Toeplitz and Asymptotic Toeplitz operators on H 2 ( D n ) Amit Maji - - PowerPoint PPT Presentation

Toeplitz and Asymptotic Toeplitz operators on H2(Dn)

Amit Maji

(Joint work with Jaydeb Sarkar & Srijan Sarkar)

Indian Statistical Institute, Bangalore Centre

OTOA, December 13-22, 2016

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 1 / 18

Aim

To characterize Toeplitz operators on H2(Dn).

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 2 / 18

Aim

To characterize Toeplitz operators on H2(Dn). To characterize asymptotically Toeplitz operators on H2(Dn).

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 2 / 18

Aim

To characterize Toeplitz operators on H2(Dn). To characterize asymptotically Toeplitz operators on H2(Dn). To generalize some of the recent results of Chalendar and Ross to vector-valued Hardy space H2

E(D) and as well as quotient spaces of H2(Dn).

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 2 / 18

Notation

Open unit polydisc Dn = {(z1, . . . , zn) ∈ Cn : |zi| < 1, i = 1, . . . , n}. Distinguished boundary of Dn Tn = {(z1, . . . , zn) ∈ Cn : |zi| = 1, i = 1, . . . , n}.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 3 / 18

Notation

Open unit polydisc Dn = {(z1, . . . , zn) ∈ Cn : |zi| < 1, i = 1, . . . , n}. Distinguished boundary of Dn Tn = {(z1, . . . , zn) ∈ Cn : |zi| = 1, i = 1, . . . , n}. Hardy space H2(D) = {f = ∞

n=0 anzn : ∞ n=0 |an|2 < ∞}.

Vector-valued Hardy space H2

E(D) = {f = ∞ n=0 anzn : an ∈ E and ∞ n=0 an2 E < ∞}.

H∞(D) = {f = ∞

n=0 anzn : sup n≥0

|an| < ∞}. Mz is the multiplication operator on H2(D) by the coordinate function z.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 3 / 18

Notation

Open unit polydisc Dn = {(z1, . . . , zn) ∈ Cn : |zi| < 1, i = 1, . . . , n}. Distinguished boundary of Dn Tn = {(z1, . . . , zn) ∈ Cn : |zi| = 1, i = 1, . . . , n}. Hardy space H2(D) = {f = ∞

n=0 anzn : ∞ n=0 |an|2 < ∞}.

Vector-valued Hardy space H2

E(D) = {f = ∞ n=0 anzn : an ∈ E and ∞ n=0 an2 E < ∞}.

H∞(D) = {f = ∞

n=0 anzn : sup n≥0

|an| < ∞}. Mz is the multiplication operator on H2(D) by the coordinate function z. Hardy space over polydisc H2(Dn) =

• f =
• k∈Nn

akzk :

• k∈Nn

|ak|2 < ∞

• ,

where k = (k1, . . . , kn) ∈ Nn and zk = zk1

1 · · · zkn n .

For j = 1, . . . , n, Mzj are the multiplication operators on H2(Dn) by the jth coordinate functions zj.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 3 / 18

Multiplication operator

For φ ∈ L∞(T), define Mφ : L2(T) → L2(T) by Mφf = φf for f ∈ L2(T).

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 4 / 18

Multiplication operator

For φ ∈ L∞(T), define Mφ : L2(T) → L2(T) by Mφf = φf for f ∈ L2(T). The matrix of Mφ with respect to the orthonormal basis {einθ}∞

n=−∞ of

L2(T) = H2(D)⊥ ⊕ H2(D) is Mφ =                 ... ... ... ... φ0 φ−1 φ−2 ... φ1 φ0 φ−1 φ−2 φ2 φ1 φ0 φ−1 φ−2 φ2 φ1 φ0 φ−1 ... φ2 φ1 φ0 ... ... ...                 where φ =

∞

• n=−∞

φneinθ is a Fourier expansion of φ.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 4 / 18

