General discussion - spaces with no reflexive subspaces Recall Bourgain’s general problem: does there exist a space X not containing c 0 , ℓ 1 , or reflexive subspaces? Theorem (W. T. Gowers (1994)) There exists a Banach space X G not containing c 0 , ℓ 1 , or reflexive subspaces. • The space X G is hereditarily indecomposable and all its infinite subspaces have non-separable dual.
General discussion - spaces with no reflexive subspaces Recall Bourgain’s general problem: does there exist a space X not containing c 0 , ℓ 1 , or reflexive subspaces? Theorem (W. T. Gowers (1994)) There exists a Banach space X G not containing c 0 , ℓ 1 , or reflexive subspaces. • The space X G is hereditarily indecomposable and all its infinite subspaces have non-separable dual.
General discussion - spaces with no reflexive subspaces Recall Bourgain’s general problem: does there exist a space X not containing c 0 , ℓ 1 , or reflexive subspaces? Theorem (W. T. Gowers (1994)) There exists a Banach space X G not containing c 0 , ℓ 1 , or reflexive subspaces. • The space X G is hereditarily indecomposable and all its infinite subspaces have non-separable dual.
General discussion - L ∞ -spaces with no reflexive subspaces • Back to L ∞ -spaces: does there exist a L ∞ -space not containing c 0 , ℓ 1 , or reflexive subspaces? • Possible approach: combine Gowers’ method of obtaining HI spaces without reflexive subspaces and the Bourgain-Delbaen construction method. • Possible result: a separable L ∞ -space X with non-separable dual. • By Stegall: X contains ℓ 1 . This is undesired. • Was perhaps Bourgain rights and such spaces don’t exist?
General discussion - L ∞ -spaces with no reflexive subspaces • Back to L ∞ -spaces: does there exist a L ∞ -space not containing c 0 , ℓ 1 , or reflexive subspaces? • Possible approach: combine Gowers’ method of obtaining HI spaces without reflexive subspaces and the Bourgain-Delbaen construction method. • Possible result: a separable L ∞ -space X with non-separable dual. • By Stegall: X contains ℓ 1 . This is undesired. • Was perhaps Bourgain rights and such spaces don’t exist?
General discussion - L ∞ -spaces with no reflexive subspaces • Back to L ∞ -spaces: does there exist a L ∞ -space not containing c 0 , ℓ 1 , or reflexive subspaces? • Possible approach: combine Gowers’ method of obtaining HI spaces without reflexive subspaces and the Bourgain-Delbaen construction method. • Possible result: a separable L ∞ -space X with non-separable dual. • By Stegall: X contains ℓ 1 . This is undesired. • Was perhaps Bourgain rights and such spaces don’t exist?
General discussion - L ∞ -spaces with no reflexive subspaces • Back to L ∞ -spaces: does there exist a L ∞ -space not containing c 0 , ℓ 1 , or reflexive subspaces? • Possible approach: combine Gowers’ method of obtaining HI spaces without reflexive subspaces and the Bourgain-Delbaen construction method. • Possible result: a separable L ∞ -space X with non-separable dual. • By Stegall: X contains ℓ 1 . This is undesired. • Was perhaps Bourgain rights and such spaces don’t exist?
General discussion - L ∞ -spaces with no reflexive subspaces • Back to L ∞ -spaces: does there exist a L ∞ -space not containing c 0 , ℓ 1 , or reflexive subspaces? • Possible approach: combine Gowers’ method of obtaining HI spaces without reflexive subspaces and the Bourgain-Delbaen construction method. • Possible result: a separable L ∞ -space X with non-separable dual. • By Stegall: X contains ℓ 1 . This is undesired. • Was perhaps Bourgain rights and such spaces don’t exist?
An answer to this problem Theorem (S. Argyros - M) There exists a separable L ∞ -space X nr that is HI (and hence does not contain c 0 or ℓ 1 ), has separable dual and does not have reflexive subspaces. Moreover, this space has the scalar-plus-compact property. Definition A Banach space X has the scalar-plus compact if every bounded linear operator T : X → X is of the form T = λ I + K with λ ∈ R and K a compact operator.
An answer to this problem Theorem (S. Argyros - M) There exists a separable L ∞ -space X nr that is HI (and hence does not contain c 0 or ℓ 1 ), has separable dual and does not have reflexive subspaces. Moreover, this space has the scalar-plus-compact property. Definition A Banach space X has the scalar-plus compact if every bounded linear operator T : X → X is of the form T = λ I + K with λ ∈ R and K a compact operator.
Some remarks • Bourgain’s problem in the class of L ∞ -spaces can be related to the following. Problem (H. P . Rosenthal) Let X be a L ∞ -saturated space. Does X contain c 0 ? • Reversing the problem: does there exists a L ∞ -saturated space X not containing c 0 ? • If such a (separable L ∞ -space) X exists, then it would not contain ℓ 1 , c 0 , or reflexive subspaces • X would be a counter-example to Bourgain’s problem in the class of L ∞ -spaces. • Is X nr such an X ? It is not.
Some remarks • Bourgain’s problem in the class of L ∞ -spaces can be related to the following. Problem (H. P . Rosenthal) Let X be a L ∞ -saturated space. Does X contain c 0 ? • Reversing the problem: does there exists a L ∞ -saturated space X not containing c 0 ? • If such a (separable L ∞ -space) X exists, then it would not contain ℓ 1 , c 0 , or reflexive subspaces • X would be a counter-example to Bourgain’s problem in the class of L ∞ -spaces. • Is X nr such an X ? It is not.
Some remarks • Bourgain’s problem in the class of L ∞ -spaces can be related to the following. Problem (H. P . Rosenthal) Let X be a L ∞ -saturated space. Does X contain c 0 ? • Reversing the problem: does there exists a L ∞ -saturated space X not containing c 0 ? • If such a (separable L ∞ -space) X exists, then it would not contain ℓ 1 , c 0 , or reflexive subspaces • X would be a counter-example to Bourgain’s problem in the class of L ∞ -spaces. • Is X nr such an X ? It is not.
