# Bounded projections in weighted function spaces in a generalized unit disc

• Volume: 62, Issue: 3, page 193-218
• ISSN: 0066-2216

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## Abstract

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

## How to cite

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A. H. Karapetyan. "Bounded projections in weighted function spaces in a generalized unit disc." Annales Polonici Mathematici 62.3 (1995): 193-218. <http://eudml.org/doc/262744>.

@article{A1995,
abstract = {Let $M_\{m,n\}$ be the space of all complex m × n matrices. The generalized unit disc in $M_\{m,n\}$ is >br>    $R_\{m,n\} = \{Z ∈ M_\{m,n\}: I^\{(m)\} - ZZ* is positive definite\}$. Here $I^\{(m)\} ∈ M_\{m,m\}$ is the unit matrix. If 1 ≤ p < ∞ and α > -1, then $L^\{p\}_\{α\}(R_\{m,n\})$ is defined to be the space $L^p\{R_\{m,n\}; [det(I^\{(m)\} - ZZ*)]^α dμ_\{m,n\}(Z)\}$, where $μ_\{m,n\}$ is the Lebesgue measure in $M_\{m,n\}$, and $H^p_α(R_\{m,n\}) ⊂ L^\{p\}_\{α\}(R_\{m,n\})$ 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 Re β ≥ α (for p = 1), then     $f()= T^\{β\}_\{m,n\}(f)(), ∈ R_\{m,n\},$where $T^\{β\}_\{m,n\}$ 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 $T^\{β\}_\{m,n\}$ to be a bounded projection of $L^p_α(R_\{m,n\})$ onto $H^p_α(R_\{m,n\})$. Some applications of this result are given.},
author = {A. H. Karapetyan},
journal = {Annales Polonici Mathematici},
keywords = {generalized unit disc; holomorphic and pluriharmonic functions; weighted spaces; integral representations; bounded integral operators; weighted function spaces; integral operator; bounded projection},
language = {eng},
number = {3},
pages = {193-218},
title = {Bounded projections in weighted function spaces in a generalized unit disc},
url = {http://eudml.org/doc/262744},
volume = {62},
year = {1995},
}

TY - JOUR
AU - A. H. Karapetyan
TI - Bounded projections in weighted function spaces in a generalized unit disc
JO - Annales Polonici Mathematici
PY - 1995
VL - 62
IS - 3
SP - 193
EP - 218
AB - Let $M_{m,n}$ be the space of all complex m × n matrices. The generalized unit disc in $M_{m,n}$ is >br>    $R_{m,n} = {Z ∈ M_{m,n}: I^{(m)} - ZZ* is positive definite}$. Here $I^{(m)} ∈ M_{m,m}$ is the unit matrix. If 1 ≤ p < ∞ and α > -1, then $L^{p}_{α}(R_{m,n})$ is defined to be the space $L^p{R_{m,n}; [det(I^{(m)} - ZZ*)]^α dμ_{m,n}(Z)}$, where $μ_{m,n}$ is the Lebesgue measure in $M_{m,n}$, and $H^p_α(R_{m,n}) ⊂ L^{p}_{α}(R_{m,n})$ 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 Re β ≥ α (for p = 1), then     $f()= T^{β}_{m,n}(f)(), ∈ R_{m,n},$where $T^{β}_{m,n}$ 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 $T^{β}_{m,n}$ to be a bounded projection of $L^p_α(R_{m,n})$ onto $H^p_α(R_{m,n})$. Some applications of this result are given.
LA - eng
KW - generalized unit disc; holomorphic and pluriharmonic functions; weighted spaces; integral representations; bounded integral operators; weighted function spaces; integral operator; bounded projection
UR - http://eudml.org/doc/262744
ER -

## References

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