Multiplication operator

For φ ∈ L∞(T), define Mφ : L2(T) → L2(T) by Mφf = φf for f ∈ L2(T). The matrix of Mφ with respect to the orthonormal basis {einθ}∞

n=−∞ of

L2(T) = H2(D)⊥ ⊕ H2(D) is Mφ =                 ... ... ... ... φ0 φ−1 φ−2 ... φ1 φ0 φ−1 φ−2 φ2 φ1 φ0 φ−1 φ−2 φ2 φ1 φ0 φ−1 ... φ2 φ1 φ0 ... ... ...                 where φ =

∞

• n=−∞

φneinθ is a Fourier expansion of φ. Toeplitz operator with symbol φ ∈ L∞(T) is the operator Tφ defined by Tφf = PH2(D)(φf ) for f ∈ H2(D).

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 4 / 18

Toeplitz operator

Toeplitz operators on the Hardy space (or, on the l2 space) were first studied by O. Toeplitz (1911)(and then by P. Hartman and A. Wintner (1954)).

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 5 / 18

Toeplitz operator

Toeplitz operators on the Hardy space (or, on the l2 space) were first studied by O. Toeplitz (1911)(and then by P. Hartman and A. Wintner (1954)). A systematic study of Toeplitz operators on H2(D) was triggered by the seminal paper of Brown and Halmos: Algebraic properties of Toeplitz

• perators J.Reine Angew. Math. 213:89–102, 1963/1964.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 5 / 18

Toeplitz operator

Toeplitz operators on the Hardy space (or, on the l2 space) were first studied by O. Toeplitz (1911)(and then by P. Hartman and A. Wintner (1954)). A systematic study of Toeplitz operators on H2(D) was triggered by the seminal paper of Brown and Halmos: Algebraic properties of Toeplitz

• perators J.Reine Angew. Math. 213:89–102, 1963/1964.

Brown-Halmos theorem characterize Toeplitz operators on H2(D) as follows: Let T be a bounded linear operator on H2(D). Then T is a Toeplitz operator if and only if M∗

z TMz = T.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 5 / 18

Toeplitz operator

Toeplitz operators on the Hardy space (or, on the l2 space) were first studied by O. Toeplitz (1911)(and then by P. Hartman and A. Wintner (1954)). A systematic study of Toeplitz operators on H2(D) was triggered by the seminal paper of Brown and Halmos: Algebraic properties of Toeplitz

• perators J.Reine Angew. Math. 213:89–102, 1963/1964.

Brown-Halmos theorem characterize Toeplitz operators on H2(D) as follows: Let T be a bounded linear operator on H2(D). Then T is a Toeplitz operator if and only if M∗

z TMz = T.

The notion of Toeplitzness was extended to more general settings by Barr´ ıa and Halmos (1982) and Feintuch (1989).

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 5 / 18

Toeplitz operator

Toeplitz operators on the Hardy space (or, on the l2 space) were first studied by O. Toeplitz (1911)(and then by P. Hartman and A. Wintner (1954)). A systematic study of Toeplitz operators on H2(D) was triggered by the seminal paper of Brown and Halmos: Algebraic properties of Toeplitz

• perators J.Reine Angew. Math. 213:89–102, 1963/1964.

Brown-Halmos theorem characterize Toeplitz operators on H2(D) as follows: Let T be a bounded linear operator on H2(D). Then T is a Toeplitz operator if and only if M∗

z TMz = T.

The notion of Toeplitzness was extended to more general settings by Barr´ ıa and Halmos (1982) and Feintuch (1989). A bounded linear operator T on H2(D) is (uniformly) asymptotically Toeplitz

• perator if {M∗m

z

TMm

z }m≥1 converges in operator norm.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 5 / 18

Toeplitz operator

Toeplitz operators on the Hardy space (or, on the l2 space) were first studied by O. Toeplitz (1911)(and then by P. Hartman and A. Wintner (1954)). A systematic study of Toeplitz operators on H2(D) was triggered by the seminal paper of Brown and Halmos: Algebraic properties of Toeplitz

• perators J.Reine Angew. Math. 213:89–102, 1963/1964.

Brown-Halmos theorem characterize Toeplitz operators on H2(D) as follows: Let T be a bounded linear operator on H2(D). Then T is a Toeplitz operator if and only if M∗

z TMz = T.

