## Banach Journal of Mathematical Analysis

### Extrapolation theorems for $(p,q)$-factorable operators

Orlando Galdames-Bravo

#### Abstract

The operator ideal of $(p,q)$-factorable operators can be characterized as the class of operators that factors through the embedding $L^{q'}(\mu)\hookrightarrow L^{p}(\mu)$ for a finite measure $\mu$, where $p,q\in[1,\infty)$ are such that $1/p+1/q\ge1$. We prove that this operator ideal is included into a Banach operator ideal characterized by means of factorizations through $r$th and $s$th power factorable operators, for suitable $r,s\in[1,\infty)$. Thus, they also factor through a positive map $L^{s}(m_{1})^{\ast}\to L^{r}(m_{2})$, where $m_{1}$ and $m_{2}$ are vector measures. We use the properties of the spaces of $u$-integrable functions with respect to a vector measure and the $u$th power factorable operators to obtain a characterization of $(p,q)$-factorable operators and conditions under which a $(p,q)$-factorable operator is $r$-summing for $r\in[1,p]$.

#### Article information

Source
Banach J. Math. Anal., Volume 12, Number 2 (2018), 497-514.

Dates
Accepted: 19 October 2017
First available in Project Euclid: 7 March 2018

https://projecteuclid.org/euclid.bjma/1520413213

Digital Object Identifier
doi:10.1215/17358787-2017-0059

Mathematical Reviews number (MathSciNet)
MR3779725

Zentralblatt MATH identifier
06873512

#### Citation

Galdames-Bravo, Orlando. Extrapolation theorems for $(p,q)$ -factorable operators. Banach J. Math. Anal. 12 (2018), no. 2, 497--514. doi:10.1215/17358787-2017-0059. https://projecteuclid.org/euclid.bjma/1520413213

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