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Factorization of operators on Banach (function) spaces

Emiel Lorist

Delft University of Technology, The Netherlands

Madrid, Spain

September 9, 2019

Joint work with Nigel Kalton and Lutz Weis

Euclidean structures• Project started in the early 2000s

• Understand the role R-boundedness from anabstract operator-theoretic viewpoint

• Connected to completely bounded maps

• Project on hold since Nigel passed away

• Studied and improved/extended parts of themanuscript in 2016

• Joined the project in 2018, rewrote and modernizedthe manuscript

• Euclidean structures• Part I: Representation of operator families

on a Hilbert space• Part II: Factorization of operator families• Part III: Vector-valued function spaces• Part IV: Sectorial operators and H∞-calculus• Part V: Counterexamples

• Project to be finished this fall

Nigel Kalton

Lutz Weis

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Nigel Kalton

Lutz Weis

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Nigel Kalton

Lutz Weis

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Nigel Kalton

Lutz Weis

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R-boundedness

Definition

Let X be a Banach space and Γ ⊆ L(X ). Then we say that Γ is R-bounded if forany finite sequences (Tk)nk=1 in Γ and (xk)nk=1 in X(

E∥∥∥ n∑k=1

εkTkxk

∥∥∥2

X

)1/2

≤ C(E∥∥∥ n∑k=1

εkxk

∥∥∥2

X

)1/2

,

where (εk)nk=1 is a sequence of independent Rademacher variables.

• R-boundedness is a strengthening of uniform boundedness.• Equivalent to uniform boundedness on Hilbert spaces.

• R-boundedness plays a (key) role in e.g.• Schauder multipliers• Operator-valued Fourier multiplier theory• Functional calculus• Maximal regularity of PDE’s• Stochastic integration in Banach spaces

•(E∥∥∑n

k=1 εkxk∥∥2)1/2

is a norm on X n.• A Euclidean structure is such a norm with a left and right ideal property.

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R-boundedness

Definition


E∥∥∥ n∑k=1

εkTkxk

∥∥∥2

X

)1/2

≤ C(E∥∥∥ n∑k=1

εkxk

∥∥∥2

X

)1/2

,




•(E∥∥∑n

k=1 εkxk∥∥2)1/2


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R-boundedness

Definition


E∥∥∥ n∑k=1

εkTkxk

∥∥∥2

X

)1/2

≤ C(E∥∥∥ n∑k=1

εkxk

∥∥∥2

X

)1/2

,




•(E∥∥∑n

k=1 εkxk∥∥2)1/2


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Euclidean structures

Definition

Let X be a Banach space. A Euclidean structure α is a family of norms ‖·‖α on X n

for all n ∈ N such that

‖(x)‖α = ‖x‖X , x ∈ X ,

‖Ax‖α ≤ ‖A‖‖x‖α, x ∈ X n, A ∈ Mm,n(C),

‖(Tx1, · · · ,Txn)‖α ≤ C ‖T‖‖x‖α, x ∈ X n, T ∈ L(X ),

A Euclidean structure induces a norm on the finite rank operators from `2 to X .

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Definition



‖(x)‖α = ‖x‖X , x ∈ X ,

‖Ax‖α ≤ ‖A‖‖x‖α, x ∈ X n, A ∈ Mm,n(C),



Non-example:

• On a Banach space X :

‖x‖R :=(E∥∥ n∑k=1

εkxk∥∥2

X

) 12, x ∈ X n,

is not a Euclidean structure. It fails the right-ideal property.

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Definition



‖(x)‖α = ‖x‖X , x ∈ X ,

‖Ax‖α ≤ ‖A‖‖x‖α, x ∈ X n, A ∈ Mm,n(C),



Examples:

• On any Banach space X : The Gaussian structure

‖x‖γ :=(E∥∥ n∑k=1

γkxk∥∥2

X

) 12, x ∈ X n,

where (γk)nk=1 is a sequence of independent normalized Gaussians.

• On a Banach lattice X : The `2-structure

‖x‖`2 :=∥∥∥( n∑

k=1

|xk |2)1/2

∥∥∥X, x ∈ X n.

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Definition



‖(x)‖α = ‖x‖X , x ∈ X ,

‖Ax‖α ≤ ‖A‖‖x‖α, x ∈ X n, A ∈ Mm,n(C),



Examples:

• On any Banach space X : The Gaussian structure

‖x‖γ :=(E∥∥ n∑k=1

γkxk∥∥2

X

) 12, x ∈ X n,

where (γk)nk=1 is a sequence of independent normalized Gaussians.

• On a Banach lattice X : The `2-structure

‖x‖`2 :=∥∥∥( n∑

k=1

|xk |2)1/2

∥∥∥X, x ∈ X n.

