On the space spanned by the powers of an operator and its adjoint
Bindner, Donald J
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In this work, the re exivity properties of the space spanned by the powers of an operator and its adjoint are investigated. For an operator A, let T A be the weak-star closure of f A j ; (A ) j j j 0 g . Call A star-re exive if T A is re exive and star-transitive if T A is transitive. |The regular unilateral shift is known to be star-transitive, as are the unicel- lular shifts. We will discover that many non-injective weighted shifts are not star- transitive, but rather star-re exive. Among the operators found to be star-re exive are direct sums of non-nilpotent shifts, 0 T for injective shifts T where T has a nonzero eigenvalue, and A A where A is a Jordan matrix. |There is an explicit construction of a star-re exive weighted shift T similar to the regular unilateral shift S. This contrasts distinctly with the star-transitive nature of S. The similarity that carries S to T is of the simplest type, a diagonal matrix. This example illustrates clearly that the property of being star-re exive is not invariant under similarity. |The question of similarity and star-re exivity is also explored for n n matrices. It is shown that Mn contains an open dense set of matrices that are both re exive and star-transitive. It is also shown, for n 4, that the only star-re exive, complex- valued matrices in Mn with n distinct eigenvalues are normal matrices.