The p-approximation property in Banach spaces
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Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems The p-approximation property in Banach spaces J. M. Delgado Department of Mathematics University of Huelva V ENCUENTRO DE ANÁLISIS FUNCIONAL Y SUS APLICACIONES Salobreña 2009 J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Outline 1Introduction 2Density of finite rank operators and the p-approximation property 3A trace characterization of the p-approximation property 4Open problems J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Outline 1Introduction 2Density of finite rank operators and the p-approximation property 3A trace characterization of the p-approximation property 4Open problems J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Notation –X,YBanach spaces, BX={x∈X:kxk ≤ 1} –L(Y,X)is the space of bounded operators from Yinto X –F(Y,X) = {T∈ L(Y,X): Thas finite rank} –K(Y,X) = {T∈ L(Y,X): Tis compact} –p∈[1,∞)⇒p0=p p−1⇔1 p+1 p0=1 –`p(X) = {(xn)⊂X:Pnkxnkp<∞} k(xn)kp= (Pnkxnkp)1/p –`w p(X) = {(xn)⊂X:Pn|hx∗,xni|p<∞,for all x∗∈X∗} k(xn)kw p=supx∗∈BX∗(Pn|hx∗,xni|p)1/p –Np(Y,X) = {T∈ L(Y,X): Tis p-nuclear} –T:Y→X p-nuclear ⇔∃(y∗ n)∈`p(Y∗) ∃(xn)∈`w p0(X): J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Notation –X,YBanach spaces, BX={x∈X:kxk ≤ 1} –L(Y,X)is the space of bounded operators from Yinto X –F(Y,X) = {T∈ L(Y,X): Thas finite rank} –K(Y,X) = {T∈ L(Y,X): Tis compact} –p∈[1,∞)⇒p0=p p−1⇔1 p+1 p0=1 –`p(X) = {(xn)⊂X:Pnkxnkp<∞} k(xn)kp= (Pnkxnkp)1/p –`w p(X) = {(xn)⊂X:Pn|hx∗,xni|p<∞,for all x∗∈X∗} k(xn)kw p=supx∗∈BX∗(Pn|hx∗,xni|p)1/p –Np(Y,X) = {T∈ L(Y,X): Tis p-nuclear} –T:Y→X p-nuclear ⇔∃(y∗ n)∈`p(Y∗) ∃(xn)∈`w p0(X): J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Notation –X,YBanach spaces, BX={x∈X:kxk ≤ 1} –L(Y,X)is the space of bounded operators from Yinto X –F(Y,X) = {T∈ L(Y,X): Thas finite rank} –K(Y,X) = {T∈ L(Y,X): Tis compact} –p∈[1,∞)⇒p0=p p−1⇔1 p+1 p0=1 –`p(X) = {(xn)⊂X:Pnkxnkp<∞} k(xn)kp= (Pnkxnkp)1/p –`w p(X) = {(xn)⊂X:Pn|hx∗,xni|p<∞,for all x∗∈X∗} k(xn)kw p=supx∗∈BX∗(Pn|hx∗,xni|p)1/p –Np(Y,X) = {T∈ L(Y,X): Tis p-nuclear} –T:Y→X p-nuclear ⇔∃(y∗ n)∈`p(Y∗) ∃(xn)∈`w p0(X):T=Pny∗ n⊗xn J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Notation –X,YBanach spaces, BX={x∈X:kxk ≤ 1} –L(Y,X)is the space of bounded operators from Yinto X –F(Y,X) = {T∈ L(Y,X): Thas finite rank} –K(Y,X) = {T∈ L(Y,X): Tis compact} –p∈[1,∞)⇒p0=p p−1⇔1 p+1 p0=1 –`p(X) = {(xn)⊂X:Pnkxnkp<∞} k(xn)kp= (Pnkxnkp)1/p –`w p(X) = {(xn)⊂X:Pn|hx∗,xni|p<∞,for all x∗∈X∗} k(xn)kw p=supx∗∈BX∗(Pn|hx∗,xni|p)1/p –Np(Y,X) = {T∈ L(Y,X): Tis p-nuclear} –T:Y→X p-nuclear ⇔∃(y∗ n)∈`p(Y∗) ∃(xn)∈`w p0(X):T(·) = Pnhy∗ n,·ixn J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems The approximation property (Tn)⊂ F(Y,X) Tn k·k −→ T ⇒T∈ K(Y,X) ⇔ F(Y,X)k·k⊂ K(Y,X) Xhas Schauder basis ⇒For every Banach Y,F(Y,X)k·k =K(Y,X) J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems The approximation property (Tn)⊂ F(Y,X) Tn k·k −→ T ⇒T∈ K(Y,X)⇔ F(Y,X)k·k⊂ K(Y,X) Xhas Schauder basis ⇒For every Banach Y,F(Y,X)k·k =K(Y,X) J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Approximation property in terms of tensor products J1 Y∗ˆ ⊗πX−→ N1(Y,X) Pny∗ n⊗xn7→ Pnhy∗ n,·ixn Pnky∗ nkkxnk<∞ Theorem [Grothendieck, 1955] Xa Banach space. The following statements are equivalent: 1Xhas the AP. 2For every Banach space Y,F(Y,X)k·k =K(Y,X). 