Multinorms and Banach lattices Based on results of G.Dales, - - PowerPoint PPT Presentation

Multinorms and Banach lattices

Based on results of G.Dales, M.Polyakov, N.Laustsen, G.Pisier, L.McClaran, P.Ramsden, T.Oikhberg Vladimir Troitsky

University of Alberta

July 2014

Multinorms

Multinorms

Given a vector space X.

Multinorms

Given a vector space X. For each n, given a norm ·n on X n such that

Multinorms

Given a vector space X. For each n, given a norm ·n on X n such that (A1)

• (xσ(1), . . . , xσ(n))
• n =
• (x1, . . . , xn)
• n

Multinorms

Given a vector space X. For each n, given a norm ·n on X n such that (A1)

• (xσ(1), . . . , xσ(n))
• n =
• (x1, . . . , xn)
• n

(A2)

• (x1, . . . , xn, 0)
• n+1 =
• (x1, . . . , xn)
• n

Multinorms

Given a vector space X. For each n, given a norm ·n on X n such that (A1)

• (xσ(1), . . . , xσ(n))
• n =
• (x1, . . . , xn)
• n

(A2)

• (x1, . . . , xn, 0)
• n+1 =
• (x1, . . . , xn)
• n

(A3)

• (α1x1, . . . , αnxn)
• n max|αi| ·
• (x1, . . . , xn)
• n

Multinorms

Given a vector space X. For each n, given a norm ·n on X n such that (A1)

• (xσ(1), . . . , xσ(n))
• n =
• (x1, . . . , xn)
• n

(A2)

• (x1, . . . , xn, 0)
• n+1 =
• (x1, . . . , xn)
• n

(A3)

• (α1x1, . . . , αnxn)
• n max|αi| ·
• (x1, . . . , xn)
• n

(A4)

• (x1, . . . , xn−1, xn, xn)
• n+1 =
• (x1, . . . , xn−1, xn)
• n

Multinorms

Given a vector space X. For each n, given a norm ·n on X n such that (A1)

• (xσ(1), . . . , xσ(n))
• n =
• (x1, . . . , xn)
• n

(A2)

• (x1, . . . , xn, 0)
• n+1 =
• (x1, . . . , xn)
• n

(A3)

• (α1x1, . . . , αnxn)
• n max|αi| ·
• (x1, . . . , xn)
• n

(A4)

• (x1, . . . , xn−1, xn, xn)
• n+1 =
• (x1, . . . , xn−1, xn)
• n

Such a sequence of norms is called a multinorm on X.

Multinorms

Given a vector space X. For each n, given a norm ·n on X n such that (A1)

• (xσ(1), . . . , xσ(n))
• n =
• (x1, . . . , xn)
• n

(A2)

• (x1, . . . , xn, 0)
• n+1 =
• (x1, . . . , xn)
• n

(A3)

• (α1x1, . . . , αnxn)
• n max|αi| ·
• (x1, . . . , xn)
• n

(A4)

• (x1, . . . , xn−1, xn, xn)
• n+1 =
• (x1, . . . , xn−1, xn)
• n

Such a sequence of norms is called a multinorm on X.

Example

Let X be a normed space. Put

• (x1, . . . , xn)
• := maxxi.

Multinorms

Given a vector space X. For each n, given a norm ·n on X n such that (A1)

• (xσ(1), . . . , xσ(n))
• n =
• (x1, . . . , xn)
• n

(A2)

• (x1, . . . , xn, 0)
• n+1 =
• (x1, . . . , xn)
• n

(A3)

• (α1x1, . . . , αnxn)
• n max|αi| ·
• (x1, . . . , xn)
• n

(A4)

• (x1, . . . , xn−1, xn, xn)
• n+1 =
• (x1, . . . , xn−1, xn)
• n

Such a sequence of norms is called a multinorm on X.

Example

Let X be a normed space. Put

• (x1, . . . , xn)
• := maxxi.

Example

Let X be a Banach lattice. Put

• (x1, . . . , xn)
• :=
• n

i=1 |xi|

• .

