Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History...
Transcript of Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History...
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results of theBishop-Phelps-Bollobas point property
Sheldon DantasPostech University (포스텍)
WORKSHOP ON INFINITE DIMENSIONAL ANALYSISVALENCIA 2017
Joint work with V. Kadets, S. K. Kim, H. J. Lee and M. Martın
October, 2017, Valencia (Spain)
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 2: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/2.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Table of contents
1 Motivation & History background
2 First results about the BPBpp
3 Recent results about the BPBpp
4 The dual property
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 3: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/3.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 4: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/4.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Notation
X ,Y and Z are real or complex Banach spaces.
K is the field R or C,
BX is the closed unit ball of X ,
SX is the unit sphere of X ,
L(X ,Y ) continuous linear operators from X into Y ,
K (X ,Y ) compact linear operators from X into Y ,
X ∗ = L(X ;K) topological dual of X .
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 5: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/5.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
Definition
We say that a linear functional x∗ ∈ X ∗ attains its norm if thereexists x0 ∈ SX such that |x∗(x0)| = ‖x∗‖.
NA(X ) is the set of all norm attaining functionals.
James theorem (1957)
A Banach space X is reflexive if and only if every bounded linearfunctional is norm attaining.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 6: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/6.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
Definition
We say that a linear functional x∗ ∈ X ∗ attains its norm if thereexists x0 ∈ SX such that |x∗(x0)| = ‖x∗‖.
NA(X ) is the set of all norm attaining functionals.
James theorem (1957)
A Banach space X is reflexive if and only if every bounded linearfunctional is norm attaining.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 7: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/7.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
Definition
We say that a linear functional x∗ ∈ X ∗ attains its norm if thereexists x0 ∈ SX such that |x∗(x0)| = ‖x∗‖.
NA(X ) is the set of all norm attaining functionals.
James theorem (1957)
A Banach space X is reflexive if and only if every bounded linearfunctional is norm attaining.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 8: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/8.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
Bishop-Phelps theorem (1961)
Every element in X ∗ can be approximated by a norm attaininglinear functional. In other words, NA(X ) = X ∗.
Question (Bishop-Phelps)
Is it true for bounded linear operators?
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 9: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/9.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
Bishop-Phelps theorem (1961)
Every element in X ∗ can be approximated by a norm attaininglinear functional. In other words, NA(X ) = X ∗.
Question (Bishop-Phelps)
Is it true for bounded linear operators?
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 10: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/10.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
Definition
We say that a bounded linear operator T ∈ L(X ,Y ) attains itsnorm if there exists x0 ∈ SX such that ‖T (x0)‖ = ‖T‖.
NA(X ,Y ) is the set of all norm attaining operators.
(1963, Lindenstrauss) Counterexample
There exists a Banach space X such that
NA(X ,X ) 6= L(X ,X ),
showing that the Bishop-Phelps result does not hold for boundedlinear operators.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 11: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/11.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
Definition
We say that a bounded linear operator T ∈ L(X ,Y ) attains itsnorm if there exists x0 ∈ SX such that ‖T (x0)‖ = ‖T‖.
NA(X ,Y ) is the set of all norm attaining operators.
(1963, Lindenstrauss) Counterexample
There exists a Banach space X such that
NA(X ,X ) 6= L(X ,X ),
showing that the Bishop-Phelps result does not hold for boundedlinear operators.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 12: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/12.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
Definition
We say that a bounded linear operator T ∈ L(X ,Y ) attains itsnorm if there exists x0 ∈ SX such that ‖T (x0)‖ = ‖T‖.
NA(X ,Y ) is the set of all norm attaining operators.
(1963, Lindenstrauss) Counterexample
There exists a Banach space X such that
NA(X ,X ) 6= L(X ,X ),
showing that the Bishop-Phelps result does not hold for boundedlinear operators.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 13: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/13.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
In 1970, Bollobas improved the Bishop-Phelps theorem.
1970, Bollobas, Bishop-Phelps-Bollobas theorem(2014, M. Chica, V. Kadets, M. Martın, S. Moreno-Pulido)
Let ε ∈ (0, 2). Given x ∈ BX and x∗ ∈ BX∗ with
|x∗(x)| > 1− ε2
2,
there are elements y ∈ SX and y∗ ∈ SX∗ such that
‖y∗‖ = |y∗(y)| = 1, ‖y − x‖ < ε and ‖y∗ − x∗‖ < ε.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 14: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/14.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
In 1970, Bollobas improved the Bishop-Phelps theorem.
