Refinements of A-numerical radius inequalities and their applications

Abstract

We present sharp lower bounds for the A-numerical radius of semi-Hilbertian space operators. We also present an upper bound. Further we compute new upper bounds for the B-numerical radius of 2 × 2 operator matrices where B = diag(A, A), A being a positive operator. As an application of the A-numerical radius inequalities, we obtain a bound for the zeros of a polynomial which is quite a bit improvement of some famous existing bounds for the zeros of polynomials.

On $${\mathbb {A}}$$-numerical radius equalities and inequalities for certain operator matrices

“…Then Remark 4. 9 The inequalities in Theorems 4.7 and 4.8 become equalities if T 2 = T 3 . So, the -numerical radius inequalities in Theorems 4.7 and 4.8 are sharp.…”

“…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 ].…”

On Numerical Radius Bounds Involving Generalized Aluthge Transform

“…For example, upper and lower bounds are utilized to define the operator norm, which plays significantly in solving related problems. The study of the numerical radius of an operator defined on the Hilbert space is in the focus of researchers in these days in studying perturbation, convergence, iterative solution methods, and integrative methods, etc, see [1][2][3][4][5][6][7][8][9]. In this regard, the numerical radius inequality stated in (3) is studied extensively by various mathematicians, see [10][11][12][13][14][15][16][17][18][19][20][21].…”

“…The study of the numerical radius of an operator defined on the Hilbert space is in the focus of researchers in these days in studying perturbation, convergence, iterative solution methods, and integrative methods, etc, see [1][2][3][4][5][6][7][8][9]. In this regard, the numerical radius inequality stated in (3) is studied extensively by various mathematicians, see [10][11][12][13][14][15][16][17][18][19][20][21]. Actually, it is interesting for the researchers to get refinements and generalizations of this inequality [22][23][24][25][26][27].…”

“…If S 2 = 0, then the first inequality becomes equality and if S is normal then the second inequality becomes equality. Many authors worked on numerical radius inequalities and developed a number of numerical radius bounds [10,[13][14][15][16][17][18].…”