Multinorms

Given a vector space X. For each n, given a norm ·n on X n such that (A1)

• (xσ(1), . . . , xσ(n))
• n =
• (x1, . . . , xn)
• n

(A2)

• (x1, . . . , xn, 0)
• n+1 =
• (x1, . . . , xn)
• n

(A3)

• (α1x1, . . . , αnxn)
• n max|αi| ·
• (x1, . . . , xn)
• n

(A4)

• (x1, . . . , xn−1, xn, xn)
• n+1 =
• (x1, . . . , xn−1, xn)
• n

Such a sequence of norms is called a multinorm on X.

Example

Let X be a normed space. Put

• (x1, . . . , xn)
• := maxxi.

Example

Let X be a Banach lattice. Put

• (x1, . . . , xn)
• :=
• n

i=1 |xi|

• .

The only multinorm on R is the ℓ∞-norm.

1-multinorms

A sequence of norms on X n is a 1-multimorm if it satisfies (A1)

• (xσ(1), . . . , xσ(n))
• n =
• (x1, . . . , xn)
• n

(A2)

• (x1, . . . , xn, 0)
• n+1 =
• (x1, . . . , xn)
• n

(A3)

• (α1x1, . . . , αnxn)
• n max|αi| ·
• (x1, . . . , xn)
• n

(A4’)

• (x1, . . . , xn−1, xn, xn)
• n+1 =
• (x1, . . . , xn−1, 2xn)
• n.

1-multinorms

A sequence of norms on X n is a 1-multimorm if it satisfies (A1)

• (xσ(1), . . . , xσ(n))
• n =
• (x1, . . . , xn)
• n

(A2)

• (x1, . . . , xn, 0)
• n+1 =
• (x1, . . . , xn)
• n

(A3)

• (α1x1, . . . , αnxn)
• n max|αi| ·
• (x1, . . . , xn)
• n

(A4’)

• (x1, . . . , xn−1, xn, xn)
• n+1 =
• (x1, . . . , xn−1, 2xn)
• n.

Example

Let X be a normed space. Put

• (x1, . . . , xn)
• := n

i=1xi.

1-multinorms

A sequence of norms on X n is a 1-multimorm if it satisfies (A1)

• (xσ(1), . . . , xσ(n))
• n =
• (x1, . . . , xn)
• n

(A2)

• (x1, . . . , xn, 0)
• n+1 =
• (x1, . . . , xn)
• n

(A3)

• (α1x1, . . . , αnxn)
• n max|αi| ·
• (x1, . . . , xn)
• n

(A4’)

• (x1, . . . , xn−1, xn, xn)
• n+1 =
• (x1, . . . , xn−1, 2xn)
• n.

Example

Let X be a normed space. Put

• (x1, . . . , xn)
• := n

i=1xi.

Example

Let X be a Banach lattice. Put

• (x1, . . . , xn)
• :=
• n

i=1 |xi|

• .

1-multinorms

A sequence of norms on X n is a 1-multimorm if it satisfies (A1)

• (xσ(1), . . . , xσ(n))
• n =
• (x1, . . . , xn)
• n

(A2)

• (x1, . . . , xn, 0)
• n+1 =
• (x1, . . . , xn)
• n

(A3)

• (α1x1, . . . , αnxn)
• n max|αi| ·
• (x1, . . . , xn)
• n

(A4’)

• (x1, . . . , xn−1, xn, xn)
• n+1 =
• (x1, . . . , xn−1, 2xn)
• n.

Example

Let X be a normed space. Put

• (x1, . . . , xn)
• := n

i=1xi.

Example

Let X be a Banach lattice. Put

• (x1, . . . , xn)
• :=
• n

i=1 |xi|

• .

The only 1-multinorm on R is the ℓ1-norm.