1970, Bollobas, Bishop-Phelps-Bollobas theorem(2014, M. Chica, V. Kadets, M. Martın, S. Moreno-Pulido)
Let ε ∈ (0, 2). Given x ∈ BX and x∗ ∈ BX∗ with
|x∗(x)| > 1− ε2
2,
there are elements y ∈ SX and y∗ ∈ SX∗ such that
‖y∗‖ = |y∗(y)| = 1, ‖y − x‖ < ε and ‖y∗ − x∗‖ < ε.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 15: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/15.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
Observation 1
Bishop-Phelps-Bollobas theorem ⇒ Bishop-Phelps theorem.
Observation 2
It is not expected that there exists a Bishop-Phelps-Bollobastheorem version for bounded linear operators in general.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 16: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/16.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
Observation 1
Bishop-Phelps-Bollobas theorem ⇒ Bishop-Phelps theorem.
Observation 2
It is not expected that there exists a Bishop-Phelps-Bollobastheorem version for bounded linear operators in general.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 17: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/17.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
(2008, M. Acosta, R. Aron, D. Garcıa, M. Maestre)
Bishop-Phelps-Bollobas property (BPBp)
A pair of Banach spaces (X ,Y ) is said to have the BPBp if forevery ε ∈ (0, 1), there exists η(ε) > 0 such that if T ∈ L(X ,Y )with ‖T‖ = 1 and x ∈ SX satisfy
‖T (x)‖ > 1− η(ε),
there exist S ∈ L(X ,Y ) with ‖S‖ = 1 and x0 ∈ SX such that
‖S(x0)‖ = 1, ‖x0 − x‖ < ε and ‖T − S‖ < ε.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 18: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/18.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
(2008, M. Acosta, R. Aron, D. Garcıa, M. Maestre)
Bishop-Phelps-Bollobas property (BPBp)
A pair of Banach spaces (X ,Y ) is said to have the BPBp if forevery ε ∈ (0, 1), there exists η(ε) > 0 such that if T ∈ L(X ,Y )with ‖T‖ = 1 and x ∈ SX satisfy
‖T (x)‖ > 1− η(ε),
there exist S ∈ L(X ,Y ) with ‖S‖ = 1 and x0 ∈ SX such that
‖S(x0)‖ = 1, ‖x0 − x‖ < ε and ‖T − S‖ < ε.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 19: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/19.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
They proved that the pair (X ,Y ) has the BPBp if:
(1) X and Y are finite dimensional Banach spaces.
(2) X arbitrary and Y = c0 or Y = `∞.
(3) X = `1 and
Y = L1(µ) with µ a finite measure.
Y is uniformly convex.
Y = C (K ) for K a compact Haurdorff space.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 20: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/20.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
They proved that the pair (X ,Y ) has the BPBp if:
(1) X and Y are finite dimensional Banach spaces.
(2) X arbitrary and Y = c0 or Y = `∞.
(3) X = `1 and
Y = L1(µ) with µ a finite measure.
Y is uniformly convex.
Y = C (K ) for K a compact Haurdorff space.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 21: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/21.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
They proved that the pair (X ,Y ) has the BPBp if:
(1) X and Y are finite dimensional Banach spaces.
(2) X arbitrary and Y = c0 or Y = `∞.
(3) X = `1 and
Y = L1(µ) with µ a finite measure.
Y is uniformly convex.
Y = C (K ) for K a compact Haurdorff space.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 22: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/22.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
They proved that the pair (X ,Y ) has the BPBp if:
(1) X and Y are finite dimensional Banach spaces.
(2) X arbitrary and Y = c0 or Y = `∞.
(3) X = `1 and
Y = L1(µ) with µ a finite measure.
Y is uniformly convex.
Y = C (K ) for K a compact Haurdorff space.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 23: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/23.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
They proved that the pair (X ,Y ) has the BPBp if:
(1) X and Y are finite dimensional Banach spaces.
(2) X arbitrary and Y = c0 or Y = `∞.
(3) X = `1 and
Y = L1(µ) with µ a finite measure.
