ λ ⟩ 1 − ν .

By applying the well-known Young inequality

a ν b 1 − ν ≤ ν a + ( 1 − ν ) b , a , b ≥ 0 , ν ∈ ( 0 , 1 )

to the aforementioned relation for a = ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ and b = ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ , we obtain the following inequality:

∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ 2 ≤ ν 1 + ν ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ ⟨ ∣ W ∣ 2 k ^ λ , k ^ λ ⟩ + 1 1 + ν ( ν ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ + ( 1 − ν ) ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ ) ( ν ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ + ( 1 − ν ) ⟨ ∣ W ∣ 2 k ^ λ , k ^ λ ⟩ ) = ν 1 + ν ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ ⟨ ∣ W ∣ 2 k ^ λ , k ^ λ ⟩ + ν 2 1 + ν ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ 2 + ν ( 1 − ν ) 1 + ν ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ ⟨ ( ∣ Q ∣ 2 + ∣ W ∣ 2 ) k ^ λ , k ^ λ ⟩ + ( 1 − ν ) 2 1 + ν ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ ⟨ ∣ W ∣ 2 k ^ λ , k ^ λ ⟩ ,

which is equivalent to

( 1 + ν − ν 2 ) ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ 2 ≤ ( 1 − ν + ν 2 ) ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ ⟨ ∣ W ∣ 2 k ^ λ , k ^ λ ⟩ + ν ( 1 − ν ) ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ ⟨ ( ∣ Q ∣ 2 + ∣ W ∣ 2 ) k ^ λ , k ^ λ ⟩ ≤ 1 − ν + ν 2 2 ⟨ ( ∣ Q ∣ 4 + ∣ W ∣ 4 ) k ^ λ , k ^ λ ⟩ + ν ( 1 − ν ) ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ ⟨ ( ∣ Q ∣ 2 + ∣ W ∣ 2 ) k ^ λ , k ^ λ ⟩ .

Thus, we have

(3.2) 2 ( 1 + ν − ν 2 ) ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ 2 ≤ ( 1 − ν + ν 2 ) ⟨ ( ∣ Q ∣ 4 + ∣ W ∣ 4 ) k ^ λ , k ^ λ ⟩ + 2 ν ( 1 − ν ) ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ ⟨ ( ∣ Q ∣ 2 + ∣ W ∣ 2 ) k ^ λ , k ^ λ ⟩ .

Therefore, from the algebraic inequality ∣ z 1 + z 2 ∣ 2 ≤ 2 ( ∣ z 1 ∣ 2 + ∣ z 2 ∣ 2 ) , we deduce that

( 1 + ν − ν 2 ) ∣ ⟨ ( W * Q + W * Q ′ ) k ^ λ , k ^ λ ⟩ ∣ 2 ( 3.2 ) ≤ 2 ( 1 + ν − ν 2 ) ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ 2 + 2 ( 1 + ν − ν 2 ) ∣ ⟨ W * Q ′ k ^ λ , k ^ λ ⟩ ∣ 2 ≤ ( 1 − ν + ν 2 ) ⟨ ( ∣ Q ∣ 4 + ∣ Q ′ ∣ 4 + 2 ∣ W ∣ 4 ) k ^ λ , k ^ λ ⟩ + 2 ν ( 1 − ν ) [ ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ ⟨ ( ∣ Q ∣ 2 + ∣ W ∣ 2 ) k ^ λ , k ^ λ ⟩ + ∣ ⟨ W * Q ′ k ^ λ , k ^ λ ⟩ ∣ ⟨ ( ∣ Q ′ ∣ 2 + ∣ W ∣ 2 ) k ^ λ , k ^ λ ⟩ ] .

Taking the supremum over λ ∈ Θ in the aforementioned inequality and taking into account that the operators ∣ Q ∣ 4 + ∣ Q ′ ∣ 4 + 2 ∣ W ∣ 4 , ∣ Q ∣ 2 + ∣ W ∣ 2 , and ∣ Q ′ ∣ 2 + ∣ W ∣ 2 are positive, we obtain the inequality

( 1 + ν − ν 2 ) ber 2 ( W * Q + W * Q ′ ) ≤ ( 1 − ν + ν 2 ) ‖ ∣ Q ∣ 4 + ∣ Q ′ ∣ 4 + 2 ∣ W ∣ 4 ‖ ber + 2 ν ( 1 − ν ) ( ber ( W * Q ) ‖ ∣ Q ∣ 2 + ∣ W ∣ 2 ‖ ber + ber ( W * Q ′ ) ‖ ∣ Q ′ ∣ 2 + ∣ W ∣ 2 ‖ ber ) .

