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I am having trouble understanding $T$-cyclic subspaces. My textbook gives the following definition:

Let $T$ be a linear operator on a vector space $V$, and let $x$ be a nonzero vector in $V$. The subspace: $$W = span([x,T(x),T^2(x),...])$$ is called the $T$-cyclic subspace of $V$ generated by $x$.

I am having trouble understanding what this means and the motivation behind such a definition. Once again in linear algebra I find myself wondering: "who cares?" If anyone could provide any intuition or motivation behind this definition that would be awesome. Or even any kind of application of this definition.

Thank you!

Aweygan
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jmoore00
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    Actually, "who cares" applies already to "subspace". So if you find a subspace useful now, then a cyclic one is a very nice and special one. Otherwise really, who cares. – Dietrich Burde Mar 25 '18 at 19:02
  • One application is the cyclic decomposition Theorem, see here and here for example. – Dietrich Burde Mar 25 '18 at 19:13
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    Another application is that every bounded operator on a non-separable Banach space has an invariant subspace, thus limiting the study of the invariant subspace problem: The closure of any $T$-cyclic subspace is a proper subspace invariant under $T$. – Aweygan Mar 25 '18 at 19:14
  • There are going to be places in the book where they use the phrase "$T$-cyclic", otherwise they wouldn't give the definition. Those places in the book are applications of the concept. – David C. Ullrich Mar 25 '18 at 22:20
  • @Aweygan Proper? No. (The existence of $T$-cyclic subspaces shows that they need not be proper: Say $V$ is a $T$-cyclic subspace. Let $S$ be the restriction of $T$ to $V$. Then $V$ is an improper $S$-cyclic subspace of $V$.) – David C. Ullrich Mar 25 '18 at 23:27
  • @Aweygan I missed the non-separable, sorry. – David C. Ullrich Mar 25 '18 at 23:37

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The cyclic subspaces are very useful for one thing: they are subspaces that are invariant with respect to $T$! That means if $x\in W$, $f(T)x \in W$, where $f$ is a polynomial.

Other useful examples of T-invariant subspaces include $KerT$, $ImT$. Hope this helps!

usualwitch
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