## ABSTRACT

Over the last fifty years, several generations of operator theorists have been acquainted with the expository writings of Paul R. Halmos, John B. Conway, Ronald G. Douglas and Allen P. Shields. A common theme in these works is the ubiquity of unilateral weighted shifts acting on the Hilbert space ℓ
^{2}(ℤ) and ℓ
^{2}(ℤ_{+}), or its function-theoretic counterparts, the multiplication operators on the Hardy space, Bergman space, Dirichlet space, weighted Bergman spaces and more generally reproducing kernel Hilbert spaces. While weighted shifts represent the core of the celebrated survey article by Allen P. Shields [80], they are also explicitly mentioned in at least 29 of the 250 problems listed by P.R. Halmos in [65] (see also [64]. Weighted shifts have been, and continue to be, the source of countless examples and counterexamples in spectral theory, invariant subspace theory, C
^{*} -dynamical system theory, and subnormal and hyponormal operator theory and its generalizations. Here’s a sample elementary result.