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Definition: Let V be a vector space, and let $T:V \rightarrow V$ be linear. A subspace W of V is T-invariant if $T(x) \in W$ for every $x \in W$, that is $T(W) \subseteq W$. If W is T-invariant, define the restriction of T on W to be the function $T_W: W \rightarrow W$ defined by $T_W(x)=T(x)$ for all $x \in W$.
For $T_W(x)=T(x)$, is there any meaning of the W underneath T other than to express that it is T-invariant?
1 Answer
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$T$ is in fact $T$-invariant and the subscript is there to tell you that you consider the restriction of $T$ to $W$, which should be understood as fixed, rather than an arbitrary subspace.
