Parametric extensions of Shannon inequality and its reverse one in Hilbert space operators Original Research Article

We shall state the following parametric extensions of Shannon inequality and its reverse one in Hilbert space operators. Let pset membership, variant[0,1] and also let {A1,A2,…,An} and {B1,B2,…,Bn} be two sequences of strictly positive operators on a Hilbert space H such that ∑j=1nAjmusical sharppBjless-than-or-equals, slantI. Thenimage∑j=1nSp+1(AjBj)greater-or-equal, slanted × loggreater-or-equal, slantedloggreater-or-equal, slanted∑j=1nSp(AjBj)greater-or-equal, slanted−loggreater-or-equal, slanted− × loggreater-or-equal, slanted∑j=1nSp−1(AjBj),where image for A>0, B>0 and any real number q and image for A>0, B>0 and any real number q.