key: cord-0070934-92yxvasf
authors: Rout, Nirmal Chandra; Mishra, Debasisha
title: Further results on A-numerical radius inequalities
date: 2021-12-06
journal: Ann
DOI: 10.1007/s43034-021-00156-3
sha: 7dfb7e0bfe44e505099e047784f945297004c04f
doc_id: 70934
cord_uid: 92yxvasf

Let A be a bounded linear positive operator on a complex Hilbert space [Formula: see text] Furthermore, let [Formula: see text] denote the set of all bounded linear operators on [Formula: see text] whose A-adjoint exists, and [Formula: see text] signify a diagonal operator matrix with diagonal entries are A. Very recently, several [Formula: see text] -numerical radius inequalities of [Formula: see text] operator matrices were established. In this paper, we prove a few new [Formula: see text] -numerical radius inequalities for [Formula: see text] and [Formula: see text] operator matrices. We also provide a new proof of an existing result by relaxing a sufficient condition “A is strictly positive”. Our proofs show the importance of the theory of the Moore–Penrose inverse of a bounded linear operator in this field of study.

Throughout, H denotes a complex Hilbert space with inner product ⟨⋅, ⋅⟩ . By B(H) , we mean the C * -algebra of all bounded linear operators on H . Let ‖ ⋅ ‖ be the norm induced from ⟨⋅, ⋅⟩. operator, then we use the notation A ≥ 0 ( A > 0 ). Let be an n × n diagonal operator matrix whose diagonal entries are positive operator A for n = 1, 2, ... . Then, ∈ B( ⨁ n i=1 H) and ≥ 0 . If A ≥ 0 , then it induces a positive semidefinite sesquilinear form, ⟨⋅, ⋅⟩ A ∶ H × H → ℂ defined by ⟨x, y⟩ A = ⟨Ax, y⟩, x, y ∈ H. Let ‖ ⋅ ‖ A denote the seminorm on H induced by ⟨⋅, ⋅⟩ A , i.e., ‖x‖ A = √ ⟨x, x⟩ A for all x ∈ H. Then, ‖x‖ A is a norm if and only if A > 0 . Also, (H, ‖ ⋅ ‖ A ) is complete if and only if R(A) is closed in H. Henceforth, we use the symbol A and for positive operators on H and ⨁ n i=1 H , respectively. We retain the notations O and I for the null operator and the identity operator on H , respectively. Given T ∈ B⇐H⇒ , the A-operator seminorm ‖T‖ A is defined as follows:

We set B A ⇐H⇒ = {T ∈ B⇐H⇒ ∶ ‖T‖ A < ∞}. Then B A ⇐H⇒ is not a subalgebra of B⇐H⇒ . It is pertinent to point out that ‖T‖ A = 0 if and only if ATA = O. For T ∈ B A ⇐H⇒, we have If AT ≥ 0 , then the operator T is called A-positive. Note that if T is A-positive, then Before we proceed further, it is necessary to introduce the concept of A-adjoint operator. We say an operator X ∈ B⇐H⇒ to be A-adjoint operator of T ∈ B⇐H⇒ if ⟨Tx, y⟩ A = ⟨x, Xy⟩ A for every x, y ∈ H, i.e., AX = T * A. By Douglas Theorem [5] , the existence of an A-adjoint operator is not guaranteed. An operator T ∈ B⇐H⇒ may admit none, one or many A-adjoints. A rather well-known result states that A-adjoint of an operator T ∈ B⇐H⇒ exists if and only if R(T * A) ⊆ R(A) . Let us now denote

Note that B A ⇐H⇒ is a subalgebra of B⇐H⇒ which is neither closed nor dense in B⇐H⇒. Moreover, we have the following inclusion relations:

And the equality holds if A is injective and has a closed range.

