There he earned his PhD with the dissertation Generalized Convolution Algebras and Spectral Representations supervised by Cassius Ionescu-Tulcea.
There he worked, interrupted by guest professorships at the University of Frankfurt (in 1966–67 and 1970–71), until 1973, from 1969 on as a Full Professor.
In 1973 Maltese moved to Germany where he accepted a position as a Full Professor for mathematics at the University of Münster; there he worked until he retired in 1996.
The Mathematics Genealogy Project lists 17 PhD students of Maltese, among others Ferdinand Beckhoff (Habilitation in 1994) and Anand Srivastav (Professor of Computer Science at the Christian-Albrechts-University of Kiel).
Following his retirement Maltese went, together with his wife Marlene (née Kunz) back to Middletown and the Wesleyan University.