Sums of linear operators in Hilbert C*-moduleswork in progress, based on discussions with Bram

Matthias Lesch

Universitat Bonnml@uni-bonn.de

06.01.2017

Last Update: 2017-01.06

Outline

Hilbert C ∗-modules

Unbounded operators

Regular operators

Localization

Local Global Principle

Sums of regular selfadjoint operators

Hilbert C ∗–modulesKaplansky 1953, Paschke 1973, Rieffel 1974, Kasparov 1980

1. A C ∗-algebra

2. E Hilbert C ∗–module over A:I E A–right moduleI 〈·, ·〉 : E × E −→ A inner product (A–valued)I Banach space w.r.t. ‖x‖ := ‖〈x , x〉‖1/2 = ‖〈x , x〉1/2‖

3. Superficially looks like a Hilbert space BUTI No Projection Theorem, hence closed submodules need not be

complementableI No self-duality; unit ball is not (well, almost never) weakly compact.

4. L(E ) bounded, adjointable, A-module endomorphismsI L(E ) is a C∗-algebraI Selfadjoint elements in L(E ) do have a continuous functional calculus.

5. Why?I Important tool in Kasparov’s bivariant KK-theory.I Curiosity driven: very natural generalization of Hilbert spaces

Examples of Hilbert C ∗–modules . . . are abundant

1. Hilbert space (A = C).

2. E = A, 〈x , y〉 := x∗y .J ⊂ A closed non-trivial ∗–ideal. J ⊂ E closed non-trivial submodule.

I E.g. A = C [0, 1],J ={

f ∈ A∣∣ f (0) = 0

}.

I J⊥ = {0}, J not complementable.

3. X compact space, V → X (continuous) vector bundle, h hermitianmetric on V .

I A := C (X ); E := Γ(X ,V ) (continuous sections of V )I 〈f , g〉(x) := h(f (x), g(x))

4. H = `2(N) standard Hilbert space,

HA ={

(aj)j∈N∣∣ ∑ a∗j aj converges in A

}〈(aj), (bj)〉 :=

∞∑j=1

a∗j bj .

HA = H⊗AA standard module over A.

Unbounded operatorsAt least: Banach space theory of unbounded operators available

T operator in E , domain D(T ) dense in E

semiregular (operator affiliated with A)

I D(T ) ⊂ E dense submoduleI T ∗ densely defined

⇒ T : D(T )→ E A–module map

Superficially looks like densely defined closable operator in Hilbert space.Indeed: T closable, T ∗ = T

∗.

Pathologies

I No Functional Calculus for selfadjoint semiregular operatorsI in general T $ T ∗∗

I Exist T = T ∗ semiregular, BUT T + iλ not invertible.

Regular operators

Definition (Regular operator)

Let T semiregular. T regular if I + T ∗T has dense range.

Regular operators behave much like closable densely defined (resp.selfadjoint) operators in Hilbert space.

Proposition

Let T symmetric, densely defined, closed.

I T regular ⇔ T ± i Id have complementable range.

I T selfadjoint and regular ⇔ T ± i Id have dense range.

Proposition

Selfadjoint and regular operators admit a bounded continuous functionalcalculus, bounded imaginary powers etc. E.g. spec(T ) ⊂ R,f (T ) ∈ L(E ), f ∈ Cb(spec T ), ‖f (T )‖ = ‖f ‖∞, in particular‖(T + z)−1‖ ≤ 1/| Im z |,

Hilbert space case

A = C: semiregular ⇒ regular

LocalizationExploit that for Hilbert spaces: “semiregular” ⇔ “regular”

I (ω,Hω, ξω) cyclic representation of A w.r.t. state ω : A → C.

I E ⊗AHω Hilbert space completion of E ⊗ ξω ⊂ E ⊗A Hω w.r.t.

〈x ⊗ ξω, x ′ ⊗ ξω〉 = ω(〈x , x ′〉A

), x , x ′ ∈ E

I D(Tω0 ) := D(T )⊗A ξω ⊂ E ⊗AHω

Tω0 (x ⊗ ξω) := (Tx)⊗ ξω ∈ E ⊗AHω, x ∈ D(T ).

LemmaTω

0 densely defined and closable. (T ∗)ω0 ⊂ (Tω0 )∗.

Tω := Tω0 localization of T w.r.t. (ω,Hω, ξω).

Important: (T ∗)ω ⊂ (Tω)∗.

E Hilbert A–module.

