Characterization of invariant subspaces in the polydiscjay/OTOA/OTOA18/zAmit.pdfCharacterization of...

Characterization of invariant subspaces in the polydisc

Amit Maji

IIIT Guwahati

(Joint work with Aneesh M., Jaydeb Sarkar & Sankar T. R.)

Recent Advances in Operator Theory and Operator AlgebrasIndian Statistical Institute, Bangalore

December 13-19, 2018

Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 1 / 25

Characterization of invariant subspaces . . .

To give a complete characterization of (joint) invariant subspaces for(Mz1 , . . . ,Mzn) on the Hardy space H2(Dn) over the unit polydisc Dn in Cn,n > 1.

To discuss about a complete set of unitary invariants for invariant subspacesas well as unitarily equivalent invariant subspaces.

To classify a large class of n-tuples of commuting isometries.

Notation

Unit polydisc Dn = {(z1, . . . , zn) ∈ Cn : |zi | < 1, i = 1, . . . , n}.Hardy space H2(D) = {f =

∑∞n=0 anzn :

∑∞n=0 |an|2 <∞}.

Vector-valued Hardy space

H2E(D) = {f =

∞∑n=0

anzn : an ∈ E and∞∑n=0

‖an‖2E <∞},

where E is some Hilbert space.

Mz denote the multiplication operator on H2E(D) defined by

(Mz f )(w) = wf (w) (f ∈ H2E(D),w ∈ D).

H∞B(E)(D) denote the space of bounded B(E)-valued holomorphic functions onD.

Invariant subspaces: Motivation

A closed subspace M of a Hilbert space H is said to be invariant subspaceunder T ∈ B(H) if T (M) ⊆M.

Given an invariant subspace M of T ∈ B(H), one can view T as an operatormatrix with respect to the decomposition H =M⊕M⊥[

T |M ∗0 ∗

One of the most famous open problems in operator theory and functiontheory is the so-called invariant subspace problem: Does every bounded linearoperator have a non-trivial closed invariant subspace?

T |M ∗0 ∗

The celebrated Beurling theorem (1949) says that a non-zero closed subspaceS of H2(D) is invariant for Mz if and only if there exists an inner functionθ ∈ H∞(D) such that

S = θH2(D).

One may now ask whether an analogous characterization holds for (joint)invariant subspaces for (Mz1 , . . . ,Mzn) on H2(Dn), n > 1, i.e., if S is aninvariant subspace for (Mz1 , . . . ,Mzn) on H2(Dn), then

S = ψH2(Dn),

where ψ ∈ H∞(Dn) is an inner function.

Rudin’s pathological examples (Rudin (1969)): There exist invariantsubspaces S1 and S2 for (Mz1 ,Mz2 ) on H2(D2) such that(1) S1 is not finitely generated, and(2) S2 ∩ H∞(D2) = {0}.

S = θH2(D).

One may now ask whether an analogous characterization holds for (joint)invariant subspaces for (Mz1 , . . . ,Mzn) on H2(Dn), n > 1, i.e.,

if S is aninvariant subspace for (Mz1 , . . . ,Mzn) on H2(Dn), then

S = ψH2(Dn),

S = θH2(D).

S = ψH2(Dn),

S = θH2(D).

S = ψH2(Dn),

To understand the structure of a large class of operators we need tounderstand the structure of shift invariant subspaces for the vector-valuedHardy space over the unit disc.

Theorem (Beurling-Lax-Halmos)

A non-zero closed subspace S of H2E(D) is invariant for Mz if and only if there

exists a closed subspace F ⊆ E and an inner function Θ ∈ H∞B(F,E)(D) such that

S = ΘH2F (D).

Identify Hardy space over polydisc H2(Dn+1) to the H2(Dn)-valued Hardyspace over disc H2

H2(Dn)(D).

Represent (Mz1 ,Mz2 , . . . ,Mzn+1 ) on H2(Dn+1) to (Mz ,Mκ1 , . . . ,Mκn) onH2

H2(Dn)(D), where κi ∈ H∞B(H2(Dn))(D), i = 1, . . . , n, is a constant as well as

simple and explicit B(H2(Dn))-valued analytic function.

S = ΘH2F (D).

