Abstract
A vectorx in a Hilbert spaceH iscyclic for a bounded linear operatorT∶H→H if the closed linear span of the orbit {T n x∶n≥0} ofx underT is all ofH. Operators which have a cyclic vector are said to be cyclic.
Jordan operators are the infinite direct sums of Jordan cells acting on finite- dimensional Hilbert spaces. Necessary and sufficient conditions for a Jordan operator to be cyclic are given (see Corollary 6). In this case, a dense set of cyclic vectors is exhibited (see Corollary 4). Sufficient conditions for uncountable collections of cyclic Jordan operators to have a common cyclic vector are given and, in this case, a dense set of common cyclic vectors is exhibited (see Corollary 9).
Analogues of these cyclicity results for Jordan operators are obtained for compressions of analytic Toeplitz operatorsT A ∶F→AF on the Hardy spaceH 2 to subspaces (BH 2)⊥ invariant for the backward shiftT z * whereB is a Blaschke product by showing that such compressions are quasisimilar to Jordan operators.
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