Traces and spectral properties of shift operators Contribution to - - PowerPoint PPT Presentation

Traces

and

spectral properties of shift operators

Contribution to the Aleksander Pełczyński Memorial Conference

————————————————–

Albrecht Pietsch

1

• K. Kuratowski: A Half Century of Polish Mathematics

2

My personal intention is to improve mathematical theories by making them simpler and more elegant. It is desirable or even necessary that such modifications yield not only all well-known results, but also new ones. The today’s lecture is a typical realization of this philosophy. I going to describe a new view on traces, which could and should have been found already 25 years ago. In the meantime, trace theory and its applications to non-commutative geometry and pseudo-differential operators have developed quickly. Since all standard books are written

• n the basis of the classical technique,

it will be hard to accomplish any change. To ensure acceptance of mathematical concepts, they must be presented at the right moment. 3

Gorbachev whispers into Honecker’s ear, October 1989: Wer zu spät kommt, den bestraft das Leben. Life punishes those who come too late. 4

Part 1

5

Sequence ideals — > Operator ideals

S ∈ L(X, Y ) n ∈ N := {1, 2, 3, . . . } n-th approximation number: an(S) := inf

• S − F : rank(F) < n
• .

With every set z(N0) of sequences we associate the class Dz :=

• X,Y

Dz(X, Y )

• f operators, whose components are defined by

Dz(X, Y ) :=

• S ∈ L(X, Y ) :
• a2h(S)
• ∈ z(N0)
• ,

h ∈ N0 := {0, 1, 2, . . . }.

• an(S)
• is replaced by the dyadic subsequence
• a2h(S)
• .

6

Which properties of z(N0) ensure that Dz is an operator ideal? (1) z(N0) is an ideal in l∞(N0), the algebra of bounded sequences. (2) e0 =(1, 0, 0, . . . )∈z(N0) guarantees that Dz contains non-zero operators. (3) If z(N0) is invariant under the forward shift S+ : (α0, α1, α2, . . . ) → (0, α0, α1, . . . ), then we infer from a2h+1(S + T)≤a2h(S) + a2h(T) for X

S,T∈L

− − − − − → Y that S, T ∈Dz(X, Y ) ⇒ S + T ∈Dz(X, Y ). Since the ideal z(N0) is solid, (βh) ∈ z(N0) and |αh| ≤ |βh| ⇒ (αh) ∈ z(N0), it follows from an(BSA)≤Ban(S)A for X0

A∈L

− − − → X

S∈L

− − − → Y

B∈L

− − − → Y0 that S ∈Dz(X, Y ) ⇒ BSA∈Dz(X0, Y0). 7

Shift-monotone sequence ideals

The map z(N0) → Dz is far from being one-to-one. What condition ensures that z(N0) can be reconstructed from Dz? a=(αh)∈l∞(N0) h, k ∈ N0 := {0, 1, 2, . . . } k-th ordering number:

• k(a) := sup

h≥k

|αh|. If |α0|≥|α1|≥|α2|≥. . .≥0, then ok(a)=|αk|. ℓ2: Hilbert sequence space over N := {1, 2, 3, . . . }. Da: dyadic diagonal operator on ℓ2 associated with the sequence da := (

2k−1 terms

• α0, α1, α1, . . .
• 1≤n<2k

,

2k terms

• αk, . . . , αk
• 2k≤n<2k+1

, . . . ) for a=(αh)∈l∞(N0). Lemma

• k(a) = a2k(Da).

8

Shift-monotone sequence ideal z(N0): (1) z(N0) is an ideal in l∞(N0). (2) e0 =(1, 0, 0, . . . )∈z(N0). (3) z(N0) is invariant under the forward shift S+. (4) a ∈ z(N0), b ∈ l∞(N0), and ok(b) ≤ ok(a) imply b ∈ z(N0). Theorem The map z(N0) → Dz is one-to-one: a∈z(N0) ⇔ Da ∈ Dz(ℓ2). Proof : Substitute b=

• k(a)
• in (4).

a∈z(N0) ⇔

• k(a)
• ∈z(N0) ⇔
• a2k(Da)
• ∈z(N0) ⇔ Da ∈ Dz(ℓ2).

