key: cord-0632949-5ghcnybg
authors: Rout, Nirmal Chandra; Sahoo, Satyajit; Mishra, Debasisha
title: On A-numerical radius inequalities for $2 times 2$ operator matrices
date: 2020-04-16
journal: nan
DOI: nan
sha: f0daa3ec1577c0a45bc26291a2870aff65632501
doc_id: 632949
cord_uid: 5ghcnybg

Let ($mathcal{H}, langle . , .rangle )$ be a complex Hilbert space and $A$ be a positive bounded linear operator on it. Let $w_A(T)$ be the $A$-numerical radius and $|T|_A$ be the $A$-operator seminorm of an operator $T$ acting on the semi-Hilbertian space $(mathcal{H}, langle .,.rangle_A),$ where $langle x, yrangle_A:=langle Ax, yrangle$ for all $x,yin mathcal{H}$. In this article, we establish several upper and lower bounds for $B$-numerical radius of $2times 2$ operator matrices, where $B=begin{bmatrix} A&0 0&A end{bmatrix}$. Further, we prove some refinements of earlier $A$-numerical radius inequalities for operators.

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

One may refer [3, 6, 9, 10, 11, 19] for several generalizations, refinements and applications of numerical radius inequalities in different settings which appeared in the last decade. Let ⋅ be the norm induced from ⟨⋅, ⋅⟩. 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). Let B be a 2 × 2 diagonal operator matrix, in which each of the diagonal entries is a positive operator A. Through out this article, A is always assumed to a positive operator. Clearly, if A is a positive operator, 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 semi-norm on H induced by ⟨⋅, ⋅⟩ A , i.e., x A = ⟨x, x⟩ A for all x ∈ H. It is easy to verify that x A is a norm if and only if A is a strictly positive operator. Also, (H, ⋅ A ) is complete if and only if the range of A (R(A)) is closed in H. For T ∈ B(H), A-operator seminorm of T , denoted as T A , is defined as 

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

For T ∈ B(H), an operator R ∈ B(H) is called an A-adjoint operator of T if for every x, y ∈ H, we have ⟨T x, y⟩ A = ⟨x, Ry⟩ A , i.e., AR = T * A. By Douglas Theorem [14] , the existence of an A-adjoint operator is not guaranteed. In fact, an operator T ∈ B(H) may admit none, one or many A-adjoints. 

It is unique, and is denoted by A † . If T ∈ B A (H), the reduced solution of the equation AX = T * A is a distinguished A-adjoint operator of T, which is denoted by T # A (see [7] ).

T T # A are A-selfadjoint and A-positive operators. So,

. For further details, we refer the reader to [1, 2] . in 2012, Saddi [15] defined A−numerical radius of T, denoted as w A (T ), for T ∈ B(H) as follows

In 2019, Zamani [12] showed that if T ∈ B A (H), then

(1.

3)

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

In 2019, Moslehian et al. [8] further continued the study of A-numerical radius and established some inequalities for A-numerical radius. For a 2 × 2 operator matrix T, B-numerical radius of T is defined as

In 2019, Bhunia et al. [4] studied B-numerical radius inequalities of 2 × 2 operator matrices, where B is a 2 × 2 diagonal operator matrix whose diagonal entries are A. In this directions some authors has been studied many generalizations and refinements of Anumerical radius, for more details one can refer [13, 16, 17] . This motivates us to further study on this topic.

The objective of this paper is to present new B-numerical radius inequalities for 2 × 2 operator matrices. Further two refinements of the 1st inequality in (1.4) is addressed in this article. In this aspect, the article is structured as follows. In Section 2, we recall some upper and lower bounds for B-numerical radius inequalities for a 2 × 2 operator matrix.The next section contains our main results and is of two folds. First part establishes some upper and lower bounds for 2×2 operator matrices while the second part deals with certain refinements of (1.4).

In 2020, Pintu et al. [4] proved the following lemma for 2 × 2 operator matrices.

. Then the following results hold:

In 2019, the authors of [13] established an upper and lower bound for a 2 × 2 operator matrix.

In 2020, Feki [18] proved the following result.

Then

This section is two fold. First, we present some generalizations of A-numerical radius inequalities. Further we prove some upper and lower bounds for B-numerical radius of operator matrices. Second, we provide different refinements of A-numerical radius inequalities.

3.1. Upper and lower bounds for B-numerical radius of 2 × 2 operator matrix.

In this subsection, we establish different upper and lower bounds for B-numerical radius of a 2 × 2 block operator matrix. We start with the following lemma.

Here,

The following inequality generalizes (1.4).

Proof. By using Lemma 2.1 and Lemma 3.1

6 Therefore,

On the other hand, by using Lemma 2.1, we have

A particular case of the inequality (3.1) is the following.

We need the following lemma to prove Theorem 3.2.

Replacing T 2 by −iT 2 in the identity, we have

Then The following result demonstrates an upper bound for B-numerical radius of a 2 × 2 operator matrix.

Proof. Using similar argument as used in the previous theorem, we have

The following result is an estimate of an lower bound for B-numerical radius of a 2 × 2 operator matrix.

Proof. It follows from Lemma 3.1 that

To prove the next lemma, we need the following identities,

for any two real numbers a and b.

Then

and

Then

and

Following theorem demonstrates an upper bound for B-numerical radius of 2×2 operator matrix using (1.4) and Lemma 3.5.

where,

Proof. We know that

By using Lemma 3.5, we get

Applying the previous calculation to

Hence, we get the desired result.

Next result shows a lower bound for B-numerical radius of a 2 × 2 operator matrix in which 2nd row is zero.

Hence, we get

Further upper bound for the B-numerical radius of

is proved next using the Lemma 2.3.

Theorem 3.7. Let T 1 , T 2 ∈ B A (H).

Metric properties of projections in semi-Hilbertian spaces

Partial isometries in semi-Hilbertian spaces

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

On inequalities for A-numerical radius of operators

Generalized inverses of linear operators: representation and approximation

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

Numerical radius inequalities for operator matrices

Some extended numerical radius inequalities, Linear and Multilinear Algebra

Berezin number and numerical radius inequalities for operators on Hilbert spaces

A-Numerical radius inequalities for semi-Hilbertian space operators

Some improvements of numerical radius inequalities of operators and operator matrices

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

A-normal operators in semi Hilbertian spaces

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

Refinements of A-numerical radius inequalities and its applications

Some A-spectral radius inequalities for A-bounded Hilbert space operators

Numerical radius inequalities for 2 × 2 operator matrices

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

In particular,