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Let ( H , 〈 · , · 〉) be a complex Hilbert space and  A  be a positive bounded linear operator on  H . The semi-inner product 〈 x ,  y 〉A: = 〈 Ax ,  y 〉,  x ,  y  ∈  H , induces a semi-normon  H . Let  ωA( T ) anddenote the  A -numerical radius and the  A -operator semi-norm of an operator  T  in semi-Hilbertian space ( H , 〈 · , · 〉A), respectively. In this paper, some new bounds for the  A -numerical radius of operators in semi-Hilbertian space are obtained, which improve the existing ones. In particular, a refinement of the triangle inequality for  A -operator semi-norm is also shown.

A -numerical radius and A -norm inequalities for semi-Hilbertian space operators