Inequalities for the A-joint numerical radius of two operators and their applications

Let (3-C, (center dot, center dot)) be a complex Hilbert space and A be a positive (semidefinite) bounded linear operator on 3-C. The semi -inner product induced by A is given by (x, y)A := (Ax, y), x, y E 3-C and defines a seminorm II center dot IIA on 3-C. This makes 3-C into a semi -Hilbert space. The A -joint numerical radius of two A -bounded operators T and S is given by omega A,e(T, S) = sup IIxIIA=1 ✓⠌⠌(T x, x)A ⠌⠌2 + ⠌⠌(Sx, x)A ⠌⠌2. In this paper, we aim to prove several bounds involving omega A,e(T, S). This allows us to establish some inequalities for the A -numerical radius of A -bounded operators. In particular, we extend the well-known inequalities due to Kittaneh [Numerical radius inequalities for Hilbert space operators, Studia Math. 168 (1), 73-80, 2005]. Moreover, several bounds related to the A-Davis-Wielandt radius of semi -Hilbert space operators are also provided.