By ok(S−a) ≤ ok(a), z(N0) is invariant under the backward shift S− : (α0, α1, α2, . . . ) → (α1, α2, α3, . . . ). Monotony (4) means that every a = (αh)∈z(N0) is dominated by a monotone sequence b=(βh)∈z(N0); that is |αh|≤βh and β0 ≥ β1 ≥ β2 ≥ · · · ≥ 0. 9

The S±-invariant sequence ideal l1(N0) :=

• (αh) :

∞

• h=0

|αh| < ∞

• fails to be monotone:

(1, 0, 0, 0, 1

4, . . . , n2 terms

• 0, . . . , 0, 1

n2 , 0, . . . ),

corresponding sequence of ordering numbers (1, 1

4, 1 4, 1 4, 1 4, . . . , n2 terms

• 1

n2 , . . . , 1 n2 , 1 (n+1)2 , . . . ).

Example l1[2−h](N0) :=

• (αh) :

∞

• h=0

2h|αh| < ∞

• is a shift-monotone sequence ideal.

10

Example c0(N0) :=

• (αh) : lim

h→∞ αh = 0

• is the largest shift-monotone sequence ideal = l∞(N0).

Operators in Dc0 are called approximable. In what follows, z(N0) denotes a shift-monotone sequence ideal contained in c0(N0). 11

Dyadic representations

S =

∞

• k=0

Sk, convergent with respect to the operator norm, S ∈L(X, Y ) and Sk ∈L(X, Y ) such that rank(Sk)≤2k. Theorem An operator is approximable if and only if it admits a dyadic representation. Dk: orthogonal projection from ℓ2 onto span {en : 2k ≤n<2k+1}. Da =

∞

• k=0

αkDk for a = (αh) ∈ l∞(N0) is a dyadic representation of the dyadic diagonal operator Da,

• h(a) =
• Da −

h−1

• k=0

αkDk

• = a2h(Da).

12

z-representations

S =

∞

• k=0

Sk such that rank(Sk) ≤ 2k and

• S −

h−1

• k=0

Sk

• ∈ z(N0).

(♦) former definition:

• Sh
• ∈ z(N0).

() It follows from the S−-invariance of z(N0) that (♦) ⇒ (): Sh ≤

• S −

h−1

• k=0

Sk

• +
• S −

h

• k=0

Sk

• .

aha experience: (♦) ensures the convergence of ∞

k=0 Sk, while () does not.

13

Theorem The ideal Dz consists of all operators that admit a z-representation. Proof : Given S ∈Dz(X, Y ), there exists Fk ∈L(X, Y ) such that S − Fk ≤ 2a2k(S) and rank(Fk)<2k. Note that F0 =O. Letting S0 := O, S1 := O, and Sk+2 := Fk+1−Fk for k =0, 1, 2, . . . , we get

h+1

• k=0

Sk = Fh,

• S−

h+1

• k=0

Sk

• = S−Fh ≤ 2a2h(S),

rank(Sk) ≤ 2k. Hence the S+-invariance of z(N0) yields (♦), which means that S = ∞

k=0 Sk is a z-representation.

14

Since rank h−1

• k=0

Sk

• < 2h

whenever rank(Sk) ≤ 2k, we have a2h(S) ≤

• S −

h−1

• k=0

Sk

• .

So, conversely, (♦) implies

• a2h(S)
• ∈z(N0) or S ∈Dz(X, Y ).

15

Lemma If S =

∞

• k=0

Sk and T =

∞

• k=0

Tk are z-representations of S, T ∈Dz(X, Y ), then S+T =

∞

• k=0

Zk with Z0 :=O and Zk+1 :=Sk +Tk for k = 0, 1, 2, . . . is a z-representation of their sum.

• S + T −

h

• k=0

Zk

• =
• S + T −

h−1

• k=0

(Sk + Tk)

• ≤
• S −

h−1

• k=0

Sk

• +
• T −

h−1

• k=0

Tk

• .

