Bounded projections in weighted function spaces in a generalized unit disc

A. H. Karapetyan

Annales Polonici Mathematici, Tome 62 (1995), p. 193-218 / Harvested from The Polish Digital Mathematics Library

Résumé

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

Publié le : 1995-01-01

Zbl 0848.45009

EUDML-ID : urn:eudml:doc:262744

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     author = {A. H. Karapetyan},
     title = {Bounded projections in weighted function spaces in a generalized unit disc},
     journal = {Annales Polonici Mathematici},
     volume = {62},
     year = {1995},
     pages = {193-218},
     zbl = {0848.45009},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-apmv62z3p193bwm}
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A. H. Karapetyan. Bounded projections in weighted function spaces in a generalized unit disc. Annales Polonici Mathematici, Tome 62 (1995) pp. 193-218. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-apmv62z3p193bwm/

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