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 \|₏/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 ₁ ₈ ₍ \{ _{₉, ₕ㶁 ₕ䲛dⱼ\} }, 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.