3For every Banach space Y,Y∗ˆ ⊗πX≃ N1(Y,X). J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Approximation property of order pvia tensor products Chevet–Saphar’s tensor norm: p∈[1,∞) gp(u) = inf(k(y∗ n)kpk(xn)k`w p0(X):u= m X n=1 y∗ n⊗xn∈Y∗⊗X) Jp Y∗ˆ ⊗gpX−→ Np(Y,X) Pny∗ n⊗xn7→ Pnhy∗ n,·ixn Saphar (1970’s): p∈[1,∞] Xhas the approximation property of order p (APp) if, for every Banach space Y,Y∗ˆ ⊗gpX≃ Np(Y,X). Reinov (1980’s): p∈(0,1] Xhast approximation property of order p (APp) if, for every Banach space Y, the restriction of J1to Hpis injective, where Hp={u=Pny∗ n⊗xn:Pn(ky∗ nkkxnk)p<∞} ⊂ Y∗ˆ ⊗πX. AP1=AP J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Approximation property of order pvia tensor products Chevet–Saphar’s tensor norm: p∈[1,∞) gp(u) = inf(k(y∗ n)kpk(xn)k`w p0(X):u= m X n=1 y∗ n⊗xn∈Y∗⊗X) Jp Y∗ˆ ⊗gpX−→ Np(Y,X) Pny∗ n⊗xn7→ Pnhy∗ n,·ixn Saphar (1970’s): p∈[1,∞] Xhas the approximation property of order p (APp) if, for every Banach space Y,Y∗ˆ ⊗gpX≃ Np(Y,X). Reinov (1980’s): p∈(0,1] Xhast approximation property of order p (APp) if, for every Banach space Y, the restriction of J1to Hpis injective, where Hp={u=Pny∗ n⊗xn:Pn(ky∗ nkkxnk)p<∞} ⊂ Y∗ˆ ⊗πX. AP1=AP J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Approximation property of order pvia tensor products Chevet–Saphar’s tensor norm: p∈[1,∞) gp(u) = inf(k(y∗ n)kpk(xn)k`w p0(X):u= m X n=1 y∗ n⊗xn∈Y∗⊗X) Jp Y∗ˆ ⊗gpX−→ Np(Y,X) Pny∗ n⊗xn7→ Pnhy∗ n,·ixn Saphar (1970’s): p∈[1,∞] Xhas the approximation property of order p (APp) if, for every Banach space Y,Y∗ˆ ⊗gpX≃ Np(Y,X). Reinov (1980’s): p∈(0,1] Xhast approximation property of order p (APp) if, for every Banach space Y, the restriction of J1to Hpis injective, where Hp={u=Pny∗ n⊗xn:Pn(ky∗ nkkxnk)p<∞} ⊂ Y∗ˆ ⊗πX. AP1=AP J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Approximation property of order pvia tensor products Chevet–Saphar’s tensor norm: p∈[1,∞) gp(u) = inf(k(y∗ n)kpk(xn)k`w p0(X):u= m X n=1 y∗ n⊗xn∈Y∗⊗X) Jp Y∗ˆ ⊗gpX−→ Np(Y,X) Pny∗ n⊗xn7→ Pnhy∗ n,·ixn Saphar (1970’s): p∈[1,∞] Xhas the approximation property of order p (APp) if, for every Banach space Y,Y∗ˆ ⊗gpX≃ Np(Y,X). Reinov (1980’s): p∈(0,1] Xhast approximation property of order p (APp) if, for every Banach space Y, the restriction of J1to Hpis injective, where Hp={u=Pny∗ n⊗xn:Pn(ky∗ nkkxnk)p<∞} ⊂ Y∗ˆ ⊗πX. AP1=AP J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Approximation property of order pvia tensor products Chevet–Saphar’s tensor norm: p∈[1,∞) gp(u) = inf(k(y∗ n)kpk(xn)k`w p0(X):u= m X n=1 y∗ n⊗xn∈Y∗⊗X) Jp Y∗ˆ ⊗gpX−→ Np(Y,X) Pny∗ n⊗xn7→ Pnhy∗ n,·ixn Saphar (1970’s): p∈[1,∞] Xhas the approximation property of order p (APp) if, for every Banach space Y,Y∗ˆ ⊗gpX≃ Np(Y,X). Reinov (1980’s): p∈(0,1] Xhast approximation property of order p (APp) if, for every Banach space Y, the restriction of J1to Hpis injective, where Hp={u=Pny∗ n⊗xn:Pn(ky∗ nkkxnk)p<∞} ⊂ Y∗ˆ ⊗πX. AP1=AP J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Approximation property of order pvia tensor products Theorem [Grothendieck, 1955] Xa Banach space. The following statements are equivalent: 1Xhas the AP (≡IXcan be approximated by finite rank operators uniformly on compact subsets K⊂X). 