key: cord-0511487-6puwdrsg
authors: Sahoo, Satyajit
title: On $mathbb{A}$-numerical radius inequalities for $2times2$ operator matrices-II
date: 2020-07-06
journal: nan
DOI: nan
sha: 7c1b76a985da7b0e69f00d79ae8da7eb3272fec5
doc_id: 511487
cord_uid: 6puwdrsg

The main goal of this article is to establish several new upper and lower bounds for the $mathbb{A}$-numerical radius of $2times 2$ operator matrices, where $mathbb{A}$ be the $2times 2$ diagonal operator matrix whose diagonal entries are positive bounded operator $A$.

Let H be a complex Hilbert space with inner product ⟨⋅, ⋅⟩ and L(H) be the C * -algebra of all bounded linear operators on H. The numerical range of T ∈ B(H) is defined as

The numerical radius of T , denoted by w(T ), is defined as w(T ) = sup{ z ∶ z ∈ W (T )}. It is well-known that w(⋅) defines a norm on H, and is equivalent to the usual operator norm T = sup{ T x ∶ x ∈ H, x = 1}. In fact, for every T ∈ B(H), Here onward, we fix A and A for positive operators on H and H ⊕ H, respectively. We also reserve the notation I and O for the identity operator and the null operator on H in this paper. T A denotes the A-operator seminorm of T ∈ L(H). This is defined as follows: 

If AT ≥ 0, then the operator T is called A-positive. Note that if T is A-positive, then

An operator X ∈ B(H) is called an A-adjoint operator of T ∈ B(H) if ⟨T x, y⟩ A = ⟨x, Xy⟩ A for every x, y ∈ H, i.e., AX = T * A. By Douglas Theorem [9] , the existence of an A-adjoint operator is not guaranteed. An operator T ∈ B(H) may admit none, one or many A-adjoints. hold with equality if A is injective and has a closed range. The Moore-Penrose inverse of A ∈ B(H) [16] 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, 14] ). Note that [9] ). An operator

By applying Douglas theorem, one can easily see that the subspace of all operators admitting A 1 2 -adjoints, denoted by L A 1 2 (H), is equal the collection of all A-bounded operators, i.e.,

Notice that L A (H) and L A 1 2 (H) are two subalgebras of L(H) which are, in general, neither closed nor dense in L(H). Moreover, we have L A (H) ⊂ L A 1 2 (H) (see [2, 3] ).

In 2012, Saddi [19] introduced A-numerical radius of T for T ∈ B(H), which is denoted as w A (T ), and is defined as follows:

A fundamental inequality for the A-numerical radius is the power inequality (see [15] ) which says that for T ∈ B(H),

Notice that the A-numerical radius of semi-Hilbertian space operators satisfies the weak A-unitary invariance property which asserts that

(1.5) 3 for every T ∈ L A (H) and every A-unitary operator U ∈ L A (H) (see [7, Lemma 3.8] ).

An interested reader may refer [1, 2] for further properties of operators on Semi-Hilbertian space.

Let

for any arbitrary operator T ∈ B A (H). Recently, in 2019 Zamani [24, Theorem 2.5] showed that if T ∈ L A (H), then

In 2019, Zamani [24] showed that if T ∈ L A (H), then

(1.7)

The author then extended the inequality (1.1) using A-numerical radius of T , and the same is produced below:

In 2019, Moslehian et al. [15] again continued the study of A-numerical radius and established some inequalities for A-numerical radius. Further generalizations and refinements of A-numerical radius are discussed in [5, 6, 17] . In 2020, Bhunia et al. [8] obtained several A-numerical radius inequalities. For more results on A-numerical radius inequalities we refer the reader to visit [10, 18, 23, 12] . In 2020, the concept of the A-spectral radius of A-bounded operators was introduced by Feki in [11] as follows:

(1.9)

Here we want to mention that the proof of the second equality in (1.9) can also be found in [11, Theorem 1] . Like the classical spectral radius of Hilbert space operators, it was shown in [11] that r A (⋅) satisfies the commutativity property, i.e.

for all T, S ∈ L A 1 2 (H). For the sequel, if A = I, then T , r(T ) and ω(T ) denote respectively the classical operator norm, the spectral radius and the numerical radius of an operator T . The objective of this paper is to present a few new A-numerical radius inequalities for 2 × 2 operator matrices. In this aspect, the rest of the paper is broken down as follows. In 4 section 2, we collect a few results about A-numerical radius inequalities which are required to state and prove the results in the subsequent section. Section 3 contains our main results, and is of two parts. Motivated by the work of Hirzallah et al. [13] , the first part presents several A-numerical radius inequalities of 2 × 2 operator matrices while the next part focuses on some A-numerical radius inequalities.