Y is uniformly convex.
Y = C (K ) for K a compact Haurdorff space.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 24: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/24.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
They proved that the pair (X ,Y ) has the BPBp if:
(1) X and Y are finite dimensional Banach spaces.
(2) X arbitrary and Y = c0 or Y = `∞.
(3) X = `1 and
Y = L1(µ) with µ a finite measure.
Y is uniformly convex.
Y = C (K ) for K a compact Haurdorff space.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 25: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/25.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
They proved that the pair (X ,Y ) has the BPBp if:
(1) X and Y are finite dimensional Banach spaces.
(2) X arbitrary and Y = c0 or Y = `∞.
(3) X = `1 and
Y = L1(µ) with µ a finite measure.
Y is uniformly convex.
Y = C (K ) for K a compact Haurdorff space.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 26: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/26.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
Since 2008, there has been a lot of attention on this topic:
(L1[0, 1], L∞[0, 1]) has the BPBp.(2011, R. Aron, Y. S. Choi, D. Garcıa, M. Maestre)
(X ,A) has the BPBp (X Asplund and A uniform algebra).(2013, B. Cascales, A. Guirao, V. Kadets)
(L1(µ), L1(ν)) has the BPBp.(2014, Y. S. Choi, S. K. Kim, H. J. Lee, M. Martın)
(X ,Y ) has the BPBp whenever X uniformly convex.(2014, S. K. Kim, H. J. Lee)
(C (K ), L1(µ)) has the BPBp.(2016, M. Acosta)
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 27: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/27.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
Since 2008, there has been a lot of attention on this topic:
(L1[0, 1], L∞[0, 1]) has the BPBp.(2011, R. Aron, Y. S. Choi, D. Garcıa, M. Maestre)
(X ,A) has the BPBp (X Asplund and A uniform algebra).(2013, B. Cascales, A. Guirao, V. Kadets)
(L1(µ), L1(ν)) has the BPBp.(2014, Y. S. Choi, S. K. Kim, H. J. Lee, M. Martın)
(X ,Y ) has the BPBp whenever X uniformly convex.(2014, S. K. Kim, H. J. Lee)
(C (K ), L1(µ)) has the BPBp.(2016, M. Acosta)
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
![Page 28: Recent results of the Bishop-Phelps-Bollob as point property · Motivation & History backgroundFirst results about the BPBppRecent results about the BPBppThe dual property Notation](https://reader034.fdocument.pub/reader034/viewer/2022042313/5edcad34ad6a402d666771dd/html5/thumbnails/28.jpg)
Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
Since 2008, there has been a lot of attention on this topic:
(L1[0, 1], L∞[0, 1]) has the BPBp.(2011, R. Aron, Y. S. Choi, D. Garcıa, M. Maestre)
(X ,A) has the BPBp (X Asplund and A uniform algebra).(2013, B. Cascales, A. Guirao, V. Kadets)
(L1(µ), L1(ν)) has the BPBp.(2014, Y. S. Choi, S. K. Kim, H. J. Lee, M. Martın)
(X ,Y ) has the BPBp whenever X uniformly convex.(2014, S. K. Kim, H. J. Lee)
(C (K ), L1(µ)) has the BPBp.(2016, M. Acosta)
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
Since 2008, there has been a lot of attention on this topic:
(L1[0, 1], L∞[0, 1]) has the BPBp.(2011, R. Aron, Y. S. Choi, D. Garcıa, M. Maestre)
(X ,A) has the BPBp (X Asplund and A uniform algebra).(2013, B. Cascales, A. Guirao, V. Kadets)
(L1(µ), L1(ν)) has the BPBp.(2014, Y. S. Choi, S. K. Kim, H. J. Lee, M. Martın)
(X ,Y ) has the BPBp whenever X uniformly convex.(2014, S. K. Kim, H. J. Lee)
(C (K ), L1(µ)) has the BPBp.(2016, M. Acosta)
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
Since 2008, there has been a lot of attention on this topic:
(L1[0, 1], L∞[0, 1]) has the BPBp.(2011, R. Aron, Y. S. Choi, D. Garcıa, M. Maestre)
