## Banach Journal of Mathematical Analysis

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

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

#### Abstract

The operator ideal of $(p,q)$-factorable operators can be characterized as the class of operators that factors through the embedding ${L}^{q\text{'}}\left(\mu \right)\hookrightarrow {L}^{p}\left(\mu \right)$ for a finite measure $\mu $, where $p,q\in [1,\infty )$ are such that $1/p+1/q\ge 1$. 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 ${L}^{s}({m}_{1}{)}^{*}\to {L}^{r}({m}_{2})$, 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**

Received: 11 July 2017

Accepted: 19 October 2017

First available in Project Euclid: 7 March 2018

**Permanent link to this document**

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

**Digital Object Identifier**

doi:10.1215/17358787-2017-0059

**Mathematical Reviews number (MathSciNet)**

MR3779725

**Zentralblatt MATH identifier**

06873512

**Subjects**

Primary: 47B10: Operators belonging to operator ideals (nuclear, p-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20]

Secondary: 46G10: Vector-valued measures and integration [See also 28Bxx, 46B22]

**Keywords**

operator ideal $(p,q)$-factorable operator pth power factorable operator vector measure

#### 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