Some remarks • Bourgain’s problem in the class of L ∞ -spaces can be related to the following. Problem (H. P . Rosenthal) Let X be a L ∞ -saturated space. Does X contain c 0 ? • Reversing the problem: does there exists a L ∞ -saturated space X not containing c 0 ? • If such a (separable L ∞ -space) X exists, then it would not contain ℓ 1 , c 0 , or reflexive subspaces • X would be a counter-example to Bourgain’s problem in the class of L ∞ -spaces. • Is X nr such an X ? It is not.
Some remarks • Bourgain’s problem in the class of L ∞ -spaces can be related to the following. Problem (H. P . Rosenthal) Let X be a L ∞ -saturated space. Does X contain c 0 ? • Reversing the problem: does there exists a L ∞ -saturated space X not containing c 0 ? • If such a (separable L ∞ -space) X exists, then it would not contain ℓ 1 , c 0 , or reflexive subspaces • X would be a counter-example to Bourgain’s problem in the class of L ∞ -spaces. • Is X nr such an X ? It is not.
Some remarks • Bourgain’s problem in the class of L ∞ -spaces can be related to the following. Problem (H. P . Rosenthal) Let X be a L ∞ -saturated space. Does X contain c 0 ? • Reversing the problem: does there exists a L ∞ -saturated space X not containing c 0 ? • If such a (separable L ∞ -space) X exists, then it would not contain ℓ 1 , c 0 , or reflexive subspaces • X would be a counter-example to Bourgain’s problem in the class of L ∞ -spaces. • Is X nr such an X ? It is not.
Some remarks • Bourgain’s problem in the class of L ∞ -spaces can be related to the following. Problem (H. P . Rosenthal) Let X be a L ∞ -saturated space. Does X contain c 0 ? • Reversing the problem: does there exists a L ∞ -saturated space X not containing c 0 ? • If such a (separable L ∞ -space) X exists, then it would not contain ℓ 1 , c 0 , or reflexive subspaces • X would be a counter-example to Bourgain’s problem in the class of L ∞ -spaces. • Is X nr such an X ? It is not.
Some remarks on the construction of X nr • The space X nr is constructed with the Bourgain-Delbaen method, but not in combination with Gowers’ method. • Instead, it uses a new variation of the method of saturation under constraints, a Tsirelson-type saturation method. • This version of saturation under constraints provides a new way of defining HI spaces, where the conditional structure is provided by certain averages, and not special functionals. Intereasting feature: it provides a unified approach for constructing reflexive HI spaces and HI spaces with no boundedly complete sequences.
Some remarks on the construction of X nr • The space X nr is constructed with the Bourgain-Delbaen method, but not in combination with Gowers’ method. • Instead, it uses a new variation of the method of saturation under constraints, a Tsirelson-type saturation method. • This version of saturation under constraints provides a new way of defining HI spaces, where the conditional structure is provided by certain averages, and not special functionals. Intereasting feature: it provides a unified approach for constructing reflexive HI spaces and HI spaces with no boundedly complete sequences.
Some remarks on the construction of X nr • The space X nr is constructed with the Bourgain-Delbaen method, but not in combination with Gowers’ method. • Instead, it uses a new variation of the method of saturation under constraints, a Tsirelson-type saturation method. • This version of saturation under constraints provides a new way of defining HI spaces, where the conditional structure is provided by certain averages, and not special functionals. Intereasting feature: it provides a unified approach for constructing reflexive HI spaces and HI spaces with no boundedly complete sequences.
Some remarks on the construction of X nr • The space X nr is constructed with the Bourgain-Delbaen method, but not in combination with Gowers’ method. • Instead, it uses a new variation of the method of saturation under constraints, a Tsirelson-type saturation method. • This version of saturation under constraints provides a new way of defining HI spaces, where the conditional structure is provided by certain averages, and not special functionals. Intereasting feature: it provides a unified approach for constructing reflexive HI spaces and HI spaces with no boundedly complete sequences.
Some results of new saturation under constraints • A result based on this new method in a classical (Tsirelson, non- L ∞ ) setting. Theorem (S. Argyros - M. (2015)) There exists a hereditarily indecomposable Banach space X U that has separable dual and does not contain boundedly complete sequences. In particular is has no reflexive subspaces. (i) Every bounded linear operator T defined on X U is of the form T = λ I + S with S strictly singular. (ii) The ideal of strictly singular operators on X U is non-separable. (iii) The composition of any two strictly singular operators defined on X U is compact. (iv) Every bounded linear operator defined on the space has a non-trivial closed invariant subspace. give short explanation Furthermore, there exists a reflexive HI space X T satisfying (i), (ii), (iii), and (iv).
Some results of new saturation under constraints • A result based on this new method in a classical (Tsirelson, non- L ∞ ) setting. Theorem (S. Argyros - M. (2015)) There exists a hereditarily indecomposable Banach space X U that has separable dual and does not contain boundedly complete sequences. In particular is has no reflexive subspaces. (i) Every bounded linear operator T defined on X U is of the form T = λ I + S with S strictly singular. (ii) The ideal of strictly singular operators on X U is non-separable. (iii) The composition of any two strictly singular operators defined on X U is compact. (iv) Every bounded linear operator defined on the space has a non-trivial closed invariant subspace. give short explanation Furthermore, there exists a reflexive HI space X T satisfying (i), (ii), (iii), and (iv).
Some results of new saturation under constraints • A result based on this new method in a classical (Tsirelson, non- L ∞ ) setting. Theorem (S. Argyros - M. (2015)) There exists a hereditarily indecomposable Banach space X U that has separable dual and does not contain boundedly complete sequences. In particular is has no reflexive subspaces. (i) Every bounded linear operator T defined on X U is of the form T = λ I + S with S strictly singular. (ii) The ideal of strictly singular operators on X U is non-separable. (iii) The composition of any two strictly singular operators defined on X U is compact. (iv) Every bounded linear operator defined on the space has a non-trivial closed invariant subspace. give short explanation Furthermore, there exists a reflexive HI space X T satisfying (i), (ii), (iii), and (iv).