(X ,A) has the BPBp (X Asplund and A uniform algebra).(2013, B. Cascales, A. Guirao, V. Kadets)
(L1(µ), L1(ν)) has the BPBp.(2014, Y. S. Choi, S. K. Kim, H. J. Lee, M. Martın)
(X ,Y ) has the BPBp whenever X uniformly convex.(2014, S. K. Kim, H. J. Lee)
(C (K ), L1(µ)) has the BPBp.(2016, M. Acosta)
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Motivation & History background
Since 2008, there has been a lot of attention on this topic:
(L1[0, 1], L∞[0, 1]) has the BPBp.(2011, R. Aron, Y. S. Choi, D. Garcıa, M. Maestre)
(X ,A) has the BPBp (X Asplund and A uniform algebra).(2013, B. Cascales, A. Guirao, V. Kadets)
(L1(µ), L1(ν)) has the BPBp.(2014, Y. S. Choi, S. K. Kim, H. J. Lee, M. Martın)
(X ,Y ) has the BPBp whenever X uniformly convex.(2014, S. K. Kim, H. J. Lee)
(C (K ), L1(µ)) has the BPBp.(2016, M. Acosta)
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
The Bishop-Phelps-Bollobaspoint property
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
The Bishop-Phelps-Bollobas point property
Bishop-Phelps-Bollobas point property (BPBpp)
A pair of Banach spaces (X ,Y ) is said to have the BPBpp if forevery ε ∈ (0, 1), there exists η(ε) > 0 such that if T ∈ L(X ,Y )with ‖T‖ = 1 and x ∈ SX satisfy
‖T (x)‖ > 1− η(ε),
there exists S ∈ L(X ,Y ) with ‖S‖ = 1 such that
‖S(x)‖ = 1 and ‖T − S‖ < ε.
It is clear that BPBpp ⇒ BPBp.
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The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
The Bishop-Phelps-Bollobas point property
Bishop-Phelps-Bollobas point property (BPBpp)
A pair of Banach spaces (X ,Y ) is said to have the BPBpp if forevery ε ∈ (0, 1), there exists η(ε) > 0 such that if T ∈ L(X ,Y )with ‖T‖ = 1 and x ∈ SX satisfy
‖T (x)‖ > 1− η(ε),
there exists S ∈ L(X ,Y ) with ‖S‖ = 1 such that
‖S(x)‖ = 1 and ‖T − S‖ < ε.
It is clear that BPBpp ⇒ BPBp.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
First results about the BPBpp
(2016, D., S. K. Kim and H. J. Lee)
(X ,K) has the BPBpp if and only if X is uniformly smooth.
(X ,Y ) has the BPBpp for some Y ⇒ X is uniformly smooth.
(H,Y ) has the BPBpp for all Hilbert spaces H and any Y .
(X ,Y ) has the BPBpp for X uniformly smooth and Y property β.
there are uniformly smooth Banach spaces X such that thepair (X ,Y ) fails the BPBpp for some Y .
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
First results about the BPBpp
(2016, D., S. K. Kim and H. J. Lee)
(X ,K) has the BPBpp if and only if X is uniformly smooth.
(X ,Y ) has the BPBpp for some Y ⇒ X is uniformly smooth.
(H,Y ) has the BPBpp for all Hilbert spaces H and any Y .
(X ,Y ) has the BPBpp for X uniformly smooth and Y property β.
there are uniformly smooth Banach spaces X such that thepair (X ,Y ) fails the BPBpp for some Y .
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
First results about the BPBpp
(2016, D., S. K. Kim and H. J. Lee)
(X ,K) has the BPBpp if and only if X is uniformly smooth.
(X ,Y ) has the BPBpp for some Y ⇒ X is uniformly smooth.
(H,Y ) has the BPBpp for all Hilbert spaces H and any Y .
(X ,Y ) has the BPBpp for X uniformly smooth and Y property β.
there are uniformly smooth Banach spaces X such that thepair (X ,Y ) fails the BPBpp for some Y .
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
First results about the BPBpp
(2016, D., S. K. Kim and H. J. Lee)
(X ,K) has the BPBpp if and only if X is uniformly smooth.
(X ,Y ) has the BPBpp for some Y ⇒ X is uniformly smooth.
(H,Y ) has the BPBpp for all Hilbert spaces H and any Y .
(X ,Y ) has the BPBpp for X uniformly smooth and Y property β.
there are uniformly smooth Banach spaces X such that thepair (X ,Y ) fails the BPBpp for some Y .
