Norms of idempotent Schur multipliers Rupert Levene University - - PowerPoint PPT Presentation

Norms of idempotent Schur multipliers

Rupert Levene

University College Dublin

Banach Algebras and Applications 29 July 2013

Rupert Levene (Dublin) Norms of idempotent Schur multipliers 1 / 14

Outline

1

Schur multipliers

2

Some norm calculations

3

Gaps in the set of norms

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Schur multipliers

The Schur product of A, B ∈ Mm×n is A • B = [aijbij] (a.k.a. the Hadamard product) The Schur multiplier corresponding to B is the linear map SB : Mm×n → Mm×n, SB(A) = A • B. We also consider “m = n = ∞”: change Mm×n to B(ℓ2) and take infinite matrices B that give bounded maps SB. These form a commutative semisimple Banach algebra.

Rupert Levene (Dublin) Norms of idempotent Schur multipliers

• 1. Schur multipliers

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Idempotent Schur multipliers

In this talk B will be a matrix of 0s and 1s. Then B • B = B = ⇒ SB ◦ SB = SB, so SB is idempotent

Motivating question

What are the possible values of SB : Mm×n → Mm×n? Trivially, 0 and 1 are possible values, but nothing in between: SB = S2

B ≤ SB2 =

⇒ SB ∈ {0} ∪ [1, ∞). We can have SB > 1:

Example

B = 1 1

0 1

• has

SB ≥ U • B =

• 4

3

for U =

1 √ 3

√

2 1 −1 √ 2

• .

C = B⊗k has SC = SBk → ∞ as k → ∞

Rupert Levene (Dublin) Norms of idempotent Schur multipliers

• 1. Schur multipliers

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Showing that S 1 1

0 1 =

• 4

3

Theorem (Grothendieck)

SB ≤ 1 ⇐ ⇒ ∃ vi, wj ∈ ball(ℓ2): B = [vi, wj]

w1 v1 w2 v2

[vi, wj] =

• 3

4

1 1

0 1

• So S 1 1

0 1 =

• 4

3S

3 4

1 1 0 1 ≤

• 4

3.

Have unitary U with S 1 1

0 1 (U) =

• 4

3, so we have equality.

w1 v1 w2 v2 w3

In fact, S 1 1 0

0 1 1 =

• 4

3 too:

Rupert Levene (Dublin) Norms of idempotent Schur multipliers

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“Diagonal + superdiagonal” idempotents

Let B =   

1 1 1 1

... ...

1 1 1

   ∈ Mn×n and C =   

1 1 1 1

... ...

1 1 1 1

   ∈ Mn×(n+1).

Theorem (L., 2012)

SB = SC = 2 n + 1 cot π 2(n + 1).

Question

Given a matrix Y, which submatrices X have SX = SY?

Example [Davidson–Donsig 2007]

For n odd, D =   

1 1 1 1

... ...

1 1 1 1

  ∈ M(n+1)×(n+1) has SD =

2 n+1 cot π 2(n+1).

Note that B and C are both submatrices.

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Livshits’ two gaps theorem

Theorem (Livshits, 1995)

For any 0–1 matrix B, we have SB ∈ {0, 1} ∪

• 4

3, ∞

• .

We can say more: there are at least six gaps.

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Bipartite graphs

{0–1 matrices in Mm×n} ← → {(m,n) bipartite graphs} rows and columns ← → vertices entries equal to 1 ← → edges

Examples

B = 1 1

0 1

• ←

→ GB = C = 1 1 0 0 0

0 1 1 1 0 0 0 0 1 1

• ←

→ GC = B ⊕ C = B 0

0 C

• ←

→ GB⊕C =

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Dictionary of operations

0–1 matrix B bipartite graph GB norm SB shuffle rows

• r columns

bipartite graph isomorphism equal duplicate rows

• r columns

duplicate vertices and their edges equal submatrix induced subgraph decreases direct sum disjoint union max

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A proof of Livshits’ theorem in this language

Theorem (Livshits, 1995)

For any 0–1 matrix B, we have SB ∈ {0, 1} ∪

• 4

3, ∞

• .

Proof.

Let G = GB. WLOG: G is connected. If G is complete bipartite then SB ∈ {0, 1}. Otherwise, take vertices c and r so that:

◮ c and r are in different parts of the bipartition; ◮ (r, c) is not an edge of G; and ◮ the distance from c to r is as small as possible

A minimal path joining c to r starts with the subgraph Minimality = ⇒ c and r are the ends of this path (c, r) not an edge of G = ⇒ is an induced subgraph of G = ⇒ SB ≥

• 4

3.

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Small idempotent Schur multipliers

Define ηk, Ek, Fk for 1 ≤ k ≤ 6 as follows: k 1 2 3 4 5 6 ηk 1

• 4

3 1+ √ 2 2 1 15

• 169 + 38

√ 19

• 3

2 2 5

• 5 + 2

√ 5 Ek Fk

Theorem (L., 2012)

If G = GB is a connected, duplicate-free bipartite graph and 1 ≤ k ≤ 6, then the following are equivalent:

1

SB = ηk

2

ηk−1 < SB ≤ ηk

3

Ek ≤ G ≤ Fk

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Small idempotent Schur multipliers

Define ηk, Ek, Fk for 1 ≤ k ≤ 6 as follows: k 1 2 3 4 5 6 ηk 1

• 4

3 1+ √ 2 2 1 15

• 169 + 38

√ 19

• 3

2 2 5

• 5 + 2

√ 5 Ek Fk

Corollary

If G = GB is any bipartite graph and 1 ≤ k ≤ 6, then the following are equivalent:

1

SB = ηk

2

ηk−1 < SB ≤ ηk

3

(i) Some component H of df(G) has Ek ≤ H ≤ Fk, and (ii) Every component H of df(G) has Ej ≤ H ≤ Fj for some j ≤ k.

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Six gaps

Corollary

If SB is any idempotent Schur multiplier, then SB ∈ {η0, η1, η2, η3, η4, η5} ∪ [η6, ∞). Using tools of Katavolos–Paulsen (2005), this generalises:

Theorem

The same is true if we replace SB with any idempotent normal masa-bimodule map S : B(H) → B(H) where H is a separable Hilbert space.

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Some natural questions

N = {SB: B ∈ {0, 1}m×n, m, n ∈ N ∪ {∞} } contains left accumulation points, such as 4/π = limn→∞

2 n+1 cot( π 2(n+1)).

Is 4/π the smallest accumulation point in N? Is N countable? Does N contain an open interval? Which graphs give Schur idempotents of the same norm? Find a combinatorial characterisation of the idempotent Schur multipliers on B(ℓ2).

Rupert Levene (Dublin) Norms of idempotent Schur multipliers

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Thank you!