Bessel-type inequalities in semi-Hilbert spaces with applications to operator tuples

Consider a non-zero positive operator A on a complex Hilbert space $(\mathfrak{H}, \langle \cdot , \cdot \rangle )$. This operator induces an A-semi-inner product defined by $(u \mid v)_{\mathbf{A}} := \langle \mathbf{A}u, v \rangle $. The space $(\mathfrak{H}, \|\cdot \|_{\mathbf{A}})$ then becomes a semi-Hilbert space, where $\|\cdot \|_{\mathbf{A}}$ is the seminorm generated by this A-semi-inner product. The primary focus of this work is to establish novel additive bounds for Bessel’s inequality within the framework of semi-Hilbert spaces. Furthermore, we explore applications of these new bounds to several A-seminorms that are associated with n-tuples of operators.