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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.

In this paper, a new definition of numerical radius for adjointable operators in Hilbert-module space will be introduced. We also give a new proof of numerical radius inequalities for Hilbert space operators.

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.

In this paper, recommended spiral passive micromixer was designed and simulated. spiral design has the potential to create and strengthen the centrifugal force and the secondary flow. A series of simulations were carried out to evaluate the effects of channel width, channel depth, the gap between loops, and flowrate on the micromixer performance. These features impact the contact area of the two fluids and ultimately lead to an increment in the quality of the mixture. In this study, for the flow rate of 25 μl/min and molecular diffusion coefficient of 1×10-10 m2/s, mixing efficiency of more than 90% is achieved after 30 (approximately one-third of the total channel length). Finally, the optimized design fabricated using proposed 3D printing method.