Some results of new saturation under constraints • A result based on this new method in a classical (Tsirelson, non- L ∞ ) setting. Theorem (S. Argyros - M. (2015)) There exists a hereditarily indecomposable Banach space X U that has separable dual and does not contain boundedly complete sequences. In particular is has no reflexive subspaces. (i) Every bounded linear operator T defined on X U is of the form T = λ I + S with S strictly singular. (ii) The ideal of strictly singular operators on X U is non-separable. (iii) The composition of any two strictly singular operators defined on X U is compact. (iv) Every bounded linear operator defined on the space has a non-trivial closed invariant subspace. give short explanation Furthermore, there exists a reflexive HI space X T satisfying (i), (ii), (iii), and (iv).
Some results of new saturation under constraints • A result based on this new method in a classical (Tsirelson, non- L ∞ ) setting. Theorem (S. Argyros - M. (2015)) There exists a hereditarily indecomposable Banach space X U that has separable dual and does not contain boundedly complete sequences. In particular is has no reflexive subspaces. (i) Every bounded linear operator T defined on X U is of the form T = λ I + S with S strictly singular. (ii) The ideal of strictly singular operators on X U is non-separable. (iii) The composition of any two strictly singular operators defined on X U is compact. (iv) Every bounded linear operator defined on the space has a non-trivial closed invariant subspace. give short explanation Furthermore, there exists a reflexive HI space X T satisfying (i), (ii), (iii), and (iv).
Some results of new saturation under constraints • A result based on this new method in a classical (Tsirelson, non- L ∞ ) setting. Theorem (S. Argyros - M. (2015)) There exists a hereditarily indecomposable Banach space X U that has separable dual and does not contain boundedly complete sequences. In particular is has no reflexive subspaces. (i) Every bounded linear operator T defined on X U is of the form T = λ I + S with S strictly singular. (ii) The ideal of strictly singular operators on X U is non-separable. (iii) The composition of any two strictly singular operators defined on X U is compact. (iv) Every bounded linear operator defined on the space has a non-trivial closed invariant subspace. give short explanation Furthermore, there exists a reflexive HI space X T satisfying (i), (ii), (iii), and (iv).
Some results of new saturation under constraints • A result based on this new method in a classical (Tsirelson, non- L ∞ ) setting. Theorem (S. Argyros - M. (2015)) There exists a hereditarily indecomposable Banach space X U that has separable dual and does not contain boundedly complete sequences. In particular is has no reflexive subspaces. (i) Every bounded linear operator T defined on X U is of the form T = λ I + S with S strictly singular. (ii) The ideal of strictly singular operators on X U is non-separable. (iii) The composition of any two strictly singular operators defined on X U is compact. (iv) Every bounded linear operator defined on the space has a non-trivial closed invariant subspace. give short explanation Furthermore, there exists a reflexive HI space X T satisfying (i), (ii), (iii), and (iv).
Some results of new saturation under constraints • A nice application: Definition Let X be a Banach space. The Calkin algebra of X is the quotient algebra C αℓ ( X ) = L ( X ) / K ( X ) . Theorem (R. Skillicorn (2015)) The Calkin algebra of X U is non-separable and it admits two non-equivalent complete algebra norms. • The above uses that C αℓ ( X U ) = ( R [ Id ]) ⊕ ( S ( X U ) / K ( X U )) and that for every [ S ] , [ T ] ∈ S ( X U ) / K ( X U ) we have [ S ][ T ] = 0.
Some results of new saturation under constraints • A nice application: Definition Let X be a Banach space. The Calkin algebra of X is the quotient algebra C αℓ ( X ) = L ( X ) / K ( X ) . Theorem (R. Skillicorn (2015)) The Calkin algebra of X U is non-separable and it admits two non-equivalent complete algebra norms. • The above uses that C αℓ ( X U ) = ( R [ Id ]) ⊕ ( S ( X U ) / K ( X U )) and that for every [ S ] , [ T ] ∈ S ( X U ) / K ( X U ) we have [ S ][ T ] = 0.
Some results of new saturation under constraints • A nice application: Definition Let X be a Banach space. The Calkin algebra of X is the quotient algebra C αℓ ( X ) = L ( X ) / K ( X ) . Theorem (R. Skillicorn (2015)) The Calkin algebra of X U is non-separable and it admits two non-equivalent complete algebra norms. • The above uses that C αℓ ( X U ) = ( R [ Id ]) ⊕ ( S ( X U ) / K ( X U )) and that for every [ S ] , [ T ] ∈ S ( X U ) / K ( X U ) we have [ S ][ T ] = 0.
Saturation under constraints - some history Method of saturation under constraints • Most important feature: allows heterogeneous asymptotic structure to appear hereditarily. • It was used for the first time by E. Odell and Th. Schlumprecht and it was further developed by S. Argyros and his collaborators (e.g. Beanland, Freeman, M. etc) • It can be used to address problems concerning local block structures, asymptotic structures, and operators defined on certain spaces.
Saturation under constraints - some history Method of saturation under constraints • Most important feature: allows heterogeneous asymptotic structure to appear hereditarily. • It was used for the first time by E. Odell and Th. Schlumprecht and it was further developed by S. Argyros and his collaborators (e.g. Beanland, Freeman, M. etc) • It can be used to address problems concerning local block structures, asymptotic structures, and operators defined on certain spaces.
Saturation under constraints - some history Method of saturation under constraints • Most important feature: allows heterogeneous asymptotic structure to appear hereditarily. • It was used for the first time by E. Odell and Th. Schlumprecht and it was further developed by S. Argyros and his collaborators (e.g. Beanland, Freeman, M. etc) • It can be used to address problems concerning local block structures, asymptotic structures, and operators defined on certain spaces.
Saturation under constraints - some history Method of saturation under constraints • Most important feature: allows heterogeneous asymptotic structure to appear hereditarily. • It was used for the first time by E. Odell and Th. Schlumprecht and it was further developed by S. Argyros and his collaborators (e.g. Beanland, Freeman, M. etc) • It can be used to address problems concerning local block structures, asymptotic structures, and operators defined on certain spaces.