H2(Dn)(D).

S = ΘH2F (D).

H2(Dn)(D).

Basic definitions

A closed subspace S ⊆ H2E(Dn) is called a (joint) invariant subspace for

(Mz1 , . . . ,Mzn) on H2E(Dn) if

ziS ⊆ S,for all i = 1, . . . , n.

An isometry V on H is called a pure isometry (or shift) if V ∗m → 0 in SOT.

Let V be a pure isometry on H. Then

H =∞⊕

m=0V mW,

where W = ker V ∗ = H VH.

The natural map ΠV : H → H2W(D) defined by

ΠV (V mη) = zmη,

for all m ≥ 0 and η ∈ W, is a unitary operator and

ΠV V = MzΠV .

We call ΠV the Wold-von Neumann decomposition of the shift V .

Basic definitions

ziS ⊆ S,for all i = 1, . . . , n.

H =∞⊕

m=0V mW,

ΠV (V mη) = zmη,

ΠV V = MzΠV .

Basic definitions

ziS ⊆ S,for all i = 1, . . . , n.

H =∞⊕

m=0V mW,

ΠV (V mη) = zmη,

ΠV V = MzΠV .

Basic definitions

ziS ⊆ S,for all i = 1, . . . , n.

H =∞⊕

m=0V mW,

ΠV (V mη) = zmη,

ΠV V = MzΠV .

Commutator of shift

Theorem 1 (Maji, Sarkar, & Sankar’ 18)

Let V be a pure isometry on H, and let C be a bounded operator on H. Let PiVbe the Wold-von Neumann decomposition of V . Let W = Ker(V ∗) and assumethat M = ΠV C Π∗V . Then

CV = VC ,

if and only ifM = MΘ,

whereΘ(z) = PW(IH − zV ∗)−1C |W (z ∈ D).

RemarkIn addition if CV ∗ = V ∗C , then

Θ(z) = C |W = Θ(0) (z ∈ D),

as C (I − VV ∗) = (I − VV ∗)C and V ∗m |W= 0 for all m ≥ 1.

Commutator of shift

Theorem 1 (Maji, Sarkar, & Sankar’ 18)

Let V be a pure isometry on H, and let C be a bounded operator on H. Let PiVbe the Wold-von Neumann decomposition of V . Let W = Ker(V ∗) and assumethat M = ΠV C Π∗V . Then

CV = VC ,

if and only ifM = MΘ,

whereΘ(z) = PW(IH − zV ∗)−1C |W (z ∈ D).

RemarkIn addition if CV ∗ = V ∗C , then

Θ(z) = C |W = Θ(0) (z ∈ D),

as C (I − VV ∗) = (I − VV ∗)C and V ∗m |W= 0 for all m ≥ 1.

Preparation for main result

For the sake of simplicity we discuss firstly for n = 2, i.e., vector-valuedHardy space over the bidisc H2

E(D2).

We now identify H2E(D2) with H2

H2E (D)

(D) by the canonical unitaries

H2E(D2)

U−→ H2(D)⊗ H2E(D)

U−→ H2H2E (D)(D)

whereU(zk1

1 zk22 η) = zk1 ⊗ (zk2

1 η), ( k1, k2 ≥ 0, η ∈ E )

andU(zk ⊗ ζ) = zkζ, ( k ≥ 0, ζ ∈ H2

E(D) ).

Set U = UU. Then it follows that U : H2E(D2)→ H2

H2E (D)

(D) is a unitary

operator. Since

UMz1 = (Mz ⊗ IH2E (D) ))U and UMz2 = (IH2(D) ⊗Mz1 )U,

we have UMz1 = MzU, and UMz2 = Mκ1 U, where κ1(w) = Mz1 for w ∈ D.

E(D2).

H2E (D)

H2E(D2)

U−→ H2(D)⊗ H2E(D)

U−→ H2H2E (D)(D)

whereU(zk1

1 zk22 η) = zk1 ⊗ (zk2

1 η), ( k1, k2 ≥ 0, η ∈ E )

andU(zk ⊗ ζ) = zkζ, ( k ≥ 0, ζ ∈ H2

E(D) ).

