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.