Integral operator's norm on space with 1-norm

Asked Nov 19 '22 at 22:40

Active Nov 19 '22 at 22:40

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I'm trying to find the norm of this operator:

$$\begin{aligned} T:(C[0,1], &||\cdot||_1) &&\rightarrow \quad (C[0,1], ||\cdot||_1) \\ &f &&\mapsto \quad Tf(t):= \int_0^t f(s)ds \end{aligned}$$

And here's my current progress: $$\begin{aligned} ||Tf||_1 = \int_0^1 \left| \int_0^tf(s)ds\right|dt &\leq \int_0^1 \left( \int_0^t|f(s)|ds\right) dt \\ &\leq \int_0^1 ||f||_1 dt =||f||_1. 1 \end{aligned}$$

I'm having difficulties to find a function such that $||Tf||_1 = ||f||_1.1$

Any guidance is appreciated, thanks!

asked Nov 19 '22 at 22:40

Alper

Be impulsive, subject to continuity constraints. – copper.hat Nov 19 '22 at 22:44
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You can't find a function for which the bound is attained, but you can come arbitrarily close. – copper.hat Nov 19 '22 at 22:51
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This should help $$\int\limits_0^1 \left( \int\limits_0^t|f(s)|ds\right) dt
=\int\limits_0^1 (1-s)|f(s)|,ds$$ – Ryszard Szwarc Nov 20 '22 at 00:26
See here: https://math.stackexchange.com/questions/151425/spectrum-of-indefinite-integral-operators – Math1000 Nov 20 '22 at 15:03

Integral operator's norm on space with 1-norm

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