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
First results about the BPBpp
(2016, D., S. K. Kim and H. J. Lee)
(X ,K) has the BPBpp if and only if X is uniformly smooth.
(X ,Y ) has the BPBpp for some Y ⇒ X is uniformly smooth.
(H,Y ) has the BPBpp for all Hilbert spaces H and any Y .
(X ,Y ) has the BPBpp for X uniformly smooth and Y property β.
there are uniformly smooth Banach spaces X such that thepair (X ,Y ) fails the BPBpp for some Y .
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results
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The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Stability results
Proposition
Let X1 be a one-complemented subspace of X . If (X ,Y ) has theBPBpp, then (X1,Y ) has the BPBpp.
Questions
(a) Is this true for the BPBp? (201?, D., Garcıa, Maestre, Martın)
(b) Is this true for norm attaining operators?
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Stability results
Proposition
Let X1 be a one-complemented subspace of X . If (X ,Y ) has theBPBpp, then (X1,Y ) has the BPBpp.
Questions
(a) Is this true for the BPBp? (201?, D., Garcıa, Maestre, Martın)
(b) Is this true for norm attaining operators?
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Stability results
Proposition
Let X1 be a one-complemented subspace of X . If (X ,Y ) has theBPBpp, then (X1,Y ) has the BPBpp.
Questions
(a) Is this true for the BPBp? (201?, D., Garcıa, Maestre, Martın)
(b) Is this true for norm attaining operators?
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Stability results
Proposition ((201?, D., Garcıa, Maestre, Martın) adapted)
If Y = Y1 ⊕a Y2 and (X ,Y ) has the BPBpp, then (X ,Yj) has theBPBpp.
Proposition ((2015, Aron, Choi, Kim, Lee, Martın) adapted)
If (X ,C (K ,Y )) has the BPBpp, then (X ,Y ) has the BPBpp.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Stability results
Proposition ((201?, D., Garcıa, Maestre, Martın) adapted)
If Y = Y1 ⊕a Y2 and (X ,Y ) has the BPBpp, then (X ,Yj) has theBPBpp.
Proposition ((2015, Aron, Choi, Kim, Lee, Martın) adapted)
If (X ,C (K ,Y )) has the BPBpp, then (X ,Y ) has the BPBpp.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Universal properties
Definition (2014, Aron, Choi, Kim, Lee, Martın)
(a) X is universal BPBpp domain space if (X ,Y ) has theBPBpp for all Y .
(b) Y is universal BPBpp range space if (X ,Y ) has theBPBpp for all X uniformly smooth.
Examples (2016, D., S. K. Kim, H. J. Lee)
Hilbert spaces are universal BPBpp domain spaces.
Uniform algebras and Banach spaces with property β areuniversal BPBpp range spaces.
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The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Universal properties
Definition (2014, Aron, Choi, Kim, Lee, Martın)
(a) X is universal BPBpp domain space if (X ,Y ) has theBPBpp for all Y .
(b) Y is universal BPBpp range space if (X ,Y ) has theBPBpp for all X uniformly smooth.
Examples (2016, D., S. K. Kim, H. J. Lee)
Hilbert spaces are universal BPBpp domain spaces.
Uniform algebras and Banach spaces with property β areuniversal BPBpp range spaces.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Universal properties
Definition (2014, Aron, Choi, Kim, Lee, Martın)
(a) X is universal BPBpp domain space if (X ,Y ) has theBPBpp for all Y .
(b) Y is universal BPBpp range space if (X ,Y ) has theBPBpp for all X uniformly smooth.
Examples (2016, D., S. K. Kim, H. J. Lee)
Hilbert spaces are universal BPBpp domain spaces.
Uniform algebras and Banach spaces with property β areuniversal BPBpp range spaces.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Universal properties
Definition (2014, Aron, Choi, Kim, Lee, Martın)
(a) X is universal BPBpp domain space if (X ,Y ) has theBPBpp for all Y .
(b) Y is universal BPBpp range space if (X ,Y ) has theBPBpp for all X uniformly smooth.
Examples (2016, D., S. K. Kim, H. J. Lee)
Hilbert spaces are universal BPBpp domain spaces.
