Suppose ##V## is a complex vector space of dimension ##n## and ##T## an operator in it. Furthermore, suppose ##v\in V##. Then I form a list of vectors in ##V##, ##(v,Tv,T^2v,\ldots,T^mv)## where ##m>n##. Due to the last inequality, the vectors in that list must be linearly dependent. This implies that the equation(adsbygoogle = window.adsbygoogle || []).push({});

$$

0=a_0v+a_1Tv+a_2T^2v+\ldots+a_mT^mv

$$

are satisfied by some nonzero coefficients. For a particular case, assume that the above equation is satisfied by some choice of coefficients where all of them are nonzero.

Now since the equation above forms a polynomial, I can write the factorized form

$$

0=A(T-\mu_1)\ldots(T-\mu_m)v

$$

The last equation suggests that ##T## has ##m## eigenvalues. But this contradicts the fact that ##T## is an operator in a vector space of dimension ##n<m##. Where is my mistake?

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# Confusion about eigenvalues of an operator

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