Theorem

A sequence of norms is a multinorm iff A¯ xm A¯ xn for every ¯ x ∈ X n and every A ∈ Mm,n;

Theorem

A sequence of norms is a multinorm iff A¯ xm A¯ xn for every ¯ x ∈ X n and every A ∈ Mm,n; where (A¯ x)i = n

j=1 aijxj

Theorem

A sequence of norms is a multinorm iff A¯ xm A¯ xn for every ¯ x ∈ X n and every A ∈ Mm,n; where (A¯ x)i = n

j=1 aijxj and A = A: ℓn ∞ → ℓm ∞

Theorem

A sequence of norms is a multinorm iff A¯ xm A¯ xn for every ¯ x ∈ X n and every A ∈ Mm,n; where (A¯ x)i = n

j=1 aijxj and A = A: ℓn ∞ → ℓm ∞

Theorem

A sequence of norms is a 1-multinorm iff A¯ xm A: ℓn

1 → ℓm 1 · ¯

xn for every ¯ x ∈ X n and A ∈ Mm,n.

Theorem

A sequence of norms is a multinorm iff A¯ xm A¯ xn for every ¯ x ∈ X n and every A ∈ Mm,n; where (A¯ x)i = n

j=1 aijxj and A = A: ℓn ∞ → ℓm ∞

Theorem

A sequence of norms is a 1-multinorm iff A¯ xm A: ℓn

1 → ℓm 1 · ¯

xn for every ¯ x ∈ X n and A ∈ Mm,n.

Definition

Given 1 p ∞, we say that a sequence of norms on X n is a p-multinorm if A¯ xm A: ℓn

p → ℓm p · ¯

xn for every ¯ x ∈ X n and A ∈ Mm,n.

Theorem

A sequence of norms is a multinorm iff A¯ xm A¯ xn for every ¯ x ∈ X n and every A ∈ Mm,n; where (A¯ x)i = n

j=1 aijxj and A = A: ℓn ∞ → ℓm ∞

Theorem

A sequence of norms is a 1-multinorm iff A¯ xm A: ℓn

1 → ℓm 1 · ¯

xn for every ¯ x ∈ X n and A ∈ Mm,n.

Definition

Given 1 p ∞, we say that a sequence of norms on X n is a p-multinorm if A¯ xm A: ℓn

p → ℓm p · ¯

xn for every ¯ x ∈ X n and A ∈ Mm,n. multinorm = ∞-multinorm

Theorem

A sequence of norms is a multinorm iff A¯ xm A¯ xn for every ¯ x ∈ X n and every A ∈ Mm,n; where (A¯ x)i = n

j=1 aijxj and A = A: ℓn ∞ → ℓm ∞

Theorem

A sequence of norms is a 1-multinorm iff A¯ xm A: ℓn

1 → ℓm 1 · ¯

xn for every ¯ x ∈ X n and A ∈ Mm,n.

Definition

Given 1 p ∞, we say that a sequence of norms on X n is a p-multinorm if A¯ xm A: ℓn

p → ℓm p · ¯

xn for every ¯ x ∈ X n and A ∈ Mm,n. multinorm = ∞-multinorm p-multinorms satisfy (A1), (A2), and (A3).

Subspaces and quotients

Let X be a p-multinormed space and Y be a linear subspace of X.

Subspaces and quotients

Let X be a p-multinormed space and Y be a linear subspace of X. Then Y is p-multinormed.

Subspaces and quotients

Let X be a p-multinormed space and Y be a linear subspace of X. Then Y is p-multinormed. X/Y is p-multinormed under

• x1 + Y , . . . , xn + Y
• :=

inf

y1,...,yn∈Y

• (x1 + y1, . . . , xn + yn)

Duality

Identify (X ∗)n with (X n)∗ as follows

Duality

Identify (X ∗)n with (X n)∗ as follows ¯ f = (f1, . . . , fn) f1, . . . , fn ∈ X ∗

Duality

Identify (X ∗)n with (X n)∗ as follows ¯ f = (f1, . . . , fn) f1, . . . , fn ∈ X ∗ ¯ f , ¯ x = n

i=1fi, xi

Duality

Identify (X ∗)n with (X n)∗ as follows ¯ f = (f1, . . . , fn) f1, . . . , fn ∈ X ∗ ¯ f , ¯ x = n

i=1fi, xi

This induces a norm on (X ∗)n for every n.

