An in nite dimensional Schur-Horn Theorem and majorization theory: the nonzero kernel … · 2013....

An infinite dimensional Schur-Horn Theoremand majorization theory: the nonzero kernel case

Jireh Loreauxjoint work with Gary Weiss

Great Plains Operator Theory Symposium, UC Berkeley 2013

Jireh Loreaux joint work with Gary Weiss A Schur-Horn Theorem in B(H) GPOTS ‘13 1 / 14

Introduction History

Majorization and the Schur-Horn Theorem

Definition

Given a sequence ξ ∈ Rn, ξ∗ will denote the nonincreasing monotonization of ξ.

Definition

Given sequences ξ, η ∈ Rn, we say that η majorizes ξ, written ξ ≺ η, if

k∑j=1

ξ∗j ≤k∑

η∗j , for k ≤ n andn∑

ξj =n∑

Theorem (Schur-Horn)

If X is a Hermitian matrix with eigenvalue list λ ∈ Rn (with multiplicity), thenthe diagonal list d ∈ Rn of X satisfies d ≺ λ. Conversely, for any sequence dmajorized by λ, there exists a basis with respect to which d is the diagonal list ofX .

Introduction History

Majorization and the Schur-Horn Theorem

Definition

Given a sequence ξ ∈ Rn, ξ∗ will denote the nonincreasing monotonization of ξ.

Definition

Given sequences ξ, η ∈ Rn, we say that η majorizes ξ, written ξ ≺ η, if

k∑j=1

ξ∗j ≤k∑

η∗j , for k ≤ n andn∑

ξj =n∑

Theorem (Schur-Horn)

If X is a Hermitian matrix with eigenvalue list λ ∈ Rn (with multiplicity), thenthe diagonal list d ∈ Rn of X satisfies d ≺ λ. Conversely, for any sequence dmajorized by λ, there exists a basis with respect to which d is the diagonal list ofX .

Introduction Generalization

Generalizing the Schur-Horn Theorem

There are a few prerequisites to generalizing the Schur-Horn theorem to othercontexts. In each context we would need:

A notion of majorization;

A notion of “taking the diagonal.”

In B(H), the latter is easy, since we can fix a basis {en}n∈N and then viewoperators in B(H) as matrices acting on `2(N). However, the diagonal matriceswith respect to this fixed basis form a masa A of B(H), and “taking thediagonal” of some operator X ∈ B(H), amounts to taking EA(X ), whereEA : B(H)→ A is the faithful normal conditional expectation. As Arveson andKadison point out, the situation with a faithful normal trace-preservingconditional expectation also arises in type II1 factors. In contrast, majorization inother contexts takes slightly more work.

Majorization in c+0

Definition

Given a sequence ξ ∈ c+0 , let ξ∗ denote the monotone decreasing rearrangement

of ξ. Succinctly, ξ∗ is obtained by listing the nonzero elements of ξ in monotonedecreasing order, which can be done since ξn → 0.

Definition

Given sequences ξ, η ∈ c+0 , we say that ξ is majorized by η if for all n ∈ N.

n∑j=1

ξ∗j ≤n∑

η∗j and∞∑j=1

ξ∗j =∞∑j=1

η∗j .

Majorization in K (H)+

Instead of defining majorization in B(H), let us restrict our attention to K (H)+,the compact positive operators. We have the association:

Ks-numbers−−−−−−→ s(K ) = 〈sn(K )〉n∈N .

Since K is positive and compact, s(K ) is precisely the sequence of nonzeroeigenvalues (when K has infinite rank) of K listed in monotone decreasing orderand repeated according to multiplicity.

Definition

Given two positive operators K ,K ′, we say that K is majorized by K ′, denotedK ≺ K ′, if s(K ) ≺ s(K ′).

The Schur-Horn theorem (finite dimensional) can be restated succinctly as

EA(U(X )) = {A ∈ A | A ≺ X}.

Equations which are similar in form to this we will call Schur-Horn Theorems.

Definition

EA(U(X )) = {A ∈ A | A ≺ X}.

Definition

EA(U(X )) = {A ∈ A | A ≺ X}.

Definition

EA(U(X )) = {A ∈ A | A ≺ X}.