Theorem A (Local–Global Principle; Pierrot 2006; Kaad-L 2012)

Let T closed semiregular operator in E .

T regular ⇔ ∀ state ω ∈ S(A) : (T ∗)ω = (Tω)∗.

Let additionally T be symmetric (〈Tx , y〉 = 〈x ,Ty〉 for x , y ∈ D(T ))

T selfadjoint and regular

⇔ ∀ state ω ∈ S(A) : localization Tω is selfadjoint.

Theorem B (implicit in Pierot 2006; Kaad-L 2012)

Let L ⊂ E , L 6= E closed nontrivial submodule; x0 ∈ E \ L.

∃ state ω ∈ S(A) : x0 ⊗ ξω 6∈ L⊗ ξω.

In particular ∃ state ω:(L⊗ ξω

)⊥ 6= {0}.Short: Submodule L ⊂ E is dense ⇔ ∀ state ω: L⊗ ξω dense in E ⊗Hω.

Application: weak cores

Proposition

T closed operator in Hilbert space H(A) Let (xn) ⊂ D(T ) with xn ⇀ x (weak convergence), supn ‖Txn‖ <∞.Then x ∈ D(T ) and Txn ⇀ Tx.(B) Let E ⊂ D(T ) subspace. Let E space of x ∈ D(T ) admitting anapproximating sequence (xn) ⊂ E as in (A). If E is a core for T then E is acore for T .

Proposition

Let T be a semi-regular operator in the Hilbert A-module E .In general (A) fails, even if T is regular.(B) holds true if E is a submodule

Proof 1

1. For y ∈ D(T ∗):

|〈x ,Ty〉| = limn|〈xn,Ty〉| = lim

n|〈Txn, y〉| ≤

(supn‖Txn‖

)· ‖y‖,

thus x ∈ D(T ) and

〈Tx , y〉 = 〈x ,Ty〉 = limn〈xn,Ty〉 = lim

n〈Txn, y〉, y ∈ D(T ),

thus Txn ⇀ Tx .2. Γ(E) :=graph of T over E , similarly Γ(E).

Γ(E)strong ⊂ Γ(E) ⊂ Γ(E)

weak= Γ(E)

strong

E core ⇒ Γ(E)strong

= Γ(E) = Γ(T ).

Proof 2

3. Counterexample to (A) for Hilbert modules: A = Cb(N),E = C0(N),〈f , g〉(k) := f (k) · g(k).

I Tf (k) := k · f (k) if (k · f (k))k is bounded.

I T is selfadjoint and regular.

I Fix F ∈ Cb(N) such that limk→∞

F (k) does not exist.

fn(k) :=

{1k · F (k) k ≤ n,

0 k > n.

I Then fn ∈ D(T ), fn → f := 1id F ∈ E , ‖Tfn‖ ≤ ‖F‖∞. BUT f 6∈ D(T ).

I Nevertheless: Cc(N) is a core for T .

Proof 3

4. Proof of (B) using Theorem (B):Fix state ω with cyclic representation (Hω, ξω).

I D(Tω) = D(T )⊗Hω.

I Fix x ⊗ ξω ∈ D(T )⊗ ξω, choose sequence (xn) ⊂ E with xn ⇀ x andsupn ‖Txn‖ <∞. For any η ∈ E ⊗Hω

E 3 z 7→ 〈η, z ⊗ ξω〉E⊗Hω

is continuous linear form, hence

〈x ⊗ ξω, η〉E⊗Hω= lim

n〈xn ⊗ ξω, η〉E⊗Hω

,

thus xn ⊗ ξω ⇀ x ⊗ ξω in E ⊗Hω.

I supn ‖Tω(xn ⊗ ξω)‖ = supn ‖ω(〈Txn,Txn〉

)‖ ≤ supn ‖Txn‖2 <∞.

I Result: ∀ω : E ⊗ ξω is dense in D(Tω) = D(T )⊗Hω.With Theorem B: E is dense in D(T ).

