UNIFORM OPERATORS.

摘要

Uniform operator is a newly-defined operator on complex Hilbert space. The long-standing invariant subspace problem partially motivates this research; and a natural example is the unilateral shift of multiplicity one. The definition and main results are as follows:;Definition: A bounded linear operator T on a separable Hilbert space is said to be uniform if for each infinite-dimensional invariant subspace, T(VBAR) is unitarily equivalent to T. Examples: (1) Unilateral shift of multiplicity one. (2) Donoghue operator.;Theorem: If T is a uniform operator and T = (lamda)I, then T is cyclic.;Theorem: If T is a non-scalar uniform operator, then T is not quasi-similar to any normal operator.;Theorem: If T is a non-zero compact operator, then T is uniform if and only if T has no non-trivial infinite-dimensional invariant subspace.;Theorem: If T is a non-scalar uniform operator with a non-empty compression spectrum, then T is unitarily equivalent to an analytic Toeplitz operator.;Theorem: The set of uniform operators is neither open nor closed in the norm topology of B(H).