Recent work Approximate Schur-Horn theorems

The work of Arveson-Kadison

Arveson and Kadison proved an approximate Schur-Horn theorem for positivetrace-class operators in [AK06].

Theorem (Arveson-Kadison 2006)

Let A be a discrete (atomic) masa in B(H), and EA the faithful normaltrace-preserving conditional expectation. Let K be a positive trace-class operator(i.e., K ∈ L+

1 ). Then

EA(U(K )

‖·‖1)

= {A ∈ A ∩ L+1 | A ≺ K}.

We call such a theorem an approximate Schur-Horn Theorem because it involvestaking the closure in some topology on the left-hand side. In [Kad02], Kadisonhighlights how sometimes fine detail may be lost in an approximate Schur-Horntheorem instead of an exact one.In [AK06], Arveson and Kadison also formulated an approximate Schur-Hornconjecture for II1 factors, which involved providing and appropriate notion ofmajorization in that context. However, this is not the context of this talk.

1 ). Then

EA(U(K )

‖·‖1)

= {A ∈ A ∩ L+1 | A ≺ K}.

We call such a theorem an approximate Schur-Horn Theorem because it involvestaking the closure in some topology on the left-hand side. In [Kad02], Kadisonhighlights how sometimes fine detail may be lost in an approximate Schur-Horntheorem instead of an exact one.

In [AK06], Arveson and Kadison also formulated an approximate Schur-Hornconjecture for II1 factors, which involved providing and appropriate notion ofmajorization in that context. However, this is not the context of this talk.

1 ). Then

EA(U(K )

‖·‖1)

= {A ∈ A ∩ L+1 | A ≺ K}.

We call such a theorem an approximate Schur-Horn Theorem because it involvestaking the closure in some topology on the left-hand side. In [Kad02], Kadisonhighlights how sometimes fine detail may be lost in an approximate Schur-Horntheorem instead of an exact one.In [AK06], Arveson and Kadison also formulated an approximate Schur-Hornconjecture for II1 factors, which involved providing and appropriate notion ofmajorization in that context. However, this is not the context of this talk.

Recent work Exact Schur-Horn theorems

The work of Kaftal-Weiss

In [KW10], Kaftal and Weiss analyzed the situation in B(H) and gave extensionsof Arveson and Kadison’s results to positive compact compact operators, as wellas provided some exact Schur-Horn results.

Definition

For an operator X ∈ B(H), the partial isometry orbit of X , denoted V(X ) if givenby

V(X ) = {VXV ∗ | V ∈ B(H),V ∗V = RX ∨ RX∗}.

For trace-class operators, V(X ) = U(X )‖·‖1 .

Theorem (Kaftal-Weiss 2010)

Let A be a discrete (atomic) masa in B(H), and EA the faithful normaltrace-preserving conditional expectation. Let K be a positive compact operator.Then

EA(V(K )) = {A ∈ A ∩ K (H)+ | A ≺ K}.

Definition

V(X ) = {VXV ∗ | V ∈ B(H),V ∗V = RX ∨ RX∗}.

EA(V(K )) = {A ∈ A ∩ K (H)+ | A ≺ K}.

Definition

V(X ) = {VXV ∗ | V ∈ B(H),V ∗V = RX ∨ RX∗}.

EA(V(K )) = {A ∈ A ∩ K (H)+ | A ≺ K}.

Definition

V(X ) = {VXV ∗ | V ∈ B(H),V ∗V = RX ∨ RX∗}.

EA(V(K )) = {A ∈ A ∩ K (H)+ | A ≺ K}.

The previous theorem is an exact Schur-Horn theorem, but it considers the partialisometry orbit instead of the unitary orbit. Kaftal and Weiss also provided acharacterization of EA(U(K )) under certain conditions.

If K has finite rank, then EA(V(K )) = EA(U(K )) (and hence the partialisometry orbit may be replaced by the unitary orbit in the above theorem);

If ker K = {0} (alt. R⊥K = I ), then

EA(U(K )) = {A ∈ A ∩ K (H)+ | A ≺ K ,R⊥A = I}.

