# On operator ideals related to (p,σ)-absolutely continuous operators

J. López Molina; E. Sánchez Pérez

Studia Mathematica (2000)

- Volume: 138, Issue: 1, page 25-40
- ISSN: 0039-3223

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topLópez Molina, J., and Sánchez Pérez, E.. "On operator ideals related to (p,σ)-absolutely continuous operators." Studia Mathematica 138.1 (2000): 25-40. <http://eudml.org/doc/216688>.

@article{LópezMolina2000,

abstract = {We study tensor norms and operator ideals related to the ideal $P_\{p,σ\}$, 1 < p < ∞, 0 < σ < 1, of (p,σ)-absolutely continuous operators of Matter. If α is the tensor norm associated with $P_\{p,σ\}$ (in the sense of Defant and Floret), we characterize the $(α^\{\prime \})^t$-nuclear and $(α^\{\prime \})^t$- integral operators by factorizations by means of the composition of the inclusion map $L^r(μ) → L^1(μ) + L^p(μ)$ with a diagonal operator $B_w:L^\{∞\}(μ) → L^r(μ)$, where r is the conjugate exponent of p’/(1-σ). As an application we study the reflexivity of the components of the ideal $P_\{p,σ\}$.},

author = {López Molina, J., Sánchez Pérez, E.},

journal = {Studia Mathematica},

keywords = {tensor norms; operator ideals; (p,σ)-absolutely continuous operators; $g_\{p,σ\}$-nuclear and $g_\{p,σ\}$ integral operators; Matter's ideal; -absolutely continuous operators; factorization properties; inclusion maps; diagonal operators},

language = {eng},

number = {1},

pages = {25-40},

title = {On operator ideals related to (p,σ)-absolutely continuous operators},

url = {http://eudml.org/doc/216688},

volume = {138},

year = {2000},

}

TY - JOUR

AU - López Molina, J.

AU - Sánchez Pérez, E.

TI - On operator ideals related to (p,σ)-absolutely continuous operators

JO - Studia Mathematica

PY - 2000

VL - 138

IS - 1

SP - 25

EP - 40

AB - We study tensor norms and operator ideals related to the ideal $P_{p,σ}$, 1 < p < ∞, 0 < σ < 1, of (p,σ)-absolutely continuous operators of Matter. If α is the tensor norm associated with $P_{p,σ}$ (in the sense of Defant and Floret), we characterize the $(α^{\prime })^t$-nuclear and $(α^{\prime })^t$- integral operators by factorizations by means of the composition of the inclusion map $L^r(μ) → L^1(μ) + L^p(μ)$ with a diagonal operator $B_w:L^{∞}(μ) → L^r(μ)$, where r is the conjugate exponent of p’/(1-σ). As an application we study the reflexivity of the components of the ideal $P_{p,σ}$.

LA - eng

KW - tensor norms; operator ideals; (p,σ)-absolutely continuous operators; $g_{p,σ}$-nuclear and $g_{p,σ}$ integral operators; Matter's ideal; -absolutely continuous operators; factorization properties; inclusion maps; diagonal operators

UR - http://eudml.org/doc/216688

ER -

## References

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- [6] López Molina, J. A. and Sánchez Pérez, E. A., Ideales de operadores absolutamente continuos, Rev. Real Acad. Cienc. Madrid 87 (1993), 349-378.
- [7] López Molina, J. A. and Sánchez Pérez, E. A., On ultraproducts of compositions of certain operators between some finite dimensional ${\ell}^{p}$-spaces, preprint, 1999.
- [8] Matter, U., Absolutely continuous operators and super-reflexivity, Math. Nachr. 130 (1987), 193-216. Zbl0622.47045
- [9] Matter, Factoring through interpolation spaces and super-reflexive Banach spaces, Rev. Roumaine Math. Pures Appl. 34 (1989), 147-156. Zbl0675.46032
- [10] Niculescu, C. P., Absolute continuity in Banach space theory, ibid. 24 (1979), 413-422. Zbl0406.47010
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- [12] Saphar, P., Produits tensoriels d'espaces de Banach et classes d'applications linéaires, Studia Math. 38 (1970), 71-100. Zbl0213.14201

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