Lifting of nest approximation properties and related principles of local reflexivity

Transcription

1 Lifting of nest approximation properties and related principles of local reflexivity Eve Oja Conference on NonLinear Functional Analysis Valencia, October 18, 2017

2 Life in Banach spaces with certain approximation properties is much easier. (Pietsch, History..., 2007, p. 287) Recall: A Banach space X has AP: (S ν ) F(X, X) such that S ν I X uniformly on compact subsets of X. X has λ-bap: (S ν ) as above, and S ν λ ν BAP: λ-bap ( λ); MAP= 1-BAP basis BAP; monotone basis MAP MAP BAP AP in X Converses do not hold even in X c 0 : Johnson, Schechtman 1996, Godefroy, Saphar 1988; Figiel, Johnson, Pełczyński 2011

3 AP BAP MAP in X Problem 1 (goes back to Grothendieck s Memoir 1955) : Does AP MAP in X? AP MAP problem Grothendieck: Yes for X reflexive or X separable This mysterious X...

4 AP BAP MAP in X Problem 1 (goes back to Grothendieck s Memoir 1955) : Does AP MAP in X? AP MAP problem Grothendieck: Yes for X reflexive or X separable This mysterious X... Some recent variants of (B)AP: weaker forms, e.g. weak MAP [Lima, Oja; Math. Ann. 2005] for this direction, see [Lassalle, Oja, Turco; J. Appr. Th. 2016] stronger forms = refinements of the classical APs: APs of pairs [Figiel, Johnson, Pełczyński; Israel J. Math. 2011] nest APs [Figiel, Johnson; J. Funct. Anal. 2016] Our topic: lifting of nest APs between X and X

5 Classics (Grothendieck, Johnson; Enflo, James, Lindenstrauss): X (λ-b)ap X (λ-b)ap X AP X MAP X MAP in all equivalent norms Problem 2:?? What about X = l 1? Pr. 1 Yes Pr. 2 Yes [LO2005]: X AP X weak MAP in all equivalent norms [Oja2006]: if X or X Asplund X MAP X MAP in all equivalent norms

6 Classics (Grothendieck, Johnson; Enflo, James, Lindenstrauss): X (λ-b)ap X (λ-b)ap X AP X MAP X MAP in all equivalent norms Problem 2:?? What about X = l 1? Pr. 1 Yes Pr. 2 Yes [LO2005]: X AP X weak MAP in all equivalent norms [Oja2006]: if X or X Asplund X MAP X MAP in all equivalent norms Classics ((Grothendieck), Reinov; many different proofs): If X or X Asplund, then X AP X MAP Problem 3 [LO2005]: Does weak MAP MAP? Pr. 3 Yes Pr. 1 Yes

7 Nests X Banach space, N {U X : U closed subspace} Recall: N is a nest in X: N is linearly ordered by. (For U, W N, one has U W or W U.) Examples N = {U} N = {U 1, U 2,..., U n }, e.g., U 1 U 2... U n U with a basis e 1, e 2,..., U n := span{e 1, e 2,..., e n } N = {{0}, U, X, U n : n N} or N = {U n : n N} Volterra nest V in L p [0, 1], 1 p <, V = {L p [0, t] : 0 t 1} N is complete: {0}, X N, and N is closed under intersections and closures of unions.

8 Nest approximation properties Recall: X has (λ-b)ap: (S ν ) F(X, X) such that S ν I X uniformly on compact subsets of X (and S ν λ). U X, U closed subspace The pair (X, U) has (λ-b)ap: S ν (U) U for all ν. E.g.: X has AP (X, X) has AP (X, {0}) has AP N nest in X The pair (X, N ) has (λ-b)ap: S ν (U) U for all ν and U N. E.g.: (X, U) has AP (X, N ) has AP, where N = {U} (X, U) has AP X, U, X/U all have AP. Problem 4:?? [FJP2011]: X λ-bap (X, U) λ-bap for all U, codim U <.

9 [J. A. Erdos; J. London Math. Soc. 1968]: (H, N ) has AP for all nests N in a Hilbert space H. [FJ2016]: For X over C, (X, N ) has AP for all N X has the Lidskii trace property: If a nuclear op. T on X has absolutely summable eigenvalues λ n, then tracet = n λ n. (X, U) has AP for all U X has the hereditary AP (HAP) Problem 5 (Johnson, Szankowski; Ann. Math. 2012, Isr. J. Math. 2014): Does HAP Lidskii trace property? Problem 6: Does (X, U) has AP for all U (X, N ) has AP for all N? Pr. 5 Yes Pr. 6 Yes

10 Lifting down classics: X (λ-b)ap X (λ-b)ap General reason: (1) PLR + (2) convex combination argument (1) PLR (λ-b)ap of X is given with conjugate operators (S ν). (2) Convex combinations of S ν give (λ-b)ap of X. This works for convex APs.

