On inequalities satisfied by tuples of operators in semi-Hilbert spaces

In this paper, we study $(\alpha ,\beta )$-A-normal tuples of operators acting on semi-Hilbertian spaces, that is, Hilbert-like spaces endowed with a positive bounded operator A inducing a semi-inner product. By exploiting the geometric structure associated with A, we establish several operator inequalities and norm estimates that characterize this class of operator tuples. An A-characterization of $(\alpha ,\beta )$-A-normal tuples is obtained, and their stability properties are investigated. In particular, we show that this class is stable under the A-adjoint, invariant under similarity transformations induced by A-unitary operators, and stable under sums and products under suitable conditions. These results extend a number of classical inequalities from the Hilbert space setting to the semi-Hilbertian framework and contribute to the development of multivariable operator inequalities, with potential applications to joint spectral theory and numerical radius estimates.