Uniform algebras and Banach spaces with property β areuniversal BPBpp range spaces.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Universal properties
Definition (2014, Aron, Choi, Kim, Lee, Martın)
(a) X is universal BPBpp domain space if (X ,Y ) has theBPBpp for all Y .
(b) Y is universal BPBpp range space if (X ,Y ) has theBPBpp for all X uniformly smooth.
Examples (2016, D., S. K. Kim, H. J. Lee)
Hilbert spaces are universal BPBpp domain spaces.
Uniform algebras and Banach spaces with property β areuniversal BPBpp range spaces.
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The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Universal properties
Question
We know that Hilbert spaces are universal BPBpp domain spaces.
Is it possible to extend the result for Lp-spaces with 1 < p <∞?
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The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Universal properties
Question
We know that Hilbert spaces are universal BPBpp domain spaces.
Is it possible to extend the result for Lp-spaces with 1 < p <∞?
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Universal properties
Theorem
If X is universal BPBpp domain space, then X is uniformly convex.
Theorem
If X is universal BPBpp domain space and X is isomorphic to aHilbert space, then δX (ε) ≥ Cε2.
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The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Universal properties
Theorem
If X is universal BPBpp domain space, then X is uniformly convex.
Theorem
If X is universal BPBpp domain space and X is isomorphic to aHilbert space, then δX (ε) ≥ Cε2.
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The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Universal properties
Corollary
Lp(µ) is not a BPBpp domain space for p > 2.
Question
Is Lp(µ) a BPBpp domain space for 1 < p < 2?
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Universal properties
Corollary
Lp(µ) is not a BPBpp domain space for p > 2.
Question
Is Lp(µ) a BPBpp domain space for 1 < p < 2?
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The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Universal properties
ACKρ-structure (2017, Cascales, Guirao, Kadets, Soloviova)
Theorem
If Y has ACKρ-structure, then Y is universal BPBpp range space.
C (K ) and C0(L) and, more in general, uniform algebras.
Banach spaces with property β.
finite `∞-sums of Banach spaces with ACKρ-structure.
c0(Y ), `∞(Y ) when Y has ACKρ-structure.
C (K ,Y ) when Y has ACKρ-structure.
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Universal properties
ACKρ-structure (2017, Cascales, Guirao, Kadets, Soloviova)
Theorem
If Y has ACKρ-structure, then Y is universal BPBpp range space.
C (K ) and C0(L) and, more in general, uniform algebras.
Banach spaces with property β.
finite `∞-sums of Banach spaces with ACKρ-structure.
c0(Y ), `∞(Y ) when Y has ACKρ-structure.
C (K ,Y ) when Y has ACKρ-structure.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Universal properties
Counterexample
For p ≥ 2, there is a Banach space Xp which is uniformly convexand uniformly smooth such that (Xp, `
2p) fails the BPBpp.
Note that (Xp, `2p) has the BPBp since Xp is uniformly convex.
Questions
(1) If Y is universal BPBp range space, then Y is universalBPBpp range for uniformly smooth X?
(2) It is not known whether all finite dimensional spaces areuniversal BPBp range spaces or even if they haveLindenstrauss property B.
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Universal properties
Counterexample
For p ≥ 2, there is a Banach space Xp which is uniformly convexand uniformly smooth such that (Xp, `
2p) fails the BPBpp.
Note that (Xp, `2p) has the BPBp since Xp is uniformly convex.
Questions
(1) If Y is universal BPBp range space, then Y is universalBPBpp range for uniformly smooth X?
(2) It is not known whether all finite dimensional spaces areuniversal BPBp range spaces or even if they haveLindenstrauss property B.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Universal properties
Counterexample
For p ≥ 2, there is a Banach space Xp which is uniformly convexand uniformly smooth such that (Xp, `
2p) fails the BPBpp.
Note that (Xp, `2p) has the BPBp since Xp is uniformly convex.
Questions
(1) If Y is universal BPBp range space, then Y is universalBPBpp range for uniformly smooth X?
(2) It is not known whether all finite dimensional spaces areuniversal BPBp range spaces or even if they haveLindenstrauss property B.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
Universal properties
Counterexample
For p ≥ 2, there is a Banach space Xp which is uniformly convexand uniformly smooth such that (Xp, `
2p) fails the BPBpp.
Note that (Xp, `2p) has the BPBp since Xp is uniformly convex.
