An Ostrowski Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

Year 2024, Volume: 6 Issue: 1, 1 - 14, 03.05.2024

Sever Dragomır

https://doi.org/10.47087/mjm.1362713

Cited By: 1

Abstract

Let H be a Hilbert space. Assume that f is continuously differentiable on I with ‖f′‖_{I,∞}:=sup_{t∈I}|f′(t)|<∞ and A, B are selfadjoint operators with Sp(A), Sp(B)⊂I, then

‖f((1-λ)A⊗1+λ1⊗B)-∫₀¹f((1-u)A⊗1+u1⊗B)du‖
≤‖f′‖_{I,∞}[(1/4)+(λ-(1/2))²]‖1⊗B-A⊗1‖

for λ∈[0,1]. In particular, we have the midpoint inequality

‖f(((A⊗1+1⊗B)/2))-∫₀¹f((1-u)A⊗1+u1⊗B)du‖
≤(1/4)‖f′‖_{I,∞}‖1⊗B-A⊗1‖.

Keywords

Tensorial product, Selfadjoint operators, Operator norm, Ostrowski’s inequality

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