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Let Mm,n be the space of all complex m × n matrices. The generalized unit disc in Mm,n is Rm,n = {Z ∈Mm,n : I − ZZ∗ is positive definite}. Here I∈Mm,m is the unit matrix. If 1≤p<∞ and α > −1, then Lα(Rm,n) is defined to be the space L{Rm,n; [det(I(m) − ZZ∗)]α dμm,n(Z)}, where μm,n is the Lebesgue measure in Mm,n, and H p α(Rm,n) ⊂ Lα(Rm,n) is the subspace of holomorphic functions. In [8, 9] M. M. Djrbashian and A. H. Karapetyan proved that, if Reβ > (α + 1)/p − 1 (for 1 < p <∞) and Reβ ≥ α (for p = 1), then f(Z) = T m,n(f)(Z), Z ∈ Rm,n, where T m,n is the integral operator defined by (0.13)–(0.14). In the present paper, given 1 ≤ p < ∞, we find conditions on α and β for T m,n to be a bounded projection of Lα(Rm,n) onto H p α(Rm,n). Some applications of this result are given. 0. Introduction 0.1. In the fourties M. M. Djrbashian [4, 5] introduced the classes H(α) (1 ≤ p < ∞, α > −1) of functions f(z) holomorphic in the unit disc D = {z ∈ C : |z| < 1}, with (0.1) ∫ ∫ D |f(ζ)|(1− |ζ|) dξ dη <∞ (ζ = ξ + iη). In the same papers [4, 5] the following result was established. 1991 Mathematics Subject Classification: 30E20, 31C10, 32A07, 32A10, 32A25, 32M15, 45P05.

Bounded projections in weighted function spaces in a generalized unit disc