H2E (D)

(D) is a unitary

operator. Since

we have UMz1 = MzU, and UMz2 = Mκ1 U, where κ1(w) = Mz1 for w ∈ D.

E(D2).

H2E (D)

H2E(D2)

U−→ H2(D)⊗ H2E(D)

U−→ H2H2E (D)(D)

whereU(zk1

1 zk22 η) = zk1 ⊗ (zk2

1 η), ( k1, k2 ≥ 0, η ∈ E )

andU(zk ⊗ ζ) = zkζ, ( k ≥ 0, ζ ∈ H2

E(D) ).

H2E (D)

(D) is a unitary

operator. Since

we have UMz1 = MzU, and UMz2 = Mκ1 U, where κ1(w) = Mz1 for w ∈ D.Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 9 / 25

Let E be a Hilbert space and let En = H2(Dn)⊗ E for n ≥ 1. Let κi ∈ H∞B(En)(D)

denote the B(En)-valued constant function on D defined by

κi (w) = Mzi ∈ B(En),

for all w ∈ D, and let Mκi denote the multiplication operator on H2En(D) defined by

Mκi f = κi f ,

for all f ∈ H2En(D) and i = 1, . . . , n. Then

Theorem 2 (Maji, Aneesh, Sarkar, & Sankar’ 18)

(i) (Mz1 ,Mz2 . . . ,Mzn+1 ) on H2E(Dn+1) and (Mz ,Mκ1 , . . . ,Mκn) on H2

En(D) areunitarily equivalent.

Let S ⊆ H2H2E (D)

(D) be a closed invariant subspace for (Mz ,Mκ1 ) on

H2H2E (D)

(D). Set

V = Mz |S and V1 = Mκ1 |S .

Let ΠV : S → H2W(D) be the Wold-von Neumann decomposition of V on S.

ThenΠV V Π∗V = Mz and ΠV V1Π∗V = MΦ1 ,

whereΦ1(w) = PW(IS − wV ∗)−1V1|W ,

for all w ∈ D, Φ1 ∈ H∞B(W)(D).

Let iS denote the inclusion map iS : S ↪→ H2En(D). Then

H2W(D)

Π∗V−−→ S iS−→ H2H2E (D)(D)

i.e., ΠS = iS ◦ Π∗V : H2W(D)→ H2

H2E (D)

(D) is an isometry and

ran ΠS = S.

Let S ⊆ H2H2E (D)

H2H2E (D)

(D). Set

H2W(D)

Π∗V−−→ S iS−→ H2H2E (D)(D)

i.e., ΠS = iS ◦ Π∗V : H2W(D)→ H2

H2E (D)

ran ΠS = S.

Let S ⊆ H2H2E (D)

H2H2E (D)

(D). Set

H2W(D)

Π∗V−−→ S iS−→ H2H2E (D)(D)

i.e., ΠS = iS ◦ Π∗V : H2W(D)→ H2

H2E (D)

ran ΠS = S.

Main result

We have invariant subspace result on vector-valued Hardy space over polydiscsetting:

Theorem 2 (Maji, Aneesh, Sarkar, & Sankar’ 18)

(ii) Let E be a Hilbert space, S ⊆ H2En(D) be a closed subspace, and let

W = S zS. Then S is invariant for (Mz ,Mκ1 , . . . ,Mκn) if and only if(MΦ1 , . . . ,MΦn) is an n-tuple of commuting shifts on H2

W(D) and there exists aninner function Θ ∈ H∞B(W,En)(D) such that

S = ΘH2W(D),

andκiΘ = ΘΦi ,

whereΦi (w) = PW(IS − wPSM∗z )−1Mκi |W ,

for all w ∈ D and i = 1, . . . , n.

Remarks

One obvious necessary condition for a closed subspace S of H2En(D) to be

(joint) invariant for (Mz ,Mκ1 , . . . ,Mκn) is that S is invariant for Mz , and,consequently

S = ΘH2W(D),

where W = S zS and Θ ∈ H∞B(W,En)(D) is the Beurling, Lax and Halmosinner function.

Again κiS ⊆ S, =⇒κiΘ = ΘΓi ,

for some Γi ∈ B(H2W(D)), i = 1, . . . , n (by Douglas’s range inclusion theorem

In the above theorem, we prove that Γi is explicit, that is

Γi = Φi ∈ H∞B(W)(D),

for all i = 1, . . . , n, and (Γ1, . . . , Γn) is an n-tuple of commuting shifts onH2W(D).