Saturation under constraints - norming sets • A way of defining norms: norming sets • If X is a vector space and W ⊂ X # , W is called a norming set if the function � · � W : X → R with � x � W = sup {| f ( x ) | : f ∈ W } is a norm. • Commonly, X = c 00 ( N ) and W ⊂ c 00 ( N ) . • This makes sense: if x = ( a k ) k , f = ( b k ) k are in c 00 ( N ) then ∞ � f ( x ) = a k b k k = 1 (this is a finite sum).
Saturation under constraints - norming sets • A way of defining norms: norming sets • If X is a vector space and W ⊂ X # , W is called a norming set if the function � · � W : X → R with � x � W = sup {| f ( x ) | : f ∈ W } is a norm. • Commonly, X = c 00 ( N ) and W ⊂ c 00 ( N ) . • This makes sense: if x = ( a k ) k , f = ( b k ) k are in c 00 ( N ) then ∞ � f ( x ) = a k b k k = 1 (this is a finite sum).
Saturation under constraints - norming sets • A way of defining norms: norming sets • If X is a vector space and W ⊂ X # , W is called a norming set if the function � · � W : X → R with � x � W = sup {| f ( x ) | : f ∈ W } is a norm. • Commonly, X = c 00 ( N ) and W ⊂ c 00 ( N ) . • This makes sense: if x = ( a k ) k , f = ( b k ) k are in c 00 ( N ) then ∞ � f ( x ) = a k b k k = 1 (this is a finite sum).
Saturation under constraints - norming sets • A way of defining norms: norming sets • If X is a vector space and W ⊂ X # , W is called a norming set if the function � · � W : X → R with � x � W = sup {| f ( x ) | : f ∈ W } is a norm. • Commonly, X = c 00 ( N ) and W ⊂ c 00 ( N ) . • This makes sense: if x = ( a k ) k , f = ( b k ) k are in c 00 ( N ) then ∞ � f ( x ) = a k b k k = 1 (this is a finite sum).
Saturation under constraints - norming sets • A way of defining norms: norming sets • If X is a vector space and W ⊂ X # , W is called a norming set if the function � · � W : X → R with � x � W = sup {| f ( x ) | : f ∈ W } is a norm. • Commonly, X = c 00 ( N ) and W ⊂ c 00 ( N ) . • This makes sense: if x = ( a k ) k , f = ( b k ) k are in c 00 ( N ) then ∞ � f ( x ) = a k b k k = 1 (this is a finite sum).
Saturation under constraints - norming sets • Two vectors x , y are called successive ( x < y ) if max supp ( x ) < min supp ( y ) , where their supports are taken with respect to the unit vector basis of c 00 ( N ) • The Schreier familiers: S 1 = { F ⊂ N : # F � min ( F ) } S n + 1 = {∪ m k = 1 F k : m ∈ N and m � F 1 < · · · < F m } . • A sequence of vectors x 1 , x 2 , · · · , x m is S n -admissible if x 1 < x 2 < · · · < x m and { min supp ( x k ) : 1 � k � m } ∈ S n . • In particular, x 1 < · · · < x n are S 1 -admissible means n � min supp ( x 1 ) .
Saturation under constraints - norming sets • Two vectors x , y are called successive ( x < y ) if max supp ( x ) < min supp ( y ) , where their supports are taken with respect to the unit vector basis of c 00 ( N ) • The Schreier familiers: S 1 = { F ⊂ N : # F � min ( F ) } S n + 1 = {∪ m k = 1 F k : m ∈ N and m � F 1 < · · · < F m } . • A sequence of vectors x 1 , x 2 , · · · , x m is S n -admissible if x 1 < x 2 < · · · < x m and { min supp ( x k ) : 1 � k � m } ∈ S n . • In particular, x 1 < · · · < x n are S 1 -admissible means n � min supp ( x 1 ) .
Saturation under constraints - norming sets • Two vectors x , y are called successive ( x < y ) if max supp ( x ) < min supp ( y ) , where their supports are taken with respect to the unit vector basis of c 00 ( N ) • The Schreier familiers: S 1 = { F ⊂ N : # F � min ( F ) } S n + 1 = {∪ m k = 1 F k : m ∈ N and m � F 1 < · · · < F m } . • A sequence of vectors x 1 , x 2 , · · · , x m is S n -admissible if x 1 < x 2 < · · · < x m and { min supp ( x k ) : 1 � k � m } ∈ S n . • In particular, x 1 < · · · < x n are S 1 -admissible means n � min supp ( x 1 ) .
Saturation under constraints - norming sets • Two vectors x , y are called successive ( x < y ) if max supp ( x ) < min supp ( y ) , where their supports are taken with respect to the unit vector basis of c 00 ( N ) • The Schreier familiers: S 1 = { F ⊂ N : # F � min ( F ) } S n + 1 = {∪ m k = 1 F k : m ∈ N and m � F 1 < · · · < F m } . • A sequence of vectors x 1 , x 2 , · · · , x m is S n -admissible if x 1 < x 2 < · · · < x m and { min supp ( x k ) : 1 � k � m } ∈ S n . • In particular, x 1 < · · · < x n are S 1 -admissible means n � min supp ( x 1 ) .
Saturation under constraints - norming sets • Two vectors x , y are called successive ( x < y ) if max supp ( x ) < min supp ( y ) , where their supports are taken with respect to the unit vector basis of c 00 ( N ) • The Schreier familiers: S 1 = { F ⊂ N : # F � min ( F ) } S n + 1 = {∪ m k = 1 F k : m ∈ N and m � F 1 < · · · < F m } . • A sequence of vectors x 1 , x 2 , · · · , x m is S n -admissible if x 1 < x 2 < · · · < x m and { min supp ( x k ) : 1 � k � m } ∈ S n . • In particular, x 1 < · · · < x n are S 1 -admissible means n � min supp ( x 1 ) .
