Invariant Subspace Problem
The invariant subspace problem concerns the case where V is a separable Hilbert space over the complex numbers, of dimension > 1, and T is a bounded operator. The problem is to decide whether every such T has a non-trivial, closed, invariant subspace. This problem is unsolved as of 2012.
In the more general case where V is hypothesized to be a Banach space, there is an example of an operator without an invariant subspace due to Per Enflo (1976). A concrete example of an operator without an invariant subspace was produced in 1985 by Charles Read.
Read more about this topic: Invariant Subspace
Famous quotes containing the word problem:
“The problem is simply this: no one can feel like CEO of his or her life in the presence of the people who toilet trained her and spanked him when he was naughty. We may have become Masters of the Universe, accustomed to giving life and taking it away, casually ordering people into battle or out of their jobs . . . and yet we may still dirty our diapers at the sound of our mommy’s whimper or our daddy’s growl.”
—Frank Pittman (20th century)