Hence, we have proved the inequality of the statement.□

Remark 3.1

For ν ∈ { 0 , 1 } in Inequality (3.1), we have ∣ ⟨ x , y ⟩ ∣ 2 ≤ ‖ x ‖ 2 ‖ y ‖ 2 , and so ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ 2 ≤ ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ ⟨ ∣ W ∣ 2 k ^ λ , k ^ λ ⟩ . Therefore, from the algebraic inequality ∣ z 1 + z 2 ∣ 2 ≤ 2 ( ∣ z 1 ∣ 2 + ∣ z 2 ∣ 2 ) , we obtain

∣ ⟨ ( W * Q + W * Q ′ ) k ^ λ , k ^ λ ⟩ ∣ 2 ≤ 2 ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ 2 + 2 ∣ ⟨ W * Q ′ k ^ λ , k ^ λ ⟩ ∣ 2 ≤ ⟨ ( ∣ Q ∣ 4 + ∣ Q ′ ∣ 4 + 2 ∣ W ∣ 4 ) k ^ λ , k ^ λ ⟩ .

Thus, we deduce the following:

(3.3) ber 2 ( W * Q + W * Q ′ ) ≤ ‖ ∣ Q ∣ 4 + ∣ Q ′ ∣ 4 + 2 ∣ W ∣ 4 ‖ ber .

Theorem 3.2

Let Q , Q ′ , W ∈ L ( ℰ Θ ) and ν ∈ [ 0 , 1 ] . Then,

ber 2 ( W * Q + W * Q ′ ) ≤ ν ∥ ∣ Q ∣ 4 + ∣ Q ′ ∣ 4 + 2 ∣ W ∣ 4 ∥ ber + ( 1 − ν ) ber ( W * Q ) ∥ ∣ Q ∣ 2 + ∣ W ∣ 2 ∥ ber + ( 1 − ν ) ber ( W * Q ′ ) ∥ ∣ Q ′ ∣ 2 + ∣ W ∣ 2 ∥ ber .

Proof

We consider the first inequality from Lemma 2.2:

∣ ⟨ x , y ⟩ ∣ 2 ≤ ν ‖ x ‖ 2 ‖ y ‖ 2 + ( 1 − ν ) ∣ ⟨ x , y ⟩ ∣ ‖ x ‖ ‖ y ‖ ,

for any x , y ∈ ℰ and ν ∈ [ 0 , 1 ] . The Hölder-McCarthy inequality can only be applied to positive operators, and the operators ∣ Q ∣ 2 and ∣ W ∣ 2 are positive. Now, we replace x and y by Q k ^ λ and W k ^ λ , in the aforementioned inequality, and we know that ‖ k ^ λ ‖ = 1 , then

∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ 2 ≤ ν ‖ Q k ^ λ ‖ 2 ‖ W k ^ λ ‖ 2 + ( 1 − ν ) ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ ‖ Q k ^ λ ‖ ‖ W k ^ λ ‖ ≤ ν ⟨ Q k ^ λ , Q k ^ λ ⟩ ⟨ W k ^ λ , W k ^ λ ⟩ + ( 1 − ν ) ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ ⟨ Q k ^ λ , Q k ^ λ ⟩ ⟨ W k ^ λ , W k ^ λ ⟩ = ν ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ ⟨ ∣ W ∣ 2 k ^ λ , k ^ λ ⟩ + ( 1 − ν ) ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ ⟨ ∣ W ∣ 2 k ^ λ , k ^ λ ⟩ ≤ ν 4 ( ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ + ⟨ ∣ W ∣ 2 k ^ λ , k ^ λ ⟩ ) 2 + 1 − ν 2 ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ ( ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ + ⟨ ∣ W ∣ 2 k ^ λ , k ^ λ ⟩ ) ≤ ν 2 ( ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ 2 + ⟨ ∣ W ∣ 2 k ^ λ , k ^ λ ⟩ 2 ) + 1 − ν 2 ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ ( ⟨ ( ∣ Q ∣ 2 + ∣ W ∣ 2 ) k ^ λ , k ^ λ ⟩ ) ≤ McCarthy ν 2 ( ⟨ ∣ Q ∣ 4 k ^ λ , k ^ λ ⟩ + ⟨ ∣ W ∣ 4 k ^ λ , k ^ λ ⟩ ) + 1 − ν 2 ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ ( ⟨ ( ∣ Q ∣ 2 + ∣ W ∣ 2 ) k ^ λ , k ^ λ ⟩ ) = ν 2 ( ⟨ ( ∣ Q ∣ 4 + ∣ W ∣ 4 ) k ^ λ , k ^ λ ⟩ ) + 1 − ν 2 ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ ( ⟨ ( ∣ Q ∣ 2 + ∣ W ∣ 2 ) k ^ λ , k ^ λ ⟩ ) .