For T ∈ B⇐H⇒ , w A (T) , the A-numerical radius of T was proposed by Saddi [13] . And is defined as follows:

When T = (T ij ) is an n × n operator matrix with T ij ∈ B⇐H⇒, then (1) can be written as

Very recently, Zamani [16] obtained the following A-numerical radius inequality for T ∈ B A (H):

The first inequality in (2) becomes an equality if T 2 = O and the second inequality becomes an equality if T is A-selfadjoint. The A-Crawford number of T ∈ B A (H) is defined as This terminology was introduced by Zamani [16] . Furthermore, if T is A-selfadjoint, then w A (T) = ‖T‖ A . Moslehian et al. [8] continued the study of A-numerical radius and obtained some new A-numerical radius inequalities. In this year, Bhunia et al. [3] presented several -numerical radius inequalities for a strictly positive operator A. We refer the interested reader to [12, [14] [15] [16] and the references cited therein for further generalizations and refinements of A-numerical radius inequalities.

The objective of this paper is to present a few new -numerical radius inequalities for 2 × 2 and n × n operator matrices. Besides these, we aim to establish some existing -numerical radius inequalities by relaxing sufficient condition A > 0. To this end, the paper is sectioned as follows. In Sect. 2, we define additional mathematical constructs including the definition of the Moore-Penrose inverse of an operator, A-adjoint, A-selfadjoint and A-unitary operator, that are required to state and prove the results in the subsequent sections. Section 3 contains several new A-numerical radius inequalities. More interestingly, it also provides some recent existing results in the literature on A-numerical radius inequalities by dropping a sufficient condition.

This section gathers a few more definitions and results that are useful in proving our main results. It starts with the definition of the Moore-Penrose inverse of a bounded operator A in H . The Moore-Penrose inverse of A ∈ B⇐H⇒ [9] is the operator X ∶ R(A) ⨁ R(A) ⟂ ⟶ H which satisfies the following four equations:

Here, N(A) and P L denote the null space of A and the orthogonal projection onto L, respectively. The Moore-Penrose inverse is unique, and is denoted by A † . In general, [2, 7] ). Note that [5] ). One can observe that

Besides, we derive below two new properties of A-adjoint of an operator T ∈ B A (H), which are crucial in providing some new proofs of the existing results and in proving new results on A-numerical radius inequalities.

and

For any T 1 , T 2 ∈ B A (H), we have However, the above proof is a very simple one and directly follows from the definition of A-norm.

The real and imaginary part of an operator

2i . An interested reader may refer [1, 2] for further properties of operators on semi-Hilbertian space. From the definition of A-numerical radius of an operator, it follows that Some interesting results are collected hereunder for further use.

The next result is a combination of [3, Lemma 2.4(i)] and [11, Lemma 2.2].

The other parts of [3, Lemma 2.4] assume the condition A is strictly positive. Rout et al. [11] proved the same result for positive A, and the same is stated below.

The next result establishes upper and lower bounds for the -numerical radius of a particular type of 2 × 2 operator matrix that is a generalization of (2).

This section begins with the power inequality for semi-Hilbert space that has been proved by Moslehian et al. [8] , which states that for T ∈ B⇐H⇒, w A (T n ) ≤ w n A (T) for n ∈ ℕ. Using this, we prove the following theorem.

This implies Remark 3.2 Using Lemma 2.1 and a known inequality w (T) ≤ 1 2 (‖T‖ + ‖T 2 ‖ 1∕2 ),

We generalize some of the results of [6] now. Using Lemma 2.2, one can now state [3, Theorem 3.1] without assuming the condition A > 0 . Its proof takes the

same steps as in the [3, proof of Theorem 3.1] after the use of Lemma 2.2, and hence the proof is omitted.

In particular, putting Y = X Considering X = Y = Q and T = I in Lemma 3.3, we get the following corollary. 

We, therefore, have ◻ The next result provides an estimate for lower bound of -numerical radius of a 2 × 2 operator matrix.

