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

Let $A$ be a positive bounded operator on a Hilbert space $\big($\mathcal{H}$, \langle \cdot, \cdot\rangle \big)$\mathrm{.}Thesemi-innerproduct${\langle x, y\rangle}A := \langle Ax, y\rangle$,$x, y\in$\mathcal{H}$,$inducesaseminorm${|\cdot|}A$on$$\mathcal{H}$$\mathrm{.}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 $\big($\mathcal{H}$, {|\cdot|}A\big)$, 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 \begin{align*} \frac{1}{16} {|TT^{\sharp{A}} + T^{\sharp{A}}T|}^{2}{A} + \frac{1}{16}c{A}\Big(\big(T^2 + (T^{\sharp_{A}})^2\big)^2\Big) \leq w^4_{A}(T) \leq \frac{1}{8} {|TT^{\sharp_{A}} + T^{\sharp_{A}}T|}^{2}{A} + \frac{1}{2}w^2{A}(T^2), \end{align*} where $T^{{\mathrm{♯}}_A}$ is a distinguished $A$-adjoint operator of $T$. Some applications of our inequalities are also provided.