key: cord-0065942-74me7kqg
authors: Kittaneh, Fuad; Sahoo, Satyajit
title: On [Formula: see text] -numerical radius equalities and inequalities for certain operator matrices
date: 2021-07-21
journal: Ann
DOI: 10.1007/s43034-021-00137-6
sha: 376d081527398a5623d8b82aa1864a662fcbfc8b
doc_id: 65942
cord_uid: 74me7kqg

The main goal of this article is to establish several new [Formula: see text] -numerical radius equalities for [Formula: see text] circulant, skew circulant, imaginary circulant, imaginary skew circulant, tridiagonal, and anti-tridiagonal operator matrices, where [Formula: see text] is the [Formula: see text] diagonal operator matrix whose diagonal entries are positive bounded operator A. Some special cases of our results lead to the results of earlier works in the literature, which shows that our results are more general. Further, some pinching type [Formula: see text] -numerical radius inequalities for [Formula: see text] block operator matrices are given. Some equality conditions are also given. We also provide a concluding section, which may lead to several new problems in this area.

The operator matrices such as circulant, reverse circulant, symmetric circulant, k-circulant, Toeplitz matrices etc. [13, 24] play a crucial role in pure as well as applied mathematical researches such as graph theory, image processing, block filtering design, signal processing, regular polygon solutions, encoding, control and system theory, network, etc. The norm estimation for the operator matrices [6, 27] is extensively carried out in the past and it is widely used in operator theory, quantum information theory, mathematical physics, numerical analysis, etc. The norms of some If T i ∈ B(H), i = 1, … , n , then the circulant operator matrix circ = circ (T 1 , … , T n ) is the n × n matrix whose first row has entries T 1 , … , T n and the other rows are obtained by successive cyclic permutations of these entries, i.e.,

The skew circulant operator matrix scirc = scirc (T 1 , … , T n ) is the n × n circulant matrix followed by a change in sign to all the elements below the main diagonal. Thus,

. It is well-known that every skew circulant operator matrix is unitarily equivalent to a circulant operator matrix. Details discussion on circulant, skew-circulant and their properties are given in [13] .

If T i ∈ B(H), i = 1, … , n , then the imaginary circulant operator matrix circ i = circ i (T 1 , … , T n ) is the n × n matrix whose first row has entries T 1 , … , T n and the other rows are obtained by successive cyclic permutations of i-multiplies of these entries, i.e., circ i =

. Every imaginary circulant operator matrix is unitarily equivalent to a circulant operator matrix. The imaginary skew circulant operator matrix scirc i = scirc i (T 1 , … , T n ) is the n × n imaginary circulant followed by a change in sign to all the elements below the main diagonal.

Here L is the norm closure of the linear subspace L in the norm topology of H , P M is the orthogonal projection onto the closed linear subspace M of H , I is identity operator and O is the null operator on H , respectively. For any A ∈ B(H) , the range, null space and adjoint of A are denoted by R(A) , N(A) and A * , respectively. An 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 ≥ O ( A > O ). Throughout this paper, we assume that A ∈ B(H) is a positive operator, and ∈ B( ⨁ n i=1 H) is an n × n diagonal operator matrix whose diagonal entries are positive operator A. Then, any such A defines a positive semidefinite sesquilinear form: 

An operator X ∈ B(H) is called an A-adjoint operator of T ∈ B(H) if ⟨Tx, y⟩ A = ⟨x, Xy⟩ A for every x, y ∈ H, i.e., if AX = T * A. By [14, 29] , the existence of an A-adjoint operator is not guaranteed. An operator T ∈ B(H) can have none, one or many A-adjoints. Extensive studies on different generalizations, refinements and applications of numerical radius inequalities have been conducted [3, 21, 22, 30, [38] [39] [40] . Saddi [36] introduced the A-numerical radius of T for T ∈ B(H) , which is denoted as w A (T) , and is defined as follows: [42] developed a new formula for computing the numerical radius of T ∈ B A (H):

The inequality (2) is also studied using A-numerical radius of T, and is given as