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α-boundedness

Definition

Let X be a Banach space, α an Euclidean structure and Γ ⊆ L(X ). Then we say thatΓ is α-bounded if for any T = diag(T1, · · · ,Tn) with T1, · · · ,Tn ∈ Γ

‖Tx‖α ≤ C‖x‖α, x ∈ X n.

• α-boundedness implies uniform boundedness

• On a Banach space X with finite cotype

‖x‖γ =(E∥∥ n∑k=1

γkxk∥∥2

X

) 12 '

(E∥∥ n∑k=1

εkxk∥∥2

X

) 12

= ‖x‖R,

so γ-boundedness is equivalent to R-boundedness

• On a Banach lattice X with finite cotype

‖x‖`2 =∥∥( n∑

k=1

|xk |2)1/2∥∥

X'(E∥∥ n∑k=1

εkxk∥∥2

X

) 12

= ‖x‖R,

so `2-boundedness is equivalent to R-boundedness.

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α-boundedness

Definition


‖Tx‖α ≤ C‖x‖α, x ∈ X n.



‖x‖γ =(E∥∥ n∑k=1

γkxk∥∥2

X

) 12 '

(E∥∥ n∑k=1

εkxk∥∥2

X

) 12

= ‖x‖R,



‖x‖`2 =∥∥( n∑

k=1

|xk |2)1/2∥∥

X'(E∥∥ n∑k=1

εkxk∥∥2

X

) 12

= ‖x‖R,


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α-boundedness

Definition


‖Tx‖α ≤ C‖x‖α, x ∈ X n.



‖x‖γ =(E∥∥ n∑k=1

γkxk∥∥2

X

) 12 '

(E∥∥ n∑k=1

εkxk∥∥2

X

) 12

= ‖x‖R,



‖x‖`2 =∥∥( n∑

k=1

|xk |2)1/2∥∥

X'(E∥∥ n∑k=1

εkxk∥∥2

X

) 12

= ‖x‖R,


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Factorization through a Hilbert space

Theorem (Kwapien ’72 and Maurey ’74)

Let X and Y be a Banach spaces and T ∈ L(X ,Y ). If X has type 2 and Y cotype2, then there is a Hilbert space H and operators S ∈ L(X ,H) and U ∈ L(H,Y ) s.t.

T = US .

Corollary (Kwapien ’72)

Let X be a Banach space. X has type 2 and cotype 2 if and only if X is isomorphicto a Hilbert space.

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T = US .



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T = US .



Theorem (Kalton–L.–Weis ’19)

Let X and Y be a Banach spaces, Γ1 ⊆ L(X ,Y ) . If X has type 2, Ycotype 2 and Γ1 is γ-bounded, then there is a Hilbert space H, aT ∈ L(X ,H) for every T ∈ Γ1 and U ∈ L(H,Y ) s.t.the following diagram commutes:

X Y

H

T

T

U

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T = US .




Let X and Y be a Banach spaces, Γ1 ⊆ L(X ,Y ) and Γ2 ⊆ L(Y ). If X has type 2, Ycotype 2 and Γ1 and Γ2 are γ-bounded, then there is a Hilbert space H, aT ∈ L(X ,H) for every T ∈ Γ1, a S ∈ L(H) for every S ∈ Γ2 and U ∈ L(H,Y ) s.t.the following diagram commutes:

X Y Y

H H

T

T S

U

S

U

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Factorization through weighted L2

For a measure space (S ,Σ, µ) and a weight w : S → R+ let L2(S ,w) be the space ofall measurable f : S → C such that

‖f ‖L2(S,w) :=(∫

S

|f |2w dµ)1/2

<∞


Let X be an order-continuous Banach function space over (S , µ) and let Γ ⊆ L(X ).Γ is `2-bounded if and only if for any g0, g1 ∈ X there is a weight w : S → R+ s.t.

‖Tf ‖L2(S,w) ≤ C ‖f ‖L2(S,w), f ∈ X ∩ L2(S ,w), T ∈ Γ

‖g0‖L2(S,w) ≤ C ‖g0‖X ,

‖g1‖X ≤ C ‖g1‖L2(S,w).

The if statement is trivial. Indeed for f1, · · · , fn ∈ X and T1, · · · ,Tn ∈ Γ set

g0 :=( n∑k=1

|fk |2) 1

2 , g1 :=( n∑k=1

|Tk fk |2) 1

2

Then we have

‖Tf ‖2`2 = ‖g1‖2

X ≤ C 2n∑

k=1

∫S

|Tk fk |2w dµ ≤ C 4n∑

k=1

∫S

|fk |2w dµ ≤ C 6‖g0‖2X = ‖f ‖2

`2

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‖f ‖L2(S,w) :=(∫

S

|f |2w dµ)1/2

<∞




‖g0‖L2(S,w) ≤ C ‖g0‖X ,

‖g1‖X ≤ C ‖g1‖L2(S,w).


