Abstract
We will use the author’s Two Nonzero Component Lemma to give a new proof for the Greub-Reinboldt Inequality. This method has the advantage of showing exactly when the inequality becomes equality. It also provides information about vectors for which the inequality becomes equality. Furthermore, using the Two Nonzero Component Lemma, we will generalize Greub-Reinboldt Inequality to operators on infinite dimensional separable Hilbert spaces.
Highlights
Many authors have established Kantorovich inequality and its generalizations such as Greub-Reinboldt Inequality by variational methods
One differentiates the functional involved to arrive at an “Euler Equation” and solves the Euler Equating to obtain the minimizing or maximizing vectors of the functional involved
Others have established Kantorovich-type inequalities for positive operators by going through a two-step process which consists of first computing upper bounds for suitable functions on intervals containing the spectrum of suitable matrix and applying the standard operational calculus to that matrix for an example of this method
Summary
Many authors have established Kantorovich inequality and its generalizations such as Greub-Reinboldt Inequality by variational methods. Others have established Kantorovich-type inequalities for positive operators by going through a two-step process which consists of first computing upper bounds for suitable functions on intervals containing the spectrum of suitable matrix and applying the standard operational calculus to that matrix (see [2]) for an example of this method. This method, which we refer to as “the operational calculus method”, has the following two limitations: First, it does not provide any information about vectors for which the established inequalities become equalities. In this paper we use the author’s Two Nonzero Component Lemma to prove, improve and extend matrix form of Greub-Reinboldt Inequality
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