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Thread: Prove T has infinitely many LI eigenvectors

  1. #1
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    Prove T has infinitely many LI eigenvectors

    Let $\displaystyle X$ be an infinite-dimensional complex Banach space and $\displaystyle T:X\to X$ a continuous/bounded linear operator. Suppose that for every nonzero $\displaystyle e\in X$, the "orbit space" $\displaystyle \mathcal{O}_T(e):=\text{span}(T^ne)_{n=0}^\infty$ is finite-dimensional, say has dimension $\displaystyle 1\leq d(e)<\infty$.

    I would like to prove or disprove the following conjecture:

    Conjecture 1: $\displaystyle T$ has infinitely many linearly independent eigenvectors.

    Conjecture 1 easily follows from the following:

    Conjecture 2: There is a sequence $\displaystyle (e_n)_{n=0}^\infty$ of nonzero vectors in $\displaystyle X$ such that $\displaystyle \mathcal{O}_T(e_{N+1})\cap(\mathcal{O}_T(e_0)+ \cdots +\mathcal{O}_T(e_N))=\{0\}$ for all $\displaystyle N$.

    Proof that Conjecture 2 implies Conjecture 1: First we claim that for each nonzero $\displaystyle e\in X$, $\displaystyle T$ has an eigenvector in $\displaystyle \mathcal{O}_T(e)$. For proof, notice that since $\displaystyle O_T(e)$ is $\displaystyle T$-invariant then the restriction $\displaystyle T|_{\mathcal{O}_T(e)}$ of $\displaystyle T$ to $\displaystyle \mathcal{O}_T(e)$ is a linear operator on a finite-dimensional vector space. Being defined on a nonzero but finite-dimensional vector space, this means $\displaystyle T|_{\mathcal{O}_T(e)}$ has eigenvalues. Thus we can find an eigenvector $\displaystyle v\in\mathcal{O}_T(e)$ under $\displaystyle T|_{\mathcal{O}_T(e)}$, and the claim is proved. In particular, for each $\displaystyle n$ we can find an eigenvector $\displaystyle v_n\in\mathcal{O}_T(e_n)$. Next we claim that the resulting sequence $\displaystyle (v_n)_{n=0}^\infty$ is linearly independent. For suppose otherwise, towards a contradiction. Then we can find $\displaystyle a_0,\cdots,a_{N+1}\in\mathbb{C}$, not all zero, such that $\displaystyle 0=\sum_{n=0}^{N+1} a_nv_n$. Without loss of generality suppose $\displaystyle a_{N+1}=-1$. Then $\displaystyle v_{N+1}=\sum_{n=0}^Na_nv_n\in\mathcal{O}_T(e_0)+ \cdots +\mathcal{O}_T(e_N)$. However $\displaystyle v_{N+1}\in\mathcal{O}_T(e_{N+1})$, contradicting the hypothesis that $\displaystyle \mathcal{O}_T(e_{N+1})\cap(\mathcal{O}_T(e_0)+ \cdots +\mathcal{O}_T(e_N))=\{0\}$. $\displaystyle \square$

    So if anyone can help me prove either conjecture, that would be great. Thanks!
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  2. #2
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    Re: Prove T has infinitely many LI eigenvectors

    Hey hatsoff.

    Does the operator have to be Hermitian? (Forgive me, it's been a while since I looked at this stuff)?
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  3. #3
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    Re: Prove T has infinitely many LI eigenvectors

    Quote Originally Posted by chiro View Post
    Hey hatsoff.

    Does the operator have to be Hermitian? (Forgive me, it's been a while since I looked at this stuff)?
    Unfortunately no---I must prove this for a diverse class of operators on non-Hilbert Banach spaces. However I might be able to assume it has a finite spectrum. For what it's worth, I can also assume $\displaystyle X$ is not reflexive. (The reflexive case doesn't really need this result.)
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  4. #4
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    Re: Prove T has infinitely many LI eigenvectors

    I don't think I can help you out on this one: This is a lot more general than the stuff I've been exposed to.
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