A Note on Lower and Upper Bounds of Matrices
نویسنده
چکیده
k=1 |ak| . Hardy’s inequality can be interpreted as the lp operator norm of the Cesàro matrix C, given by cj,k = 1/j, k ≤ j, and 0 otherwise, is bounded on l p and has norm ≤ p/(p − 1) (The norm is in fact p/(p − 1)). It is known that the Cesàro operator is not bounded below, or the converse of inequality (1.1) does not hold for any positive constant. However, if one assumes C acting only on non-increasing non-negative sequences in lp, then such a lower bound does exist, and this is first obtained by Lyons in [15] for the case of l2 with the best possible constant. For the general case concerning the lower bounds for an arbitrary non-negative matrix acting on non-increasing non-negative sequences in lp when p ≥ 1, Bennett [2] determined the best possible constant. When 0 < p ≤ 1, one can also consider a dual question and this has been studied in [3], [6] and [4]. Let A = (aj,k), 1 ≤ j ≤ m, 1 ≤ k ≤ n with aj,k ≥ 0, we can summarize the main results in this area in the following Theorem 1.1 ([2, Theorem 2], [4, Theorem 4]). Let x = (x1, . . . , xn), x1 ≥ . . . ≥ xn ≥ 0, p ≥ 1, 0 < q ≤ p, then (1.2) ||Ax||q ≥ λ||x||p, where ||Ax||q = m
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تاریخ انتشار 2009