The notion of Toeplitzness was extended to more general settings by Barr´ ıa and Halmos (1982) and Feintuch (1989). A bounded linear operator T on H2(D) is (uniformly) asymptotically Toeplitz

• perator if {M∗m

z

TMm

z }m≥1 converges in operator norm.

Feintuch (1989) gives a remarkable characterization of asymptotically Toeplitz operators: A bounded linear operator T on H2(D) is asymptotically Toeplitz if and only if T = compact + Toeplitz.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 5 / 18

Basic Definitions

A closed subspace S of H is said to be invariant subspace of T ∈ B(H) if T(S) ⊆ S and S is said to be co-invariant subspace if T ∗(S) ⊆ S.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 6 / 18

Basic Definitions

A closed subspace S of H is said to be invariant subspace of T ∈ B(H) if T(S) ⊆ S and S is said to be co-invariant subspace if T ∗(S) ⊆ S. An operator T ∈ B(H) is said to be contraction if T ≤ 1.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 6 / 18

Basic Definitions

A closed subspace S of H is said to be invariant subspace of T ∈ B(H) if T(S) ⊆ S and S is said to be co-invariant subspace if T ∗(S) ⊆ S. An operator T ∈ B(H) is said to be contraction if T ≤ 1. A contraction T is said to be pure contraction if T ∗m → 0 as m → ∞ in strong operator topology.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 6 / 18

Basic Definitions

A closed subspace S of H is said to be invariant subspace of T ∈ B(H) if T(S) ⊆ S and S is said to be co-invariant subspace if T ∗(S) ⊆ S. An operator T ∈ B(H) is said to be contraction if T ≤ 1. A contraction T is said to be pure contraction if T ∗m → 0 as m → ∞ in strong operator topology. An inner function is a bounded analytic function ψ on D (that is, ψ ∈ H∞(D)) such that |ψ(eiθ)| = 1 for almost everywhere on the unit circle.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 6 / 18

Basic Definitions

A closed subspace S of H is said to be invariant subspace of T ∈ B(H) if T(S) ⊆ S and S is said to be co-invariant subspace if T ∗(S) ⊆ S. An operator T ∈ B(H) is said to be contraction if T ≤ 1. A contraction T is said to be pure contraction if T ∗m → 0 as m → ∞ in strong operator topology. An inner function is a bounded analytic function ψ on D (that is, ψ ∈ H∞(D)) such that |ψ(eiθ)| = 1 for almost everywhere on the unit circle. H∞

B(E)(D): the space of all operator valued bounded analytic functions on D.

A multiplier Θ ∈ H∞

B(E)(D) is said to be inner if MΘ is an isometry on H2 E(D),

where (MΘf )(w) = Θ(w)f (w) (f ∈ H2

E(D), w ∈ D).

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 6 / 18

Model Space in H2(D)

(Beurling Theorem (1948)) Let S be a non-zero shift invariant subspace of H2(D). Then S = θH2(D) for some inner function θ ∈ H∞(D).

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 7 / 18

Model Space in H2(D)

(Beurling Theorem (1948)) Let S be a non-zero shift invariant subspace of H2(D). Then S = θH2(D) for some inner function θ ∈ H∞(D). For an inner function θ, the model space Kθ is defined as Kθ = H2(D) ⊖ θH2(D). Kθ is finite dimensional if θ is finite Blaschke product (that is, θ(z) = n

k=1 z−zk 1−¯ zkz ).

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 7 / 18

Model Space in H2(D)

(Beurling Theorem (1948)) Let S be a non-zero shift invariant subspace of H2(D). Then S = θH2(D) for some inner function θ ∈ H∞(D). For an inner function θ, the model space Kθ is defined as Kθ = H2(D) ⊖ θH2(D). Kθ is finite dimensional if θ is finite Blaschke product (that is, θ(z) = n

k=1 z−zk 1−¯ zkz ).

