We discuss recent joint work with D. Galicer, M. Mansilla, M. Mastyło, and S. Muro. The general problem we address is to develop new methods within the study of projection constants of Banach spaces of multivariate polynomials.
The relative projection constant 
of a subspace 
of a Banach space 
is the smallest norm among all possible projections on onto , and the (absolute) projection constant 
is the supremum of all relative projection constants of taken with respect to all possible super spaces . This is one of the most significant notions of modern Banach space theory, and one that has been intensively studied since the birth of abstract operator theory. We develop an abstract setting, which allows to handle in a unified way a wide variety of Banach spaces of multivariate polynomials, including polynomials on compact abelian groups, polynomials on Boolean cubes 
, Dirichlet polynomials on the complex plane, and polynomials on 
for 
Finally, we consider what we call unitary harmonics to get an explicit formula for the projection constant of the trace class on 
, together with their precise asymptotic whenever 
tends to infinity.
Andreas Defant.
Universität Oldenburg (Germany)
A. Defant has been, over the last 30 years, one of the leading experts in Functional Analysis. His research covers a wide range of interests and topics: normed tensor products, local theory in Banach spaces, vector measures, Dirichlet series or polynomials and holomorphic functions on Banach spaces. In the last 15 years he has focused his activity in the study of Dirichlet series from an analytical, exploring their close connection with polynomials, holomorphic functions and harmonic analysis. He is author of over 100 research papers and three books.