11 Lifting down classics: X (λ-b)ap X (λ-b)ap General reason: (1) PLR + (2) convex combination argument (1) PLR (λ-b)ap of X is given with conjugate operators (S ν). (2) Convex combinations of S ν give (λ-b)ap of X. This works for convex APs. A L(X, X), A convex. X has A-AP: (S ν ) A such that S ν I X uniformly on compact subsets of X, i.e., in τ c of X. X has A-AP with conjugate operators: (S ν ) A such that S ν I X in τ c of X. Example: (λ-b)ap of (X, N ) = A-AP with A = {S F(X, X) : S(U) U U N ( S λ)} (2) X A-AP with conjugate op-rs X A-AP [Lissitsin, Oja; JMAA 2011]

12 (2) X A-AP with conjugate operators X A-AP N nest in X N := {U : U N } nest in X S(U) U S (U ) U. Hence, in particular, (2 ) (X, N ) (λ-b)ap with conjugate op-rs (X, N ) (λ-b)ap All we need: (λ-b)ap of (X, N ) to be given with conjugate op-rs! In particular, (λ-b)ap of (X, U ) to be given with conjugate op-rs! Recall (1): PLR works for (X, {0} ) = (X, X ) = (λ-b)ap of X. Find a working PLR (respecting nests) for (λ-b)ap of (X, N )!

13 (2) X A-AP with conjugate operators X A-AP N nest in X N := {U : U N } nest in X S(U) U S (U ) U. Hence, in particular, (2 ) (X, N ) (λ-b)ap with conjugate op-rs (X, N ) (λ-b)ap All we need: (λ-b)ap of (X, N ) to be given with conjugate op-rs! In particular, (λ-b)ap of (X, U ) to be given with conjugate op-rs! Recall (1): PLR works for (X, {0} ) = (X, X ) = (λ-b)ap of X. Find a working PLR (respecting nests) for (λ-b)ap of (X, N )! [Oja; Adv. Math. 2014]: PLR respecting subspaces (λ-b)ap (with projections) of (X, U ) is always given with conjugate operators (projections). (X, U ) (λ-b)ap (X, U) (λ-b)ap (by (2 ))

14 PLR respecting subspaces [O2014]: X, Y Banach spaces; U X, V Y closed subspaces. Let S F(Y, X ) satisfy S(V ) U. If F Y, dim F <, and ε > 0, then T F(X, Y ) satisfying T (U) V such that 1 T S < ε, 2 T y = Sy, y F, 2 rant = rans, 3 T x = S x whenever S x Y. When Y = X and S is a projection, also T is a projection. Applying PLR resp. subsp. twice: Finite-rank operators/projections between bi-duals, who respect bi-annihilators, are locally bi-conjugate to operators/projections, who respect subspaces.

15 PLR respecting subspaces (λ-b)ap (with projections) of (X, U ) is always given with conjugate operators (projections). (X, U ) (λ-b)ap (X, U) (λ-b)ap (by (2 )) Recall: (2 ) (X, N ) (λ-b)ap with conj. op-rs (X, N ) (λ-b)ap (X, U ) λ-bap with proj. (X, U) (λ 2 + 2λ)-BAP with proj. (X, U ) λ-bap with proj. (X, U) λ-bap with proj.

16 PLR respecting subspaces (λ-b)ap (with projections) of (X, U ) is always given with conjugate operators (projections). (X, U ) (λ-b)ap (X, U) (λ-b)ap (by (2 )) Recall: (2 ) (X, N ) (λ-b)ap with conj. op-rs (X, N ) (λ-b)ap (X, U ) λ-bap with proj. (X, U) (λ 2 + 2λ)-BAP with proj. (X, U ) λ-bap with proj. (X, U) λ-bap with proj. NoLiFA: [Chávez-Domínguez; arxiv 2017] uses PLR resp. subsp.: If X is separable and if (X, U) has the Lipschitz lifting property ( i.e. R L(X, F(X)), βr = I X (β can. quotient map), R(U) F(U)), then: (X, U) Lipschitz BAP (X, U) BAP [Godefroy, Kalton; Studia Math. 2003]: For arbitrary X: X Lipschitz λ-bap X λ-bap

17 X, Y Banach spaces; U nest in X, N U = {V U : U U} nest in Y. N U is increasing on U: if U W and U W in U, then V U V W and V U V W. PLR respecting nests [Oja, Veidenberg; J. Funct. Anal. 2017]: Assume {0} U, NU is complete. Let S F(Y, X ) satisfy S(VU ) U for all U U. If K X, L Y are compact sets, and ε > 0, then T F(X, Y ) satisfying T (U) V U for all U U such that 1 T S < ε, 2 T y Sy < ε for all y L, 3 T x S x < ε whenever x K and S x Y. Applying PLR resp. nests twice (assuming U is complete): Finite-rank operators between bi-duals, who respect bi-annihilators, are locally bi-conjugate to operators, who respect subspaces.