Therefore, the inequality

(3.4) 2 ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ 2 ≤ ν ( ⟨ ( ∣ Q ∣ 4 + ∣ W ∣ 4 ) k ^ λ , k ^ λ ⟩ ) + ( 1 − ν ) ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ ( ⟨ ( ∣ Q ∣ 2 + ∣ W ∣ 2 ) k ^ λ , k ^ λ ⟩ )

is true. Consequently, from the algebraic inequality ∣ z 1 + z 2 ∣ 2 ≤ 2 ( ∣ z 1 ∣ 2 + ∣ z 2 ∣ 2 ) , we deduce

∣ ⟨ W * Q + W * Q ′ k ^ λ , k ^ λ ⟩ ∣ 2 ≤ 2 ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ 2 + 2 ∣ ⟨ W * Q ′ k ^ λ , k ^ λ ⟩ ∣ 2 ≤ ( 3.4 ) ν ⟨ ( ∣ Q ∣ 4 + ∣ Q ′ ∣ 4 + 2 ∣ W ∣ 4 ) k ^ λ , k ^ λ ⟩ + ( 1 − ν ) ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ ⟨ ( ∣ Q ∣ 2 + ∣ W ∣ 2 ) k ^ λ , k ^ λ ⟩ + ( 1 − ν ) ∣ ⟨ W * Q ′ k ^ λ , k ^ λ ⟩ ∣ ⟨ ( ∣ Q ′ ∣ 2 + ∣ W ∣ 2 ) k ^ λ , k ^ λ ⟩ .

Taking the supremum over λ ∈ Θ in the aforementioned inequality and taking into account that the operators ∣ Q ∣ 4 + ∣ Q ′ ∣ 4 + 2 ∣ W ∣ 4 , ∣ Q ∣ 2 + ∣ W ∣ 2 and ∣ Q ′ ∣ 2 + ∣ W ∣ 2 are positive, we obtain the inequality of the statement.□

Remark 3.2

Through various particular cases of ν in Theorem 3.2, we obtain some known results; thus, for ν = 1 in Theorem 3.2, we deduce Inequality (3.3), and for ν = 0 , we find

ber 2 ( W * Q + W * Q ′ ) ≤ ber ( W * Q ) ‖ ∣ Q ∣ 2 + ∣ W ∣ 2 ‖ ber + ber ( W * Q ′ ) ‖ ∣ Q ′ ∣ 2 + ∣ W ∣ 2 ‖ ber .

In particular, if Q = Q ′ in the last relation, we deduce the following inequality:

(3.5) ber ( W * Q ) ≤ 1 2 ‖ ∣ Q ∣ 2 + ∣ W ∣ 2 ‖ ber .

Theorem 3.3

Let Q , W ∈ L ( ℰ Θ ) and ν ∈ [ 0 , 1 ] . Then,

ber 2 ( Q + W ) ≤ ν ∥ ∣ Q ∣ 2 + ∣ W ∣ 2 + ∣ Q * ∣ 2 + ∣ W * ∣ 2 ∥ ber + ( 1 − ν ) ber ( Q ) ∥ ∣ Q ∣ + ∣ Q * ∣ ∥ ber + ( 1 − ν ) ber ( W ) ∥ ∣ W ∣ + ∣ W * ∣ ∥ ber .

Proof

Applying Inequality (2.3), we deduce the following relation:

∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ ≤ ν ⟨ ∣ Q ∣ k ^ λ , k ^ λ ⟩ ⟨ ∣ Q * ∣ k ^ λ , k ^ λ ⟩ + ( 1 − ν ) ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ ,

for any ν ∈ [ 0 , 1 ] . Multiplying by ⟨ ∣ Q ∣ k ^ λ , k ^ λ ⟩ ⟨ ∣ Q * ∣ k ^ λ , k ^ λ ⟩ and taking into account that

∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ 2 ≤ ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ ⟨ ∣ Q ∣ x , x ⟩ ⟨ ∣ Q * ∣ k ^ λ , k ^ λ ⟩ ,

we find the following inequality:

(3.6) ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ 2 ≤ ν ⟨ ∣ Q ∣ k ^ λ , k ^ λ ⟩ ⟨ ∣ Q * ∣ k ^ λ , k ^ λ ⟩ + ( 1 − ν ) ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ ⟨ ∣ Q ∣ k ^ λ , k ^ λ ⟩ ⟨ ∣ Q * ∣ k ^ λ , k ^ λ ⟩ .

Because the operators ∣ Q ∣ and ∣ Q * ∣ are positive, we can apply the McCarthy inequality; thus, ⟨ ∣ Q ∣ k ^ λ , k ^ λ ⟩ 2 ≤ ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ , ⟨ ∣ Q * ∣ k ^ λ , k ^ λ ⟩ 2 ≤ ⟨ ∣ Q * ∣ 2 k ^ λ , k ^ λ ⟩ . Therefore, Inequality (3.6) becomes

∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ 2 ≤ ν 4 ( ⟨ ∣ Q ∣ k ^ λ , k ^ λ ⟩ + ⟨ ∣ Q * ∣ k ^ λ , k ^ λ ⟩ ) 2 + 1 − ν 2 ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ ⟨ ( ∣ Q ∣ + ∣ Q * ∣ ) k ^ λ , k ^ λ ⟩ ≤ ν 2 ( ⟨ ∣ Q ∣ k ^ λ , k ^ λ ⟩ 2 + ⟨ ∣ Q * ∣ k ^ λ , k ^ λ ⟩ 2 ) + 1 − ν 2 ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ ⟨ ( ∣ Q ∣ + ∣ Q * ∣ ) k ^ λ , k ^ λ ⟩ ≤ ν 2 ⟨ ( ∣ Q ∣ 2 + ∣ Q * ∣ 2 ) k ^ λ , k ^ λ ⟩ + 1 − ν 2 ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ ⟨ ( ∣ Q ∣ + ∣ Q * ∣ ) k ^ λ , k ^ λ ⟩ .