g0 :=( n∑k=1

|fk |2) 1

2 , g1 :=( n∑k=1

|Tk fk |2) 1

2

Then we have

‖Tf ‖2`2 = ‖g1‖2

X ≤ C 2n∑

k=1

∫S


k=1

∫S

|fk |2w dµ ≤ C 6‖g0‖2X = ‖f ‖2

`2

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‖f ‖L2(S,w) :=(∫

S

|f |2w dµ)1/2

<∞




‖g0‖L2(S,w) ≤ C ‖g0‖X ,

‖g1‖X ≤ C ‖g1‖L2(S,w).


g0 :=( n∑k=1

|fk |2) 1

2 , g1 :=( n∑k=1

|Tk fk |2) 1

2

Then we have

‖Tf ‖2`2 = ‖g1‖2

X ≤ C 2n∑

k=1

∫S


k=1

∫S

|fk |2w dµ ≤ C 6‖g0‖2X = ‖f ‖2

`2

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‖f ‖L2(S,w) :=(∫

S

|f |2w dµ)1/2

<∞




‖g0‖L2(S,w) ≤ C ‖g0‖X ,

‖g1‖X ≤ C ‖g1‖L2(S,w).


g0 :=( n∑k=1

|fk |2) 1

2 , g1 :=( n∑k=1

|Tk fk |2) 1

2

Then we have

‖Tf ‖2`2 = ‖g1‖2

X ≤ C 2n∑

k=1

∫S


k=1

∫S

|fk |2w dµ ≤ C 6‖g0‖2X = ‖f ‖2

`2

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Application: Harmonic analysis

Let p ∈ [1,∞). For f ∈ Lp(R) define the Hilbert transform

Hf (x) := p. v.1

π

∫R

1

x − yf (y) dy , x ∈ R,

= F−1(ξ 7→ −i sgn(ξ)F f (ξ))(x).

For f ∈ Lp(Rd) define the Hardy–Littlewood maximal operator

Mf (x) := supr>0

1

|B(x , r)|

∫Rd

|f (y)| dy , x ∈ Rd .

• Very important operators in harmonic analysis.

• Both H and M are bounded on Lp for p ∈ (1,∞).

• We are interested in the tensor extensions of H and M onthe Bochner space Lp(Rd ;X ).

• These have numerous applications in both harmonic analysis and PDE.

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Hf (x) := p. v.1

π

∫R

1

x − yf (y) dy , x ∈ R,

= F−1(ξ 7→ −i sgn(ξ)F f (ξ))(x).


Mf (x) := supr>0

1

|B(x , r)|

∫Rd

|f (y)| dy , x ∈ Rd .





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Hf (x) := p. v.1

π

∫R

1

x − yf (y) dy , x ∈ R,

= F−1(ξ 7→ −i sgn(ξ)F f (ξ))(x).


Mf (x) := supr>0

1

|B(x , r)|

∫Rd

|f (y)| dy , x ∈ Rd .





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Hf (x) := p. v.1

π

∫R

1

x − yf (y) dy , x ∈ R,

= F−1(ξ 7→ −i sgn(ξ)F f (ξ))(x).


Mf (x) := supr>0

1

|B(x , r)|

∫Rd

|f (y)| dy , x ∈ Rd .





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Let p ∈ [1,∞) and let X be a Banach space. For f ∈ Lp(R;X ) define thevector-valued Hilbert transform

Hf (x) := p. v.

∫R

1

x − yf (y) dy , x ∈ R

where the integral is interpreted in the Bochner sense.

Let X be a Banach function space. For f ∈ Lp(Rd ;X ) define the latticeHardy–Littlewood maximal operator

Mf (x) := supr>0

1

|B(x , r)|

∫Rd

|f (y)| dy , x ∈ Rd ,

where the supremum is taken in the lattice sense.

• Boundedness of H is independent of p ∈ (1,∞)(Calderon–Benedek–Panzone ’62).

• Boundedness of M is independent of p ∈ (1,∞) and d ∈ N(Garcıa-Cuerva–Macias–Torrea ’93).

• Boundedness of H and M depends on the geometry of X .

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Hf (x) := p. v.

∫R

1

x − yf (y) dy , x ∈ R



Mf (x) := supr>0

1

|B(x , r)|

∫Rd

|f (y)| dy , x ∈ Rd ,





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Hf (x) := p. v.

∫R

1

x − yf (y) dy , x ∈ R



Mf (x) := supr>0

1

|B(x , r)|

∫Rd

|f (y)| dy , x ∈ Rd ,





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Theorem (Burkholder ’83 and Bourgain ’83)

Let X be a Banach space and p ∈ (1,∞). The following are equivalent:

(i) X has the UMD property.

(ii) H is bounded on Lp(R;X ).

