Abstrakt: |
Let 퓗 be a complex Hilbert space and A be a non-zero positive bounded linear operator on 퓗. The main aim of this paper is to discuss a general method to develop A-operator seminorm and A-numerical radius inequalities of semi-Hilbertian space operators using the existing corresponding inequalities of bounded linear operators on 퓗. Among many other inequalities we prove that if S, T, X ∈ 퓑A(퓗), i.e., if A-adjoint of S, T, X exist, then 2 ∥ S ♯ A X T ∥ A ≤ ∥ S S ♯ A X + X T T ♯ A ∥ A. Further, we prove that if T ∈ 퓑A(퓗), then 1 4 ∥ T ♯ A T + T T ♯ A ∥ A ≤ 1 8 (∥ T + T ♯ A ∥ A 2 + ∥ T − T ♯ A ∥ A 2) ≤ 1 8 (∥ T + T ♯ A ∥ A 2 + ∥ T − T ♯ A ∥ A 2) + 1 8 c A 2 (T + T ♯ A ) + 1 8 c A 2 (T − T ♯ A ) ≤ w A 2 (T). Here wA(⋅), cA(⋅) and ∥⋅∥A denote A-numerical radius, A-Crawford number and A-operator seminorm, respectively. [ABSTRACT FROM AUTHOR] |