Closed ideals in L ( p q ) . Andr as Zs ak Peterhouse, - PowerPoint PPT Presentation

Closed ideals in L ( ℓ p ⊕ ℓ q ) . Andr´ as Zs´ ak Peterhouse, Cambridge (Joint work with Thomas Schlumprecht.) August 2014, Maresias, Brazil

Some old results Calkin [1941]: { 0 } � K ( ℓ 2 ) � L ( ℓ 2 ).

Some old results Calkin [1941]: { 0 } � K ( ℓ 2 ) � L ( ℓ 2 ). Gohberg, Markus, Feldman [1960]: Same holds for ℓ p , 1 ≤ p < ∞ , and c 0 .

Some old results Calkin [1941]: { 0 } � K ( ℓ 2 ) � L ( ℓ 2 ). Gohberg, Markus, Feldman [1960]: Same holds for ℓ p , 1 ≤ p < ∞ , and c 0 . If T ∈ L ( ℓ p ) \ K ( ℓ p ), then Id ℓ p = ATB for some A , B ∈ L ( ℓ p ).

Some old results Calkin [1941]: { 0 } � K ( ℓ 2 ) � L ( ℓ 2 ). Gohberg, Markus, Feldman [1960]: Same holds for ℓ p , 1 ≤ p < ∞ , and c 0 . If T ∈ L ( ℓ p ) \ K ( ℓ p ), then Id ℓ p = ATB for some A , B ∈ L ( ℓ p ). It is natural to consider L ( ℓ p ⊕ ℓ q ). Early results by Pietsch and Milman in 70s.

Some old results Calkin [1941]: { 0 } � K ( ℓ 2 ) � L ( ℓ 2 ). Gohberg, Markus, Feldman [1960]: Same holds for ℓ p , 1 ≤ p < ∞ , and c 0 . If T ∈ L ( ℓ p ) \ K ( ℓ p ), then Id ℓ p = ATB for some A , B ∈ L ( ℓ p ). It is natural to consider L ( ℓ p ⊕ ℓ q ). Early results by Pietsch and Milman in 70s. Pietsch [Operator Ideals, 1977]: L ( ℓ p ⊕ ℓ q ) has exactly two maximal ideals, and all other proper, closed ideals are in one-to-one correspondence with closed ideals in L ( ℓ p , ℓ q ).

Some old results Calkin [1941]: { 0 } � K ( ℓ 2 ) � L ( ℓ 2 ). Gohberg, Markus, Feldman [1960]: Same holds for ℓ p , 1 ≤ p < ∞ , and c 0 . If T ∈ L ( ℓ p ) \ K ( ℓ p ), then Id ℓ p = ATB for some A , B ∈ L ( ℓ p ). It is natural to consider L ( ℓ p ⊕ ℓ q ). Early results by Pietsch and Milman in 70s. Pietsch [Operator Ideals, 1977]: L ( ℓ p ⊕ ℓ q ) has exactly two maximal ideals, and all other proper, closed ideals are in one-to-one correspondence with closed ideals in L ( ℓ p , ℓ q ). Question (Pietsch): Are there infinitely many closed ideals in L ( ℓ p , ℓ q )?

Some old results Calkin [1941]: { 0 } � K ( ℓ 2 ) � L ( ℓ 2 ). Gohberg, Markus, Feldman [1960]: Same holds for ℓ p , 1 ≤ p < ∞ , and c 0 . If T ∈ L ( ℓ p ) \ K ( ℓ p ), then Id ℓ p = ATB for some A , B ∈ L ( ℓ p ). It is natural to consider L ( ℓ p ⊕ ℓ q ). Early results by Pietsch and Milman in 70s. Pietsch [Operator Ideals, 1977]: L ( ℓ p ⊕ ℓ q ) has exactly two maximal ideals, and all other proper, closed ideals are in one-to-one correspondence with closed ideals in L ( ℓ p , ℓ q ). Question (Pietsch): Are there infinitely many closed ideals in L ( ℓ p , ℓ q )? Answer (Schlumprecht, Z): Yes for 1 < p < q < ∞ .

Some definitions

Some definitions An ideal in L ( X , Y ) is a subspace J of L ( X , Y ) such that ATB ∈ J whenever T ∈ J and A ∈ L ( Y ), B ∈ L ( X ).

Some definitions An ideal in L ( X , Y ) is a subspace J of L ( X , Y ) such that ATB ∈ J whenever T ∈ J and A ∈ L ( Y ), B ∈ L ( X ). T : X → Y is strictly singular if for all Z ⊂ X with dim Z = ∞ and for all ε > 0, there exists x ∈ Z such that � Tx � < ε � x � .