2For every Banach space Y,Y∗ˆ ⊗πX≃ N1(Y,X). A set K⊂Xis relatively compact if and only if there exists (xn)∈c0(X)such that K⊂aco (xn) := {Pnanxn: (an)∈B`1}. Theorem [Bourgain and Reinov, 1984-85] A Banach space Xhas the APp(p∈(0,1)) if and only if IXcan be approximated by finite rank operators uniformly on subsets K⊂Xfor which there exists (xn)∈`q(X)such that K⊂ {Pnanxn: (an)∈B`1} (p−1−q−1=1). J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Approximation property of order pvia tensor products Theorem [Grothendieck, 1955] Xa Banach space. The following statements are equivalent: 1Xhas the AP (≡IXcan be approximated by finite rank operators uniformly on compact subsets K⊂X). 2For every Banach space Y,Y∗ˆ ⊗πX≃ N1(Y,X). A set K⊂Xis relatively compact if and only if there exists (xn)∈c0(X)such that K⊂aco (xn) := {Pnanxn: (an)∈B`1}. Theorem [Bourgain and Reinov, 1984-85] A Banach space Xhas the APp(p∈(0,1)) if and only if IXcan be approximated by finite rank operators uniformly on subsets K⊂Xfor which there exists (xn)∈`q(X)such that K⊂ {Pnanxn: (an)∈B`1} (p−1−q−1=1). J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Approximation property of order pvia tensor products Theorem [Grothendieck, 1955] Xa Banach space. The following statements are equivalent: 1Xhas the AP (≡IXcan be approximated by finite rank operators uniformly on compact subsets K⊂X). 2For every Banach space Y,Y∗ˆ ⊗πX≃ N1(Y,X). A set K⊂Xis relatively compact if and only if there exists (xn)∈c0(X)such that K⊂aco (xn) := {Pnanxn: (an)∈B`1}. Theorem [Bourgain and Reinov, 1984-85] A Banach space Xhas the APp(p∈(0,1)) if and only if IXcan be approximated by finite rank operators uniformly on subsets K⊂Xfor which there exists (xn)∈`q(X)such that K⊂ {Pnanxn: (an)∈B`1} (p−1−q−1=1). J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems p-compact sets and the p-approximation property Definition [Sinha and Karn, 2002] Let p≥1. K⊂Xis relatively p-compact if there exists (xn)∈`p(X)such that K⊂p-co (xn):=nPnanxn: (an)∈B`p0o. A Banach space Xhas the p-approximation property (p-AP) if the identity map IXcan be approximated by finite rank operators uniformly on every p-compact subset of X. ∞-AP=AP. All Banach spaces have the p-AP for all p∈[1,2]. For every p>2, there exist Banach spaces without the p-AP. A necessary condition in terms of the trace is obtained for Banach spaces having the p-AP. J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems p-approximation property and p-compact operators –Πd p(Y,X) = {T∈ L(Y,X): T∗is p-summing} T∈Πd p(Y,X)⇔Tmaps relatively compact sets in Yto relatively p-compact sets in X. Theorem [Oja, Piñeiro, Serrano and Delgado, 2009] Xa Banach space, p∈[1,+∞]. The following statements are equivalent: 1Xhas the p-AP. 2For every Banach Y,F(Y,X)is k·k-dense in Kp(Y,X). 