Duality

Identify (X ∗)n with (X n)∗ as follows ¯ f = (f1, . . . , fn) f1, . . . , fn ∈ X ∗ ¯ f , ¯ x = n

i=1fi, xi

This induces a norm on (X ∗)n for every n. ¯ f n = sup

¯ xn1

¯ f , ¯ x

Duality

Identify (X ∗)n with (X n)∗ as follows ¯ f = (f1, . . . , fn) f1, . . . , fn ∈ X ∗ ¯ f , ¯ x = n

i=1fi, xi

This induces a norm on (X ∗)n for every n. ¯ f n = sup

¯ xn1

¯ f , ¯ x This is a q-multinorm on X ∗, where q = p∗.

Operators

A linear operator T : X → Y between two p-multinormed spaces is multibounded if ∃ C > 0 such that

Operators

A linear operator T : X → Y between two p-multinormed spaces is multibounded if ∃ C > 0 such that

• (Tx1, . . . , Txn)
• C
• (x1, . . . , xn)
• for any x1, . . . , xn ∈ X.

Operators

A linear operator T : X → Y between two p-multinormed spaces is multibounded if ∃ C > 0 such that

• (Tx1, . . . , Txn)
• C
• (x1, . . . , xn)
• for any x1, . . . , xn ∈ X.

multibounded ⇒ bounded

Operators

A linear operator T : X → Y between two p-multinormed spaces is multibounded if ∃ C > 0 such that

• (Tx1, . . . , Txn)
• C
• (x1, . . . , xn)
• for any x1, . . . , xn ∈ X.

multibounded ⇒ bounded T is a multiisometry if

• (Tx1, . . . , Txn)
• =
• (x1, . . . , xn)
• for any x1, . . . , xn ∈ X.

Operators

A linear operator T : X → Y between two p-multinormed spaces is multibounded if ∃ C > 0 such that

• (Tx1, . . . , Txn)
• C
• (x1, . . . , xn)
• for any x1, . . . , xn ∈ X.

multibounded ⇒ bounded T is a multiisometry if

• (Tx1, . . . , Txn)
• =
• (x1, . . . , xn)
• for any x1, . . . , xn ∈ X.

X and Y are multiisometric if there is a surjective multiisometry from X onto Y .

p-multinorms and tensor products

p-multinorms and tensor products

A p-multinorm on X can be viewed as a norm on c00(X).

p-multinorms and tensor products

A p-multinorm on X can be viewed as a norm on c00(X). c00(X) = c00 ⊗ X

p-multinorms and tensor products

A p-multinorm on X can be viewed as a norm on c00(X). c00(X) = c00 ⊗ X (xi) →

n

• i=1

ei ⊗ xi

p-multinorms and tensor products

A p-multinorm on X can be viewed as a norm on c00(X). c00(X) = c00 ⊗ X (xi) →

n

• i=1

ei ⊗ xi This induces a norm on c00 ⊗ X.

p-multinorms and tensor products

A p-multinorm on X can be viewed as a norm on c00(X). c00(X) = c00 ⊗ X (xi) →

n

• i=1

ei ⊗ xi This induces a norm on c00 ⊗ X.

Theorem

There is a one-to-one correspondence between

◮ the p-multinorms on X;

p-multinorms and tensor products

A p-multinorm on X can be viewed as a norm on c00(X). c00(X) = c00 ⊗ X (xi) →

n

• i=1

ei ⊗ xi This induces a norm on c00 ⊗ X.

Theorem

There is a one-to-one correspondence between

◮ the p-multinorms on X; ◮ the cross-norms on c00 ⊗ X such that A ⊗ IX A for

every matrix A viewed as an operator A: ℓp → ℓp;

p-multinorms and tensor products

A p-multinorm on X can be viewed as a norm on c00(X). c00(X) = c00 ⊗ X (xi) →

n

• i=1

ei ⊗ xi This induces a norm on c00 ⊗ X.

