key: cord-0561167-qobbbrj2
authors: Rout, Nirmal Chandra; Mishra, Debasisha
title: Further results on $mathbb{A}$-numerical radius inequalities
date: 2020-07-08
journal: nan
DOI: nan
sha: 8aabbcd1e218b9b10148e88a9eac5c5bd84e891d
doc_id: 561167
cord_uid: qobbbrj2

Let $mathcal{H}$ be a complex Hilbert space, and $A$ be a positive bounded linear operator on $mathcal{H}.$ Let $mathcal{B}_Amathcal{(H)}$ denotes the set of all bounded linear operators on $mathcal{H}$ whose $A$-adjoint exists. Let $mathbb{A}$ denotes a diagonal operator matrix with diagonal entries are $A.$ In this paper, we prove a few new $mathbb{A}$-numerical radius inequalities for $2times 2$ and $ntimes n$ operator matrices. We also provide some new proofs of the existing results by relaxing different sufficient conditions like"$A$ is strictly positive"and"$mathcal{N}(A)^perp$ is invariant subspace for different operators". Our proofs show the importance of the theory of the Moore-Penrose inverse of bounded operators in this field of study.

Let H be a complex Hilbert space with inner product ⟨⋅, ⋅⟩ and B(H) be the C * -algebra of all bounded linear operators on H. Let ⋅ be the norm induced from ⟨⋅, ⋅⟩. An operator A ∈ B(H) is called selfadjoint if A = A * , where A * denotes the adjoint of A. A selfadjoint operator A ∈ B(H) is called positive if ⟨Ax, x⟩ ≥ 0 for all x ∈ H, and is called strictly positive if ⟨Ax, x⟩ > 0 for all non-zero x ∈ H. We denote a positive (strictly positive) operator A by A ≥ 0 (A > 0). We denote R(A) as the range space of A and R(A) as the norm closure of R(A) in H. Let A be a n × n diagonal operator matrix whose diagonal entries are positive operator A for n = 1, 2, .... Then A ∈ B(⊕ n i=1 H) and A ≥ 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 Furthermore, if T is A-selfadjoint, then w A (T ) = T A . In 2019, Moslehian et al. [14] again continued the study of A-numerical radius and established some inequalities for A-numerical radius. In 2020, Bhunia et al. [4] and [5] obtained several A-numerical radius inequalities for strictly positive operator A. Feki [8] and Feki et al. [9] obtained several A-numerical radius inequalities under the assumption N (A) ⊥ is invariant subspace for different operators. Further generalizations and refinements of A-numerical radius are discussed in [10, 17] . The objective of this paper is to present a few new A-numerical radius inequalities for n × n and 2 × 2 operator matrices. Besides this, we also aim to establish some existing A-numerical radius inequalities without using the condition A > 0 and N (A) ⊥ is invariant subspace for different operators. To this end, the paper is sectioned as follows. In Section 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 new proof to the very recent existing results in the literature on A-numerical radius inequalities by dropping a few sufficient conditions.

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) [15] 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 [2, 13] ). Note that [6] ). We can observe that

and

operators, and

This fact is same as Lemma 2.8 of [9] . However, the above proof is a very simple one and directly follows using the definition of A-norm. . An interested reader may refer [1, 2] for further properties of operators on Semi-Hilbertian space. From (1.1), it follows that

Some interesting results are collected hereunder for further use.

The next result is a combination of Lemma 2.4 (i) [4] and Lemma 2.2 [16] .

The other parts of Lemma 2.4 [4] assumes the condition A is strictly positive. Rout et al. [16] proved the same result for positive A, and the same is stated below.

The next result establishes upper and lower bounds for the A-numerical radius of a particular type of 2 × 2 operator matrix that is a generalization of (1.2). 

Theorem 2.4 [8] for operators T 1 , T 2 ∈ B A (H) is stated as follows.

It is well known that P R(A) T ≠ T P R(A) for any T ∈ B A (H) (even if A and T are finite matrices). And the equality holds if N (A) ⊥ is invariant for T. The first result shows that the A-numerical radius of P R(A) T and P R(A) T are same for any T ∈ B A (H).

Proof.

We therefore have

We demonstrate an interesting property of A−numerical radius of an n × n operator matrix which is a generalization of Lemma 2.1 [18] .

