Vol. 16, No. 3, 1966

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Solution of an invariant subspace problem of K. T. Smith and P. R. Halmos

Allen Richard Bernstein and Abraham Robinson

Vol. 16 (1966), No. 3, 421–431

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.

Mathematical Subject Classification
Primary: 47.35
Secondary: 02.57
Received: 5 July 1964
Revised: 10 December 1964
Published: 1 March 1966
Allen Richard Bernstein
Abraham Robinson