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

Consider a non-zero positive operator A on a complex Hilbert space (H, · , · ). This operator induces an A-semi-inner product defined by (u v)A := Au, v. The space (H, \|· \|A) then becomes a semi-Hilbert space, where \|· \|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.