 
Summary: [Page 1]
Invariant symbolic calculi and eigenvalues
of invariant opertors on symmetric
domains
Jonathan Arazy and Harald Upmeier \Lambda
Abstract. We study the structure of invariant symbolic calculi A in the context of
weighted Bergman spaces on symmetric domains D = G=K and the eigenvalues of the
associated link transforms A 0 A. We parametrize all such calculi by Kinvariants maps
which have very simple description. We also introduce and study the properties of the
fundamental function aA(–) associated with an invariant symbolic calculus A. Our main
result is the formula for the eigenvalues of the associated link transform A 0 A:
]
A 0 A(–) = aA(–) aA(–)
aT (–) ;
where T is the Toeplitz calculus.
1991 Mathematics Subject Classification:
32M15, 46E22, 47B35, 81T70
0. Introduction
Let D j G=K be a hermitian symmetric domain in C d and let H be a Hilbert
space of holomorphic functions on D with reproducing kernel K(z; w), which is
