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

Authors : Sever Dragomır

Pages : 1-14

Doi:10.47087/mjm.1362713

View : 73 | Download : 112

Publication Date : 2024-05-03

Article Type : Research Paper

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, Ostrowskis inequality