Let A be a positive bounded operator on a Hilbert space . The semi-inner product , , induces a seminorm on . Let , and denote the A-operator seminorm, the A-numerical radius, and the A-Crawford number of an operator T in the semi-Hilbertian space , 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 where is the A-adjoint operator of T. Some applications of the newly obtained inequalities are also provided.

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