The Shift Operators

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is the intersection of all subspaces containing S. It is obvious that ! S is always a subspace. Definition . If A is an operator and M is a subspace, we say that M is an invariant subspace of A if AM M. That is, M is invariant under A if f M implies Af M. The trivial subspaces, {0} and H, are invariant under every operator. One of the most famous unsolved problems in analysis (the invariant subspace problem) is the question whether every bounded linear operator on an innite- dimensional Hilbert space has a nontrivial invariant subspace. Notation . If M is an invariant subspace of the operator A, then A
is the restriction of the operator A to M. Definition . Given a vector f and a bounded linear operator A, the invariant subspace generated by f is the...

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Published On:
04/19/2021
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04/19/2021
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