Theorem

There is a one-to-one correspondence between

◮ the p-multinorms on X; ◮ the cross-norms on c00 ⊗ X such that A ⊗ IX A for

every matrix A viewed as an operator A: ℓp → ℓp; (A ⊗ IX) k

• i=1

ui ⊗ xi

• =

k

• i=1

Aui ⊗ xi

p-multinorms and tensor products

A p-multinorm on X can be viewed as a norm on c00(X). c00(X) = c00 ⊗ X (xi) →

n

• i=1

ei ⊗ xi This induces a norm on c00 ⊗ X.

Theorem

There is a one-to-one correspondence between

◮ the p-multinorms on X; ◮ the cross-norms on c00 ⊗ X such that A ⊗ IX A for

every matrix A viewed as an operator A: ℓp → ℓp;

◮ the cross-norms on ℓp ⊗ X such that T ⊗ IX T for

every operator T : ℓp → ℓp. (A ⊗ IX) k

• i=1

ui ⊗ xi

• =

k

• i=1

Aui ⊗ xi

p-multinorms and tensor products

A p-multinorm on X can be viewed as a norm on c00(X). c00(X) = c00 ⊗ X (xi) →

n

• i=1

ei ⊗ xi This induces a norm on c00 ⊗ X.

Theorem

There is a one-to-one correspondence between

◮ the p-multinorms on X; ◮ the cross-norms on c00 ⊗ X such that A ⊗ IX A for

every matrix A viewed as an operator A: ℓp → ℓp;

◮ the cross-norms on ℓp ⊗ X such that T ⊗ IX T for

every (compact) operator T : ℓp → ℓp. (A ⊗ IX) k

• i=1

ui ⊗ xi

• =

k

• i=1

Aui ⊗ xi

Canonical p-multinorms on Banach lattices

Given a Banach lattice E.

Canonical p-multinorms on Banach lattices

Given a Banach lattice E.

• (x1, . . . , xn)
• =
• n
• i=1

|xi|

• is an ∞-multinorm

Canonical p-multinorms on Banach lattices

Given a Banach lattice E.

• (x1, . . . , xn)
• =
• n
• i=1

|xi|

• is an ∞-multinorm
• (x1, . . . , xn)
• =
• n
• i=1

|xi|

• is a 1-multinorm

Canonical p-multinorms on Banach lattices

Given a Banach lattice E.

• (x1, . . . , xn)
• =
• n
• i=1

|xi|

• is an ∞-multinorm
• (x1, . . . , xn)
• =
• n
• i=1

|xi|

• is a 1-multinorm
• (x1, . . . , xn)
• =
• n
• i=1

|xi|p 1

p

• is a p-multinorm

Canonical p-multinorms on Banach lattices

Given a Banach lattice E.

• (x1, . . . , xn)
• =
• n
• i=1

|xi|

• is an ∞-multinorm
• (x1, . . . , xn)
• =
• n
• i=1

|xi|

• is a 1-multinorm
• (x1, . . . , xn)
• =
• n
• i=1

|xi|p 1

p

• is a p-multinorm

— the canonical p-multinorm on a Banach lattice E.

Representation theorem, case p = ∞

Representation theorem, case p = ∞

Theorem

Every multinormed space is multiisometric to a subspace of a Banach lattice (with the canonical multinorm).

Representation theorem, case p = ∞

Theorem

Every multinormed space is multiisometric to a subspace of a Banach lattice (with the canonical multinorm). That is, for every multinormed space X there exists a Banach lattice E and an operator T : X → E such that

• (x1, . . . , xn)
• =
• n
• i=1

|Txi|

• for any x1, . . . , xn ∈ X.

Representation theorem, case p = ∞

Theorem

Every multinormed space is multiisometric to a subspace of a Banach lattice (with the canonical multinorm). That is, for every multinormed space X there exists a Banach lattice E and an operator T : X → E such that

• (x1, . . . , xn)
• =
• n
• i=1

|Txi|

• for any x1, . . . , xn ∈ X.

That is, there is a one-to-one correspondence between multinorms and subspaces of Banach lattices.