Remarks

S = ΘH2W(D),

Remarks

S = ΘH2W(D),

Remarks

Uniqueness

Let S be an invariant subspace for (Mz ,Mκ1 , . . . ,Mκn) on H2En(D). Then

S = ΘH2W(D) and

κiΘ = ΘΦi (i = 1, . . . , n),

from the above Theorem.

Now suppose S = ΘH2W(D) and κi Θ = ΘΦi for some Hilbert space W, inner

function Θ ∈ H∞B(W)(D) and shift MΦi

on H2W(D), i = 1, . . . , n.

Then there exists a unitary operator τ :W → W such that

Θ = Θτ,

andτΦi = Φiτ,

for all i = 1, . . . , n.

Remarks

Uniqueness

S = ΘH2W(D) and

κiΘ = ΘΦi (i = 1, . . . , n),

from the above Theorem.Now suppose S = ΘH2

W(D) and κi Θ = ΘΦi for some Hilbert space W, inner

on H2W(D), i = 1, . . . , n.

Θ = Θτ,

andτΦi = Φiτ,

for all i = 1, . . . , n.

Remarks

Uniqueness

S = ΘH2W(D) and

κiΘ = ΘΦi (i = 1, . . . , n),

from the above Theorem.Now suppose S = ΘH2

W(D) and κi Θ = ΘΦi for some Hilbert space W, inner

on H2W(D), i = 1, . . . , n.

Θ = Θτ,

andτΦi = Φiτ,

for all i = 1, . . . , n.

Nested invariant subspaces

Theorem (Maji, Aneesh, Sarkar, & Sankar’ 18)

Let E be a Hilbert space, and let S1 = Θ1H2W1

(D) and S2 = Θ2H2W2

(D) be twoinvariant subspaces for (Mz ,Mκ1 , . . . ,Mκn) on H2

En(D) and Wj = Sj zSj forj = 1, 2. Let

Φj,i (w) = PWj (ISj − wPSj M∗z )−1Mκi |Wj ,

for all w ∈ D, j = 1, 2, and i = 1, . . . , n.

Then S1 ⊆ S2 if and only if there exists an inner multiplier Ψ ∈ H∞B(W1,W2)(D)such that Θ1 = Θ2Ψ and ΨΦ1,i = Φ2,iΨ for all i = 1, . . . , n.

Nested invariant subspaces

Let E be a Hilbert space, and let S1 = Θ1H2W1

(D) and S2 = Θ2H2W2

(D) be twoinvariant subspaces for (Mz ,Mκ1 , . . . ,Mκn) on H2

En(D) and Wj = Sj zSj forj = 1, 2. Let

Φj,i (w) = PWj (ISj − wPSj M∗z )−1Mκi |Wj ,

for all w ∈ D, j = 1, 2, and i = 1, . . . , n.Then S1 ⊆ S2 if and only if there exists an inner multiplier Ψ ∈ H∞B(W1,W2)(D)such that Θ1 = Θ2Ψ and ΨΦ1,i = Φ2,iΨ for all i = 1, . . . , n.

Unitarily equivalent invariant subspaces

Definition

Let S and S be invariant subspaces for the (n + 1)-tuples of multiplicationoperators (Mz ,Mκ1 , . . . ,Mκn) on H2

En(D) and H2En

(D), respectively. We say that Sand S are unitarily equivalent, and write S ∼= S, if there is a unitary mapU : S → S such that

UMz |S = Mz |SU and UMκi |S = Mκi |SU,

for all i = 1, . . . , n.

Identification

There exists a unitary operator UE : H2E(Dn+1)→ H2

En(D) such that

UEMz1 = MzUE ,

andUEMzi+1 = Mκi UE ,

for all i = 1, . . . , n.

Definition

Let S and S be invariant subspaces for the (n + 1)-tuples of multiplicationoperators (Mz ,Mκ1 , . . . ,Mκn) on H2

En(D) and H2En

(D), respectively. We say that Sand S are unitarily equivalent, and write S ∼= S, if there is a unitary mapU : S → S such that

UMz |S = Mz |SU and UMκi |S = Mκi |SU,

for all i = 1, . . . , n.

