- Thread starter
- Moderator
- #1
- Jan 26, 2012
- 995
Here's this week's problem.
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Problem: Let $V:L^2([0,1])\rightarrow L^2([0,1])$ be the operator defined by $Vf(x) = \displaystyle\int_0^x f(t)\,dt$ for all $f\in L^2([0,1])$. This is known as the Volterra operator.
(a) Show that $V$ has no nonzero eigenvalues.
(b) Compute $V^{\ast}$, the adjoint of $V$.
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Remember to read the POTW submission guidelines to find out how to submit your answers!
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Problem: Let $V:L^2([0,1])\rightarrow L^2([0,1])$ be the operator defined by $Vf(x) = \displaystyle\int_0^x f(t)\,dt$ for all $f\in L^2([0,1])$. This is known as the Volterra operator.
(a) Show that $V$ has no nonzero eigenvalues.
(b) Compute $V^{\ast}$, the adjoint of $V$.
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Remember to read the POTW submission guidelines to find out how to submit your answers!