## Halmos Sections 39 and 40: Invariance and Reducibility

**Point of post: **In this post we complete the problems at the end of sections 39 and 40 in Halmos.

1.

**Problem: **Find a finite dimensional vector space and an endomorphism which leaves only and invariant.

**Proof: **Take and the transformation identified with the matrix . So, let . Then, since and we know that . So, let . Then, for to leave invariant we must have that

for some . Note though that

Comparing this though we see that if there existed such a then

and

Or,

But, this then implies that or that and thus . But, then this implies that contradictory to assumption.

2.

**Problem:** Prove that the differentiation operator leaves invariant the subspaces for every . Is there a complement of which reduces ?

**Proof: **The first part follows immediately since . For the second part we merely need notice that

3.

**Problem: **Prove that if is an -space and and is such that are invariant under prove that is invariant under .

**Proof: **This follows immediately since

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