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By taking into account that the computation of the Numerical radius is an optimization problem, we prove, in this paper, several refinements of the Numerical radius inequalities for Hilbert space operators. It is shown, among other inequalities, that if A is a bounded linear operator on a complex Hilbert space, then ω,(A) ≤,1 2 r |A|2 + |A∗, |2 + ∥, |A| |A∗, | + |A∗, | |A|∥, , where ω,(A), ∥, A∥, , and |A| are the Numerical radius, the usual operator norm, and the absolute value of A, respectively. This inequality provides a refinement of an earlier Numerical radius inequality due to Kittaneh, namely, ω,(A) ≤,1 2 ,∥, A∥,+ A2 12 , . Some related inequalities are also discussed.

We introduce some Numerical radius inequalities for prod-ucts of two Hilbert space operators. Among other inequalities, it is shown that if S,T 2 B(H) and ST = TS , , then! (ST) , ! (S)! (T) + 1 2 DS sup , 2R Dei, T+e􀀀, i, T, , where DS = inf , 2C ∥, S 􀀀,, I∥, . Also, we show that if S,T 2 B(H) and S be self-adjointable, then! (ST) ,( 2∥, S∥,􀀀,min , 2, (S) j, j )! (T):

We present some new Numerical radius inequalities of Hilbert space operators. We improve and generalize some inequalities with respect to Specht’, s ratio. Let A and B be two positive invertible operators on a Hilbert space H and let X be a bounded operator on H. Then ω, ((A♮, B)X) ≤,1 2S( √,h) ∥, X ∗,BX + A∥, , (h > 0, h ̸, = 1) where ∥,·,∥, , ω, (·, ), S(·, ), and ♮,denote the usual operator norm, Numerical radius, the Specht’, s ratio, and the operator geometric mean, respectively.

We apply Numerical radius and spectral radius estimates to the Frobenius companion matrices of monic polynomials to derive new bounds for their zeros and give different proofs of some known bounds.