Invariant and Reducing Subspaces

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32 1 Introduction

. Assume that there is a bounded operator A with the following property: there exist subspaces N and M invariant under A such that N M, the dimension of M N is greater than 1, and the only subspaces L invariant under A that satisfy N L M are L N and L M. Show that this assumption implies the existence of an operator B on a Hilbert space whose only invariant subspaces are {0} and the entire space.

. Show that the operator A has a nontrivial invariant subspace if and only if the operator equation XAX AX has a solution other than zero and the identity.

. Show that every operator of nite rank can be written in the form

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04/19/2021
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invariant-and-reducing-subspaces-75.pdf
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