Schatten p-norm and numerical radius inequalities with applications

We develop a new refinement of the Kato's inequality and using this refinement we obtain several upper bounds for the numerical radius of a bounded linear operator as well as the product of operators, which improve the well known existing bounds. Further, we obtain a necessary and sufficient condition for the positivity of 2× 2 certain block matrices and using this condition we deduce an upper bound for the numerical radius involving a contraction operator. Furthermore, we study the Schatten p-norm inequalities for the sum of two n× n complex matrices via singular values and from the inequalities we obtain the p-numerical radius and the classical numerical radius bounds. We show that for every $p>0$, the p-numerical radius wₚ(·): Mₙ( C)→ R satisfies wₚ(T) ≤ √12\| |T|²⁽¹⁻ᵗ⁾+|T^*|²⁽¹⁻ᵗ⁾ \|^ \, \||T|²ᵗ+|T^*|²ᵗ \|p/2^ for all t∈ [0,1]. Considering p→ ∞, we get a nice refinement of the well known classical numerical radius bound w(T) ≤ √12 \| T^*T+TT^* \|. As an application of the Schatten p-norm inequalities we develop a bound for the energy of graph. We show that E(G) ≥ 2m √ max1≤ i ≤ n \ ∑j, vᵢ ~ vⱼdⱼ\ , where E(G) is the energy of a simple graph G with m edges and n vertices v₁,v₂,…,vₙ such that degree of vᵢ is dᵢ for each i=1,2,…,n.