# Abstract Nonsense

## Reducibility and the Fundamental Theorem of Reducible Subspaces

Point of post: In this post we discuss the concepts covered in section 40 of Halmos, but extend it further to discuss reducibility by arbitrary (finite) number of subspaces. Moreover, we discuss what is sometimes called that ‘Fundamental Theorem of Reducible Subspaces’.

Motivation

In our last post we discussed the concepts of invariant subspaces, and some characterizations of them. In this post we discuss a stronger version of invariant subspaces and show how a reducible endomorphism can be canonically decomposed into the direct sum of other endomorphisms. In essence, a reducible transformation is as nice as one can reasonably expect a endomorphism to be.

Reducible Transformations

Let $\mathscr{V}$ be an $F$-space. Then, for $T\in\text{End}\left(\mathscr{V}\right)$ we call $T$ $m$-reducible if there exists $m$ subspaces $\mathscr{W}_1,\cdots,\mathscr{W}_m\leqslant\mathscr{V}$ such that

$\displaystyle \mathscr{V}=\bigoplus_{k=1}^{m}\mathscr{W}_k$

and $\mathscr{W}_k$ is invariant under $T$ for $k\in[m]$. Alternatively ,we say that $T$ is $m$-reduced by $\mathscr{W}_1,\cdots,\mathscr{W}_m$.

Considering our last few posts on the direct sums of matrices we have relatively little work to do now, except to notice that:

Theorem: Let $\mathscr{V}$ be an $F$ space and suppose that $T\in\text{End}\left(\mathscr{V}\right)$ is $m$-reduced by $\mathscr{W}_1,\cdots,\mathscr{W}_m$. Then,

$\displaystyle T=\bigoplus_{k=1}^{m}T_{\mid\mathscr{W}_k}$

Proof: We evidently have, by prior theorem that $T_{\mid\mathscr{W}_k}\in\text{End}\left(\mathscr{W}_k\right),\text{ }k\in[m]$ and the rest follows by noticing that

$\displaystyle T\left(\sum_{k=1}^{m}w_k\right)=\sum_{k=1}^{m}T(w_k)=\sum_{k=1}^{m}T_{\mid\mathscr{W}_k}(w_k)$

from where the conclusion follows. $\blacksquare$

References:

1. Halmos, Paul R.  Finite-dimensional Vector Spaces,. New York: Springer-Verlag, 1974. Print