Sums of regular selfadjoint operatorsMotivation Unbounded KK-product

I D1⊗D2 = D1 ⊗ 1 + 1⊗∇ D2

I Example: (A(t))t∈R family of selfadjoint Fredholm operators in H1

(single operator in H = H1⊗C0(R)C0(R)). InH⊗C0(R)L2(R) = H1⊗L2(R):(

0 A(t)A(t) 0

)︸ ︷︷ ︸

S

+

(0 d

dx

− ddx 0

)︸ ︷︷ ︸

T

S ,T selfadjoint regular operators in E

ProblemAppropriate smallness condition on [S ,T ] = ST + TS should imply S + Tselfadjoint and regular

I Both operators are sectorial with spectral angle π (hyperbolic case)

I S2,T 2 are nonnegative (sectorial) operators

Banach space results

Theorem (Da Prato-Grisvard)

A,B sectorial operators in a Banach space X with spectral angle < π.Assume (A + λ)−1D(B) ⊂ D(B) and∥∥B(A + λ)−1 − (A + λ)−1B

)(µ+ B)−1

∥∥≤ c

(1 + |λ|)α|µ|β, α, β > 0, β < 1, α + β > 1

Then for λ large enough (outside spectral sector) A + B + λ is invertible.

I Labbas-Terreni: same conclusion under

‖A(A + λ)−1(A−1(B + µ)−1 − (B + µ)−1A−1

)‖

≤ c

(1 + |λ|)1−α|µ|1+β; 0 ≤ α < β < 1

Banach space results II

I Closedness of A + B on D(A) ∩ D(B) proved under additionalassumptions on the Banach space X (HT ) and that A,B admit BIP.

Dore-Venni 1987 A,B, resolvent commutingMonnieux-Pruss 1997 A,B satisfy Labbas-Terreni commutator

conditionPruss-Simonett 2007 Da Prato-Grisvard or Labbas-Terreni commutator

condition; emphasis on H∞ calculus.

I Important pattern:

Pλ :=1

2πi

∫Γ(z + λ+ A)−1 · (z − B)−1dz

“approximates” (A + B + λ)−1.

Main Result

Theorem CS ,T selfadjoint and regular in Hilbert A–module E .Assumptions:

1. (S + λ)−1(D(T )

)⊂ F for λ ∈ iR large enough.

F := F(S ,T ) :={

x ∈ D(S) ∩ D(T )∣∣ Sx ∈ D(T ),Tx ∈ D(S)

}2. For x ∈ F :∥∥[S ,T ]x := (S · T + T · S)x‖ ≤ C1 · ‖Sx‖+ C2 · ‖Tx‖+ C3 · ‖x‖.

Then S + T with domain D(S) ∩ D(T ) is selfadjoint and regular.

In equation solver speak (more impressive): for z ∈ C \ R and y ∈ E theequation

Sx + Tx + z · x = y

has a unique solution x ∈ D(S) ∩ D(T ).

Main Result

Theorem CS ,T selfadjoint and regular in Hilbert A–module E .Assumptions:

1. (S + λ)−1(D(T )

)⊂ F for λ ∈ iR large enough.

F := F(S ,T ) :={

x ∈ D(S) ∩ D(T )∣∣ Sx ∈ D(T ),Tx ∈ D(S)

}2. For x ∈ F :∥∥[S ,T ]x := (S · T + T · S)x‖ ≤ C1 · ‖Sx‖+ C2 · ‖Tx‖+ C3 · ‖x‖.

Then S + T with domain D(S) ∩ D(T ) is selfadjoint and regular.

Remark

1. The assumptions are symmetric in S ,T (exchange roles of S ,T ).

2. Suffices that (S + λ)−1(E) ⊂ F and (2) holds on (S + λ)−1(E) for acore E of T .

Comparison to da Prato Grisvard / Labbas-Terreni

1. ‖[S ,T ](S + λ)−1(T + µ)−1‖ ≤ c(

1|λ| + 1

|µ|).

2. ⇒

[T 2, (S2 + λ)−1](T 2 + µ)−1 � 1

|λ|( 1√|λ|

+1√|µ|),

BUT (S2 + λ)−1D(T 2) 6⊂ D(T 2).

3.

Pλ :=1

2πi

∫Γ(z + λ+ S2)−1(S + T − iλ)(z − T 2)−1dz

For y ∈ D(S) ∩ D(T ), λ large

(S + T + iλ)Pλy = (I + Rλ)y , ‖Rλ‖ < 1,

hence ran(S + T + iλ) dense.

Main Result: Consequences

1. (S + λ)−1D(T ) ⊂ F and (T + λ)−1D(S) ⊂ F for all λ ∈ iR, |λ| ≥ λ0.

