Universal operators and invariant subspaces

Abstract

For any Banach space X, let B(X) denote the space of continuous endomorphisms of X. An operator U in B(X) will be called universal if, given any T in B(X), then some nonzero multiple of T is similar to a part of U i.e. there exists XEC, X 0, a closed subspace Xo of X such that UX0CXo and a linear homeomorphism q of X onto Xo such thatXT=q-'(Uj Xo)q. The first example of a universal operator (or model) was constructed by G.-C. Rota [1] for the Hilbert space case. In that instance, U is (unitarily equivalent to) the direct sum of countably many copies of the reverse shift , ( 2, 23, * * * ) > (2i 4, 4, * . . ). Such a direct sum obviously defines an operator whose nullspace is infinite-dimensional and whose range is the whole space. In this note, we show that all such operators are universal (when X is a separable Hilbert space) and that, with rather obvious modifications, the arguments extend to arbitrary Banach spaces.

Locations

• Proceedings of the American Mathematical Society - View - PDF