Saturation under constraints - norming sets • Two vectors x , y are called successive ( x < y ) if max supp ( x ) < min supp ( y ) , where their supports are taken with respect to the unit vector basis of c 00 ( N ) • The Schreier familiers: S 1 = { F ⊂ N : # F � min ( F ) } S n + 1 = {∪ m k = 1 F k : m ∈ N and m � F 1 < · · · < F m } . • A sequence of vectors x 1 , x 2 , · · · , x m is S n -admissible if x 1 < x 2 < · · · < x m and { min supp ( x k ) : 1 � k � m } ∈ S n . • In particular, x 1 < · · · < x n are S 1 -admissible means n � min supp ( x 1 ) .
Saturated norms - Tsirelson space • First, a typical norming set. • We define W T = ∪ ∞ n = 0 W n where W 0 ⊂ W 1 ⊂ W 2 ⊂ · · · . Set W 0 = {± e i : i ∈ N } and � m 1 � f k : d ∈ N and ( f k ) m W n + 1 = W n ∪ k = 1 is an 2 k = 1 � � draw � S 1 -admissible sequence of W n . picture • If T is the completion of c 00 ( N ) endowed with � · � W T , then T is Tsirelson space.
Saturated norms - Tsirelson space • First, a typical norming set. • We define W T = ∪ ∞ n = 0 W n where W 0 ⊂ W 1 ⊂ W 2 ⊂ · · · . Set W 0 = {± e i : i ∈ N } and � m 1 � f k : d ∈ N and ( f k ) m W n + 1 = W n ∪ k = 1 is an 2 k = 1 � � draw � S 1 -admissible sequence of W n . picture • If T is the completion of c 00 ( N ) endowed with � · � W T , then T is Tsirelson space.
Saturated norms - Tsirelson space • First, a typical norming set. • We define W T = ∪ ∞ n = 0 W n where W 0 ⊂ W 1 ⊂ W 2 ⊂ · · · . Set W 0 = {± e i : i ∈ N } and � m 1 � f k : d ∈ N and ( f k ) m W n + 1 = W n ∪ k = 1 is an 2 k = 1 � � draw � S 1 -admissible sequence of W n . picture • If T is the completion of c 00 ( N ) endowed with � · � W T , then T is Tsirelson space.
Saturated norms - Tsirelson space • First, a typical norming set. • We define W T = ∪ ∞ n = 0 W n where W 0 ⊂ W 1 ⊂ W 2 ⊂ · · · . Set W 0 = {± e i : i ∈ N } and � m 1 � f k : d ∈ N and ( f k ) m W n + 1 = W n ∪ k = 1 is an 2 k = 1 � � draw � S 1 -admissible sequence of W n . picture • If T is the completion of c 00 ( N ) endowed with � · � W T , then T is Tsirelson space.
Saturated norms - Tsirelson space • Such constructions always yield boundedly complete bases, e.g. Tsirelson space has the stronger property of being asymptotic- ℓ 1 . • That is, for S 1 -admissible vector x 1 < · · · < x m : � � m m � 1 � � � � x k � x k � T . � � � � 2 � � k = 1 k = 1 T • Because, for any S 1 -admissible f 1 < · · · < f m in W we have ( 1 / 2 ) � m k = 1 f k is in W . (picture)
Saturated norms - Tsirelson space • Such constructions always yield boundedly complete bases, e.g. Tsirelson space has the stronger property of being asymptotic- ℓ 1 . • That is, for S 1 -admissible vector x 1 < · · · < x m : � � m m � 1 � � � � x k � x k � T . � � � � 2 � � k = 1 k = 1 T • Because, for any S 1 -admissible f 1 < · · · < f m in W we have ( 1 / 2 ) � m k = 1 f k is in W . (picture)
Saturated norms - Tsirelson space • Such constructions always yield boundedly complete bases, e.g. Tsirelson space has the stronger property of being asymptotic- ℓ 1 . • That is, for S 1 -admissible vector x 1 < · · · < x m : � � m m � 1 � � � � x k � x k � T . � � � � 2 � � k = 1 k = 1 T • Because, for any S 1 -admissible f 1 < · · · < f m in W we have ( 1 / 2 ) � m k = 1 f k is in W . (picture)
Saturation under constraints - the norming set W α • We now define a simple norming set saturated under constraints. • Let G be a subset of c 00 ( N ) . A vector α 0 ∈ c 00 ( N ) is called an α -average of G of size s ( α 0 ) = n if there are f 1 < · · · < f n in G with α 0 = 1 n ( f 1 + · · · + f n ) . • A sequence of α -averages α 1 < α 2 < · · · is called very fast growing if s ( α k + 1 ) > max supp ( α k ) .
Saturation under constraints - the norming set W α • We now define a simple norming set saturated under constraints. • Let G be a subset of c 00 ( N ) . A vector α 0 ∈ c 00 ( N ) is called an α -average of G of size s ( α 0 ) = n if there are f 1 < · · · < f n in G with α 0 = 1 n ( f 1 + · · · + f n ) . • A sequence of α -averages α 1 < α 2 < · · · is called very fast growing if s ( α k + 1 ) > max supp ( α k ) .
Saturation under constraints - the norming set W α • We now define a simple norming set saturated under constraints. • Let G be a subset of c 00 ( N ) . A vector α 0 ∈ c 00 ( N ) is called an α -average of G of size s ( α 0 ) = n if there are f 1 < · · · < f n in G with α 0 = 1 n ( f 1 + · · · + f n ) . • A sequence of α -averages α 1 < α 2 < · · · is called very fast growing if s ( α k + 1 ) > max supp ( α k ) .
Saturation under constraints - the norming set W α • We define W α = ∪ ∞ n = 0 W α n where W α 0 ⊂ W α 1 ⊂ W α 2 ⊂ · · · . Set W α 0 = {± e i : i ∈ N } and � m 1 � W α n + 1 = W α α k : d ∈ N and ( α k ) m n ∪ k = 1 is an S d -admissible 2 d k = 1 � sequence of very fast growing α -averages of W n . • If T ( 1 / 2 n , S n ,α ) is the completion of c 00 ( N ) endowed with � · � W , then T ( 1 / 2 n , S n ,α ) is a (Tsirelson-type) space saturated under constraints. Remark: for any d ∈ N and S d -admissible and very fast growing α -averages α 1 < · · · < α m in W α , f = ( 1 / 2 d ) � m k = 1 α k is in W α . Such an f is said to have weight d .