2. For λ, µ ∈ iR, |λ, µ| ≥ λ0

ran(T + µ)−1 · (S + λ)−1 = ran(S + λ)−1 · (T + µ) = F

3. D(S) ∩ D(T ) is dense in E and F is dense in D(S) ∩ D(T ) in thefollowing sense: for x ∈ D(S) ∩ D(T ),

xλ := λ2(T + λ)−1 · (S + λ)−1x ∈ F ,

and xλ → x ,Sxλ → Sx ,Txλ → Tx , as iR 3 λ→∞.

TheoremS2 + T 2 is selfadjoint and regular on D(S2) ∩ D(T 2) = D((S + T )2). Forx ∈ D(S2) ∩ D(T 2) one has automatically Sx ∈ D(T ),Tx ∈ D(S).

Application: Iteration

TheoremS1, S2,S3 selfadjoint and regular. Assume that (S1, S2), (S2,S3), (S1,S3)satisfy the assumptions of Theorem C. Then also (S1 + S2,S3) satisfies theseassumptions and S1 + S2 + S3 is selfadjoint and regular onD(S1) ∩ D(S2) ∩ D(S3).

Hard

(S1 + S2 + λ)−1D(S3) ⊂ F(S1 + S2,S3).

Easy

(S3 + λ)−1(D(S1) ∩ D(S2)

)⊂ F(S1 + S2,S3)

Structure of Proof

1. Extend domain inclusion to all |λ, µ| ≥ |λ0|

(S + λ)(T + µ)− (T + µ)(S + λ) = ST − TS

Use Clifford Algebra trick.

2. Closedness of Sum operator:

c1

(‖Sx‖+‖Tx‖+‖x‖

)≤ ‖(S +T )x‖+‖x‖ ≤ ‖Sx‖+‖Tx‖+‖x‖

3. Selfadjointness: λ2(S + λ)−1(T + λ)−1

4. Regularity: Local Global Principle

Clifford Algebra tool

σ1 :=

(0 11 0

), σ2 :=

(0 i−i 0

), ω := iσ1 · σ2 =

(1 00 −1

)generators of C`(2). Replace E by E ⊗ C2 (ungraded), S ,T by

S ⊗ I =

(S 00 S

), T ⊗ I =

(T 00 T

).

⇒W.l.o.g. C`(2) ⊂ L(E ) acts unitarily on E and commutes with S ,T .Relations:

(Sσj)∗ = Sσj , (Tσj)

∗ = Tσj , (6.1)

Sσ1 · Tσ2 − Tσ2 · Sσ1 = (ST + TS)σ1σ2 (6.2)

Sω · T + T · Sω = (ST + TS) · ω (6.3)

(S · ω + T ) · Sσ1 + Sσ1 · (S · ω + T ) = (ST + TS) · σ1 (6.4)

Closedness of the sum operator

〈(S + T )x , (S + T )x〉 = 〈Sx , Sx〉+ 〈Tx ,Tx〉+ 〈Sx ,Tx〉+ 〈Tx , Sx〉︸ ︷︷ ︸≤〈Sx ,Sx〉+〈Tx ,Tx〉

〈Sx , Sx〉+ 〈Tx ,Tx〉 =1

2

(〈 1

µ[S ,T ]x , µx〉+ 〈µx ,

1

µ[S ,T ]x〉

)≤ µ−2C1‖Sx‖2 + µ−2C2‖Tx‖2 + µ2C3‖x‖2

≤ 1

4‖〈Sx ,Sx〉‖+

1

4‖〈Tx ,Tx〉‖+ C‖x‖2

≤ 1

2‖〈Sx ,Sx〉+ 〈Tx ,Tx〉‖+ C‖x‖2

⇒

‖〈Sx ,Sx〉‖+ ‖〈Tx ,Tx〉‖ ≤ 2‖〈Sx , Sx〉+ 〈Tx ,Tx〉‖≤ 4‖〈(S + T )x , (S + T )x〉‖+ C‖x‖2

≤ 8‖〈Sx ,Sx〉‖+ 8‖〈Tx ,Tx〉‖+ C‖x‖2.

Proof of Selfadjointness and Regularity

Selfadjointness x ∈ D((S + T )∗);

xλ := λ2(S + λ)−1(T + λ)−1x ∈ Fxλ → x

(S + T )xλ = Commutator Term

+ λ2(S + λ)−1(T + λ)−1(S + T )∗x → (S + T )∗x .

Regularity All Localizations Sω + Tω = (S + T )ω selfadjoint Local Global

Principle ⇒ S + T regular.