16

1 2S+-invariant linear forms on z(N0)

1 2S+-invariance:

λ 1

2S+a

• = λ(a)

for all a ∈ z(N0). In the optimal case, linear forms λ on sequence ideals admit an analytic representation in the form λ(a) =

∞

• h=0

λhαh for a=(αh). λ(eh)=λh for all unit sequences eh. If λ is 1

2S+-invariant, then

λh+1 = λ(eh+1) = λ(S+eh) = 2λ(eh) = 2λh ⇒ λh =2hλ0 ⇒ λ(a) = λ0

∞

• h=0

2hαh. 17

Such linear forms can only live on shift-monotone sequence ideals z(N0) that are contained in l1[2−h](N0) :=

• (αh) :

∞

• h=0

2h|αh| < ∞

• .

The existence of further 1

2S+-invariant linear forms is based on the

axiom of choice. So we are mostly dealing with esoteric objects that are floating somewhere in the air. The value of λ at a sequence a=(αh) will be denoted by λ(αh), instead of λ(a)=λ

• (αh)
• .

18

Traces

The concept of a trace makes sense only for square matrices. Similarly, in the setting of operators, we must assume that the

• perators act on one and the same Banach space.

A trace on an operator ideal A is a linear form τ, defined on all components A(X):=A(X, X), for which the following holds: τ(AS) = τ(SA) whenever X

✲ X

AS∈A

Y

✲ Y

SA∈A

❅ ❅ ❅ ❘ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❘ ❅ ❅ ❅ ❘

• ✒
• ✒
• ✒
• ✒
• ✒

S∈A S∈A A∈L

. 19

The smallest non-trivial ideal F consists of all finite rank operators. Every trace on F is a scalar multiple of the usual trace defined by trace(F) :=

n

• m=1

xm, x∗

m,

where F =n

m=1 x∗ m ⊗ xm with x1, . . . , xn ∈X and x∗ 1, . . . , x∗ n ∈X ∗

is any finite representation. We have | trace(F)| ≤ nF whenever rank(F) ≤ n. 20

Theorem

MAIN RESULT

Let λ be an 1

2S+-invariant linear form on z(N0). Then the expression

τ(S) := λ 1

2k trace(Sk)

• does not depend on the special choice of the z-representation

S =∞

k=0 Sk. This definition yields a trace τ on Dz.

The map λ → τ is one-to-one. Proof : It follows from

1 2k | trace(Sk)|≤Sk and (♦) ⇒ () that

λ 1

2k trace(Sk)

• makes sense for any z-representation of S.

We consider different z-representations S =

∞

• k=0

Fk and S =

∞

• k=0

Gk. Letting Z0 := O and Zk+1 := Fk − Gk for k = 0, 1, 2, . . . yields a z-representation of the zero operator. 21

Define a = (αh) and b = (βh) by αh := 1

2h trace(Zh)

and βh := 1

2h trace

• h
• k=0

Zk

• .

Since | trace(F)| ≤ nF whenever rank(F) ≤ n, |αh| ≤ Zh and |βh| = 1

2h

• trace
• h
• k=0

Zk

• ≤ 2
• O −

h

• k=0

Zk

• .

Hence a, b∈z(N0). It follows from α0 = β0 = 0 and βh− 1

2βh−1 = 1 2h trace(Zh) = αh

for h=1, 2, . . . that S+S−a=a and a=b − 1

2S+b. We now obtain

λ 1

2k trace(Fk)

• − λ

1

2k trace(Gk)

• = λ

1

2k trace(Zk+1)

• =

= 2λ(αk+1)=2λ(S−a)=2λ( 1

2S+S−a)=λ(a)=λ(b) − λ( 1 2S+b)=0,

which shows that τ(S) is well-defined. 22

S =

∞

• k=0

Sk and T =

∞

• k=0

Tk S + T =

∞

• k=0

Zk with Z0 :=O and Zk+1 :=Sk +Tk for k = 0, 1, 2, . . . τ(S) = λ

• trace(S0), 1

2 trace(S1), . . .

• = λ
• , 1

2 trace(S0), 1 4 trace(S1), . . .

• τ(T) = λ
• trace(T0), 1

2 trace(T1), . . .

• = λ
• , 1

2 trace(T0), 1 4 trace(T1), . . .