Saturation under constraints - the norming set W α • We define W α = ∪ ∞ n = 0 W α n where W α 0 ⊂ W α 1 ⊂ W α 2 ⊂ · · · . Set W α 0 = {± e i : i ∈ N } and � m 1 � W α n + 1 = W α α k : d ∈ N and ( α k ) m n ∪ k = 1 is an S d -admissible 2 d k = 1 � sequence of very fast growing α -averages of W n . • If T ( 1 / 2 n , S n ,α ) is the completion of c 00 ( N ) endowed with � · � W , then T ( 1 / 2 n , S n ,α ) is a (Tsirelson-type) space saturated under constraints. Remark: for any d ∈ N and S d -admissible and very fast growing α -averages α 1 < · · · < α m in W α , f = ( 1 / 2 d ) � m k = 1 α k is in W α . Such an f is said to have weight d .
Saturation under constraints - the norming set W α • We define W α = ∪ ∞ n = 0 W α n where W α 0 ⊂ W α 1 ⊂ W α 2 ⊂ · · · . Set W α 0 = {± e i : i ∈ N } and � m 1 � W α n + 1 = W α α k : d ∈ N and ( α k ) m n ∪ k = 1 is an S d -admissible 2 d k = 1 � sequence of very fast growing α -averages of W n . • If T ( 1 / 2 n , S n ,α ) is the completion of c 00 ( N ) endowed with � · � W , then T ( 1 / 2 n , S n ,α ) is a (Tsirelson-type) space saturated under constraints. Remark: for any d ∈ N and S d -admissible and very fast growing α -averages α 1 < · · · < α m in W α , f = ( 1 / 2 d ) � m k = 1 α k is in W α . Such an f is said to have weight d .
Saturation under constraints - the norming set W α • We define W α = ∪ ∞ n = 0 W α n where W α 0 ⊂ W α 1 ⊂ W α 2 ⊂ · · · . Set W α 0 = {± e i : i ∈ N } and � m 1 � W α n + 1 = W α α k : d ∈ N and ( α k ) m n ∪ k = 1 is an S d -admissible 2 d k = 1 � sequence of very fast growing α -averages of W n . • If T ( 1 / 2 n , S n ,α ) is the completion of c 00 ( N ) endowed with � · � W , then T ( 1 / 2 n , S n ,α ) is a (Tsirelson-type) space saturated under constraints. Remark: for any d ∈ N and S d -admissible and very fast growing α -averages α 1 < · · · < α m in W α , f = ( 1 / 2 d ) � m k = 1 α k is in W α . Such an f is said to have weight d .
Saturation under constraints - the norming set W α • We define W α = ∪ ∞ n = 0 W α n where W α 0 ⊂ W α 1 ⊂ W α 2 ⊂ · · · . Set W α 0 = {± e i : i ∈ N } and � m 1 � W α n + 1 = W α α k : d ∈ N and ( α k ) m n ∪ k = 1 is an S d -admissible 2 d k = 1 � sequence of very fast growing α -averages of W n . • If T ( 1 / 2 n , S n ,α ) is the completion of c 00 ( N ) endowed with � · � W , then T ( 1 / 2 n , S n ,α ) is a (Tsirelson-type) space saturated under constraints. Remark: for any d ∈ N and S d -admissible and very fast growing α -averages α 1 < · · · < α m in W α , f = ( 1 / 2 d ) � m k = 1 α k is in W α . Such an f is said to have weight d .
Saturation under constraints - the space T ( 1 / 2 n , S n ,α ) • The space T ( 1 / 2 n , S n ,α ) is reflexive and it has an unconditional basis (that is boundedly complete) . • Every normalized block sequence in T α has a subsequence that admits an ℓ 1 spreading model or a subsequence that admits a c 0 -spreading model. Both of these types of sequences exist in every block subspace of T ( 1 / 2 n , S n ,α ) . Definition Let ( x k ) k be a sequence in some Banach space. We say that ( x k ) k generates a c 0 spreading model if there exists C � 1 so that for every n � k 1 < · · · < k n the sequence ( x k i ) n i = 1 is C -equivalent to the unit vector basis of ℓ n ∞ . Other spreading models, e.g. ℓ 1 or the summing basis of c 0 are similarly defined.
Saturation under constraints - the space T ( 1 / 2 n , S n ,α ) • The space T ( 1 / 2 n , S n ,α ) is reflexive and it has an unconditional basis (that is boundedly complete) . • Every normalized block sequence in T α has a subsequence that admits an ℓ 1 spreading model or a subsequence that admits a c 0 -spreading model. Both of these types of sequences exist in every block subspace of T ( 1 / 2 n , S n ,α ) . Definition Let ( x k ) k be a sequence in some Banach space. We say that ( x k ) k generates a c 0 spreading model if there exists C � 1 so that for every n � k 1 < · · · < k n the sequence ( x k i ) n i = 1 is C -equivalent to the unit vector basis of ℓ n ∞ . Other spreading models, e.g. ℓ 1 or the summing basis of c 0 are similarly defined.
Saturation under constraints - the space T ( 1 / 2 n , S n ,α ) • The space T ( 1 / 2 n , S n ,α ) is reflexive and it has an unconditional basis (that is boundedly complete) . • Every normalized block sequence in T α has a subsequence that admits an ℓ 1 spreading model or a subsequence that admits a c 0 -spreading model. Both of these types of sequences exist in every block subspace of T ( 1 / 2 n , S n ,α ) . Definition Let ( x k ) k be a sequence in some Banach space. We say that ( x k ) k generates a c 0 spreading model if there exists C � 1 so that for every n � k 1 < · · · < k n the sequence ( x k i ) n i = 1 is C -equivalent to the unit vector basis of ℓ n ∞ . Other spreading models, e.g. ℓ 1 or the summing basis of c 0 are similarly defined.