Identification

There exists a unitary operator UE : H2E(Dn+1)→ H2

En(D) such that

UEMz1 = MzUE ,

andUEMzi+1 = Mκi UE ,

for all i = 1, . . . , n.Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 16 / 25

Intertwining maps

Let F be another Hilbert space, and let X : H2E(Dn+1)→ H2

F (Dn+1) be a boundedlinear operator such that

XMzi = Mzi X , (0.1)

for all i = 1, . . . , n + 1. SetXn = UFXU∗E .

Then Xn : H2En(D)→ H2

Fn(D) is bounded and

XnMz = MzXn and XnMκi = Mκi Xn, (0.2)

for all i = 1, . . . , n.

Definition

Any map satisfying (0.2) will be referred to module maps.

Let S ⊆ H2Fn

(D) be a closed invariant subspace for (Mz ,Mκ1 , . . . ,Mκn) onH2Fn

(D). Then S ∼= H2En(D) if and only if there exists an isometric module map

Xn : H2En(D)→ H2

Fn(D) such that

S = XnH2En(D).

Moreover, in this casedim E ≤ dim F .

Corollary

Let S ⊆ H2Hn

(D) be a closed invariant subspace for (Mz ,Mκ1 , . . . ,Mκn) on

(D). Then S ∼= H2Hn

(D) if and only if there exists an isometric module map

Xn : H2Hn

(D)→ H2Hn

(D) such that

S = Xn(H2Hn

The above corollary was first observed by Agrawal, Clark and Douglas (1986).

Let S ⊆ H2Fn

(D) be a closed invariant subspace for (Mz ,Mκ1 , . . . ,Mκn) onH2Fn

(D). Then S ∼= H2En(D) if and only if there exists an isometric module map

Xn : H2En(D)→ H2

Fn(D) such that

S = XnH2En(D).

Moreover, in this casedim E ≤ dim F .

Corollary

Let S ⊆ H2Hn

(D) be a closed invariant subspace for (Mz ,Mκ1 , . . . ,Mκn) on

(D). Then S ∼= H2Hn

(D) if and only if there exists an isometric module map

Xn : H2Hn

(D)→ H2Hn

(D) such that

S = Xn(H2Hn

The above corollary was first observed by Agrawal, Clark and Douglas (1986).Amit Maji ( IIIT G ) Characterization of invariant subspaces in the polydisc 18 / 25

A complete set of unitary invariants

Definition

Let E and E be Hilbert spaces, and let {Ψ1, . . . ,Ψn} ⊆ H∞B(E)(D) and

{Ψ1, . . . , Ψn} ⊆ H∞B(E)(D). We say that {Ψ1, . . . ,Ψn} and {Ψ1, . . . , Ψn} coincide

if there exists a unitary operator τ : E → E such that

τΨi (z) = Ψi (z)τ,

for all z ∈ D and i = 1, . . . , n.

Let E and E be Hilbert spaces. Let S ⊆ H2En(D) and S ⊆ H2

En(D) be invariant

subspaces for (Mz ,Mκ1 , . . . ,Mκn) on H2En(D) and H2

En(D), respectively. Then

S ∼= S if and only if {Φ1, . . . ,Φn} and {Φ1, . . . , Φn} coincide.

A complete set of unitary invariants

Definition

Let E and E be Hilbert spaces, and let {Ψ1, . . . ,Ψn} ⊆ H∞B(E)(D) and

{Ψ1, . . . , Ψn} ⊆ H∞B(E)(D). We say that {Ψ1, . . . ,Ψn} and {Ψ1, . . . , Ψn} coincide

if there exists a unitary operator τ : E → E such that

τΨi (z) = Ψi (z)τ,

for all z ∈ D and i = 1, . . . , n.

Let E and E be Hilbert spaces. Let S ⊆ H2En(D) and S ⊆ H2

En(D) be invariant

subspaces for (Mz ,Mκ1 , . . . ,Mκn) on H2En(D) and H2

En(D), respectively. Then

S ∼= S if and only if {Φ1, . . . ,Φn} and {Φ1, . . . , Φn} coincide.