Dual version

Dual version

Theorem

Every 1-multinormed space is multiisometric to a quotient of a Banach lattice (with the canonical 1-multinorm).

Dual version

Theorem

Every 1-multinormed space is multiisometric to a quotient of a Banach lattice (with the canonical 1-multinorm). 1-multinormed spaces = quotients of Banach lattices

Representation theorems for other values of p? As a subspace of a Banach lattice?

Representation theorems for other values of p? As a subspace of a Banach lattice? Partial success: true under additional assumptions.

Representation theorems for other values of p? As a subspace of a Banach lattice? Partial success: true under additional assumptions.

Theorem

Every convex strong p-multinormed space is multiisometric to a subspace of a Banach lattice with the canonical p-multinorm.

Representation theorems for other values of p? As a subspace of a Banach lattice? Partial success: true under additional assumptions.

Theorem

Every convex strong p-multinormed space is multiisometric to a subspace of a Banach lattice with the canonical p-multinorm. That is, there exists a Banach lattice E and an operator T : X → E such that for any x1, . . . , xn ∈ X one has

Representation theorems for other values of p? As a subspace of a Banach lattice? Partial success: true under additional assumptions.

Theorem

Every convex strong p-multinormed space is multiisometric to a subspace of a Banach lattice with the canonical p-multinorm. That is, there exists a Banach lattice E and an operator T : X → E such that for any x1, . . . , xn ∈ X one has

• (x1, . . . , xn)
• =
• n
• i=1

|Txi|p 1

p

Representation theorems for other values of p? As a subspace of a Banach lattice? Partial success: true under additional assumptions.

Theorem

Every convex strong p-multinormed space is multiisometric to a subspace of a Banach lattice with the canonical p-multinorm. That is, there exists a Banach lattice E and an operator T : X → E such that for any x1, . . . , xn ∈ X one has

• (x1, . . . , xn)
• =
• n
• i=1

|Txi|p 1

p

• Without the extra assumptions, there are counterexamples.

Strong p-multinorms

Definition

A sequence of norms on powers of X is called a strong p-multinorm if the following condition is satisfied.

Strong p-multinorms

Definition

A sequence of norms on powers of X is called a strong p-multinorm if the following condition is satisfied. Given ¯ x ∈ X n and ¯ y ∈ X m.

Strong p-multinorms

Definition

A sequence of norms on powers of X is called a strong p-multinorm if the following condition is satisfied. Given ¯ x ∈ X n and ¯ y ∈ X m. If

• f (¯

x)

• ℓn

p

• f (¯

y)

• ℓm

p for every f ∈ X ∗ then ¯

xn ¯ ym.

Strong p-multinorms

Definition

A sequence of norms on powers of X is called a strong p-multinorm if the following condition is satisfied. Given ¯ x ∈ X n and ¯ y ∈ X m. If

• f (¯

x)

• ℓn

p

• f (¯

y)

• ℓm

p for every f ∈ X ∗ then ¯

xn ¯ ym. Here f (¯ x) =

• f (x1), . . . , f (xn)

Strong p-multinorms

Definition

A sequence of norms on powers of X is called a strong p-multinorm if the following condition is satisfied. Given ¯ x ∈ X n and ¯ y ∈ X m. If

• f (¯

x)

• ℓn

p

• f (¯

y)

• ℓm

p for every f ∈ X ∗ then ¯

xn ¯ ym. Here f (¯ x) =

• f (x1), . . . , f (xn)
• Facts:

◮ Strong p-multinorm ⇒ p-multinorm;

Strong p-multinorms

Definition

A sequence of norms on powers of X is called a strong p-multinorm if the following condition is satisfied. Given ¯ x ∈ X n and ¯ y ∈ X m. If

• f (¯

x)

• ℓn

p

• f (¯

y)

• ℓm

p for every f ∈ X ∗ then ¯

xn ¯ ym. Here f (¯ x) =

• f (x1), . . . , f (xn)
• Facts:

