| Popis: |
Let 𝓗 be a complex Hilbert space and A be a non-zero positive bounded linear operator on 𝓗. 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 𝓗. Among many other inequalities we prove that if S, T, X ∈ 𝓑 A (𝓗), 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 . $$2\|S^{\sharp_A}XT\|_A \leq \|SS^{\sharp_A}X+XTT^{\sharp_A}\|_A.$$ Further, we prove that if T ∈ 𝓑 A (𝓗), 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 ) . $$\begin{align*} & \frac{1}{4}\|T^{\sharp_{A}}T+TT^{\sharp_{A}}\|_A\\ \leq & \frac{1}{8}\bigg( \|T+T^{\sharp_{A}}\|_A^2+\|T-T^{\sharp_{A}}\|_A^2\bigg)\\ \leq & \frac{1}{8}\bigg( \|T+T^{\sharp_{A}}\|_A^2+\|T-T^{\sharp_{A}}\|_A^2\bigg) +\frac{1}{8}c_A^2\big(T+T^{\sharp_{A}}\big)+\frac{1}{8}c_A^2\big(T-T^{\sharp_{A}}\big)\leq w^2_A(T). \end{align*}$$ Here w A (⋅), c A (⋅) and ∥⋅∥ A denote A-numerical radius, A-Crawford number and A-operator seminorm, respectively. |