Some definitions An ideal in L ( X , Y ) is a subspace J of L ( X , Y ) such that ATB ∈ J whenever T ∈ J and A ∈ L ( Y ), B ∈ L ( X ). T : X → Y is strictly singular if for all Z ⊂ X with dim Z = ∞ and for all ε > 0, there exists x ∈ Z such that � Tx � < ε � x � . T is finitely strictly singular if for all ε > 0 there exists n ∈ N such that for all E ⊂ X with dim E ≥ n , there exists x ∈ E such that � Tx � < ε � x � .

Some definitions An ideal in L ( X , Y ) is a subspace J of L ( X , Y ) such that ATB ∈ J whenever T ∈ J and A ∈ L ( Y ), B ∈ L ( X ). T : X → Y is strictly singular if for all Z ⊂ X with dim Z = ∞ and for all ε > 0, there exists x ∈ Z such that � Tx � < ε � x � . T is finitely strictly singular if for all ε > 0 there exists n ∈ N such that for all E ⊂ X with dim E ≥ n , there exists x ∈ E such that � Tx � < ε � x � . { 0 } � K ( X , Y ) ⊂ FS ( X , Y ) ⊂ S ( X , Y ) ⊂ L ( X , Y ) .

Some definitions An ideal in L ( X , Y ) is a subspace J of L ( X , Y ) such that ATB ∈ J whenever T ∈ J and A ∈ L ( Y ), B ∈ L ( X ). T : X → Y is strictly singular if for all Z ⊂ X with dim Z = ∞ and for all ε > 0, there exists x ∈ Z such that � Tx � < ε � x � . T is finitely strictly singular if for all ε > 0 there exists n ∈ N such that for all E ⊂ X with dim E ≥ n , there exists x ∈ E such that � Tx � < ε � x � . { 0 } � K ( X , Y ) ⊂ FS ( X , Y ) ⊂ S ( X , Y ) ⊂ L ( X , Y ) . Fix T : U → V . For any X , Y we let J T = J T ( X , Y ) be the closed ideal in L ( X , Y ) generated by operators fectoring through T .

Some definitions An ideal in L ( X , Y ) is a subspace J of L ( X , Y ) such that ATB ∈ J whenever T ∈ J and A ∈ L ( Y ), B ∈ L ( X ). T : X → Y is strictly singular if for all Z ⊂ X with dim Z = ∞ and for all ε > 0, there exists x ∈ Z such that � Tx � < ε � x � . T is finitely strictly singular if for all ε > 0 there exists n ∈ N such that for all E ⊂ X with dim E ≥ n , there exists x ∈ E such that � Tx � < ε � x � . { 0 } � K ( X , Y ) ⊂ FS ( X , Y ) ⊂ S ( X , Y ) ⊂ L ( X , Y ) . Fix T : U → V . For any X , Y we let J T = J T ( X , Y ) be the closed ideal in L ( X , Y ) generated by operators fectoring through T . J T ( X , Y ) = span { ATB : A ∈ L ( V , Y ) , B ∈ L ( X , U ) } .

Some definitions An ideal in L ( X , Y ) is a subspace J of L ( X , Y ) such that ATB ∈ J whenever T ∈ J and A ∈ L ( Y ), B ∈ L ( X ). T : X → Y is strictly singular if for all Z ⊂ X with dim Z = ∞ and for all ε > 0, there exists x ∈ Z such that � Tx � < ε � x � . T is finitely strictly singular if for all ε > 0 there exists n ∈ N such that for all E ⊂ X with dim E ≥ n , there exists x ∈ E such that � Tx � < ε � x � . { 0 } � K ( X , Y ) ⊂ FS ( X , Y ) ⊂ S ( X , Y ) ⊂ L ( X , Y ) . Fix T : U → V . For any X , Y we let J T = J T ( X , Y ) be the closed ideal in L ( X , Y ) generated by operators fectoring through T . J T ( X , Y ) = span { ATB : A ∈ L ( V , Y ) , B ∈ L ( X , U ) } . If U = V and T = Id U , then J U = J Id U .

Closed ideals in L ( ℓ p ⊕ ℓ q )

Closed ideals in L ( ℓ p ⊕ ℓ q ) Recall: if T ∈ L ( ℓ p ) \ K ( ℓ p ), then Id ℓ p = ATB .