• τ(S + T) = λ
• trace(Z0), 1

2 trace(Z1), 1 4 trace(Z2), . . .

• Hence τ is additive.

Substituting Da =∞

k=0 αkDk into τ(S) := λ

1

2k trace(Sk)

• yields

τ(Da) = λ 1

2k trace(αkDk)

• = λ(αk).

Thus the map λ → τ is one-to-one. 23

Continuous traces

So far, the nature of operator ideals and traces was purely algebraic. To reduce the huge amount of 1

2S+-invariant linear forms

that may exist on some shift-monotone sequence ideal it is natural to require continuity with respect to a suitable topology. The best way to obtain such topologies is via quasi-norms · |z that have the following property: If a, b ∈ z(N0) and

• k(b) ≤ ok(a),

then b|z ≤ a|z. It is convenient to arrange that e0|z=1 for e0 :=(1, 0, 0, . . . ). Since ok(S−a)≤ok(a), we obtain S−a|z ≤ a|z. S+ is assumed to be continuous, as well. 24

Every quasi-norm on a shift-monotone sequence ideal z(N0) generates a quasi-norm on the associated operator ideal Dz, S|Dz :=

• a2k(S)
• z
• .

In particular, Da|Dz := a|z. As recently shown by Levitina, Pietsch, Sukochev, and Zanin, completeness carries over from z(N0) to Dz and, of course, conversely. Theorem Every trace τ on Dz that is generated by a continuous

1 2S+-invariant linear form λ on z(N0) is continuous, as well.

25

Summary of the Banach space setting

The construction z(N0) → Dz only yields a special class of operator ideals. Moreover, it is unknown whether passing from λ to τ gives

all traces on the operator ideal Dz.

These shortcomings disappear if the considerations are restricted to the Hilbert space. 26

The Hilbert space setting

H: infinite-dimensional separable Hilbert space (identified with ℓ2), A modified version of Calkin’s theorem. Theorem The rules z(N0) → Dz(H) :=

• S ∈ L(H) :
• a2k(S)
• ∈ z(N0)
• and

A(H) → z(N0) :=

• a ∈ l∞(N0) : Da ∈ A(H)
• yield a one-to-one correspondence between

all shift-monotone sequence ideals z(N0) and all of operator ideals A(H). The ideals z(N0) and A(H)=Dz(H) are said to be associated to each other. 27

Theorem

MAIN RESULT

If the ideals z(N0) and A(H) are associated, then letting τ(S) := λ 1

2k trace(Sk)

• ,

where S = ∞

k=0 Sk is any z-representation, and

λ(a) := τ(Da) yields a one-to-one correspondence between all 1

2S+-invariant linear forms λ on z(N0)

and all traces τ on A(H). 28

Proof : It remains to look at the inverse map τ → λ. If λ(a) := τ(Da) is obtained from a trace τ, then λ( 1

2S+a) = 1 2τ(DS+a) !

= τ(Da) = λ(a) tells us that λ is 1

2S+-invariant.

Partial isometries on ℓ2: U : en → e2n, U⋆ : e2n → en, V : en → e2n+1, V ⋆ : e2n+1 → en, U⋆U = I, V ⋆V = I, and DS+a = UDaU⋆ + VDaV ⋆. Hence τ(DS+a) = τ(UDaU⋆)+τ(VDaV ⋆) = τ(U⋆UDa)+τ(V ⋆VDa) = 2τ(Da). 29

Finally, we ask whether τ → λ is one-to-one. Lemma (Figiel, unpublished ∼ 1983) Every trace on an operator ideal A(ℓ2) is uniquely determined by its values at all dyadic diagonal operators Da. The proof is based on a Lemma of Steinitz (1913); see also Kwapień (1984) and a paper of Dykema, Figiel, Weiss, and Wodzicki (2004) on ‘Commutator structure of operator ideals’. Let me stress the fact that (apart from the above) commutators do not play any role in my approach. 30

Example

The shift-monotone sequence ideal l1[2−h](N0) :=

• (αh) :

∞

• h=0

2h|αh| < ∞

• and the operator ideal (trace class)

L1(H) :=

• S ∈ L(H) :

∞

• n=1

an(S) < ∞

• are associated.