◮ Strong p-multinorm ⇒ p-multinorm; ◮ The converse is true when p = ∞ or p = 2;

Strong p-multinorms

Definition

A sequence of norms on powers of X is called a strong p-multinorm if the following condition is satisfied. Given ¯ x ∈ X n and ¯ y ∈ X m. If

• f (¯

x)

• ℓn

p

• f (¯

y)

• ℓm

p for every f ∈ X ∗ then ¯

xn ¯ ym. Here f (¯ x) =

• f (x1), . . . , f (xn)
• Facts:

◮ Strong p-multinorm ⇒ p-multinorm; ◮ The converse is true when p = ∞ or p = 2; ◮ The canonical p-multinorm on a Banach lattice is strong.

Convex p-multinorms

A p-multinorm is convex if

• (x1, . . . , xn)
• p
• (x1, . . . , xk)
• p +
• (xk+1, . . . , xn)
• p

for any x1, . . . , xn ∈ X and k n.

Convex p-multinorms

A p-multinorm is convex if

• (x1, . . . , xn)
• p
• (x1, . . . , xk)
• p +
• (xk+1, . . . , xn)
• p

for any x1, . . . , xn ∈ X and k n. Facts:

◮ If p = 1, trivial.

Convex p-multinorms

A p-multinorm is convex if

• (x1, . . . , xn)
• p
• (x1, . . . , xk)
• p +
• (xk+1, . . . , xn)
• p

for any x1, . . . , xn ∈ X and k n. Facts:

◮ If p = 1, trivial. ◮ The canonical p-multinorm on a Banach lattice is convex iff

the Banach lattice is p-convex.

p-multibounded operators

For an operator T : E → F between two Banach lattices, we say that T is p-multibounded if it is multibounded w.r.t. the canonical p-multinorms on E and F.

p-multibounded operators

For an operator T : E → F between two Banach lattices, we say that T is p-multibounded if it is multibounded w.r.t. the canonical p-multinorms on E and F. That is,

• (Tx1, . . . , Txn)
• C
• (x1, . . . , xn)
• for any x1, . . . , xn ∈ X.

p-multibounded operators

For an operator T : E → F between two Banach lattices, we say that T is p-multibounded if it is multibounded w.r.t. the canonical p-multinorms on E and F. That is,

• (Tx1, . . . , Txn)
• C
• (x1, . . . , xn)
• for any x1, . . . , xn ∈ X.
• n
• i=1

|Txi|p 1

p

• C
• n
• i=1

|xi|p 1

p

• for any x1, . . . , xn ∈ X.

p-multibounded operators

For an operator T : E → F between two Banach lattices, we say that T is p-multibounded if it is multibounded w.r.t. the canonical p-multinorms on E and F. That is,

• (Tx1, . . . , Txn)
• C
• (x1, . . . , xn)
• for any x1, . . . , xn ∈ X.
• n
• i=1

|Txi|p 1

p

• C
• n
• i=1

|xi|p 1

p

• for any x1, . . . , xn ∈ X.

Easy fact: if T 0 then n

i=1|Txi|p 1

p T

n

i=1|xi|p 1

p .

p-multibounded operators

For an operator T : E → F between two Banach lattices, we say that T is p-multibounded if it is multibounded w.r.t. the canonical p-multinorms on E and F. That is,

• (Tx1, . . . , Txn)
• C
• (x1, . . . , xn)
• for any x1, . . . , xn ∈ X.
• n
• i=1

|Txi|p 1

p

• C
• n
• i=1

|xi|p 1

p

• for any x1, . . . , xn ∈ X.

Easy fact: if T 0 then n

i=1|Txi|p 1

p T

n

i=1|xi|p 1

p . So

• n
• i=1

|Txi|p 1

p

• T
• n
• i=1

|xi|p 1

p

Regular operators

So every positive operator is p-multibounded.

Regular operators

So every positive operator is p-multibounded. It follows immediately that every regular operator is p-multibounded.

Regular operators

So every positive operator is p-multibounded. It follows immediately that every regular operator is p-multibounded. Recall: T is regular if T = U − V for some positive U and V .

