Shapiro completed his
PhD thesis entitled "Linear Functionals on Non-Locally Convex Spaces" under the supervision of
Allen Shields in 1969 at the
University of Michigan. Upon graduating, he became a research associate at
Queen's University, Canada, then from 1970 onwards was at
Michigan State University, becoming a full
professor in 1979. He stayed at Michigan State until 2006, when he retired and became an adjunct professor at
Portland State University in
Oregon. Shapiro discovered some of the properties of
composition operators, including a study of the cyclic properties of such operators and the first calculations of the essential norm for
composition operators on the
Hardy spaces of the
Unit disc. ==References==