The 1

2S+-invariant linear form

λ(a) :=

∞

• h=0

2hαh generates the usual trace on L1(H). 31

Schmidt representation

Theorem An operator S ∈L(H) is approximable if and only if it admits a Schmidt representation, Sx =

∞

• n=1

σn(x|un)vn for all x ∈ H, where (un), (vn) are orthonormal sequences in H and σ1 ≥σ2 ≥. . .≥0 is a zero sequence; an(S) = σn. Defining Skx :=

• 2k≤n<2k+1

σn(x|un)vn for all x ∈ H yields a dyadic representation such that

• S −

h−1

• k=0

Sk

• = a2h(S).

32

Theorem The trace τ on Dz(H) generated by an 1

2S+-invariant linear form λ

• n z(N0) is given by

τ(S) = λ

• 1

2k

• 2k≤n<2k+1

σn(vn|un)

• .

The Schmidt representation of S+T is in no direct way related to the Schmidt representations of S and T. Therefore, without the use of dyadic representations, it is hard to prove the additivity of τ. Not only the fact above shows that Green is the color of hope Banach space techniques may be useful for Hilbert space people. 33

Classical and singular traces

τ: trace on an operator ideal, classical: τ coincides with the usual trace

• n all finite rank operators,

singular: τ vanishes on all finite rank operators, arbitrary trace = scalar · classical trace + singular trace, provided that a classical trace exists. Otherwise, all traces are singular. λ:

1 2S+-invariant linear form on a shift-monotone sequence ideal,

e0 = (1, 0, 0, . . . ) classical: λ(e0) = 1, singular: λ(e0) = 0, 2S−-invariant linear form. 34

A classification of quasi-Banach operator ideals

Given any quasi-Banach operator ideal A(H), we may ask the following questions: Does A(H) support a continuous classical trace, a non-continuous classical trace, a continuous singular trace = o, a non-continuous singular trace? 16 different classes arise: 8 classes are non-empty, 6 classes are empty, 2 classes ? 35

Historical remark

Until the late 1980s, only classical traces were viewed as useful. The question: ‘Whether a classical trace is uniquely determined by its axiomatic properties’ led to the concept of a singular trace. Already in 1966 and motivated by the theory of C ⋆-algebras, Dixmier had constructed examples (in the Hilbert space setting), which he considered as pathological monsters. About 20 years later, under the hands of his pupil Connes that monsters became an important tool in non-commutative geometry. 36

Summary of the Hilbert space setting

Traces on operator ideals A(H) can be characterized as unitarily invariant linear forms, τ(USU⋆) = τ(S) for S ∈ A(H) and all unitary U ∈ L(H). The unitary operators on H form a huge non-commutative group. Trace theory is reduced to the study of linear forms that are invariant under one operator. The second part of my lecture will show that this is a drastic simplification. 37

Part 2

38

Preliminaries

algebraic case S: linear map on a linear space Z, S-invariant linear form λ: λ(Sa) = λ(a) for all a ∈ Z. equivalent condition: λ vanishes on the range M(I − S) := {a − Sa : a ∈ Z}. non-trivial λ’s exist if and only if M(I − S) = Z. 39

topological case S: (bounded linear) operator on a (quasi-)Banach space Z, continuous non-trivial λ’s exist if and only if M(I − S) = Z. Lemma (Kato 1958) If the range of T : X → Y is finite-codimensional, then it is closed. Hence, if M(I − S) is not closed, then there exist many non-continuous S-invariant linear forms. 40

In what follows, z(N0) denotes a shift-monotone (quasi-)Banach sequence ideal contained in c0(N0). We are mainly interested in the range of I − 1

2S+

viewed as an operator on z(N0): M(I − 1

2S+) :=

• (αh) :
• 1

2k k

• h=0

2hαh

• ∈ z(N0)
• .

This is a good counterpart of the commutator space. Lemma no classical trace ⇔ e0 ∈ M(I − 1

2S+)

⇔ 1

2k

• ∈ z(N0).

