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Norm inequalities for the difference between weighted and integral means of operator differentiable functions

Silvestru Sever Dragomir (2020)

Archivum Mathematicum

Let f be a continuous function on I and A , B 𝒮𝒜 I H , the convex set of selfadjoint operators with spectra in I . If A B and f , as an operator function, is Gateaux differentiable on [ A , B ] : = ( 1 - t ) A + t B t 0 , 1 , while p : 0 , 1 is Lebesgue integrable, then we have the inequalities 0 1 p τ f 1 - τ A + τ B d τ - 0 1 p τ d τ 0 1 f 1 - τ A + τ B d τ 0 1 τ ( 1 - τ ) | τ 1 p s d s 1 - τ - 0 τ p s d s τ | f 1 - τ A + τ B B - A d τ 1 4 0 1 | τ 1 p s d s 1 - τ - 0 τ p s d s τ | f 1 - τ A + τ B B - A d τ , where f is the Gateaux derivative of f .

Notes on some spectral radius and numerical radius inequalities

Amer Abu-Omar, Fuad Kittaneh (2015)

Studia Mathematica

We prove numerical radius inequalities for products, commutators, anticommutators, and sums of Hilbert space operators. A spectral radius inequality for sums of commuting operators is also given. Our results improve earlier well-known results.

Numerical radius inequalities for 2 × 2 operator matrices

Omar Hirzallah, Fuad Kittaneh, Khalid Shebrawi (2012)

Studia Mathematica

We derive several numerical radius inequalities for 2 × 2 operator matrices. Numerical radius inequalities for sums and products of operators are given. Applications of our inequalities are also provided.

Numerical radius inequalities for Hilbert space operators

Fuad Kittaneh (2005)

Studia Mathematica

It is shown that if A is a bounded linear operator on a complex Hilbert space, then 1/4 ||A*A + AA*|| ≤ (w(A))² ≤ 1/2 ||A*A + AA*||, where w(·) and ||·|| are the numerical radius and the usual operator norm, respectively. These inequalities lead to a considerable improvement of the well known inequalities 1/2 ||A|| ≤ w(A) ≤ || A||. Numerical radius inequalities for products and commutators of operators are also obtained.

Numerical radius inequalities for Hilbert space operators. II

Mohammad El-Haddad, Fuad Kittaneh (2007)

Studia Mathematica

We give several sharp inequalities involving powers of the numerical radii and the usual operator norms of Hilbert space operators. These inequalities, which are based on some classical convexity inequalities for nonnegative real numbers and some operator inequalities, generalize earlier numerical radius inequalities.

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