Hence, we deduce that

2 ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ 2 ≤ ν ⟨ ( ∣ Q ∣ 2 + ∣ Q * ∣ 2 ) k ^ λ , k ^ λ ⟩ + ( 1 − ν ) ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ ⟨ ( ∣ Q ∣ + ∣ Q * ∣ ) k ^ λ , k ^ λ ⟩ .

Consequently, we have

∣ ⟨ ( Q + W ) k ^ λ , k ^ λ ⟩ ∣ 2 ≤ 2 ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ 2 + 2 ∣ ⟨ W k ^ λ , k ^ λ ⟩ ∣ 2 ≤ ν ⟨ ( ∣ Q ∣ 2 + ∣ Q * ∣ 2 ) k ^ λ , k ^ λ ⟩ + ( 1 − ν ) ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ ⟨ ( ∣ Q ∣ + ∣ Q * ∣ ) k ^ λ , k ^ λ ⟩ + ν ⟨ ( ∣ W ∣ 2 + ∣ W * ∣ 2 ) k ^ λ , k ^ λ ⟩ + ( 1 − ν ) ∣ ⟨ W k ^ λ , k ^ λ ⟩ ∣ ⟨ ( ∣ W ∣ + ∣ W * ∣ ) k ^ λ , k ^ λ ⟩ .

Since ‖ k ^ λ ‖ = 1 and taking the supremum over all λ ∈ Θ and taking into account that the operators ∣ Q ∣ 2 + ∣ Q * ∣ 2 , ∣ Q ∣ + ∣ Q * ∣ , ∣ W ∣ 2 + ∣ W * ∣ 2 , and ∣ W ∣ + ∣ W * ∣ are positive, the inequality of the statement is true.□

Remark 3.3

If we take ν = 1 2 in Theorem 3.3, then we obtain the following sequence of inequalities:

2 ber 2 ( Q + W ) ≤ ‖ ∣ Q ∣ 2 + ∣ W ∣ 2 + ∣ Q * ∣ 2 + ∣ W * ∣ 2 ‖ ber + ber ( Q ) ‖ ∣ Q ∣ + ∣ Q * ∣ ‖ ber + ber ( W ) ‖ ∣ W ∣ + ∣ W * ∣ ‖ ber .

In all that follows, for any arbitrary operator Q ∈ L ( ℰ Θ ) , we write

ℜ ( Q ) ≔ Q + Q * 2 and ℑ ( Q ) ≔ Q − Q * 2 i .

Thus, we have Q = ℜ ( Q ) + i ℑ ( Q ) and ∣ ℜ ( Q ) ∣ 2 + ∣ ℑ ( Q ) ∣ 2 = ∣ Q ∣ 2 + ∣ Q * ∣ 2 .

Theorem 3.4

If Q ∈ L ( ℰ Θ ) , then the inequality

ber ( ℜ ( Q ) + ℑ ( Q ) ) ≤ 2 ber ( Q )

holds.

Proof

It is easy to see that

∣ ⟨ ( ℜ ( Q ) + ℑ ( Q ) ) k ^ λ , k ^ λ ⟩ ∣ ≤ 2 ∣ ⟨ ( ℜ ( Q ) + i ℑ ( Q ) ) k ^ λ , k ^ λ ⟩ ∣ = 2 ∣ ⟨ ( Q ) k ^ λ , k ^ λ ⟩ ∣ .

Taking the supremum over all λ ∈ Θ , we find that the inequality of the statement holds.□

Remark 3.4

If Q ∈ L ( ℰ Θ ) , then from Relation (3.5) and knowing that ℑ ( Q ) * = ℑ ( Q ) , we find

ber ( ℑ ( Q ) ℜ ( Q ) ) ≤ 1 2 ∥ ∣ Q ∣ 2 + ∣ Q * ∣ 2 ∥ ber .

Theorem 3.5

If Q , Q 1 , W , W 1 ∈ L ( ℰ Θ ) , then the inequality

∥ W 1 * ( Q + W ) Q 1 ∥ ber ≤ 2 2 ( ber ( Q 1 * ( ∣ Q ∣ + i ∣ W ∣ ) Q 1 ) + ber ( W 1 * ( ∣ Q * ∣ + i ∣ W * ∣ ) W 1 ) )

holds.

