sums of linear operators in hilbert c*-modules · sums of linear operators in hilbert c*-modules...
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Sums of linear operators in Hilbert C*-moduleswork in progress, based on discussions with Bram
Matthias Lesch
Universitat [email protected]
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.