18 PLR respecting nests easily follows from: Lemma: G nest in X, N G = {V G : G G} nest in Y, N G increasing on G. Assume {0} G, N G is complete. Let T X Y satisfy T (G) VG for all G G. Then (T α ) X Y satisfying T α (G) V G for all α and G G such that 1 T α T, 2 Tαy T y for all y Y, 3 T α x Tx for those x X for which Tx Y. Sketch of proof: R := {R X Y : R(G) V G G G} S := {S X Y : S(G) VG G G} R (X Y ) = I(X, Y ) J I(X, Y ) = (X Y ) S, where J(A)=j Y A. Using extension of Ringrose th., J(R ) S. Define Φ(A + R )=J(A) + S, A I(X, Y ), R = I(X, Y )/R Φ I(X, Y )/S = S. T S and Φ (T ) T B R (T α ) R, also 1, 2, 3 hold.

19 Extension of the Ringrose theorem [OV17]: X, Y Banach spaces; G nest in X, N G = {V G : G G} nest in Y, N G increasing on G. Assume {0} G, Y N G, N G is closed under intersections. R := {R X Y : R(G) V G G G}. Then : R = x y R G G such that x (G_) and y V G. Let R X Y have rank n. R R R = n k=1 x k y k, x k y k R. (G_ := {H G : H G, H G} if G {0}; {0}_ := {0}.) Ringrose ([Proc. London Math. Soc. 1965] and [Erdos 1968]): X = Y = H Hilbert space, N G = G complete nest.

20 N nest in X, assume N is complete. PLR resp. nests (λ-b)ap of (X, N ) is given with conjugate operators. (X, N ) (λ-b)ap (X, N ) (λ-b)ap (by (2 )) Summary: Nest APs nicely go down.

21 N nest in X, assume N is complete. PLR resp. nests (λ-b)ap of (X, N ) is given with conjugate operators. (X, N ) (λ-b)ap (X, N ) (λ-b)ap (by (2 )) Summary: Nest APs nicely go down. Lifting up from X to X is known in special cases: X has the unique extension property; then X MAP X MAP [Godefroy, Saphar; Illin. J. Math. 1988] X is extendably LR (ELR) [Johnson, Oikhberg; Illin. J. Math. 2001]: X λ-elr, X µ-bap X λµ-bap

22 (E, F, ε): E X, dime < ; F X, dimf < ; ε > 0 Recall: X is λ-extendably LR: (E, F, ε) T L(X ), T (E) X, T λ + ε, x (Tx ) = x (x ) whenever x E, x F. [OV2017]: (X, N ) is λ-elr: (E, F, ε) T L(X ), T (E) X, T λ + ε, T (U ) U for all U N, x (Tx ) x (x ) ε whenever x S E, x S F. Using PLR resp. nests and [Oja, Veidenberg; JMAA 2016] gives [OV2017]: Let N be complete. Then (X, N ) λ-elr, (X, N ) µ-bap (X, N ) λµ-bap. Rosenthal (see [JO2001]): X λ-bap X λ-elr. Using PLR resp. nests: [OV2017]: Let N be complete. Then (X, N ) λ-bap (X, N ) λ-elr.

23 Below, let X have the unique extension property: the only T L(X ) such that T 1 and T X = I X is T = I X. [GS1988]: X MAP X MAP. (The same for compact MAP.) [Oja; JMAA 2006] (X, N ) weakly compact MAP (X, N ) weakly compact MAP with conj. op-rs.

24 Below, let X have the unique extension property: the only T L(X ) such that T 1 and T X = I X is T = I X. [GS1988]: X MAP X MAP. (The same for compact MAP.) [Oja; JMAA 2006] (X, N ) weakly compact MAP (X, N ) weakly compact MAP with conj. op-rs. Lifting up of bounded nest APs from X to X is possible whenever X already enjoys a weaker metric nest AP: [OV2017]: Assume (X, N ) has weakly compact MAP. Then (X, N ) λ-bap (X, N ) λ-bap. Proof. (X, N ) has weakly compact MAP with conj. op-rs. By [OV2017], (X, N ) is 1-ELR of stronger type (with ELR operator T = S, S W(X, X)). Since X has also λ-bap, (X, N ) has (1 λ)-bap by an extension of the Johnson Oikhberg theorem.