Proof

While proving a result from [42], we found the inequality

∣ ⟨ ( Q + W ) x , y ⟩ ∣ ≤ 2 2 ( ∣ ⟨ ( ∣ Q ∣ + i ∣ W ∣ ) x , x ⟩ ∣ + ∣ ⟨ ( ∣ Q * ∣ + i ∣ W * ∣ ) y , y ⟩ ∣ ) ,

where Q and W are the operators in a Hilbert space ℰ . Now, we take x = Q 1 k ^ λ , y = W 1 k ^ μ , where λ , μ ∈ C ; then, we obtain the following inequality:

∣ ⟨ W 1 * ( Q + W ) Q 1 k ^ λ , k ^ μ ⟩ ∣ ≤ 2 2 ( ∣ ⟨ Q 1 * ( ∣ Q ∣ + i ∣ W ∣ ) Q 1 k ^ λ , k ^ λ ⟩ ∣ + ∣ ⟨ W 1 * ( ∣ Q * ∣ + i ∣ W * ∣ ) W 1 k ^ μ , k ^ μ ⟩ ∣ ) ,

for four operators Q , Q 1 , W , W 1 in a RKHS H Θ . Taking the supremum over λ , μ ∈ C , we obtain the inequality of the statement.□

Corollary 3.1

If Q ∈ L ( ℰ Θ ) , then the inequality

‖ Q ‖ ber ≤ 2 2 ( ber ( ∣ ℜ ( Q ) ∣ + i ∣ ℑ ( Q ) ∣ ) + ber ( ∣ ℜ ( Q ) * ∣ + i ∣ ℑ ( Q ) * ∣ ) )

holds.

Proof

It is easy to see that ∣ W ∣ = ∣ i W ∣ . In Theorem 3.5, we take Q 1 = W 1 = I , Q = ℜ ( Q ) and W = i ℑ ( Q ) , to obtain the relation of the statement.□

Theorem 3.6

If Q , T , W ∈ L ( ℰ Θ ) , then the inequality

ber 2 T * ( Q + W ) T ≤ ber ( T * Q 2 T ) + ber ( T * W 2 T ) + 2 2 ber ( T * ( ∣ Q ∣ 2 + ∣ W ∣ 2 + i ( ∣ Q * ∣ 2 + ∣ W * ∣ 2 ) ) T )

holds.

Proof

Another inequality in the proof from [42] is the following:

∣ ⟨ ( Q + W ) x , x ⟩ ∣ 2 ≤ ∣ ⟨ Q 2 x , x ⟩ ∣ + ∣ ⟨ W 2 x , x ⟩ + 2 2 ∣ ( ⟨ ( ∣ Q ∣ 2 + ∣ W ∣ 2 ) + i ( ∣ Q * ∣ 2 + ∣ W * ∣ 2 ) x , x ⟩ ∣ ) ∣ ,

where Q and W are the operators in a Hilbert space ℰ . If we take x = T k ^ λ , where λ ∈ C , then we deduce

∣ ⟨ T * ( Q + W ) T k ^ λ , k ^ λ ⟩ ∣ 2 ≤ ∣ ⟨ T * Q 2 T k ^ λ , k ^ λ ⟩ ∣ + ∣ ⟨ T * W 2 T k ^ λ , k ^ λ ⟩ + 2 2 ∣ ( ⟨ T * ( ∣ Q ∣ 2 + ∣ W ∣ 2 ) + i ( ∣ Q * ∣ 2 + ∣ W * ∣ 2 ) T k ^ λ , k ^ λ ⟩ ∣ ) ∣ ,

for three operators Q , T , and W in a RKHS H Θ . Consequently, taking the supremum over λ ∈ C , we obtain the inequality of the statement.□

Corollary 3.2

If Q ∈ L ( ℰ Θ ) , then the inequality

ber 2 ( Q ) ≤ ber ( ℜ ( Q ) 2 ) + ber ( ℑ ( Q ) 2 ) + 2 2 ber ( ∣ ℜ ( Q ) ∣ 2 + ∣ ℑ ( Q ) ∣ 2 + i ( ∣ ℜ ( Q ) * ∣ 2 + ∣ ℑ ( Q ) * ∣ 2 ) )

holds.

Proof

In Theorem 3.6, we take T = I , Q = ℜ ( Q ) , and W = i ℑ ( Q ) , to prove the relation of the statement.□

Now, we will prove several results related to the sum of operators.

Theorem 3.7

Let Q k ∈ L ( ℰ Θ ) for all k ∈ { 1 , 2 , … , d } with d ∈ N * . Then, for every r ≥ 1 , we have

ber 2 r ∑ k = 1 d Q k ≤ d 2 r − 1 2 ∑ k = 1 d ber ( ∣ Q k ∣ 2 r + i ∣ Q k * ∣ 2 r ) .

Proof

Let λ ∈ Θ and k ^ λ be the normalized reproducing kernel of the space ℰ Θ . Using Lemmas 2.3 and 2.5, and Inequality (2.3), we see that

∑ k = 1 d Q k k ^ λ , k ^ λ 2 r ≤ ∑ k = 1 d ∣ ⟨ Q k k ^ λ , k ^ λ ⟩ ∣ 2 r ≤ d 2 r − 1 ∑ k = 1 d ∣ ⟨ Q k k ^ λ , k ^ λ ⟩ ∣ 2 r ≤ d 2 r − 1 ∑ k = 1 d ⟨ ∣ Q k ∣ k ^ λ , k ^ λ ⟩ r ⟨ ∣ Q k * ∣ k ^ λ , k ^ λ ⟩ r ≤ d 2 r − 1 ∑ k = 1 d ⟨ ∣ Q k ∣ r k ^ λ , k ^ λ ⟩ ⟨ ∣ Q k * ∣ r k ^ λ , k ^ λ ⟩ .

