Let A be a Banach algebra and let p and q be two positive integers. We show that if
A has a left bounded sequential approximate identity (en)n⩾1 such that\slim, infn→+∞|epn‐e{p+q}n| ⩽ (p/p+q)p/qq/p+q} then
A has a left‐bounded sequential identity (fn){n⩾1} such that f2n = fn for n⩾1. A simple example shows that the constant (p/p+q) p/q q/p+q is best possible.
This result is based on some algebraic or integral formulae which associate an idempotent to elements of a Banach algebra satisfying some inequalities involving polynomials or entire functions.
In this work, we propose an algorithm for finding an approximate global minimum of a concave quadratic function with a negative semi-definite matrix, subject to linear equality and inequality constraints, where the variables are bounded with finite or infinite bounds. The proposed algorithm starts with an initial extreme point, then it moves from the current extreme point to a new one with a better objective function value. The passage from one basic feasible solution to a new one is done by the construction of certain approximation sets and solving a sequence of linear programming problems. In order to compare our algorithm with the existing approaches, we have developed an implementation with MATLAB and conducted numerical experiments on numerous collections of test problems. The obtained numerical results show the accuracy and the efficiency of our approach.
Let A be a Banach algebra which does not contain any nonzero idempotent element, let γ > 0, and letWe also show, assuming a suitable spectral condition on x, that if x ≥ 1 − 1 (γ + 1)
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