Uniform approximation on ideals of multilinear mappings

Botelho, Geraldo; Galindo, Pablo; Pellegrini, Leonardo

doi:10.7146/math.scand.a-15139

Cited by 11 publications

(22 citation statements)

References 20 publications

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“…We start by stating two lemmas on the injective hull and the closed hull of an n ‐ideal

[{scriptI}_{1}, \dots, {scriptI}_{n}]

that are known results (see [, p. 309]). Because this type of n ‐ideal will play an important role in this paper, and for the sake of completeness, we include such lemmas and their proofs.…”

Section: Measures Associated To Ideals Of Multilinear Operatorsmentioning

confidence: 99%

“…Let

{scriptI}_{1}, \dots, {scriptI}_{n}

be linear operator ideals. By [, p. 309] we trivially have

{\bar{scriptL false(I_{1}, \dots, I_{n} false)}}^{i n j} \subset L ({\bar{I_{1}}}^{i n j}, \dots, {\bar{I_{n}}}^{i n j}) .

…”

Section: Measures Associated To Ideals Of Multilinear Operatorsmentioning

confidence: 99%

“…It is well‐known that

{\bar{A}}^{i n j} = K

(see [, Proposition 4.2.5, Remarks 4.6.13 and 4.7.13]). By [, Example 3.4] (see also [, proof of Theorem 4.5]), there exists a Banach space E without the approximation property and an operator

u \in L (E; E^{*})

that is compact, symmetric and non‐approximable. Let A be the bilinear form on

E \times E

considered in [, Example 3.4], defined by

A (x, y) = u (x) (y)

.…”

Section: Measures Associated To Ideals Of Multilinear Operatorsmentioning

confidence: 99%

“…By [, Example 3.4] (see also [, proof of Theorem 4.5]), there exists a Banach space E without the approximation property and an operator

u \in L (E; E^{*})

that is compact, symmetric and non‐approximable. Let A be the bilinear form on

E \times E

considered in [, Example 3.4], defined by

A (x, y) = u (x) (y)

. It is immediate that

A \in [{\bar{scriptA}}^{i n j}, {\bar{scriptA}}^{i n j}]

since

A_{1} = A_{2} = u \in K false(E; E^{*} false)

.…”

Section: Measures Associated To Ideals Of Multilinear Operatorsmentioning

confidence: 99%

“…It is immediate that

A \in [{\bar{scriptA}}^{i n j}, {\bar{scriptA}}^{i n j}]

since

A_{1} = A_{2} = u \in K false(E; E^{*} false)

. Nevertheless, if

A \in {\bar{false[scriptA, scriptA false]}}^{i n j} false(E, E; double-struckF false)

, taking into account that

double-struckF

is an injective space and

scriptA

is closed, it would follow (see [, Corollary 2.6]) that

A \in {\bar{false[scriptA, scriptA false]}}^{i n j} false(E, E; double-struckF false) = \bar{[A, A]} false(E, E; double-struckF false) = false[scriptA, scriptA false] false(E, E; double-struckF false),

but this is a contradiction because

A_{1} = A_{2} = u \notin A false(E; E^{*} false)

.…”

Section: Measures Associated To Ideals Of Multilinear Operatorsmentioning

confidence: 99%

See 4 more Smart Citations

Closed injective ideals of multilinear operators, related measures and interpolation

Manzano

Rueda

Pérez

2020

Mathematische Nachrichten

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We introduce and discuss several ways of extending the inner measure arisen from the closed injective hull of an ideal of linear operators to the multilinear case. In particular, we consider new measures that allow to characterize the operators that belong to a closed injective ideal of multilinear operators as those having measure equal to zero. Some interpolation formulas for these measures, and consequently interpolation results involving ideals of multilinear operators, are established. Examples and applications related to summing multilinear operators are also shown.

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“…We start by stating two lemmas on the injective hull and the closed hull of an n ‐ideal

[{scriptI}_{1}, \dots, {scriptI}_{n}]

that are known results (see [, p. 309]). Because this type of n ‐ideal will play an important role in this paper, and for the sake of completeness, we include such lemmas and their proofs.…”

Section: Measures Associated To Ideals Of Multilinear Operatorsmentioning

confidence: 99%

“…Let

{scriptI}_{1}, \dots, {scriptI}_{n}

be linear operator ideals. By [, p. 309] we trivially have

{\bar{scriptL false(I_{1}, \dots, I_{n} false)}}^{i n j} \subset L ({\bar{I_{1}}}^{i n j}, \dots, {\bar{I_{n}}}^{i n j}) .

…”

Section: Measures Associated To Ideals Of Multilinear Operatorsmentioning

confidence: 99%

“…It is well‐known that

{\bar{A}}^{i n j} = K

(see [, Proposition 4.2.5, Remarks 4.6.13 and 4.7.13]). By [, Example 3.4] (see also [, proof of Theorem 4.5]), there exists a Banach space E without the approximation property and an operator

u \in L (E; E^{*})

that is compact, symmetric and non‐approximable. Let A be the bilinear form on

E \times E

considered in [, Example 3.4], defined by

A (x, y) = u (x) (y)

.…”

Section: Measures Associated To Ideals Of Multilinear Operatorsmentioning

confidence: 99%

“…By [, Example 3.4] (see also [, proof of Theorem 4.5]), there exists a Banach space E without the approximation property and an operator

u \in L (E; E^{*})

that is compact, symmetric and non‐approximable. Let A be the bilinear form on

E \times E

considered in [, Example 3.4], defined by

A (x, y) = u (x) (y)

. It is immediate that

A \in [{\bar{scriptA}}^{i n j}, {\bar{scriptA}}^{i n j}]

since

A_{1} = A_{2} = u \in K false(E; E^{*} false)

.…”

Section: Measures Associated To Ideals Of Multilinear Operatorsmentioning

confidence: 99%

“…It is immediate that

A \in [{\bar{scriptA}}^{i n j}, {\bar{scriptA}}^{i n j}]

since

A_{1} = A_{2} = u \in K false(E; E^{*} false)

. Nevertheless, if

A \in {\bar{false[scriptA, scriptA false]}}^{i n j} false(E, E; double-struckF false)

, taking into account that

double-struckF

is an injective space and

scriptA

is closed, it would follow (see [, Corollary 2.6]) that

A \in {\bar{false[scriptA, scriptA false]}}^{i n j} false(E, E; double-struckF false) = \bar{[A, A]} false(E, E; double-struckF false) = false[scriptA, scriptA false] false(E, E; double-struckF false),

but this is a contradiction because

A_{1} = A_{2} = u \notin A false(E; E^{*} false)

.…”

Section: Measures Associated To Ideals Of Multilinear Operatorsmentioning

confidence: 99%

See 3 more Smart Citations

Closed injective ideals of multilinear operators, related measures and interpolation

Manzano

Rueda

Pérez

2020

Mathematische Nachrichten

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Ideal properties of the Dunford integration operator

2015

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Let F be a Banach space. We establish necessary and sufficient conditions for the Dunford integration operator, from the space of F‐valued Dunford integrable functions to the bidual F′′ of F, to belong to a given operator ideal. We also show how this fact can be used to characterize important classes of Banach spaces, such as Banach spaces with the Banach‐Saks property, separable Banach spaces not containing c0, Banach spaces not containing c0 or ℓ1 and Asplund spaces not containing c0.

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