If a group acts via unitary operators on a Hilbert space of functions then this group action extends in an obvious way to the space of Hilbert-Schmidt operators over the given Hilbert space. Even if the action on functions is irreducible, the action on H.-S. operators need not be irreducible. It is often of considerable interest to find out what the irreducible constituents are. Such an attitude has recently been advocated in the theory of "Ha-pliz" (Hankel + Toeplitz) operators. In this paper we solve this problem [for] the space of H.-S. operators over the Hilbert space L2(Δ, μα) of square integrable functions over the unit disk Δ equipped with the Dzhrbashyan measure dμα(z) = (α + 1)(1 - |z|2)α dA(z) (α > -1). This complements the earlier results. In particular we discover many new Ha-plitz type operators. The question of their smoothness properties (Sp-estimates etc.) is however only touched upon.
@article{urn:eudml:doc:41123,
title = {Fourier analysis of a space of Hilbert-Shmidt operators. New Ha-plitz type operators.},
journal = {Publicacions Matem\`atiques},
volume = {34},
year = {1990},
pages = {181-197},
zbl = {0748.47020},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:41123}
}
Peetre, Jaak. Fourier analysis of a space of Hilbert-Shmidt operators. New Ha-plitz type operators.. Publicacions Matemàtiques, Tome 34 (1990) pp. 181-197. http://gdmltest.u-ga.fr/item/urn:eudml:doc:41123/