Representations of model isometries

Question

Given a Hilbert space E , characterize (n + 1)-tuples of commuting shifts onHilbert spaces that are unitarily equivalent to (Mz ,Mκ1 , . . . ,Mκn) on H2

En(D).

Answer

Answer to this question is related to (numerical invariant) the rank of an operatorassociated with the Szego kernel on Dn+1.

Question

Given a Hilbert space E , characterize (n + 1)-tuples of commuting shifts onHilbert spaces that are unitarily equivalent to (Mz ,Mκ1 , . . . ,Mκn) on H2

En(D).

Answer

Answer to this question is related to (numerical invariant) the rank of an operatorassociated with the Szego kernel on Dn+1.

DefinitionThe defect operator corresponding to an m-tuple of commuting contractions(T1, . . . ,Tm) on a Hilbert space H is defined (see, Guo & Yang (2004)) as

S−1m (T1, . . . ,Tm) =

∑0≤|k|≤m

(−1)|k|T k11 · · ·T

kmm T ∗k1

1 · · ·T ∗kmm ,

where |k| = k1 + k2 + . . .+ km, 0 ≤ ki ≤ 1, i = 1, . . . ,m.

Definition

We say that (T1, . . . ,Tm) is of rank p (p ∈ N ∪ {∞}) if

rank [S−1m (T1, . . . ,Tm)] = p,

and we writerank (T1, . . . ,Tm) = p.

DefinitionThe defect operator corresponding to an m-tuple of commuting contractions(T1, . . . ,Tm) on a Hilbert space H is defined (see, Guo & Yang (2004)) as

S−1m (T1, . . . ,Tm) =

∑0≤|k|≤m

(−1)|k|T k11 · · ·T

kmm T ∗k1

1 · · ·T ∗kmm ,

where |k| = k1 + k2 + . . .+ km, 0 ≤ ki ≤ 1, i = 1, . . . ,m.

Definition

We say that (T1, . . . ,Tm) is of rank p (p ∈ N ∪ {∞}) if

rank [S−1m (T1, . . . ,Tm)] = p,

and we writerank (T1, . . . ,Tm) = p.

Let (V ,V1 . . . ,Vn) be an (n + 1)-tuple of doubly commuting shifts on H.Then Sarkar (2014) proved that (V ,V1, . . . ,Vn) on H and (Mz1 , . . . ,Mzn+1 )on H2

D(Dn+1) are unitarily equivalent, where

D = ran S−1n+1(V ,V1, . . . ,Vn) =

(n∩i=1

ker V ∗i

)∩ ker V ∗.

Let (V ,V1, . . . ,Vn) be an (n + 1)-tuple of doubly commuting shifts on someHilbert space H. Let W = H VH, and let

Ψi (z) = Vi |W (i = 1, . . . , n),

for all z ∈ D. Then (V ,V1, . . . ,Vn) on H, (Mz ,MΨ1 , . . . ,MΨn) on H2W(D), and

(Mz ,Mκ1 , . . . ,Mκn) on H2En(D) are unitarily equivalent, where E is a Hilbert space

and dim E = rank (V ,V1, . . . ,Vn).

Let (V ,V1 . . . ,Vn) be an (n + 1)-tuple of doubly commuting shifts on H.Then Sarkar (2014) proved that (V ,V1, . . . ,Vn) on H and (Mz1 , . . . ,Mzn+1 )on H2

D(Dn+1) are unitarily equivalent, where

D = ran S−1n+1(V ,V1, . . . ,Vn) =

(n∩i=1

ker V ∗i

)∩ ker V ∗.

Let (V ,V1, . . . ,Vn) be an (n + 1)-tuple of doubly commuting shifts on someHilbert space H. Let W = H VH, and let

Ψi (z) = Vi |W (i = 1, . . . , n),

for all z ∈ D. Then (V ,V1, . . . ,Vn) on H, (Mz ,MΨ1 , . . . ,MΨn) on H2W(D), and

(Mz ,Mκ1 , . . . ,Mκn) on H2En(D) are unitarily equivalent, where E is a Hilbert space

and dim E = rank (V ,V1, . . . ,Vn).

References

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