We need the following lemmas to prove our results.

Lemma 2.1. [Theorem 7 and corollary 2, [11] ] If T ∈ L A 1 2 (H).Then

The following lemma is already proved by Bhunia et al. [8] for the case strictly positive operator A. Very recentely the same result proved by Rout et al. [18] without the condition A > 0 is stated next for our purpose.

The following Lemma is proved by Rout et al. [18] .

Lemma 2.6. [Lemma 2.4 and Lemma 3.1, [10, 7] ] Let T 1 , T 4 ∈ L A 1 2 (H). Then, the following assertions hold

In order to prove our main result the following identity is essential for our purpose. If

We will split our results into two subsections. The first part deals with A-numerical radius of 2 × 2 operator matrices. The second part concerns some upper bound for A numerical radius inequalities.

Here, we establish our main results dealing with different upper and lower bounds for Anumerical radius of 2 × 2 block operator matrices. The very first result is stated next.

6

Now, using identity (2.3) and Lemma 2.4, we have

Replacing T 3 by −T 3 in the inequality (3.1) and using Lemma 2.4, we get

From the inequalities (3.1) and (3.2), we have

Again, in the inequality (3.3), interchanging T 2 and T 3 and using Lemma 2.4(ii), we get

From the inequalities (3.3) and (3.4), we get

This completes the proof.

(3.6)

This implies

Which in turn implies that

Thus, using inequality (2.3) and Lemma 2.4

Replacing T 3 by −T 3 in the inequality (3.7) we have

Now from inequality (3.7) and (3.8) that

Interchanging T 2 and T 3 in the ininequality (3.9), we get

(3.10)

From inequalities (3.9) and (3.10), we have

Which proves the first inequality. Again, by identity (3.5) and inequality (2.3) that

Thus,

Replacing T 3 by −T 3 in the inequality (3.12) and using Lemma 2.4, we get

It follows from inequalities (3.12) and (3.13) that

(3.14)

Interchanging T 2 and T 3 in the inequality (3.14) and using Lemma 2.4, we get max

Now combining (3.14) and (3.15), we have max

This completes the proof.

Hence by using Lemma 2.4 we obtain

Using similar argument to (

Combining (3.17) and (3.18) we get

.

Proof. Based on Lemma 2.5, Lemma 2.4 and Theorem 3.3 we have

The following lemma is already proved by Hirzallah et al. [13] for the case of Hilbert space operators. Using similar techinque we can prove this lemma for the case of semi-Hilbert space. Now we state here the result without proof for our purpose.

The forthcoming result is analogous to Theorem 3.4

and P − Q = ((T 1 + T 2 )(T 1 − T 2 )) n . By using inequality (1.4), we have

This proves the inequality (3.20) . In order to prove the inequality (3.21

Then

(3.23)

Similarly we can show that

From inequality (2.1), combining inequality (3.23) and (3.24), we obtain

In this subsection we establish some upper bounds for A-numerical radius of operators. In the next result, we derive an upper bound for A-numerical radius of product of operators on semi-Hilbertian space.

Theorem 3.7. Let T 1 , T 2 ∈ L A (H). Then

Proof. It is not difficult to see that R A (e iθ T 1 T 2 ) is an A-selfadjoint operator. So, by Lemma 2.2 we have R A (e iθ T 1 T 2 ) A = w A (R A (e iθ T 1 T 2 )).

So,

So by applying Lemma 2.2 we see that

So, by using (1.10) we have

So by taking supremum over θ ∈ R, then using 1.6 we get our desired result. 16 

Metric properties of projections in semi-Hilbertian spaces

Partial isometries in semi-Hilbertian spaces

Lifting properties in operator ranges

Upper bounds for numerical radius inequalities involving off-diagonal operator matrices

Some improvements of numerical radius inequalities of operators and operator matrices

Refinements of A-numerical radius inequalities and its applications

A-Numerical radius orthogonality and parallelism of semi-Hilbertian space operators and their applications

On inequalities for A-numerical radius of operators

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

Some A-numerical radius inequalities for d × d operator matrices

Spectral radius of semi-Hilbertian space operators and its applications

Further inequalities for the A-numerical radius of certain 2 × 2 operator matrices

Numerical radius inequalities for certain 2 × 2 operator matrices

Positivity of 2 × 2 block matrices of operators

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

Generalized Inverses and Applications

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

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

A-normal operators in semi Hilbertian spaces

Numerical radius inequalities for operator matrices

Some extended numerical radius inequalities

Berezin number and numerical radius inequalities for operators on Hilbert spaces

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

A-Numerical radius inequalities for semi-Hilbertian space operators

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

Theorem 3.6. Let T 1 , T 2 ∈ L A (H). Thenfor n ∈ N and