John Froelich and Ben Mathes
Bi-Strictly Cyclic Operators
||April 10, 1995
||Strictly cyclic, invariant subspace, column Hilbert space, completely bounded, completely isomorphic.
||Primary 47A15; Secondary 46B28.
The genesis of this paper is the construction
of a new operator that, when combined
with a theorem of Herrero, settles a
question of Herrero. Herrero proved that a strictly
cyclic operator on an infinite
dimensional Hilbert space is never triangular.
He later asks
whether the adjoint of a strictly cyclic operator
is necessarily triangular. We settle the
question by constructing an operator T for which
both T and T* are strictly cyclic. We make a
detailed study of this bi-strictly cyclic operator
which leads to theorems about general bi-strictly cyclic
operators. We conclude the paper with a comparison of
the operator space structures of the singly
generated algebras A(S) and A(T),
when S is strictly cyclic and T is bi-strictly cyclic.
Department of Mathematics, University of Houston, University Park, Houston, Texas 77204
Department of Mathematics & Computer Science, Colby College, Waterville, ME 04901