Saturation under constraints - new method • To avoid having a boundedly complete basis, the new variation of saturation under constraints restricts the choice of averages allowed to be used in the norming set. • The choice is made with the help of an appropriate tree. • Denote Q the set of all finite sequences ( f k , x k ) n k = 1 with f 1 < · · · < f n ∈ W α and x k vectors in c 00 ( N ) with rational coefficients. • Using a coding function, choose a subtree U of Q so that for all ( f k , x k ) n k = 1 in U , the weight of f n uniquely determines the sequence ( f k , x k ) n − 1 k = 1 . • This tree is ill-founded, every maximal chain is infinite. It is used to define a space without reflexive subspaces. Remark: Appropriate well-founded subtrees of U that are well founded lead to reflexive spaces.
Saturation under constraints - new method • To avoid having a boundedly complete basis, the new variation of saturation under constraints restricts the choice of averages allowed to be used in the norming set. • The choice is made with the help of an appropriate tree. • Denote Q the set of all finite sequences ( f k , x k ) n k = 1 with f 1 < · · · < f n ∈ W α and x k vectors in c 00 ( N ) with rational coefficients. • Using a coding function, choose a subtree U of Q so that for all ( f k , x k ) n k = 1 in U , the weight of f n uniquely determines the sequence ( f k , x k ) n − 1 k = 1 . • This tree is ill-founded, every maximal chain is infinite. It is used to define a space without reflexive subspaces. Remark: Appropriate well-founded subtrees of U that are well founded lead to reflexive spaces.
Saturation under constraints - new method • To avoid having a boundedly complete basis, the new variation of saturation under constraints restricts the choice of averages allowed to be used in the norming set. • The choice is made with the help of an appropriate tree. • Denote Q the set of all finite sequences ( f k , x k ) n k = 1 with f 1 < · · · < f n ∈ W α and x k vectors in c 00 ( N ) with rational coefficients. • Using a coding function, choose a subtree U of Q so that for all ( f k , x k ) n k = 1 in U , the weight of f n uniquely determines the sequence ( f k , x k ) n − 1 k = 1 . • This tree is ill-founded, every maximal chain is infinite. It is used to define a space without reflexive subspaces. Remark: Appropriate well-founded subtrees of U that are well founded lead to reflexive spaces.
Saturation under constraints - new method • To avoid having a boundedly complete basis, the new variation of saturation under constraints restricts the choice of averages allowed to be used in the norming set. • The choice is made with the help of an appropriate tree. • Denote Q the set of all finite sequences ( f k , x k ) n k = 1 with f 1 < · · · < f n ∈ W α and x k vectors in c 00 ( N ) with rational coefficients. • Using a coding function, choose a subtree U of Q so that for all ( f k , x k ) n k = 1 in U , the weight of f n uniquely determines the sequence ( f k , x k ) n − 1 k = 1 . • This tree is ill-founded, every maximal chain is infinite. It is used to define a space without reflexive subspaces. Remark: Appropriate well-founded subtrees of U that are well founded lead to reflexive spaces.
Saturation under constraints - new method • To avoid having a boundedly complete basis, the new variation of saturation under constraints restricts the choice of averages allowed to be used in the norming set. • The choice is made with the help of an appropriate tree. • Denote Q the set of all finite sequences ( f k , x k ) n k = 1 with f 1 < · · · < f n ∈ W α and x k vectors in c 00 ( N ) with rational coefficients. • Using a coding function, choose a subtree U of Q so that for all ( f k , x k ) n k = 1 in U , the weight of f n uniquely determines the sequence ( f k , x k ) n − 1 k = 1 . • This tree is ill-founded, every maximal chain is infinite. It is used to define a space without reflexive subspaces. Remark: Appropriate well-founded subtrees of U that are well founded lead to reflexive spaces.
Saturation under constraints - new method • To avoid having a boundedly complete basis, the new variation of saturation under constraints restricts the choice of averages allowed to be used in the norming set. • The choice is made with the help of an appropriate tree. • Denote Q the set of all finite sequences ( f k , x k ) n k = 1 with f 1 < · · · < f n ∈ W α and x k vectors in c 00 ( N ) with rational coefficients. • Using a coding function, choose a subtree U of Q so that for all ( f k , x k ) n k = 1 in U , the weight of f n uniquely determines the sequence ( f k , x k ) n − 1 k = 1 . • This tree is ill-founded, every maximal chain is infinite. It is used to define a space without reflexive subspaces. Remark: Appropriate well-founded subtrees of U that are well founded lead to reflexive spaces.
Saturation under constraints - new method • Given G ⊂ W α , the tree U defines four types of α -averages, built on elements of G and collectively called α c -averages of G . • One example: if ( f k , x k ) n k = 1 is in U so that f k ∈ G and f k ( x k ) = f m ( x m ) for 1 � k � m � n , then n α 0 = 1 � ( − 1 ) k f k n k = 1 is an α c -average of G . Specifically, averages similar to the above are called compatible averages of G . Comment: this type of averages will provide HI structure to the space and not special functionals. • Other types of α c -averages if G are built on elements f 1 < · · · < f n of G that are “incomparable” or “irrelevant” with respect to U .
Saturation under constraints - new method • Given G ⊂ W α , the tree U defines four types of α -averages, built on elements of G and collectively called α c -averages of G . • One example: if ( f k , x k ) n k = 1 is in U so that f k ∈ G and f k ( x k ) = f m ( x m ) for 1 � k � m � n , then n α 0 = 1 � ( − 1 ) k f k n k = 1 is an α c -average of G . Specifically, averages similar to the above are called compatible averages of G . Comment: this type of averages will provide HI structure to the space and not special functionals. • Other types of α c -averages if G are built on elements f 1 < · · · < f n of G that are “incomparable” or “irrelevant” with respect to U .
Saturation under constraints - new method • Given G ⊂ W α , the tree U defines four types of α -averages, built on elements of G and collectively called α c -averages of G . • One example: if ( f k , x k ) n k = 1 is in U so that f k ∈ G and f k ( x k ) = f m ( x m ) for 1 � k � m � n , then n α 0 = 1 � ( − 1 ) k f k n k = 1 is an α c -average of G . Specifically, averages similar to the above are called compatible averages of G . Comment: this type of averages will provide HI structure to the space and not special functionals. • Other types of α c -averages if G are built on elements f 1 < · · · < f n of G that are “incomparable” or “irrelevant” with respect to U .
