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

Abstract

Let A be a positive bounded operator on a Hilbert space H, ·, · . The semi-inner product x, y A := Ax, y , x, y ∈ H, induces a seminorm · A on H. Let T A , w A (T ), and c A (T ) denote the A-operator seminorm, the A-numerical radius, and the A-Crawford number of an operator T in the semi-Hilbertian space H, · A , respectively. In this paper, we present some seminorm inequalities and equalities for semi-Hilbertian space operators. More precisely, we give some necessary and sufficient conditions for two orthogonal semi-Hilbertian operators satisfy Pythagoras' equality. In addition, we derive new upper and lower bounds for the numerical radius of operators in semi-Hilbertian spaces. In particular, we show that

Further results on A-numerical radius inequalities

“…This section begins with the power inequality for semi-Hilbert space that has been proved by Moslehian et al [ 8 ], which states that for for Using this, we prove the following theorem.…”

“…Furthermore, if T is A -selfadjoint, then . Moslehian et al [ 8 ] continued the study of A -numerical radius and obtained some new A -numerical radius inequalities. In this year, Bhunia et al [ 3 ] presented several -numerical radius inequalities for a strictly positive operator A .…”

On A-numerical radius inequalities for 2 x 2 operator matrices-II

“…A fundamental inequality for the A-numerical radius is the power inequality (see [20]) which says that for…”

“…Furthermore, if T is A-selfadjoint, then w A T T A . In 2019, Moslehian et al [20] 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 [5,6,22,29].…”

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

“…Zamani [ 42 ] developed a new formula for computing the numerical radius of : The inequality ( 2 ) is also studied using A -numerical radius of T , and is given as Furthermore, if T is A -selfadjoint, then . 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.…”