Moreover, since ∣ Q k ∣ r ≥ 0 and ∣ Q k * ∣ r ≥ 0 , using the arithmetic-geometric mean inequality together with Lemma 2.3, we see that

∑ k = 1 d Q k k ^ λ , k ^ λ 2 r ≤ d 2 r − 1 2 ∑ k = 1 d ( ⟨ ∣ Q k ∣ r k ^ λ , k ^ λ ⟩ 2 + ⟨ ∣ Q k * ∣ r k ^ λ , k ^ λ ⟩ 2 ) ≤ d 2 r − 1 2 ∑ k = 1 d ( ⟨ ∣ Q k ∣ 2 r k ^ λ , k ^ λ ⟩ + ⟨ ∣ Q k * ∣ 2 r k ^ λ , k ^ λ ⟩ ) ≤ d 2 r − 1 2 ∑ k = 1 d ∣ ⟨ ∣ Q k ∣ 2 r k ^ λ , k ^ λ ⟩ + i ⟨ ∣ Q k * ∣ 2 r k ^ λ , k ^ λ ⟩ ∣ ,

where we have used in the last inequality the well-known relation: ∣ α + β ∣ ≤ 2 ∣ α + i β ∣ for all real numbers α and β . Hence, we infer that

∑ k = 1 d Q k k ^ λ , k ^ λ 2 r ≤ d 2 r − 1 2 ∑ k = 1 d ∣ ⟨ ( ∣ Q k ∣ 2 r + i ∣ Q k * ∣ 2 r ) k ^ λ , k ^ λ ⟩ ∣ ≤ d 2 r − 1 2 ∑ k = 1 d ber ( ∣ Q k ∣ 2 r + i ∣ Q k * ∣ 2 r ) .

Taking the supremum over all λ ∈ Θ in the last inequality yields the desired result.□

Remark 3.5

By letting r = d = 1 in Theorem 3.7, we obtain

ber 2 ( Q ) ≤ 1 2 ber ( ∣ Q ∣ 2 + i ∣ Q * ∣ 2 ) .

In order to establish our next result, we need the following elementary equality

(3.7) max { a , b } = 1 2 ( a + b + ∣ a − b ∣ ) ,

for every a , b ∈ R .

Theorem 3.8

Let Q , W ∈ L ( ℰ Θ ) . Then, the inequality

ber ( Q + W ) ≤ ∥ ∣ Q ∣ 2 + ∣ W ∣ 2 ∥ ber + 1 2 ber ( ∣ Q ∣ 2 − ∣ W ∣ 2 ) + 2 ber ( W * Q )

holds.

Proof

Let λ ∈ Θ and k ^ λ be the normalized reproducing kernel of ℰ Θ . Then, we see that

∣ ⟨ ( Q + W ) k ^ λ , k ^ λ ⟩ ∣ 2 ≤ ( ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ + ∣ ⟨ W k ^ λ , k ^ λ ⟩ ∣ ) 2 = ∣ ⟨ k ^ λ , Q k ^ λ ⟩ ∣ 2 + ∣ ⟨ k ^ λ , W k ^ λ ⟩ ∣ 2 + 2 ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ ∣ ⟨ W k ^ λ , k ^ λ ⟩ ∣ ≤ ( Lemma 2.6 ) max { ‖ Q k ^ λ ‖ 2 , ‖ W k ^ λ ‖ 2 } + ∣ ⟨ Q k ^ λ , W k ^ λ ⟩ ∣ + 2 ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ ∣ ⟨ W k ^ λ , k ^ λ ⟩ ∣ .

Furthermore, by applying (3.7), we obtain

∣ ⟨ ( Q + W ) k ^ λ , k ^ λ ⟩ ∣ 2 ≤ 1 2 ( ‖ Q k ^ λ ‖ 2 + ‖ W k ^ λ ‖ 2 + ∣ ‖ Q k ^ λ ‖ 2 − ‖ W k ^ λ ‖ 2 ∣ ) + ∣ ⟨ Q k ^ λ , W k ^ λ ⟩ ∣ + 2 ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ ∣ ⟨ W k ^ λ , k ^ λ ⟩ ∣ = 1 2 ( ⟨ Q k ^ λ , Q k ^ λ ⟩ + ⟨ S k ^ λ , S k ^ λ ⟩ + ∣ ⟨ T k ^ λ , T k ^ λ ⟩ − ⟨ W k ^ λ , W k ^ λ ⟩ ∣ ) + ∣ ⟨ Q k ^ λ , W k ^ λ ⟩ ∣ + 2 ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ ∣ ⟨ W k ^ λ , k ^ λ ⟩ ∣ = 1 2 ( ⟨ ( ∣ Q ∣ 2 + ∣ W ∣ 2 ) k ^ λ , k ^ λ ⟩ + ∣ ⟨ ( ∣ Q ∣ 2 − ∣ W ∣ 2 ) k ^ λ , k ^ λ ⟩ ) + ∣ ⟨ W * Q k ^ λ , k ^ λ ⟩ ∣ + 2 ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ ∣ ⟨ W k ^ λ , k ^ λ ⟩ ∣ ,