Saturation under constraints - new method • Given G ⊂ W α , the tree U defines four types of α -averages, built on elements of G and collectively called α c -averages of G . • One example: if ( f k , x k ) n k = 1 is in U so that f k ∈ G and f k ( x k ) = f m ( x m ) for 1 � k � m � n , then n α 0 = 1 � ( − 1 ) k f k n k = 1 is an α c -average of G . Specifically, averages similar to the above are called compatible averages of G . Comment: this type of averages will provide HI structure to the space and not special functionals. • Other types of α c -averages if G are built on elements f 1 < · · · < f n of G that are “incomparable” or “irrelevant” with respect to U .
Saturation under constraints - new method • Given G ⊂ W α , the tree U defines four types of α -averages, built on elements of G and collectively called α c -averages of G . • One example: if ( f k , x k ) n k = 1 is in U so that f k ∈ G and f k ( x k ) = f m ( x m ) for 1 � k � m � n , then n α 0 = 1 � ( − 1 ) k f k n k = 1 is an α c -average of G . Specifically, averages similar to the above are called compatible averages of G . Comment: this type of averages will provide HI structure to the space and not special functionals. • Other types of α c -averages if G are built on elements f 1 < · · · < f n of G that are “incomparable” or “irrelevant” with respect to U .
Saturation under constraints - new method • We choose a norming set W U to be the smallest subset of W α that • contains the unit vectors e i and is symmetric and • for any n ∈ N and S n -admissible and very fast growing sequence ( α q ) d q = 1 of α c -averages of W U the functional d f = 1 � α q 2 n q = 1 is in W U . • We denote the completion of ( c 00 ( N ) , � · � W U ) as X U .
Saturation under constraints - new method • We choose a norming set W U to be the smallest subset of W α that • contains the unit vectors e i and is symmetric and • for any n ∈ N and S n -admissible and very fast growing sequence ( α q ) d q = 1 of α c -averages of W U the functional d f = 1 � α q 2 n q = 1 is in W U . • We denote the completion of ( c 00 ( N ) , � · � W U ) as X U .
Saturation under constraints - new method • We choose a norming set W U to be the smallest subset of W α that • contains the unit vectors e i and is symmetric and • for any n ∈ N and S n -admissible and very fast growing sequence ( α q ) d q = 1 of α c -averages of W U the functional d f = 1 � α q 2 n q = 1 is in W U . • We denote the completion of ( c 00 ( N ) , � · � W U ) as X U .
Saturation under constraints - new method • We choose a norming set W U to be the smallest subset of W α that • contains the unit vectors e i and is symmetric and • for any n ∈ N and S n -admissible and very fast growing sequence ( α q ) d q = 1 of α c -averages of W U the functional d f = 1 � α q 2 n q = 1 is in W U . • We denote the completion of ( c 00 ( N ) , � · � W U ) as X U .
Saturation under constraints - new method • Properties of the space X U ◮ It is hereditarily indecomposable, it has separable dual, and it has no boundedly complete sequences. ◮ Every Schauder basic sequence in X U admits either c 0 , or ℓ 1 , or the summing basis of c 0 as a spreading model. All three type of spreading models appear in every subspace of X U . ◮ For every bounded linear operator T defined on X U there is a scalar λ so that S = λ I − T is weakly compact and hence strictly singular. ◮ The composition of any two strictly singular operators defined on X U is compact. ◮ Every bounded linear operator defined on the space has a non-trivial closed invariant subspace.
Saturation under constraints - new method • Properties of the space X U ◮ It is hereditarily indecomposable, it has separable dual, and it has no boundedly complete sequences. ◮ Every Schauder basic sequence in X U admits either c 0 , or ℓ 1 , or the summing basis of c 0 as a spreading model. All three type of spreading models appear in every subspace of X U . ◮ For every bounded linear operator T defined on X U there is a scalar λ so that S = λ I − T is weakly compact and hence strictly singular. ◮ The composition of any two strictly singular operators defined on X U is compact. ◮ Every bounded linear operator defined on the space has a non-trivial closed invariant subspace.
Saturation under constraints - new method • Properties of the space X U ◮ It is hereditarily indecomposable, it has separable dual, and it has no boundedly complete sequences. ◮ Every Schauder basic sequence in X U admits either c 0 , or ℓ 1 , or the summing basis of c 0 as a spreading model. All three type of spreading models appear in every subspace of X U . ◮ For every bounded linear operator T defined on X U there is a scalar λ so that S = λ I − T is weakly compact and hence strictly singular. ◮ The composition of any two strictly singular operators defined on X U is compact. ◮ Every bounded linear operator defined on the space has a non-trivial closed invariant subspace.
Saturation under constraints - new method • Properties of the space X U ◮ It is hereditarily indecomposable, it has separable dual, and it has no boundedly complete sequences. ◮ Every Schauder basic sequence in X U admits either c 0 , or ℓ 1 , or the summing basis of c 0 as a spreading model. All three type of spreading models appear in every subspace of X U . ◮ For every bounded linear operator T defined on X U there is a scalar λ so that S = λ I − T is weakly compact and hence strictly singular. ◮ The composition of any two strictly singular operators defined on X U is compact. ◮ Every bounded linear operator defined on the space has a non-trivial closed invariant subspace.
Saturation under constraints - new method • Properties of the space X U ◮ It is hereditarily indecomposable, it has separable dual, and it has no boundedly complete sequences. ◮ Every Schauder basic sequence in X U admits either c 0 , or ℓ 1 , or the summing basis of c 0 as a spreading model. All three type of spreading models appear in every subspace of X U . ◮ For every bounded linear operator T defined on X U there is a scalar λ so that S = λ I − T is weakly compact and hence strictly singular. ◮ The composition of any two strictly singular operators defined on X U is compact. ◮ Every bounded linear operator defined on the space has a non-trivial closed invariant subspace.
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