. It is easy to see that z = z −1 = z n−1 and

This implies that

The following lemma provides an upper bound for T ∈ B A (H) to prove Theorem 3.4.

Here,

Again,

Replacing T 2 by −T 2 , we get

This implies

Thus, we obtain

The next result provides an estimate for A-operator norms of certain 2 × 2 operator matrices. 3.7. 9

Theorem 3.5. Let T ∈ B A (H) and a, b ∈ C. Then

Proof. Let α, β ∈ R such that α 2 + β 2 = 1 and

Let x n , y n ∈ H be two unit vectors in H such that lim n→∞ ⟨T y n , x n ⟩ = T A for n ∈ N. Let β n ∈ R be such that a⟨T y n , x n ⟩ A = e iβn a ⟨T y n , x n ⟩ A . Suppose that

We recall below a result of [8] to obtain Corollay 3.7.

Let T ∈ B A (H). Then

Feki [8] proved the following result with the additional assumption "N (A) ⊥ is invariant for T ∈ B A (H)." Next, we prove the same result without this assumption.

From (3.7), we now have

So, we get

Using Theorem 3.5, we also obtain

Hence, we arrive at our claim by (3.8) and (3.9) .

Following theorem provides a relation between A-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 .

Proof. Here,

We generalize some of the results of [12] now. Using Lemma 2.4 [16] , one can now prove Corollary 3.3 [4] without assuming the condition A > 0, and is stated next.

In particular, putting Y = X

Considering X = Y = Q and T = I in the previous theorem, we get Lemma 2.7, which is stated below.

Feki and Sahoo [9] established many results on A-numerical radius inequalities of 2 × 2 operator matrices, very recently. In many cases, they assumed the condition "N (A) ⊥ is invariant subspace for T 1 , T 2 , T 3 , T 4 " to show their claim. They assumed these conditions in order to get the equality P R(A) T = T P R(A) which is not true, in general. One of the objective of this paper is to achieve the same claim without assuming the additional condition "N (A) ⊥ is invariant subspace for T 1 , T 2 , T 3 , T 4 ". The next result is in this direction, and is more general than Theorem 2.7 [9] . Our proof is also completely different than the corresponding proof in [9] . And, therefore our results are superior to those results in [9] and [8] that assumes the invariant condition.

Thus, Q is an A-unitary operator. By Lemma 2.7, we obtain

Hence, we have

By (3.11) and Lemma 2.3, we obtain

Again, applying Lemma 2.7 and taking T =

It is easy to verify that Q is A-unitary. We now have

By Lemma 2.5, we therefore achieve the following:

(3.14)

From (3.12) and (3.14), we get the desired result.

We provide below the same estimate as in Theorem 2.8 [9] for A-numerical radius of an operator matrix that improves but by dropping the assumption N (A) ⊥ is an invariant subspace for T 1 , T 2 ∈ B A (H).

So, Q A = 1. Using Lemma 2.7, we get

By Lemma 2.5, we thus have

Corollary 3.14. Let T = P + iQ be the cartesian decomposition in B A (H). Then

Proof.

Again, replacing T 2 and T 3 by P and iQ, respectively in Lemma 2.12 and using Lemma 2.3 of [16] , we have 

This implies

Re(e 2iθ (P T 2 T 1 +T 2 T 1 P )) A . Now, taking supremum over θ ∈ R and using Lemma 3.15, we thus obtain

Note that the authors of [5] proved the above theorem with the assumption A > 0. Using Theorem 3.16 and Lemma 2.3, we now establish the following inequality. 

A (T 2 T 1 ) + 2w A (T 2 T 1 P + P T 2 T 1 ). The last inequality follows by Theorem 3.16.

Adopting a parallel technique as in the proof of the Theorem 3.16, one can prove the following result. 

where P = T # A 1 T 1 + T 2 T # A 2 and c A (T 2 T 1 ) = inf

Re(e iθ T 2 T 1 )x A .

N (A) ⊥ is invariant for operators

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On A-numerical radius inequalities for 2 × 2 operator matrices

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A-normal operators in semi Hilbertian spaces

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We thank the Government of India for introducing the work from home initiative during the COVID-19 pandemic.