Education
Shapiro completed his Doctor of Philosophy thesis entitled "Linear Functionals on Non-Locally Convex Spaces" under the supervision of Allen Shields in 1969 at the University of Michigan.
Shapiro completed his Doctor of Philosophy thesis entitled "Linear Functionals on Non-Locally Convex Spaces" under the supervision of Allen Shields in 1969 at the University of Michigan.
He is the author of the book "Composition Operators and Classical Function Theory" (), and the American Mathematical Society memoir "Cyclic Phenomena for Composition Operators" (Memoirs of the American Mathematics Society #596, Volume(s) 125, 1997, pp 1–105), with Paul Bourdon. Upon Graduating, he became a Research Associate at Queen"s University, Canada, then from 1970 onwards climbed the ranks at Michigan State University, becoming a full Professor in 1979. He stayed at Michigan until 2006, when he retired and became and Adjunct Professor at Portland State University in Oregon.
Shapiro discovered some of the most important 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.