Moslehian et al. [32] pursued the study of A-numerical radius and established some A-numerical radius inequalities. Bhunia et al. [11] obtained several -numerical radius inequalities. Further generalizations and refinements of A-numerical radius inequalities are discussed in [8, 9, 15, 34] . Many studies on -numerical radius inequalities are given in [15-20, 35, 37, 41] . In this aspect, the rest of the paper is organized as follows. Inspired by the work of Bani-Domi and Kittaneh [4] , we establish certain -numerical radius equalities for circulant, skew circulant, imaginary circulant, and imaginary skew circulant operator matrices in Sect. 2. Some special cases of our result have been given in this section. In Sect. 3, we apply these -numerical radius equalities to obtain pinching type -numerical radius inequalities for n × n block operator matrices. Some equality condition are also given. In Sect. 4, we extend some recent results of Bani-Domi et al. [5] to the semi-Hilbert space operators. In particular, we obtain certain -numerical radius equalities and pinching type inequalities for n × n tridiagonal and anti-tridiagonal operator matrices. Finally, we end up with a conclusion section, which may spark new problems for future research interest.

We need the following lemmas to prove our results. The first lemma is already proved by Bhunia et al. [11] for the case strictly positive operator A. Very recently the same result proved by Rout et al. [35] by dropping the assumption A is strictly positive is stated next for our purpose. For usual numerical radius versions of Lemmas 1.1-1.3, one may consult [4, 21] .

The following lemma was proved by Rout et al. [35] .

Part (i) of Lemma 1.1 can be generalized as follows.

The aim of this section is to discuss certain -numerical radius equalities for circulant, skew circulant, imaginary circulant, and imaginary skew circulant operator matrices. The very first result is a formula for the -numerical radius of a circulant operator matrix.

It can be observed that ̄= n−1 ,̄2 = n−2 , ⋯ ,̄k = n−k , k = 0, 1, … , n − 1 ,

So, ‖ x‖ = ‖x‖ . Similarly, it can be proved that ‖ # x‖ = ‖x‖ . Thus, is anunitary operator. Now, using Lemma 1.4 we have,

Using the fact that w ( ) = w ( # ) for any ∈ B A (H), we get where the last equality follows from Lemma 1.5. ◻

As a special case of Theorem 2.1, we have part (iv) of Lemma 1.1. Our next result is an estimate for -numerical radius of skew circulant operator matrices.

Proof The n roots of the equation z n = −1 are , , 2 , … , n−1 . Let

and Using a similar argument as used in the Theorem 2.1, we can show that is -unitary. Now, using Lemma 1.4, we have

Using the property w ( ) = w ( # ) for any ∈ B A (H) , we get where the last equality follows from Lemma 1.5. ◻

As a special case of the above theorem we have the following corollary which is already proved in [35] .

Theorem 2.4 provides -numerical radius equalities for imaginary circulant operator matrices. 

and Using a similar argument as used in the Theorem 2.1, we can show that is -unitary. Now, using Lemma 1.4, we have

Using the property w ( ) = w ( # ) for any ∈ B A (H) , we get where the last equality follows from Lemma 1.5. ◻

As a special case of the above theorem we have the following corollary.

In Theorem 2.6, we give an estimate for imaginary skew circulant operator matrices. As a special case of the above theorem, we have the following corollary.

The pinching type inequalities are among the most inequalities of operator matrices. Very recently, Rout et al. [35] established some pinching type -numerical radius inequalities (see Lemma 1.3) . For usual pinching type numerical radius inequalities, one may consult [21] . Our goal of this section is to establish certain pinching type -numerical radius inequalities for n × n block operator matrices. 

Similarly, it can be proved that ‖ # 2,n+2 x‖ = ‖x‖ . Thus, 2,n+2 is an -unitary operator. Similarly, it can be shown that other operator matrices k+1,k+n+1 are -unitary operators for all k = 1, 2, … , n.

Thus, using Lemma 1.4, we get

Similarly,

. Then, using the proof of Theo-

jj+i−1 , with T n+i = T i (we could say that the subscripts are modulo n), i = 1, 2, … , n and = e 2 i∕n .

# circ

Using the property w ( # ) = w ( ) and triangle inequality, we get

As a special case of the above corollary, we have

We remark here that the -numerical radius inequality in Theorem 3.1 is sharp. Using the property w ( # ) = w ( ) and triangle inequality, we get

T n+j = T j (we could say that the subscripts are modulo n), j = 1, … , n,

T n+j = T j (we could say that the subscripts are modulo n), j = 1, … , n.

