## Doctoral Dissertations

#### Title

Orbit-reflexivity

Spring 1995

Dissertation

Mathematics

#### Degree Name

Doctor of Philosophy

Suppose $H$ is a separable, infinite dimensional Hilbert space and $T$ and $S$ are bounded linear transformations on $H$. Suppose that if $Sx\in\{x, ,Tx ,T\sp2x,...\}\sp{-}$ for every $x$ implies that $S\in\{1, T, T\sp2,...\}\sp{-SOT}$ then $T$ is orbit-reflexive. Many operators are proven to be orbit-reflexive, including analytic Toeplitz operators and subnormal operators with cyclic vectors.
Suppose that if $Sx\in\{\gamma x : x\in H, \gamma\in\doubc\}\sp{-}$ for every $x$, implies that $S\in\{\gamma T\sp{n} : n\ge0, \lambda\in\doubc\}\sp{-SOT}$ then $T$ is $\doubc$-orbit-reflexive. Many operators are shown to be $\doubc$-orbit-reflexive. $\doubc$-orbit-reflexivity is shown to be the same as reflexivity for algebraic operators.