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Estimates for the norm of the \(n\)th indefinite integral. (English) Zbl 0931.47032

A Volterra operator \[ Tf(x)= \int^x_0 f(t) dt \] is considered, where \(f\in L^2[0,1]\), \(0\leq x\leq 1\).
Recently, D. Kersaw [Integral Equations Appl., to appear] has proved the following property of the norm of \(T\) in \(L^2(0, 1)\) \[ \lim_{n\to\infty}\|n! T^n\|={1\over 2}.\tag{1} \] The authors present a short and direct proof of the inequality \[ \lim_{n\to\infty}\inf\|n! T^n\|\geq {1\over 2}.\tag{2} \] The inequality \[ \lim_{n\to\infty}\sup\|n! T^n\|\leq{1\over 2} \] provides the easer part of the proof of (1).

MSC:

47G10 Integral operators
45D05 Volterra integral equations
45P05 Integral operators
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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