Bounded projections in weighted function spaces in a generalized unit disc

Let ${\displaystyle M_{m,n}}$ be the space of all complex m × n matrices. The generalized unit disc in ${\displaystyle M_{m,n}}$ is ${\displaystyle R_{m,n}=\{Z\in M_{m,n}:I^{\left(m\right)}-{\mathbb{Z}}^{\ast }\text{is positive definite}\}}$. Here ${\displaystyle I^{\left(m\right)}\in M_{m,m}}$ is the unit matrix. If 1 ≤ p < ∞ and α > -1, then ${\displaystyle L_{\alpha }^p\left(R_{m,n}\right)}$ is defined to be the space ${\displaystyle L^p\{R_{m,n};{\left[\det (I^{\left(m\right)}-{\mathbb{Z}}^{\ast })\right]}^{\alpha }d{\mu }_{m,n}\left(Z\right)\}}$, where ${\displaystyle {\mu }_{m,n}}$ is the Lebesgue measure in ${\displaystyle M_{m,n}}$, and ${\displaystyle H_{\alpha }^p\left(R_{m,n}\right)\subset L_{\alpha }^p\left(R_{m,n}\right)}$ is the subspace of holomorphic functions. In [8,9] M. M. Djrbashian and A. H. Karapetyan proved that, if ${\displaystyle Re\beta >(\alpha +1)/p-1}$ (for 1 < p < ∞) and Re β ≥ α (for p = 1), then ${\displaystyle f\left(Z\right)=T_{m,n}^{\beta }\left(f\right)\left(Z\right),Z\in R_{m,n},}$ where ${\displaystyle T_{m,n}^{\beta }}$ 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 ${\displaystyle T_{m,n}^{\beta }}$ to be a bounded projection of ${\displaystyle L_{\alpha }^p\left(R_{m,n}\right)}$ onto ${\displaystyle H_{\alpha }^p\left(R_{m,n}\right)}$. Some applications of this result are given.