Our results Two new kinds of majorization

EA(U(K )) and p-majorization

Building mainly upon the work of Kaftal and Weiss, we have tried to characterizeEA(U(K )) for any positive compact operator K . So far, we have been able todetermine explicitly the case when R⊥K is an infinite projection. Further, when R⊥Kis a nonzero finite projection, we have necessary conditions and sufficientconditions (but not necessarily simultaneously!) for membership in EA(U(K )).However, first we need some new majorization-like concepts.

Definition

For ξ, η ∈ c+0 , and p ∈ N ∪ {0}, we say that ξ is p-majorized by η, denoted

ξ ≺p η, if ξ ≺ η and there exists Np such that for all n ≥ Np,

n+p∑j=1

ξ∗j ≤n∑

η∗j .

By convention, ξ ≺∞ η means ξ ≺p η for all p ∈ N (or equivalently, for infinitelymany p ∈ N).

EA(U(K )) and p-majorization

Building mainly upon the work of Kaftal and Weiss, we have tried to characterizeEA(U(K )) for any positive compact operator K . So far, we have been able todetermine explicitly the case when R⊥K is an infinite projection. Further, when R⊥Kis a nonzero finite projection, we have necessary conditions and sufficientconditions (but not necessarily simultaneously!) for membership in EA(U(K )).However, first we need some new majorization-like concepts.

Definition

For ξ, η ∈ c+0 , and p ∈ N ∪ {0}, we say that ξ is p-majorized by η, denoted

ξ ≺p η, if ξ ≺ η and there exists Np such that for all n ≥ Np,

n+p∑j=1

ξ∗j ≤n∑

η∗j .

By convention, ξ ≺∞ η means ξ ≺p η for all p ∈ N (or equivalently, for infinitelymany p ∈ N).

Approximate p-majorization

Definition

For ξ, η ∈ c+0 and p ∈ N ∪ {0}, we say that ξ is approximately p-majorized by η,

denoted ξ -p η, if ξ ≺ η and for every ε > 0 there exists Np,ε such that for alln ≥ Np,ε,

n+p∑j=1

ξ∗j ≤ ε · η∗n+1 +n∑

η∗j .

By convention, ξ -∞ η means ξ -p η for all p ∈ N (or equivalently, for infinitelymany p ∈ N).

For all p ≤ p′, ξ ≺p′ η =⇒ ξ ≺p η and ξ -p′ η =⇒ ξ -p η. Furthermore, ifp ∈ N then ξ ≺p η =⇒ ξ -p η =⇒ ξ ≺p−1 η. These together yieldξ ≺∞ η ⇐⇒ ξ -∞ η.

Approximate p-majorization

Definition

For ξ, η ∈ c+0 and p ∈ N ∪ {0}, we say that ξ is approximately p-majorized by η,

denoted ξ -p η, if ξ ≺ η and for every ε > 0 there exists Np,ε such that for alln ≥ Np,ε,

n+p∑j=1

ξ∗j ≤ ε · η∗n+1 +n∑

η∗j .

By convention, ξ -∞ η means ξ -p η for all p ∈ N (or equivalently, for infinitelymany p ∈ N).

For all p ≤ p′, ξ ≺p′ η =⇒ ξ ≺p η and ξ -p′ η =⇒ ξ -p η. Furthermore, ifp ∈ N then ξ ≺p η =⇒ ξ -p η =⇒ ξ ≺p−1 η. These together yieldξ ≺∞ η ⇐⇒ ξ -∞ η.

Examples

Example

If ξ =⟨14 ,

116 , . . .

⟩and η = 〈2−n〉, then ξ ≺1 η, but ξ 6≺2 η.

Example

If ξ =⟨14 ,

116 , . . .

⟩and η = 〈2−n〉, then ξ ≺∞ η.

Example

Suppose Q is an infinite matrix with entries 0 ≤ qij ≤ 1 and whose column androw sums are all one (i.e., Q is doubly stochastic) and qij = 0 if and only ifi > j > 1. Then if η ∈ c∗0 we have Qη = ξ -1 η, but ξ 6≺1 η.

In the context of the last example, Kaftal and Weiss showed such matrices Qexist and furthermore, they may actually be chosen to be orthostochastic. Thisshows that the concepts of p-majorization and approximate p-majorization areactually distinct.

Examples

Example

If ξ =⟨14 ,

116 , . . .

Example

If ξ =⟨14 ,

116 , . . .