• X has the UMD property if any finite martingale (fk)nk=0 in Lp(Ω;X ) hasUnconditional Martingale Differences (dfk)nk=1.

Theorem (Bourgain ’84 and Rubio de Francia ’86)

Let X be a Banach function space and p ∈ (1,∞). The following is equivalent to (i)and (ii)

(iii) M is bounded on Lp(Rd ;X ) and on Lp(Rd ;X ∗).

• (iii)⇒(ii) is “easy” and quantitative.

• (ii)⇒(iii) is very involved and technical.

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• That is, if for some p ∈ (1,∞) and all signs εk = ±1 we have∥∥∥ n∑k=1

εkdfn∥∥∥Lp(Ω;X )

.∥∥∥ n∑k=1

dfn∥∥∥Lp(Ω;X )

,






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• Reflexive Lebesgue, Lorentz, (Musielak)-Orlicz, Sobolev, Besov spaces andSchatten classes have the UMD property.

• L1 and L∞ do not have the UMD property.

• Linear constant dependence is an open problem






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• Reflexive Lebesgue, Lorentz, (Musielak)-Orlicz, Sobolev, Besov spaces andSchatten classes have the UMD property.

• L1 and L∞ do not have the UMD property.

• Linear constant dependence is an open problem






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Let X be a Banach function space on (S , µ) and p ∈ (1,∞). If H is bounded on

Lp(R;X ), then M is bounded on Lp(R;X ) with

‖M‖Lp(R;X )→Lp(R;X ) ≤ C ‖H‖2Lp(R;X )→Lp(R;X )

• Apply factorization theorem with Γ = H and g1 = Mg0 to transfer thequestion from Lp(R;X ) to L2(R× S ,w):For any g0 ∈ Lp(R;X ) there is a weight w : R× S → R+ such that

‖Hf ‖L2(R×S,w) ≤ C ‖f ‖L2(R×S,w), f ∈ Lp(R;X ) ∩ L2(R× S ,w)

‖g0‖L2(R×S,w) ≤ C ‖g0‖Lp(R;X ),

‖Mg0‖Lp(R;X ) ≤ C ‖Mg0‖L2(R×S,w).

• By Fubini it suffices to show the boundedness of M on L2(R,w(·, s)) for s ∈ S .

• For a weight v : R→ R+, M is bounded on L2(R, v) if and only if H is boundedon L2(R, v) (Muckenhoupt ’72,...).

• As H is bounded on L2(R,w(·, s)) for s ∈ S , M is as well.

• Thus ‖Mg0‖Lp(R;X ) . ‖g0‖Lp(R;X )

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‖g0‖L2(R×S,w) ≤ C ‖g0‖Lp(R;X ),

‖Mg0‖Lp(R;X ) ≤ C ‖Mg0‖L2(R×S,w).





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‖g0‖L2(R×S,w) ≤ C ‖g0‖Lp(R;X ),

‖Mg0‖Lp(R;X ) ≤ C ‖Mg0‖L2(R×S,w).





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‖g0‖L2(R×S,w) ≤ C ‖g0‖Lp(R;X ),

‖Mg0‖Lp(R;X ) ≤ C ‖Mg0‖L2(R×S,w).





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‖g0‖L2(R×S,w) ≤ C ‖g0‖Lp(R;X ),

‖Mg0‖Lp(R;X ) ≤ C ‖Mg0‖L2(R×S,w).





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‖g0‖L2(R×S,w) ≤ C ‖g0‖Lp(R;X ),

‖Mg0‖Lp(R;X ) ≤ C ‖Mg0‖L2(R×S,w).





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Application: Banach space geometryLet X be a Banach space and p ∈ (1,∞). Let (fk)nk=0 be a finite martingale inLp(Ω;X ) with difference sequence (dfk)nk=1. Then X has the UMD property if andonly if for all signs εk = ±1 we have∥∥∥ n∑

k=1

εkdfn∥∥∥Lp(Ω;X )

(1)

.∥∥∥ n∑k=1


,

if and only if∥∥∥ n∑k=1


(2)

.∥∥∥ n∑k=1

εkdfn∥∥∥Lp(Ω×Ω′;X )

(3)

.∥∥∥ n∑k=1


,

where (ε)nk=1 is a Rademacher sequence on Ω′.

• (2) does not imply (1).

• It is an open question whether (3) implies (1)


Let X be a Banach function space. Then (3) implies (1).

• Proof similar to previous slide.

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Outlook

• In the Euclidean structures manuscript:

• More factorization theorems.

• Representation of an α-bounded family of operators on a Hilbert space.

• Applications to interpolation, function spaces, functional calculus.

• More applications of factorization through weighted L2:

• Show the necessity of UMD for the R-boundedness of certain operators

• Potentially many more applications!

• To appear on arXiv before Christmas!

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Outlook









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Outlook









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Thank you for your attention!

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