Closed ideals in L ( ℓ p ⊕ ℓ q ) Recall: if T ∈ L ( ℓ p ) \ K ( ℓ p ), then Id ℓ p = ATB . � T 11 � T 12 T ∈ L ( ℓ p ⊕ ℓ q ) is a matrix T = , where T 11 ∈ L ( ℓ p ), T 22 ∈ L ( ℓ q ), T 21 T 22

Closed ideals in L ( ℓ p ⊕ ℓ q ) Recall: if T ∈ L ( ℓ p ) \ K ( ℓ p ), then Id ℓ p = ATB . � T 11 � T 12 T ∈ L ( ℓ p ⊕ ℓ q ) is a matrix T = , where T 11 ∈ L ( ℓ p ), T 22 ∈ L ( ℓ q ), T 21 T 22 T 12 ∈ L ( ℓ q , ℓ p ) = K ( ℓ q , ℓ p ),

Closed ideals in L ( ℓ p ⊕ ℓ q ) Recall: if T ∈ L ( ℓ p ) \ K ( ℓ p ), then Id ℓ p = ATB . � T 11 � T 12 T ∈ L ( ℓ p ⊕ ℓ q ) is a matrix T = , where T 11 ∈ L ( ℓ p ), T 22 ∈ L ( ℓ q ), T 21 T 22 T 12 ∈ L ( ℓ q , ℓ p ) = K ( ℓ q , ℓ p ), T 12 ∈ L ( ℓ p , ℓ q ) = S ( ℓ p , ℓ q ).

Closed ideals in L ( ℓ p ⊕ ℓ q ) Recall: if T ∈ L ( ℓ p ) \ K ( ℓ p ), then Id ℓ p = ATB . � T 11 � T 12 T ∈ L ( ℓ p ⊕ ℓ q ) is a matrix T = , where T 11 ∈ L ( ℓ p ), T 22 ∈ L ( ℓ q ), T 21 T 22 T 12 ∈ L ( ℓ q , ℓ p ) = K ( ℓ q , ℓ p ), T 12 ∈ L ( ℓ p , ℓ q ) = S ( ℓ p , ℓ q ). Two maximal ideals: { T : T 22 ∈ K ( ℓ q ) } = J ℓ p { T : T 11 ∈ K ( ℓ p ) } = J ℓ q

Closed ideals in L ( ℓ p ⊕ ℓ q ) Recall: if T ∈ L ( ℓ p ) \ K ( ℓ p ), then Id ℓ p = ATB . � T 11 � T 12 T ∈ L ( ℓ p ⊕ ℓ q ) is a matrix T = , where T 11 ∈ L ( ℓ p ), T 22 ∈ L ( ℓ q ), T 21 T 22 T 12 ∈ L ( ℓ q , ℓ p ) = K ( ℓ q , ℓ p ), T 12 ∈ L ( ℓ p , ℓ q ) = S ( ℓ p , ℓ q ). Two maximal ideals: { T : T 22 ∈ K ( ℓ q ) } = J ℓ p { T : T 11 ∈ K ( ℓ p ) } = J ℓ q Other closed, proper ideals: { T : T jj ∈ K , T 21 ∈ J } where J is a closed ideal in L ( ℓ p , ℓ q ).

Closed ideals in L ( ℓ p ⊕ ℓ q ) Recall: if T ∈ L ( ℓ p ) \ K ( ℓ p ), then Id ℓ p = ATB . � T 11 � T 12 T ∈ L ( ℓ p ⊕ ℓ q ) is a matrix T = , where T 11 ∈ L ( ℓ p ), T 22 ∈ L ( ℓ q ), T 21 T 22 T 12 ∈ L ( ℓ q , ℓ p ) = K ( ℓ q , ℓ p ), T 12 ∈ L ( ℓ p , ℓ q ) = S ( ℓ p , ℓ q ). Two maximal ideals: { T : T 22 ∈ K ( ℓ q ) } = J ℓ p { T : T 11 ∈ K ( ℓ p ) } = J ℓ q Other closed, proper ideals: { T : T jj ∈ K , T 21 ∈ J } where J is a closed ideal in L ( ℓ p , ℓ q ). E.g., J = L ( ℓ p , ℓ q ) corresponds to J ℓ p ∩ J ℓ 2 .

Closed ideals in L ( ℓ p , ℓ q )

Closed ideals in L ( ℓ p , ℓ q ) Let 1 ≤ p < q < ∞ . { 0 } � K ( ℓ p , ℓ q ) � J I p , q ⊂ L ( ℓ p , ℓ q ) , where I p , q : ℓ p → ℓ q is the formal inclusion map.

Closed ideals in L ( ℓ p , ℓ q ) Let 1 ≤ p < q < ∞ . { 0 } � K ( ℓ p , ℓ q ) � J I p , q ⊂ L ( ℓ p , ℓ q ) , where I p , q : ℓ p → ℓ q is the formal inclusion map. Milman [1970] proved the following: