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