41

Spectral radius

spectral radius: r(S±|z) := lim

m→∞ Sm ± |z1/m,

eh: h-th unite sequence in z(N0). e(S+|z):= lim

m→∞

• sup

0≤h<∞

• eh+m|z

eh|z

• 1/m

, e(S−|z):= lim

m→∞

• sup

0≤h<∞

• eh|z

eh+m|z

• 1/m

, e(S±|z) ≤ r(S±|z) and 1 ≤ e(S−|z)e(S+|z) ≤ r(S−|z)r(S+|z). 42

Example

(ζh): zero sequence such that 1 = ζ0 ≥ ζ1 ≥ ζ2 ≥ · · · ≥ 0 and ζh ≤ cζh+1. Most important examples, ζh := 2−h/r for 0 < r < ∞. l∞[ζh](N0) :=

• (αh) : αh = O(ζh)
• is a shift-monotone Banach sequence ideal under the norm

(αh)|l∞[ζh] := sup

0≤h<∞

ζ−1

h |αh|.

e(S±|l∞[ζh]) = r(S±|l∞[ζh]). e(S±|l∞[2−h/r]) = r(S±|l∞[2−h/r]) = 2±1/r. 43

Spectrum of S+ on z(N0)

Since the spectrum σ(S+|z) is invariant under rotation about 0, it suffices to look at the non-negative real part. The operator λI −S+ behaves as follows:

✲

r(S−|z)−1

• non-closed range

e(S−|z)−1 e(S+|z) r(S+|z)

• non-closed range

Φ-operator ??? non-closed range

[on the closed interval]

??? invertible

N(λI − S+) = {o}, cod

• closed range
• M(λI − S+)
• =1, cod
• closed range
• M(λI − S+)
• >1
• ???

, cod

• closed range
• M(λI − S+)
• =0

Problem Do there exist quasi-Banach operator ideals for which the linear space of all traces is n-dimensional with n > 1? 44

Lemma If r(S+|z)<2, then Dz(H) supports no trace = o. If r(S−|z)< 1

2, then Dz(H) supports just a unique continuous

classical trace. 45

The quasi-Banach Lorentz operator ideal Dl∞[2−h/r](H) = Lr,∞(H) :=

• S ∈ L : an(S) = O(

1 n1/r )

• supports

no trace = o if 1 < r < ∞, Dixmier case: many continuous and many non-continuous singular traces = o if r =1, a unique classical trace, which is continuous if 0 < r < 1. Instead of the quasi-Banach operator ideal L1,∞(H) :=

• S ∈ L(H) : an(S) = O

1

n

• ,

Dixmier used the Banach operator ideal M1,∞(H) :=

• S ∈ L(H) :

n

• m=1

am(S) = O

• log(1 + n)
• .

Main tool: approximable S, T ≥O

n

• m=1

am(S+T) ≤

n

• m=1

am(S) +

n

• m=1

am(T) ≤

2n

• m=1

am(S+T) 46

Spaces versus operators

Real fans of spaces look at operators that act on a given Banach

• space. Interesting examples are spaces with few operators.

In my opinion it is, however, equally important to start from an algebraic or analytic expression and look for all Banach spaces

• n which this expression defines an operator.

In the present case, that standpoint leads to the concept of a shift-invariant sequence space. Unfortunately, it seems to me that there are almost no results in this direction. 47

• R. Gellar, Operators commuting with a weighted shift, Proc. Amer.
• Math. Soc. 23 (1969), 538–545.
• R. Gellar, Cyclic vectors and parts of the spectrum of a weighted shift,
• Trans. Amer. Math. Soc. 146 (1969), 69–85.
• P. R. Halmos, A Hilbert Space Problem Book, Van Nostrand, Princeton,

1967.

• W. C. Ridge, Approximate point spectrum of a weighted shift, Trans.
• Amer. Math. Soc. 147 (1970), 349–356.
• A. L. Shields, Weighted shift operators and analytic function theory,

Topics in Operator Theory, Math. Surveys 13, 49–128,

• ed. C. Pearcy, Amer. Math. Soc., Providence, 1974.

I need to know some sophisticated shift-monotone Banach sequence ideals and modern literature. 48

Many thanks for your attention

and my personal tribute to

49

• Dr. honoris causa

University of Jena September 1983 50