Let Sθ = PKθMz|Kθ, where PKθ denotes the orthogonal projection from H2(D) onto Kθ. Sθ is called a Jordan block.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 7 / 18

Model Space in H2(D)

(Beurling Theorem (1948)) Let S be a non-zero shift invariant subspace of H2(D). Then S = θH2(D) for some inner function θ ∈ H∞(D). For an inner function θ, the model space Kθ is defined as Kθ = H2(D) ⊖ θH2(D). Kθ is finite dimensional if θ is finite Blaschke product (that is, θ(z) = n

k=1 z−zk 1−¯ zkz ).

Let Sθ = PKθMz|Kθ, where PKθ denotes the orthogonal projection from H2(D) onto Kθ. Sθ is called a Jordan block.

Theorem (Chalendar and Ross (2016))

Let T ∈ B(Kθ). Then (i) S∗

θ TSθ = T if and only if T = 0

(ii){S∗m

θ TSm θ }m≥1 converges in norm if and only if T is compact.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 7 / 18

Model Operator and Model Space

Let E be a Hilbert space and Θ ∈ H∞

B(E)(D) be an inner multiplier. Then the

model operator SΘ (see Garcia et al. (2016)) corresponding to Θ is the compression of Mz on the model space KΘ := H2

E(D) ⊖ ΘH2 E(D), that is,

SΘ = PKΘMz|KΘ, where PKΘ denotes the orthogonal projection from H2

E(D) onto KΘ.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 8 / 18

Model Operator and Model Space

Let E be a Hilbert space and Θ ∈ H∞

B(E)(D) be an inner multiplier. Then the

model operator SΘ (see Garcia et al. (2016)) corresponding to Θ is the compression of Mz on the model space KΘ := H2

E(D) ⊖ ΘH2 E(D), that is,

SΘ = PKΘMz|KΘ, where PKΘ denotes the orthogonal projection from H2

E(D) onto KΘ.

Note that K⊥

Θ = ΘH2 E(D) is an Mz-invariant subspace of H2 E(D) and

S∗

Θ = M∗ z |KΘ ∈ B(KΘ).

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 8 / 18

Model Operator and Model Space

Let E be a Hilbert space and Θ ∈ H∞

B(E)(D) be an inner multiplier. Then the

model operator SΘ (see Garcia et al. (2016)) corresponding to Θ is the compression of Mz on the model space KΘ := H2

E(D) ⊖ ΘH2 E(D), that is,

SΘ = PKΘMz|KΘ, where PKΘ denotes the orthogonal projection from H2

E(D) onto KΘ.

Note that K⊥

Θ = ΘH2 E(D) is an Mz-invariant subspace of H2 E(D) and

S∗

Θ = M∗ z |KΘ ∈ B(KΘ).

Questions

Characterize those T ∈ B(KΘ) for which S∗

ΘTSΘ = T.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 8 / 18

Model Operator and Model Space

Let E be a Hilbert space and Θ ∈ H∞

B(E)(D) be an inner multiplier. Then the

model operator SΘ (see Garcia et al. (2016)) corresponding to Θ is the compression of Mz on the model space KΘ := H2

E(D) ⊖ ΘH2 E(D), that is,

SΘ = PKΘMz|KΘ, where PKΘ denotes the orthogonal projection from H2

E(D) onto KΘ.

Note that K⊥

Θ = ΘH2 E(D) is an Mz-invariant subspace of H2 E(D) and

S∗

Θ = M∗ z |KΘ ∈ B(KΘ).

Questions

Characterize those T ∈ B(KΘ) for which S∗

ΘTSΘ = T.

Characterize those T ∈ B(KΘ) for which S∗m

Θ TSm Θ → A,

in norm, for some A ∈ B(KΘ).

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 8 / 18

Results on Vector-valued Hardy space

Lemma 1(B¨

• ttcher and Silbermann)

Let A be a compact operator on a Hilbert space H and R∗m → 0 in strong

• perator topology as m → ∞, then R∗mA → 0 in norm as m → ∞.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 9 / 18

Results on Vector-valued Hardy space

Lemma 1(B¨

• ttcher and Silbermann)

Let A be a compact operator on a Hilbert space H and R∗m → 0 in strong

• perator topology as m → ∞, then R∗mA → 0 in norm as m → ∞.