Regular operators

So every positive operator is p-multibounded. It follows immediately that every regular operator is p-multibounded. Recall: T is regular if T = U − V for some positive U and V . If T is regular then T ∗ is regular.

Regular operators

So every positive operator is p-multibounded. It follows immediately that every regular operator is p-multibounded. Recall: T is regular if T = U − V for some positive U and V . If T is regular then T ∗ is regular. The converse is false in general.

Regular operators

So every positive operator is p-multibounded. It follows immediately that every regular operator is p-multibounded. Recall: T is regular if T = U − V for some positive U and V . If T is regular then T ∗ is regular. The converse is false in general.

Theorem

TFAE:

◮ T is ∞-multibounded ◮ T is 1-multibounded ◮ T ∗ is regular.

Regular operators

So every positive operator is p-multibounded. It follows immediately that every regular operator is p-multibounded. Recall: T is regular if T = U − V for some positive U and V . If T is regular then T ∗ is regular. The converse is false in general.

Theorem

TFAE:

◮ T is ∞-multibounded:

• n

i=1|Txi|

• C
• n

i=1|xi|

• ◮ T is 1-multibounded

◮ T ∗ is regular.

Regular operators

So every positive operator is p-multibounded. It follows immediately that every regular operator is p-multibounded. Recall: T is regular if T = U − V for some positive U and V . If T is regular then T ∗ is regular. The converse is false in general.

Theorem

TFAE:

◮ T is ∞-multibounded:

• n

i=1|Txi|

• C
• n

i=1|xi|

• ◮ T is 1-multibounded:
• n

i=1|Txi|

• C
• n

i=1|xi|

• ◮ T ∗ is regular.

Every operator is 2-multibounded

For p = 2, every operator is 2-multibounded.

Every operator is 2-multibounded

For p = 2, every operator is 2-multibounded. This immediately follows from Krivine’s Theorem:

Theorem

For every operator T : E → F and any x1, . . . , xn ∈ E,

• n
• i=1

|Txi|2 1

2

• KGT
• n
• i=1

|xi|2 1

2

Quotient of a canonical p-multinorm

Suppose that E is a Banach lattice. Consider the canonical p-multinorm on E.

Quotient of a canonical p-multinorm

Suppose that E is a Banach lattice. Consider the canonical p-multinorm on E. Let J ⊆ E be a closed order ideal. Then E/J is again a Banach lattice.

Quotient of a canonical p-multinorm

Suppose that E is a Banach lattice. Consider the canonical p-multinorm on E. Let J ⊆ E be a closed order ideal. Then E/J is again a Banach lattice. On E/J, there are 2 natural p-multinorms: the quotient one and the canonical one.

Quotient of a canonical p-multinorm

Suppose that E is a Banach lattice. Consider the canonical p-multinorm on E. Let J ⊆ E be a closed order ideal. Then E/J is again a Banach lattice. On E/J, there are 2 natural p-multinorms: the quotient one and the canonical one. Do they agree?

Quotient of a canonical p-multinorm

Suppose that E is a Banach lattice. Consider the canonical p-multinorm on E. Let J ⊆ E be a closed order ideal. Then E/J is again a Banach lattice. On E/J, there are 2 natural p-multinorms: the quotient one and the canonical one. Do they agree? Need to prove that for any x1, . . . , xn ∈ E, inf

y1,...,yn∈J

• n
• i=1

|xi − yi|p 1

p

• = inf

y∈J

• n
• i=1

|xi|p 1

p − y

Quotient of a canonical p-multinorm

Suppose that E is a Banach lattice. Consider the canonical p-multinorm on E. Let J ⊆ E be a closed order ideal. Then E/J is again a Banach lattice. On E/J, there are 2 natural p-multinorms: the quotient one and the canonical one. Do they agree? Need to prove that for any x1, . . . , xn ∈ E, inf

y1,...,yn∈J

• n
• i=1

|xi − yi|p 1

p

• = inf

y∈J

• n
• i=1

|xi|p 1

p − y

• Yes!