To show that Q is -unitary, we need to prove that ‖x‖ = ‖Qx‖ = ‖Q # x‖ . So, 

By (13) and Lemma 2.2, we obtain Again, applying Corollary 3.4 and taking T = T

shown that Q is -unitary and w TQ # A ± QT ≤ 2w (T). Therefore, By Lemma 2.4, we, therefore, achieve the following:

w

From (14) and (15), we get the desired result. ◻ As a special case of the above result, we obtain a corollary in Krein space setting. Let J be a non-scalar Hermitian involution operator in B(ℍ) , where ℍ is a separable Hilbert space. Then, space (ℍ, ⟨., .⟩ J ) is called Krein space [4] . Note that the J-adjoint operator of T ∈ B(ℍ) is the unique operator in B(ℍ) such that ⟨Tx, y⟩ J = ⟨x, T # J y⟩ J , for all x, y ∈ ℍ. Therefore, we have T # J = JT * J and (T # J ) # J = T. One can observe that Lemmas 2.1 and 2.2 hold in the setting of Krein spaces for T 1 , T 2 ∈ B(ℍ). We provide below a lemma for the J-adjoint operator of an n × n operator matrix.

We state below another lemma for Krein space operators whose proof follows from [3, Theorem 3.1]. Based on these results, we have the following corollary of Theorem 3.6.

Let T 1 , T 2 , T 3 , T 4 ∈ B(ℍ). Then w

We provide below a lower bound for -numerical radius inequality of an operator matrix. From (16) and (17), we have ◻ One can easily prove the next lemma proceeding as in the proof of [11, Lemma 2.9] , and is stated below.

The next corollary follows from the above lemma. We conclude this section with the following two results for n × n operator matrices. First, we demonstrate an interesting property of -numerical radius of an n × n operator matrix which is a generalization of [10, Lemma 2.1].

. It is easy to see that z = z −1 = z n−1 and |z| = 1. To show that U is -unitary, we need to prove that ‖x‖ = ‖Ux‖ = ‖U # x‖ , 

Therefore,

The last result provides a relation between -numerical radius of two diagonal operator matrices, where diag(T 1 , … , T n ) means an n × n diagonal operator matrix with entries T 1 , … , T n . 

T i ≤ nw (diag(T 1 , … , T n )).

In this paper, we have further studied A-numerical radius inequalities for operators on semi-Hilbertian space and numerical radius inequalities for operators in Krein space. The important findings are summarized as follows:

• Several upper and lower bounds for a 2 × 2 operator matrices are demonstrated. • Some existing -numerical radius inequalities are provided by relaxing sufficient condition A > 0 . This shows the importance of the theory of the Moore-Penrose inverse in this field of research. • Some numerical radius inequalities for operator matrices in Krein space setting are shown as corollaries to our main results.

This paper ends with the note that further work on -numerical radius for n × n operator matrices and numerical radius inequalities for operators in Krein space can be studied.

= nw (diag(T 1 , … , T n )).

Metric properties of projections in semi-Hilbertian spaces

Partial isometries in semi-Hilbertian spaces

On inequalities for A-numerical radius of operators

Indefinite Inner Product Spaces

On majorization, factorization, and range inclusion of operators on Hilbert space

Numerical radius inequalities for commutators of Hilbert space operators

Positivity of 2 × 2 block matrices of operators

Seminorm and numerical radius inequalities of operators in semi-Hilbertian spaces

Generalized Inverses and Applications

Some numerical radius inequalities for Hilbert space operators

On -numerical radius inequalities for 2 × 2 operator matrices

Some A-numerical radius inequalities for semi-Hilbertian space operators

A-normal operators in semi Hilbertian spaces

Some upper bounds for the -numerical radius of 2 × 2 block matrices

A-numerical radius and product of semi-Hilbertian operators

A-Numerical radius inequalities for semi-Hilbertian space operators

We would like to express our sincere thanks and gratitude to the editor and the anonymous referees for their valuable comments and suggestions in the improvement of the manuscript. We thank the Government of India for introducing the work from home initiative during the COVID-19 pandemic.