whence

∣ ⟨ ( Q + W ) k ^ λ , k ^ λ ⟩ ∣ 2 ≤ 1 2 ber ( ∣ Q ∣ 2 + ∣ W ∣ 2 ) + 1 2 ber ( ∣ Q ∣ 2 − ∣ W ∣ 2 ) + ber ( W * Q ) + 2 ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ ∣ ⟨ k ^ λ , W k ^ λ ⟩ ∣ = 1 2 ∥ ∣ Q ∣ 2 + ∣ W ∣ 2 ∥ ber + 1 2 ber ( ∣ Q ∣ 2 − ∣ W ∣ 2 ) + ber ( W * Q ) + 2 ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ ∣ ⟨ k ^ λ , W k ^ λ ⟩ ∣ ,

where we have used (1.2) in the last equality since ∣ Q ∣ 2 + ∣ W ∣ 2 ≥ 0 . Moreover, by applying Lemma 2.4 and the arithmetic-geometric mean inequality, we see that

∣ ⟨ ( Q + W ) k ^ λ , k ^ λ ⟩ ∣ 2 ≤ 1 2 ∥ ∣ Q ∣ 2 + ∣ W ∣ 2 ∥ ber + 1 2 ber ( ∣ Q ∣ 2 − ∣ W ∣ 2 ) + ber ( W * Q ) + ‖ Q k ^ λ ‖ ‖ W k ^ λ ‖ + ∣ ⟨ Q k ^ λ , W k ^ λ ⟩ ∣ ≤ 1 2 ∥ ∣ Q ∣ 2 + ∣ W ∣ 2 ∥ ber + 1 2 ber ( ∣ Q ∣ 2 − ∣ W ∣ 2 ) + ber ( W * Q ) + 1 2 ⟨ ( ∣ Q ∣ 2 + ∣ W ∣ 2 ) k ^ λ , k ^ λ ⟩ + ∣ ⟨ Q k ^ λ , W k ^ λ ⟩ ∣ ≤ 1 2 ∥ ∣ Q ∣ 2 + ∣ W ∣ 2 ∥ ber + 1 2 ber ( ∣ Q ∣ 2 − ∣ W ∣ 2 ) + ber ( W * Q ) + 1 2 ber ( ∣ Q ∣ 2 + ∣ W ∣ 2 ) + ber ( W * Q ) .

So, another application of (1.2) gives

∣ ⟨ ( Q + W ) k ^ λ , k ^ λ ⟩ ∣ 2 ≤ ∥ ∣ Q ∣ 2 + ∣ W ∣ 2 ∥ ber + 1 2 ber ( ∣ Q ∣ 2 − ∣ W ∣ 2 ) + 2 ber ( W * Q ) ,

for every λ , μ ∈ Θ . Therefore, by taking the supremum over all λ , μ ∈ Θ in the aforementioned inequality, we obtain the desired result.□

In order to prove our next result, we need to recall the following inequality:

(3.8) ( a 1 a 2 + b 1 b 2 ) 2 ≤ ( a 1 2 + b 1 2 ) ( a 2 2 + b 2 2 ) , ∀ a 1 , a 2 , b 1 , b 2 ∈ R .

Our next result reads as follows.

Theorem 3.9

Let Q , W ∈ L ( ℰ Θ ) . Then, the following inequality

ber 4 ( Q + W ) ≤ 4 min { ∥ ∣ Q ∣ 2 + ∣ W ∣ 2 ∥ ber ∥ ∣ Q * ∣ 2 + ∣ W * ∣ 2 ∥ ber , ∥ ∣ Q ∣ 2 + ∣ W * ∣ 2 ∥ ber ∥ ∣ Q * ∣ 2 + ∣ W ∣ 2 ∥ ber }

holds.

Proof

Let λ ∈ Θ and k ^ λ be the normalized reproducing kernel of ℰ Θ . Using the convexity of the function t ↦ t 2 , we see that

∣ ⟨ ( Q + W ) k ^ λ , k ^ λ ⟩ ∣ 2 ≤ ( ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ + ∣ ⟨ W k ^ λ , k ^ λ ⟩ ∣ ) 2 ≤ 2 ( ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ 2 + ∣ ⟨ W k ^ λ , k ^ λ ⟩ ∣ 2 ) = 2 ( ∣ ⟨ Q k ^ λ , k ^ λ ⟩ ∣ ∣ ⟨ k ^ λ , Q * k ^ λ ⟩ ∣ + ∣ ⟨ W k ^ λ , k ^ λ ⟩ ∣ ∣ ⟨ k ^ λ , W * k ^ λ ⟩ ∣ ) ≤ 2 ( ∥ Q k ^ λ ∥ ∥ Q * k ^ λ ∥ + ∥ W k ^ λ ∥ ∥ W * k ^ λ ∥ ) = 2 ( ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ ⟨ ∣ Q * ∣ 2 k ^ λ , k ^ λ ⟩ + ⟨ ∣ W ∣ 2 k ^ λ , k ^ λ ⟩ ⟨ ∣ W * ∣ 2 k ^ λ , k ^ λ ⟩ ) .

