Trapezoid Inequality for Operator-Valued Functions in Hilbert Spaces

Salma Aljawi, Ahad Hamoud Alotaibi, Silvestru Sever Dragomir and Kais Feki

Journal of Mathematics, 2026, vol. 2026, 1-14

Abstract: Let K;·,· denote a complex Hilbert space, and let LK represent the Banach C∗-algebra of bounded linear operators acting on K. For any operator A∈LK, the modulus is defined by A:=A∗A1/2. The primary contribution of this work is the derivation of the following significant result: Assuming ζ:δ1,δ2⟶C is an integrable function and T:δ1,δ2⟶LK is a strongly differentiable operator-valued function satisfying T′∈L2δ1,δ2,LK, we have ∫δ1δ2ζsdsTδ1+Tδ2/2−∫δ1δ2ζtTtdt2≤1/4∫δ1δ2∫δ1δ2sgns−tζsds2dt∫δ1δ2T′t2dt≤1/4δ2−δ12∫δ1δ2ζs2ds∫δ1δ2T′t2dt . Specifically, in the case where ζ≡1, we obtain the inequality δ2−δ1Tδ1+Tδ2/2−∫δ1δ2Ttdt2≤1/12δ2−δ13∫δ1δ2T′t2dt, where the constant 1/12 is proven to be sharp. Furthermore, we provide error estimates for operator quadrature rules, accompanied by applications involving the operator inverse, the exponential function, and the finite Fourier transform.

Date: 2026
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jjmath:6641185

DOI: 10.1155/jom/6641185

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