3For every Banach Y,F(Y,X)is τc-dense in Kp(Y,X). 4For every Banach Y,F(Y,X)is τc-dense in Πd p(Y,X). J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems p-approximation property and p-compact operators –Πd p(Y,X) = {T∈ L(Y,X): T∗is p-summing} T∈Πd p(Y,X)⇔Tmaps relatively compact sets in Yto relatively p-compact sets in X. Theorem [Oja, Piñeiro, Serrano and Delgado, 2009] Xa Banach space, p∈[1,+∞]. The following statements are equivalent: 1Xhas the p-AP. 2For every Banach Y,F(Y,X)is k·k-dense in Kp(Y,X). 3For every Banach Y,F(Y,X)is τc-dense in Kp(Y,X). 4For every Banach Y,F(Y,X)is τc-dense in Πd p(Y,X). J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems p-approximation property and p-compact operators –Πd p(Y,X) = {T∈ L(Y,X): T∗is p-summing} T∈Πd p(Y,X)⇔Tmaps relatively compact sets in Yto relatively p-compact sets in X. Theorem [Oja, Piñeiro, Serrano and Delgado, 2009] Xa Banach space, p∈[1,+∞]. The following statements are equivalent: 1Xhas the p-AP. 2For every Banach Y,F(Y,X)is k·k-dense in Kp(Y,X). 3For every Banach Y,F(Y,X)is τc-dense in Kp(Y,X). 4For every Banach Y,F(Y,X)is τc-dense in Πd p(Y,X). J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems p-approximation property and p-compact operators 1Xhas the p-AP. 3For every Banach Y,F(Y,X)is τc-dense in Kp(Y,X). 3⇒1 ε > 0 K=p-co(xn),(xn)∈`p(X) ⇒ ∃(αn)&0: (zn) := (α−1 nxn)∈`p(X) `p0 φx −→ X Dα↓ ↑b φz `p0 Q −→ Y:= `p0/Ker φz φx(en) = xn φz(en) = zn Dα(βn) = (αnβn) b φz[(βn)] = φz(βn) H:= Q◦Dα(B`p0) compact in Y S=PN k=1ψk⊗uk∈ F(Y,X): suph∈HkSh −b φzhk< ε b φzinjective ⇒ ∃u∗ k∈X∗:suph∈Hhb φ∗ zu∗ k−ψk,hi< ε, k=1, . . . , N J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems p-approximation property and p-compact operators 1Xhas the p-AP. 3For every Banach Y,F(Y,X)is τc-dense in Kp(Y,X). 3⇒1 ε > 0 K=p-co(xn),(xn)∈`p(X) ⇒ ∃(αn)&0: (zn) := (α−1 nxn)∈`p(X) `p0 φx −→ X Dα↓ ↑b φz `p0 Q −→ Y:= `p0/Ker φz φx(en) = xn φz(en) = zn Dα(βn) = (αnβn) b φz[(βn)] = φz(βn) H:= Q◦Dα(B`p0) compact in Y S=PN k=1ψk⊗uk∈ F(Y,X): suph∈HkSh −b φzhk< ε b φzinjective ⇒ ∃u∗ k∈X∗:suph∈Hhb φ∗ zu∗ k−ψk,hi< ε, k=1, . . . , N R=PN k=1u∗ k⊗uk R∈ F(X,X)satisfying supx∈KkRx −xk< ε J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems p-approximation property and p-compact operators 1Xhas the p-AP. 3For every Banach Y,F(Y,X)is τc-dense in Kp(Y,X). 3⇒1 ε > 0 K=p-co(xn),(xn)∈`p(X) ⇒ ∃(αn)&0: (zn) := (α−1 nxn)∈`p(X) `p0 φx −→ X Dα↓ ↑b φz `p0 Q −→ Y:= `p0/Ker φz φx(en) = xn φz(en) = zn Dα(βn) = (αnβn) b φz[(βn)] = φz(βn) H:= Q◦Dα(B`p0) compact in Y S=PN k=1ψk⊗uk∈ F(Y,X): suph∈HkSh −b φzhk< ε b φzinjective ⇒ ∃u∗ k∈X∗:suph∈Hhb φ∗ zu∗ k−ψk,hi< ε, k=1, . . . , N R=PN k=1u∗ k⊗uk R∈ F(X,X)satisfying supx∈KkRx −xk< ε J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems p-approximation property and p-compact operators 1Xhas the p-AP. 3For every Banach Y,F(Y,X)is τc-dense in Kp(Y,X). 