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The following theorem is
proved.
Let T be a bounded linear operator on an infinite-dimensional Hilbert
space H over the complex numbers and let p(z)≠0 be a polynomial with
complex coefficients such that p(T) is completely continuous (compact). Then T
leaves invariant at least one closed linear subspace of H other than H or
{0}.
For p(z) = z2 this settles a problem raised by P. R. Halmos and K. T.
Smith.
The proof is within the framework of Nonstandard Analysis. That is to say, we
associate with the Hilbert space H (which, ruling out trivial cases, may be
supposed separable) a larger space, ∗H, which has the same formal properties
within a language L. L is a higher order language but ∗H still exists if we
interpret the sentences of L in the sense of Henkin. The system of natural
numbers which is associated with ∗H is a nonstandard model of arithmetic,
i.e., it contains elements other than the standard natural numbers. The
problem is solved by reducing it to the consideration of invariant subspaces in a
subspace of ∗H the number of whose dimensions is a nonstandard positive
integer.
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