Questions
(1) If Y is universal BPBp range space, then Y is universalBPBpp range for uniformly smooth X?
(2) It is not known whether all finite dimensional spaces areuniversal BPBp range spaces or even if they haveLindenstrauss property B.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
The BPBpp for compact operators
BPBpp for compact operators
A pair of Banach spaces (X ,Y ) is said to have the BPBpp forcompact operators if for every ε ∈ (0, 1), there exists η(ε) > 0such that if T ∈ K (X ,Y ) with ‖T‖ = 1 and x ∈ SX satisfy
‖T (x)‖ > 1− η(ε),
there exists S ∈ K (X ,Y ) with ‖S‖ = 1 such that
‖S(x)‖ = 1 and ‖T − S‖ < ε.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
The BPBpp for compact operators
BPBpp for compact operators
A pair of Banach spaces (X ,Y ) is said to have the BPBpp forcompact operators if for every ε ∈ (0, 1), there exists η(ε) > 0such that if T ∈ K (X ,Y ) with ‖T‖ = 1 and x ∈ SX satisfy
‖T (x)‖ > 1− η(ε),
there exists S ∈ K (X ,Y ) with ‖S‖ = 1 such that
‖S(x)‖ = 1 and ‖T − S‖ < ε.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
The BPBpp for compact operators
(H,Y ) for H Hilbert spaces and any Y .
(X ,Y ) for X is uniformly smooth and Y has ACKρ-structure.
(2017, D., Garcıa, Maestre, Martın)
(X , `p(Y )) ⇒ (X , Lp(µ,Y )) for 1 ≤ p <∞.
(X ,Y ) ⇒ (X , L∞(µ,Y ))
(X ,Y ) ⇒ (X ,C (K ,Y )).
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
The BPBpp for compact operators
(H,Y ) for H Hilbert spaces and any Y .
(X ,Y ) for X is uniformly smooth and Y has ACKρ-structure.
(2017, D., Garcıa, Maestre, Martın)
(X , `p(Y )) ⇒ (X , Lp(µ,Y )) for 1 ≤ p <∞.
(X ,Y ) ⇒ (X , L∞(µ,Y ))
(X ,Y ) ⇒ (X ,C (K ,Y )).
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
The BPBpp for compact operators
(H,Y ) for H Hilbert spaces and any Y .
(X ,Y ) for X is uniformly smooth and Y has ACKρ-structure.
(2017, D., Garcıa, Maestre, Martın)
(X , `p(Y )) ⇒ (X , Lp(µ,Y )) for 1 ≤ p <∞.
(X ,Y ) ⇒ (X , L∞(µ,Y ))
(X ,Y ) ⇒ (X ,C (K ,Y )).
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
The BPBpp for compact operators
(H,Y ) for H Hilbert spaces and any Y .
(X ,Y ) for X is uniformly smooth and Y has ACKρ-structure.
(2017, D., Garcıa, Maestre, Martın)
(X , `p(Y )) ⇒ (X , Lp(µ,Y )) for 1 ≤ p <∞.
(X ,Y ) ⇒ (X , L∞(µ,Y ))
(X ,Y ) ⇒ (X ,C (K ,Y )).
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
The BPBpp for compact operators
(H,Y ) for H Hilbert spaces and any Y .
(X ,Y ) for X is uniformly smooth and Y has ACKρ-structure.
(2017, D., Garcıa, Maestre, Martın)
(X , `p(Y )) ⇒ (X , Lp(µ,Y )) for 1 ≤ p <∞.
(X ,Y ) ⇒ (X , L∞(µ,Y ))
(X ,Y ) ⇒ (X ,C (K ,Y )).
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Recent results about the BPBpp
The BPBpp for compact operators
(H,Y ) for H Hilbert spaces and any Y .
(X ,Y ) for X is uniformly smooth and Y has ACKρ-structure.
(2017, D., Garcıa, Maestre, Martın)
(X , `p(Y )) ⇒ (X , Lp(µ,Y )) for 1 ≤ p <∞.
(X ,Y ) ⇒ (X , L∞(µ,Y ))
(X ,Y ) ⇒ (X ,C (K ,Y )).
