We give refinements of the classical Young inequality for positive real numbers and we use these refinements to establish improved Young and Heinz inequalities for matrices.
Abstract. It is shown that ifA is a bounded linear operator on a complex Hilbert space, thenwhere w(A) and A are the numerical radius and the usual operator norm of A, respectively. An application of this inequality is given to obtain a new estimate for the numerical radius of the Frobenius companion matrix. Bounds for the zeros of polynomials are also given.
Abstract. It is shown that ifA is a bounded linear operator on a complex Hilbert space, then 1 4where w(·) and · are the numerical radius and the usual operator norm, respectively. These inequalities lead to a considerable improvement of the well known inequalitiesNumerical radius inequalities for products and commutators of operators are also obtained.
We give reverses of the classical Young inequality for positive real numbers and we use these to establish reverse Young and Heinz inequalities for matrices.
Several inequalities for Hilbert space operators are extended. These include results of Furuta, Halmos, and Kato on the mixed Schwarz inequality, the generalized Reid inequality as proved by Halmos and a classical inequality in the theory of compact non-self-adjoint operators which is essentially due to Weyl. Some related inequalities are also discussed.
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