A Trapezoid Type Tensorial Norm Inequality for Continuous Functions of Selfadjoint Operators in Hilbert Spaces

Abstract

Generalized trapezoid and trapezoid rules play an important role in approximating the Lebesgue integral in the case of scalarvalued functions defined on a finite interval. Motivated by this reason, in this paper we provided several norm error bounds in approximation the integral of continuous function of the convex combination of some tensorial products in terms of the corresponding tensorial generalized and trapezoid rules. The case of continuously differentiable functions is analysed in detail in the case when the derivative is bounded on a finite interval. Related results for the case when the absolute value of the derivative is convex is also provided. The important particular case for the operator exponential function is also considered and the corresponding norm inequalities revealed.

Keywords

• tensorial product
• selfadjoint operators
• operator norm
• trapezoid rule
• convex functions

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