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
The dual property
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
The dual property
Recall that (X ,Y ) has the BPBpp if ∀ ε ∈ (0, 1), ∃ η(ε) > 0:
T ∈ SL(X ,Y ), x ∈ SX with ‖T (x)‖ > 1− η(ε),
⇒ ∃ S ∈ SL(X ,Y ) with
‖S(x)‖ = 1 and ‖T − S‖ < ε.
A possible dual property: (2016, D.) ∀ ε ∈ (0, 1), ∃ η(ε) > 0:
T ∈ SL(X ,Y ), x ∈ SX with ‖T (x)‖ > 1− η(ε),
⇒ ∃ x0 ∈ SX with
‖T (x0)‖ = 1 and ‖x0 − x‖ < ε.
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
The dual property
Recall that (X ,Y ) has the BPBpp if ∀ ε ∈ (0, 1), ∃ η(ε) > 0:
T ∈ SL(X ,Y ), x ∈ SX with ‖T (x)‖ > 1− η(ε),
⇒ ∃ S ∈ SL(X ,Y ) with
‖S(x)‖ = 1 and ‖T − S‖ < ε.
A possible dual property: (2016, D.)
∀ ε ∈ (0, 1), ∃ η(ε) > 0:
T ∈ SL(X ,Y ), x ∈ SX with ‖T (x)‖ > 1− η(ε),
⇒ ∃ x0 ∈ SX with
‖T (x0)‖ = 1 and ‖x0 − x‖ < ε.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
The dual property
Recall that (X ,Y ) has the BPBpp if ∀ ε ∈ (0, 1), ∃ η(ε) > 0:
T ∈ SL(X ,Y ), x ∈ SX with ‖T (x)‖ > 1− η(ε),
⇒ ∃ S ∈ SL(X ,Y ) with
‖S(x)‖ = 1 and ‖T − S‖ < ε.
A possible dual property: (2016, D.) ∀ ε ∈ (0, 1), ∃ η(ε) > 0:
T ∈ SL(X ,Y ), x ∈ SX with ‖T (x)‖ > 1− η(ε),
⇒ ∃ x0 ∈ SX with
‖T (x0)‖ = 1 and ‖x0 − x‖ < ε.
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
The dual property
A possible dual property: ∀ ε ∈ (0, 1), ∃ η(ε) > 0:
T ∈ SL(X ,Y ), x0 ∈ SX with ‖T (x)‖ > 1− η(ε),
⇒ ∃ x0 ∈ SX with
‖T (x0)‖ = 1 and ‖x0 − x‖ < ε.
Theorem (2014, S. K. Kim, H. J. Lee)
X is uniformly convex if and only (X ,K) has the dual property.
Counterexample (D., 2016)
There are many pairs (X ,Y ) for which this property does not hold.
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
The dual property
A possible dual property: ∀ ε ∈ (0, 1), ∃ η(ε) > 0:
T ∈ SL(X ,Y ), x0 ∈ SX with ‖T (x)‖ > 1− η(ε),
⇒ ∃ x0 ∈ SX with
‖T (x0)‖ = 1 and ‖x0 − x‖ < ε.
Theorem (2014, S. K. Kim, H. J. Lee)
X is uniformly convex if and only (X ,K) has the dual property.
Counterexample (D., 2016)
There are many pairs (X ,Y ) for which this property does not hold.
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
The dual property
The dual property is not possible for dimensions greater than 1!
Theorem
If dim(X ), dim(Y ) > 1, then the pair (X ,Y ) fails it.
Proof.
Reducing the proof for 2-dimensional spaces.
Dividing the proof in two cases:
X is Hilbert (John’s maximal ellipsoid theorem)X is not Hilbert (Day’s and Nordlander’s theorems)
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
The dual property
The dual property is not possible for dimensions greater than 1!
Theorem
If dim(X ), dim(Y ) > 1, then the pair (X ,Y ) fails it.
Proof.
Reducing the proof for 2-dimensional spaces.
Dividing the proof in two cases:
X is Hilbert (John’s maximal ellipsoid theorem)X is not Hilbert (Day’s and Nordlander’s theorems)
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Motivation & History background First results about the BPBpp Recent results about the BPBpp The dual property
Thank youfor your attention
Sheldon Dantas Postech University (포스텍) WORKSHOP ON INFINITE DIMENSIONAL ANALYSIS VALENCIA 2017
The BPBpp - WidaVa 2017