This implies that

∣ ⟨ ( Q + W ) k ^ λ , k ^ λ ⟩ ∣ 4 ≤ 4 ( ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ ⟨ ∣ Q * ∣ 2 k ^ λ , k ^ λ ⟩ + ⟨ ∣ W ∣ 2 k ^ λ , k ^ λ ⟩ ⟨ ∣ W * ∣ 2 k ^ λ , k ^ λ ⟩ ) 2 .

By choosing a 1 = ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ , a 2 = ⟨ ∣ Q * ∣ 2 k ^ λ , k ^ λ ⟩ , b 1 = ⟨ ∣ W ∣ 2 k ^ λ , k ^ λ ⟩ , and b 2 = ⟨ ∣ W * ∣ 2 k ^ λ , k ^ λ ⟩ in the above inequality and then applying (3.8), we obtain

∣ ⟨ ( Q + W ) k ^ λ , k ^ λ ⟩ ∣ 4 ≤ 4 ( ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ + ⟨ ∣ W ∣ 2 k ^ λ , k ^ λ ⟩ ) ( ⟨ ∣ Q * ∣ 2 k ^ λ , k ^ λ ⟩ + ⟨ ∣ W * ∣ 2 k ^ λ , k ^ λ ⟩ ) = 4 ⟨ ( ∣ Q ∣ 2 + ∣ W ∣ 2 ) k ^ λ , k ^ λ ⟩ ⟨ ( ∣ Q * ∣ 2 + ∣ W * ∣ 2 ) k ^ λ , k ^ λ ⟩ ≤ 4 ber ( ∣ Q ∣ 2 + ∣ W ∣ 2 ) ber ( ∣ Q * ∣ 2 + ∣ W * ∣ 2 ) = 4 ∥ ∣ Q ∣ 2 + ∣ W ∣ 2 ∥ ber ∥ ∣ Q * ∣ 2 + ∣ W * ∣ 2 ∥ ber ,

where we have used (1.2) in the last equality since we have ∣ Q ∣ 2 + ∣ W ∣ 2 ≥ 0 and ∣ Q * ∣ 2 + ∣ W * ∣ 2 ≥ 0 . By taking the supremum over all λ ∈ Θ in the last inequality, we obtain

(3.9) ber 4 ( Q + W ) ≤ 4 ∥ ∣ Q ∣ 2 + ∣ W ∣ 2 ∥ ber ∥ ∣ Q * ∣ 2 + ∣ W * ∣ 2 ∥ ber .

By choosing a 1 = ⟨ ∣ Q ∣ 2 k ^ λ , k ^ λ ⟩ , a 2 = ⟨ ∣ Q * ∣ 2 k ^ λ , k ^ λ ⟩ , b 1 = ⟨ ∣ W * ∣ 2 k ^ λ , k ^ λ ⟩ and b 2 = ⟨ ∣ W ∣ 2 k ^ λ , k ^ λ ⟩ and then proceeding as above, we see that

(3.10) ber 4 ( Q + W ) ≤ 4 ∥ ∣ Q ∣ 2 + ∣ W * ∣ 2 ∥ ber ∥ ∣ Q * ∣ 2 + ∣ W ∣ 2 ∥ ber .

Combining (3.9) together with (3.10) yields to the desired result.□

Acknowledgments

The authors would like to thank the reviewers for their valuable comments and thorough review, which greatly helped enhance the quality of the final version of this manuscript. Additionally, the first author would like to express gratitude for the support received from the Distinguished Scientist Fellowship Program at King Saud University in Riyadh, Saudi Arabia. This program, under Researchers Supporting Project Number (RSP2024R187), funded this research.

Funding information: This article received financial support from the Distinguished Scientist Fellowship Program at King Saud University in Riyadh, Saudi Arabia, under Project Number (RSP2024R187).
Author contributions: The work presented here was carried out in collaboration between all authors. All authors contributed equally and significantly in writing this article. All authors have contributed to the manuscript. All authors have read and agreed to the published version of the manuscript.
Conflict of interest: The authors declare that they have no competing interests.
Data availability statement: No data were used to support this study.

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Received: 2023-07-06

Revised: 2024-01-26

Accepted: 2024-02-09

Published Online: 2024-05-08

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

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https://doi.org/10.1515/dema-2023-0159

Keywords for this article

Berezin number; Berezin norm; sum; inequality

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