3⇒1 ε > 0 K=p-co(xn),(xn)∈`p(X)⇒ ∃(αn)&0: (zn) := (α−1 nxn)∈`p(X) `p0 φx −→ X Dα↓ ↑b φz `p0 Q −→ Y:= `p0/Ker φz φx(en) = xn φz(en) = zn Dα(βn) = (αnβn) b φz[(βn)] = φz(βn) H:= Q◦Dα(B`p0) compact in Y S=PN k=1ψk⊗uk∈ F(Y,X): suph∈HkSh −b φzhk< ε b φzinjective ⇒ ∃u∗ k∈X∗:suph∈Hhb φ∗ zu∗ k−ψk,hi< ε, k=1, . . . , N R=PN k=1u∗ k⊗uk R∈ F(X,X)satisfying supx∈KkRx −xk< ε J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems p-approximation property and p-compact operators 1Xhas the p-AP. 3For every Banach Y,F(Y,X)is τc-dense in Kp(Y,X). 3⇒1 ε > 0 K=p-co(xn),(xn)∈`p(X)⇒ ∃(αn)&0: (zn) := (α−1 nxn)∈`p(X) `p0 φx −→ X Dα↓φz% ↑b φz `p0 Q −→ Y:= `p0/Ker φz φx(en) = xn φz(en) = zn Dα(βn) = (αnβn) b φz[(βn)] = φz(βn) H:= Q◦Dα(B`p0) compact in Y S=PN k=1ψk⊗uk∈ F(Y,X): suph∈HkSh −b φzhk< ε b φzinjective ⇒ ∃u∗ k∈X∗:suph∈Hhb φ∗ zu∗ k−ψk,hi< ε, k=1, . . . , N R=PN k=1u∗ k⊗uk R∈ F(X,X)satisfying supx∈KkRx −xk< ε J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems p-approximation property and p-compact operators 1Xhas the p-AP. 3For every Banach Y,F(Y,X)is τc-dense in Kp(Y,X). 3⇒1 ε > 0 K=p-co(xn),(xn)∈`p(X)⇒ ∃(αn)&0: (zn) := (α−1 nxn)∈`p(X) `p0 φx −→ X Dα↓φz%↑b φz `p0 Q −→ Y:= `p0/Ker φz φx(en) = xn φz(en) = zn Dα(βn) = (αnβn) b φz[(βn)] = φz(βn) H:= Q◦Dα(B`p0) compact in Y S=PN k=1ψk⊗uk∈ F(Y,X): suph∈HkSh −b φzhk< ε b φzinjective ⇒ ∃u∗ k∈X∗:suph∈Hhb φ∗ zu∗ k−ψk,hi< ε, k=1, . . . , N R=PN k=1u∗ k⊗uk R∈ F(X,X)satisfying supx∈KkRx −xk< ε J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems p-approximation property and p-compact operators 1Xhas the p-AP. 3For every Banach Y,F(Y,X)is τc-dense in Kp(Y,X). 3⇒1 ε > 0 K=p-co(xn),(xn)∈`p(X)⇒ ∃(αn)&0: (zn) := (α−1 nxn)∈`p(X) `p0 φx −→ X Dα↓φz%↑b φz `p0 Q −→ Y:= `p0/Ker φz φx(en) = xn φz(en) = zn Dα(βn) = (αnβn) b φz[(βn)] = φz(βn) H:= Q◦Dα(B`p0) compact in Y S=PN k=1ψk⊗uk∈ F(Y,X): suph∈HkSh −b φzhk< ε b φzinjective ⇒ ∃u∗ k∈X∗:suph∈Hhb φ∗ zu∗ k−ψk,hi< ε, k=1, . . . , N R=PN k=1u∗ k⊗uk R∈ F(X,X)satisfying supx∈KkRx −xk< ε J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems p-approximation property for dual Banach spaces Theorem [Grothendieck, 1955] Xa Banach space. The following statements are equivalent: 1X∗has the AP, i.e., for every Banach Y,F(Y,X∗)k·k =K(Y,X∗). 2For every Banach Y,F(X,Y)k·k =K(X,Y). T∈ QNp(X,Y) (Tquasi p-nuclear) ⇔There exists (x∗ n)∈`p(X∗)such that kTxk ≤ (Pn|hx∗ n,xi|p)1/p,∀x∈X. T∈ Kp(X,Y)⇔T∗∈ QNp(Y∗,X∗). T∈ QNp(X,Y)⇔T∗∈ Kp(Y∗,X∗). Theorem [Oja, Piñeiro, Serrano and Delgado, 2009] Xa Banach space. The following statements are equivalent: 1X∗has the p-AP. 2For every Banach Y,F(X,Y)is k·k-dense in QNp(X,Y). J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Relation to Saphar’s approximation property APp Theorem [Oja, Piñeiro, Serrano and Delgado, 2009] Xa Banach space. The following statements are equivalent: 1X∗has the p-AP. 2For every Banach Y,F(X,Y)is k·k-dense in QNp(X,Y). Saphar (1970-72): For every p6=2, there exist Banach spaces without APp. X∗∗ has the APp0⇒For every Banach Y, F(X,Y)is πp-dense in QNp(X,Y). Bourgain and Reinov (1985): H∗∗ ∞,H∗∗∗∗ ∞, . . . have the APp,p∈(1,∞) Corollary X∗∗ has the APp0⇒X∗has the p-AP, p∈(1,∞) H∗ ∞,H∗∗∗ ∞, . . . have the p-AP, p∈[1,∞) J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Relation to Saphar’s approximation property APp Theorem [Oja, Piñeiro, Serrano