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Let $V$ be the vector space of all real valued continuous functions. Prove that the linear operator $\displaystyle\int_{0}^{x}f(t)dt$ has no eigenvalues.

Grobber
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2 Answers2

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Suppose that $\lambda \in \mathbb{R}^*$ is an eigenvalue of this operator associated to the eigenvector $f \neq 0$. Then, for all $x$, you have :

$$ \int_{0}^{x} f(t) \: dt = \lambda f(x) $$

with the condition that $f(0)=0$. If you differentiate the previous equality, you have $f(x) = \lambda f'(x)$ with $f(0) = 0$. You can easily solve the differential equation $f = \lambda f'$ but the only solution which satisfies $f(0)=0$ is the function $x \, \mapsto \, 0$.

pitchounet
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  • Why $f(0) = 0$ ? – Rising Star Oct 06 '15 at 10:02
  • @RisingStar : An eigenvector $f$ satisfies the relation :

    $$ \forall x \in \mathbb{R}, , \int_{0}^{x} f(t) , dt = \lambda f(x) $$

    with $\lambda \neq 0$. Evaluate this at $x=0$.

     – pitchounet Oct 06 '15 at 10:27
  • Actually why you have taken $\lambda$ to be non zero ? – Rising Star Oct 06 '15 at 10:37
  • @RisingStar : The case $\lambda = 0$ is quite straightforward. An eigenvector $f$ associated to $\lambda = 0$ would satisfy :

    $$ \forall x \in \mathbb{R}, , \int_{0}^{x} f(t) , dt = 0 $$

    As a consequence, $f \equiv 0$, which is not possible since an eigenvector is not identically zero.

     – pitchounet Oct 06 '15 at 11:31
  • what do you mean by f is similar to 0? or you meant f=0? – james black Nov 16 '20 at 03:50
  • @jamesblack: $f \equiv 0$ means that $f$ is the null function (equal to 0 everywhere). – pitchounet Nov 16 '20 at 09:05
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Hint: Suppose you have $\int_{0}^{x} f(t)dt = \lambda \cdot f(t)$. Differentiate both sides - you should easily be able to solve the resultant differential equation. Is your solution truly an eigenvector if it is nontrivial?

Alex Wertheim
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  • You can't differentiate $f$ as it is assumed to be only continuous. – Marra May 09 '16 at 21:23
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    @Marra: there is no assumption that $f$ is differentiable. The LHS is differentiable, and so too must be $\lambda f(t)$. This is a consequence of $f$ being an eigenvector. – Alex Wertheim May 09 '16 at 21:30
  • Indeed! But for that you must assume that $f$ is an eigenvector. thanks! – Marra May 09 '16 at 21:32