⟩and η = 〈2−n〉, then ξ ≺∞ η.

Example

Examples

Example

If ξ =⟨14 ,

116 , . . .

Example

If ξ =⟨14 ,

116 , . . .

⟩and η = 〈2−n〉, then ξ ≺∞ η.

Example

Examples

Example

If ξ =⟨14 ,

116 , . . .

Example

If ξ =⟨14 ,

116 , . . .

⟩and η = 〈2−n〉, then ξ ≺∞ η.

Example

Our results New majorization and unitary orbits

A necessary condition and a sufficient conditions

Theorem (Sufficiency of p-majorization)

Suppose K ∈ K (H)+ and A is a discrete (atomic) masa of B(H). IfA ∈ A ∩ K (H)+, and for some p ∈ N ∪ {0,∞} we have A ≺p K andTr(R⊥A ) ≤ Tr(R⊥K ) ≤ Tr(R⊥A ) + p, then A ∈ EA(U(K )).

Theorem (Necessity of approximate p-majorization)

Suppose K ∈ K (H)+ and A is a discrete (atomic) masa of B(H). IfA ∈ EA(U(K )), then with

p = inf{n ∈ N ∪ {0} | Tr(R⊥A ) ≤ Tr(R⊥K ) ≤ Tr(R⊥A ) + n},

we have A -p K .

Due to a counterexample of Kaftal and Weiss, we know that the above sufficientcondition is not necessary when K has infinite rank and 0 < Tr(R⊥K ) <∞. It isnot yet known whether or not the above necessary condition is sufficient in thecase.

we have A -p K .

Conclusions

The fact that ξ ≺∞ η ⇐⇒ ξ -∞ η in conjunction with the previous two resultsyields the following characterization.

Theorem

If K ∈ K (H)+ has infinite rank and Tr(R⊥K ) =∞, then

EA(U(K )) =

{A ∈ A ∩ K (H)+

∣∣∣∣∣{

A ≺ K if Tr(R⊥A ) =∞A ≺∞ K if Tr(R⊥A ) <∞

Theorem

If K (H)+ has infinite rank and 0 < Tr(R⊥K ) <∞, then⋃p

{A ∈ A ∩ K (H)+ | A ≺p K ,Tr(R⊥A ) ≤ Tr(R⊥K ) ≤ Tr(R⊥A ) + p} ( EA(U(K ))

{A ∈ A ∩ K (H)+ | A -p K ,Tr(R⊥A ) ≤ Tr(R⊥K ) ≤ Tr(R⊥A ) + p} ⊇ EA(U(K ))

Conclusions

The fact that ξ ≺∞ η ⇐⇒ ξ -∞ η in conjunction with the previous two resultsyields the following characterization.

Theorem

If K ∈ K (H)+ has infinite rank and Tr(R⊥K ) =∞, then

EA(U(K )) =

{A ∈ A ∩ K (H)+

∣∣∣∣∣{

A ≺ K if Tr(R⊥A ) =∞A ≺∞ K if Tr(R⊥A ) <∞

Theorem

If K (H)+ has infinite rank and 0 < Tr(R⊥K ) <∞, then⋃p

{A ∈ A ∩ K (H)+ | A ≺p K ,Tr(R⊥A ) ≤ Tr(R⊥K ) ≤ Tr(R⊥A ) + p} ( EA(U(K ))

{A ∈ A ∩ K (H)+ | A -p K ,Tr(R⊥A ) ≤ Tr(R⊥K ) ≤ Tr(R⊥A ) + p} ⊇ EA(U(K ))

References

William Arveson and Richard V. Kadison, Diagonals of self-adjoint operators,Operator Theory, Operator Algebras, and Applications 414 (2006), 247–263.

Richard Kadison, The Pythagorean Theorem II: the infinite discrete case,Proceedings of the National Academy of Sciences (2002).

Victor Kaftal and Gary Weiss, An infinite dimensional Schur-Horn Theoremand majorization theory, Journal of Functional Analysis 259 (2010), no. 12,3115–3162.

An in nite dimensional Schur-Horn Theorem and majorization theory: the nonzero kernel … · 2013....

Documents

Transcript of An in nite dimensional Schur-Horn Theorem and majorization theory: the nonzero kernel … · 2013....