and Delgado, 2009] Xa Banach space. The following statements are equivalent: 1X∗has the p-AP. 2For every Banach Y,F(X,Y)is k·k-dense in QNp(X,Y). Saphar (1970-72): For every p6=2, there exist Banach spaces without APp. X∗∗ has the APp0⇒For every Banach Y, F(X,Y)is πp-dense in QNp(X,Y). Bourgain and Reinov (1985): H∗∗ ∞,H∗∗∗∗ ∞, . . . have the APp,p∈(1,∞) Corollary X∗∗ has the APp0⇒X∗has the p-AP, p∈(1,∞) H∗ ∞,H∗∗∗ ∞, . . . have the p-AP, p∈[1,∞) J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Relation to Saphar’s approximation property APp Theorem [Oja, Piñeiro, Serrano and Delgado, 2009] Xa Banach space. The following statements are equivalent: 1X∗has the p-AP. 2For every Banach Y,F(X,Y)is k·k-dense in QNp(X,Y). Saphar (1970-72): For every p6=2, there exist Banach spaces without APp. X∗∗ has the APp0⇒For every Banach Y, F(X,Y)is πp-dense in QNp(X,Y). Bourgain and Reinov (1985): H∗∗ ∞,H∗∗∗∗ ∞, . . . have the APp,p∈(1,∞) Corollary X∗∗ has the APp0⇒X∗has the p-AP, p∈(1,∞) H∗ ∞,H∗∗∗ ∞, . . . have the p-AP, p∈[1,∞) J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Relation to Saphar’s approximation property APp Theorem [Oja, Piñeiro, Serrano and Delgado, 2009] Xa Banach space. The following statements are equivalent: 1X∗has the p-AP. 2For every Banach Y,F(X,Y)is k·k-dense in QNp(X,Y). Saphar (1970-72): For every p6=2, there exist Banach spaces without APp. X∗∗ has the APp0⇒For every Banach Y, F(X,Y)is πp-dense in QNp(X,Y). Bourgain and Reinov (1985): H∗∗ ∞,H∗∗∗∗ ∞, . . . have the APp,p∈(1,∞) Corollary X∗∗ has the APp0⇒X∗has the p-AP, p∈(1,∞) H∗ ∞,H∗∗∗ ∞, . . . have the p-AP, p∈[1,∞) J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Relation to Saphar’s approximation property APp Theorem [Oja, Piñeiro, Serrano and Delgado, 2009] Xa Banach space. The following statements are equivalent: 1X∗has the p-AP. 2For every Banach Y,F(X,Y)is k·k-dense in QNp(X,Y). Saphar (1970-72): For every p6=2, there exist Banach spaces without APp. X∗∗ has the APp0⇒For every Banach Y, F(X,Y)is πp-dense in QNp(X,Y). Bourgain and Reinov (1985): H∗∗ ∞,H∗∗∗∗ ∞, . . . have the APp,p∈(1,∞) Corollary X∗∗ has the APp0⇒X∗has the p-AP, p∈(1,∞) H∗ ∞,H∗∗∗ ∞, . . . have the p-AP, p∈[1,∞) J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Relation to Saphar’s approximation property APp Theorem [Oja, Piñeiro, Serrano and Delgado, 2009] Xa Banach space. The following statements are equivalent: 1X∗has the p-AP. 2For every Banach Y,F(X,Y)is k·k-dense in QNp(X,Y). Saphar (1970-72): For every p6=2, there exist Banach spaces without APp. X∗∗ has the APp0⇒For every Banach Y, F(X,Y)is πp-dense in QNp(X,Y). Bourgain and Reinov (1985): H∗∗ ∞,H∗∗∗∗ ∞, . . . have the APp,p∈(1,∞) Corollary X∗∗ has the APp0⇒X∗has the p-AP, p∈(1,∞) H∗ ∞,H∗∗∗ ∞, . . . have the p-AP, p∈[1,∞) J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Outline 1Introduction 2Density of finite rank operators and the p-approximation property 3A trace characterization of the p-approximation property 4Open problems J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems The trace functional –N1(X,Y)is a quotient of X∗ˆ ⊗πY⇒ N1(X,Y)∗,→ L(Y,X∗∗). –T= m X n=1 x∗ n⊗x∗∗ n∈ F(X,X∗∗)(trace(T):=Pm n=1hx∗∗ n,x∗ ni |trace(T)| ≤ Pm n=1kx∗∗ nkkx∗ nk –Finite-nuclear norm of T∈ F(X,X∗∗): ν0(T) := inf{Pm n=1kx∗∗ nkkx∗ nk:T=Pm n=1x∗ n⊗x∗∗ n} Theorem [Grothendieck, 1955] Xa Banach space. The following statements are equivalent: 1Xhas the AP. 2If (xn)⊂Xand (x∗ n)⊂X∗are such that Pnkx∗ nkkxnk<∞and Pnhx∗ n,xixn=0 for all x∈X, then Pnhx∗ n,xni=0. J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems The trace functional –N1(X,Y)is a quotient of X∗ˆ ⊗πY⇒ N1(X,Y)∗,→ L(Y,X∗∗). –T= m X n=1 x∗ n⊗x∗∗ n∈ F(X,X∗∗)(trace(T):=Pm n=1hx∗∗ n,x∗ ni |trace(T)| ≤ Pm n=1kx∗∗ nkkx∗ nk –Finite-nuclear norm of T∈ F(X,X∗∗): ν0(T) := inf{Pm n=1kx∗∗ nkkx∗ nk:T=Pm n=1x∗ n⊗x∗∗ n} Theorem [Grothendieck, 1955] Xa Banach space. The following statements are equivalent: 1Xhas the AP. 2If (xn)⊂Xand (x∗ n)⊂X∗are such that Pnkx∗ nkkxnk<∞and Pnhx∗ n,xixn=0 for all x∈X, then Pnhx∗ n,xni=0. J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems A trace characterization of the p-AP Proposition [Sinha and Karn, 2002] If Xhas the p-AP then the following holds: For every (xn)∈`p(X)and every (x∗ n)∈`1(X∗)such that Pnhx∗ n,xixn=0 for all x∈X, we have Pnhx∗ n,xni=0. Proposition [Oja, Piñeiro, Serrano and Delgado, 2009] The following statements are equivalent: 1Xhas the p-AP. 2For every relatively p-compact sequence (xn)⊂Xand every (x∗ n)∈`1(X∗)such that Pnhx∗ n,xixn=0 for all x∈X, we have Pnhx∗ n,xni=0. Corollary X∗∗ has the p-AP ⇒Xhas the p-AP J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems A trace characterization of the p-AP Proposition [Sinha and Karn, 2002] If Xhas the p-AP then the following holds: For every (xn)∈`p(X)and every (x∗ n)∈`1(X∗)such that Pnhx∗ n,xixn=0 for all x∈X, we have Pnhx∗ n,xni=0. Proposition [Oja, Piñeiro, Serrano and Delgado, 2009] The following statements are equivalent: 1Xhas the p-AP. 2For every relatively p-compact sequence (xn)⊂Xand every (x∗ n)∈`1(X∗)such that Pnhx∗ n,xixn=0 for all x∈X, we have Pnhx∗ n,xni=0. Corollary X∗∗ has the p-AP ⇒Xhas the p-AP J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems About the subspace structure of N1(X,Y)∗ N(p)(X,X∗∗) := T=Pnx∗ n⊗x∗∗ n:(x∗∗ n)⊂X∗∗ relatively p-compact (x∗ n)∈`1(X∗) N(p)(X,X∗∗)⊂ N1(X,X∗∗) If X∗∗ has the p-AP, the map T∈N(p)(X,X∗∗)7→ trace(T)∈Ris well-defined, linear and continuous. hS,ui= trace(S◦u), ∀u∈ N1(X,Y) Πp(Y,X∗∗),Πd p(Y,X∗∗),→ N1(X,Y)∗(p∈[1,2]). J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems About the subspace structure of N1(X,Y)∗ N(p)(X,X∗∗) := T=Pnx∗ n⊗x∗∗ n:(x∗∗ n)⊂X∗∗ relatively p-compact (x∗ n)∈`1(X∗) N(p)(X,X∗∗)⊂ N1(X,X∗∗) If X∗∗ has the p-AP, the map T∈N(p)(X,X∗∗)7→ trace(T)∈Ris well-defined, linear and continuous. hS,ui= trace(S◦u), ∀u∈ N1(X,Y) Πp(Y,X∗∗),Πd p(Y,X∗∗),→ N1(X,Y)∗(p∈[1,2]). J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems About the subspace structure of N1(X,Y)∗ N(p)(X,X∗∗) := T=Pnx∗ n⊗x∗∗ n:(x∗∗ n)⊂X∗∗ relatively p-compact (x∗ n)∈`1(X∗) N(p)(X,X∗∗)⊂ N1(X,X∗∗) If X∗∗ has the p-AP, the map T∈N(p)(X,X∗∗)7→ trace(T)∈Ris well-defined, linear and continuous. Corollary p∈[1,∞). If X∗∗ has the p-AP, then Πd p(Y,X∗∗)⊂ N1(X,Y)∗. If Y∗has the p-AP, then Πp(Y,X∗∗)⊂ N1(X,Y)∗. hS,ui= trace(S◦u), ∀u∈ N1(X,Y) Πp(Y,X∗∗),Πd p(Y,X∗∗),→ N1(X,Y)∗(p∈[1,2]). J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems About the subspace structure of N1(X,Y)∗ N(p)(X,X∗∗) := T=Pnx∗ n⊗x∗∗ n:(x∗∗ n)⊂X∗∗ relatively p-compact (x∗ n)∈`1(X∗) N(p)(X,X∗∗)⊂ N1(X,X∗∗) If X∗∗ has the p-AP, the map T∈N(p)(X,X∗∗)7→ trace(T)∈Ris well-defined, linear and continuous. Corollary p∈[1,∞). If X∗∗ has the p-AP, then Πd p(Y,X∗∗)⊂ N1(X,Y)∗. If Y∗has the p-AP, then Πp(Y,X∗∗)⊂ N1(X,Y)∗. hS,ui= trace(S◦u), ∀u∈ N1(X,Y) Πp(Y,X∗∗),Πd p(Y,X∗∗),→ N1(X,Y)∗(p∈[1,2]). J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems About the subspace structure of N1(X,Y)∗ N(p)(X,X∗∗) := T=Pnx∗ n⊗x∗∗ n:(x∗∗ n)⊂X∗∗ relatively p-compact (x∗ n)∈`1(X∗) N(p)(X,X∗∗)⊂ N1(X,X∗∗) If X∗∗ has the p-AP, the map T∈N(p)(X,X∗∗)7→ trace(T)∈Ris well-defined, linear and continuous. Corollary p∈[1,∞). If X∗∗ has the p-AP, then Πd p(Y,X∗∗)⊂ N1(X,Y)∗. If Y∗has the p-AP, then Πp(Y,X∗∗)⊂ N1(X,Y)∗. S∈Πd p(Y,X∗∗) hS,ui= trace(S◦u), ∀u∈ N1(X,Y) Πp(Y,X∗∗),Πd p(Y,X∗∗),→ N1(X,Y)∗(p∈[1,2]). J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems About the subspace structure of N1(X,Y)∗ N(p)(X,X∗∗) := T=Pnx∗ n⊗x∗∗ n:(x∗∗ n)⊂X∗∗ relatively p-compact (x∗ n)∈`1(X∗) N(p)(X,X∗∗)⊂ N1(X,X∗∗) If X∗∗ has the p-AP, the map T∈N(p)(X,X∗∗)7→ trace(T)∈Ris well-defined, linear and continuous. Corollary p∈[1,∞). If X∗∗ has the p-AP, then Πd p(Y,X∗∗)⊂ N1(X,Y)∗. If Y∗has the p-AP, then Πp(Y,X∗∗)⊂ N1(X,Y)∗. S∈Πd p(Y,X∗∗) u=Pnx∗ n⊗yn (x∗ n)∈`1(X∗),kynk → 0 hS,ui= trace(S◦u), ∀u∈ N1(X,Y) Πp(Y,X∗∗),Πd p(Y,X∗∗),→ N1(X,Y)∗(p∈[1,2]). J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems About the subspace structure of N1(X,Y)∗ N(p)(X,X∗∗) := T=Pnx∗ n⊗x∗∗ n:(x∗∗ n)⊂X∗∗ relatively p-compact (x∗ n)∈`1(X∗) N(p)(X,X∗∗)⊂ N1(X,X∗∗) If X∗∗ has the p-AP, the map T∈N(p)(X,X∗∗)7→ trace(T)∈Ris well-defined, linear and continuous. Corollary p∈[1,∞). If X∗∗ has the p-AP, then Πd p(Y,X∗∗)⊂ N1(X,Y)∗. If Y∗has the p-AP, then Πp(Y,X∗∗)⊂ N1(X,Y)∗. S∈Πd p(Y,X∗∗) u=Pnx∗ n⊗yn (x∗ n)∈`1(X∗),kynk → 0 ⇒S◦u=Pnx∗ n⊗Syn∈ N(p)(X,X∗∗) hS,ui= trace(S◦u), ∀u∈ N1(X,Y) Πp(Y,X∗∗),Πd p(Y,X∗∗),→ N1(X,Y)∗(p∈[1,2]). J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems About the subspace structure of N1(X,Y)∗ N(p)(X,X∗∗) := T=Pnx∗ n⊗x∗∗ n:(x∗∗ n)⊂X∗∗ relatively p-compact (x∗ n)∈`1(X∗) N(p)(X,X∗∗)⊂ N1(X,X∗∗) If X∗∗ has the p-AP, the map T∈N(p)(X,X∗∗)7→ trace(T)∈Ris well-defined, linear and continuous. Corollary p∈[1,∞). If X∗∗ has the p-AP, then Πd p(Y,X∗∗)⊂ N1(X,Y)∗. If Y∗has the p-AP, then Πp(Y,X∗∗)⊂ N1(X,Y)∗. S∈Πd p(Y,X∗∗) u=Pnx∗ n⊗yn (x∗ n)∈`1(X∗),kynk → 0 ⇒S◦u=Pnx∗ n⊗Syn∈ N(p)(X,X∗∗) hS,ui=trace(S◦u),∀u∈ N1(X,Y) Πp(Y,X∗∗),Πd p(Y,X∗∗),→ N1(X,Y)∗(p∈[1,2]). J. M. Delgado The p-approximation property
Introduction Density of finite rank operators and the p-approximation property A trace characterization of the p-approximation property Open problems Open problems F(Y,X)κp=Kp(Y,X),for all Y⇐⇒ ? ? A general definition including most of the mentioned approximation properties of order p. J. M. Delgado The p-approximation property