As a special case of the above corollary we have

We remark here that the -numerical radius inequality in the Theorem 3.5 is sharp.

If is an imaginary circulant operator matrix, then the inequality in Theorem 3.5 becomes equality. So, the -numerical radius inequality in Theorem 3.5 is sharp.

Based on Theorem 2.6, one can employ a similar argument as used in Theorem 3.5 to obtain a pinching type inequality analogous to that in Theorem 3.5.

In this section, inspired by the work of Bani-Domi et al. [5] , we extend some recent results of [5] to semi-Hilbert space. Using a similar analysis as used in the previous section, one can employ certain -numerical radius equalities for n × n tridiagonal and anti-tridiagonal operator matrices. We present here the results without proofs. 

be an n × n tridiagonal operator matrix, and let = e Finally, at the end of this section, we establish certain pinching inequalities for n × n tridiagonal and anti-tridiagonal operator matrices. 

be an n × n tridiagonal operator matrix. Then w ( ) = max w A T 1 + 2 cos k n + 1 T 2 ∶ k = 1, … , n .

w ( ) = max w A (−1) k+1 T 1 + 2 cos k n + 1 T 2 ∶ k = 1, … , n .

max w A T 1 + cos k n + 1 (T 2 + T 3 ) ∶ k = 1, … , n ≤ w ( ). 

In this paper, we have presented different -numerical radius equalities and inequalities, which depend on the nice structure of circulant, skew circulant, imaginary circulant, imaginary skew circulant, tridiagonal, and anti-tridiagonal operator matrices. By employing similar analysis to different special operator matrices, it is possible to obtain further -numerical radius equalities and inequalities. We now conclude the article by remarking that further study on this topic may develop an interesting area for future research.

Partial isometries in semi-Hilbertian spaces

A norm compression inequality for block partitioned positive semidefinite matrices

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

Norm equalities and inequalities for operator matrices

New norm equalities and inequalities for certain operator matrices

Norm inequalities for partitioned operators and an application

Pinchings and norms of block scaled triangular 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

Spectral norm of circulant-type matrices

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

Some -numerical radius inequalities for d × d operator matrices

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

Some numerical radius inequalities for semi-Hilbertian space operators

Generalized numerical radius inequalities of operators in Hilbert spaces

A note on the A-numerical radius of operators in semi-Hilbert spaces

Spectral radius of semi-Hilbertian space operators and its applications

Numerical radius inequalities for certain 2 × 2 operator matrices

Numerical radius inequalities for 2 × 2 operator matrices

Norm estimates of -circulant operator matrices and isomorphic operators for -circulant algebra

Inequalities for trace norms of 2 × 2 block matrices

New trace norm inequalities for 2 × 2 blocks of diagonal matrices

On Clarkson-McCarthy inequalities for n-tuples of operators

Determinants, norms, and the spread of circulant matrices with Tribonacci and Generalized Lucas numbers

Douglas factorization theorem revisited

Inequalities for operator space numerical radius of 2 × 2 block 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 -numerical radius inequalities for 2 × 2 operator matrices

A-normal operators in semi Hilbertian spaces

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

Numerical radius inequalities for operator matrices

Some extended numerical radius inequalities. Linear Multilinear Algebra

Berezin number and numerical radius inequalities for operators on Hilbert spaces

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

A-Numerical radius inequalities for semi-Hilbertian space operators

The second author is thankful to the Government of India for introducing the work from home initiative during the COVID-19 pandemic.

be an n × n tridiagonal operator matrix. Then Remark 4.2 By setting = I in Theorem 4.1, we get a recent result proved by Bani-Domi et al. [5] .Some special cases of Theorem 4.1 are described in the following table.Results on -numerical radius Results on usual numerical radiusIn particular, w( ) = w(T 1 ) see [5] .see [5] .If T 2 = iT 1 in Theorem 4.1, we havew( ) = max w 1 + 2 cos k n+1 i T 1 ∶ 1 ≤ k ≤ n , see [5] .be an n × n tridiagonal operator matrix, and let = e i 2n . Then w ( ) = max w A T 1 + 2 cos k n + 1 T 2 ∶ k = 1, … , n .w ( ) = max w A T 1 + 2 n−1 cos k n + 1 T 2 ∶ k = 1, … , n .