Theorem 2

Let E be a Hilbert space and T ∈ B(H2

E(D)). Then T is a Toeplitz operator if

and only if M∗

z TMz = T.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 9 / 18

Results on Vector-valued Hardy space

Lemma 1(B¨

• ttcher and Silbermann)

Let A be a compact operator on a Hilbert space H and R∗m → 0 in strong

• perator topology as m → ∞, then R∗mA → 0 in norm as m → ∞.

Theorem 2

Let E be a Hilbert space and T ∈ B(H2

E(D)). Then T is a Toeplitz operator if

and only if M∗

z TMz = T.

Theorem 3

Let T, A ∈ B(H2

Cp(D)) and M∗m z

TMm

z → A in norm. Then A is a Toeplitz

• perator and (T − A) is compact. Conversely, if A is a Toeplitz operator and

T − A is a compact operator, then T is asymptotically Toeplitz.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 9 / 18

Results on Vector-valued Hardy space

Proposition 4

Let Θ ∈ H∞

B(E)(D) be an inner multiplier and T ∈ B(KΘ). Assume that Θ(eiθ) is

invertible a.e. Then S∗

ΘTSΘ = T if and only if T = 0.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 10 / 18

Results on Vector-valued Hardy space

Proposition 4

Let Θ ∈ H∞

B(E)(D) be an inner multiplier and T ∈ B(KΘ). Assume that Θ(eiθ) is

invertible a.e. Then S∗

ΘTSΘ = T if and only if T = 0.

Theorem 5

Let Θ ∈ H∞

B(Cp)(D) be an inner multiplier and T ∈ B(KΘ). Assume that Θ(eiθ) is

invertible a.e. Then T is compact if and only if {S∗m

Θ TSm Θ }m≥1 converges in norm.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 10 / 18

Results on Vector-valued Hardy space

Proposition 4

Let Θ ∈ H∞

B(E)(D) be an inner multiplier and T ∈ B(KΘ). Assume that Θ(eiθ) is

invertible a.e. Then S∗

ΘTSΘ = T if and only if T = 0.

Theorem 5

Let Θ ∈ H∞

B(Cp)(D) be an inner multiplier and T ∈ B(KΘ). Assume that Θ(eiθ) is

invertible a.e. Then T is compact if and only if {S∗m

Θ TSm Θ }m≥1 converges in norm.

Theorem 6

Let Θ ∈ H∞

B(Cp)(D) be an inner multiplier and Θ(eiθ) is invertible a.e. and

T ∈ B(KΘ). Then TFAE: (i) {S∗m

Θ TSm Θ }m≥1 converges in norm;

(ii) S∗m

Θ TSm Θ → 0 in norm;

(iii) T is a compact operator.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 10 / 18

Results on H2(Dn)

Theorem 7

Let T ∈ B

• H2(Dn)
• . Then T is a Toeplitz operator if and only if M∗

zjTMzj = T

for all j = 1, . . . n.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 11 / 18

Results on H2(Dn)

Theorem 7

Let T ∈ B

• H2(Dn)
• . Then T is a Toeplitz operator if and only if M∗

zjTMzj = T

for all j = 1, . . . n.

Proof.

Let ϕ ∈ L∞(Tn) and T = PH2(Dn)Mϕ|H2(Dn). Then for f , g ∈ H2(Dn) and j = 1, . . . n, we have (M∗

zjTMzj)f , gH2(Dn) = ϕeiθjf , eiθjgL2(Tn) = ϕf , gL2(Tn),

that is, (M∗

zjTMzj)f , gH2(Dn) = PH2(Dn)Mϕ|H2(Dn)f , gH2(Dn),

and therefore M∗

zjTMzj = T for all j = 1, . . . n.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 11 / 18

Proof Cont.

Conversely, for each k ∈ N, define kd ∈ Nn by kd = (k, . . . , k). From M∗

zjTMzj = T, j = 1, . . . n, we obtain that

M∗kd

z

TMkd

z

= T (k ∈ N). Setting Ak = M∗kd

eiθ TPH2(Dn)Mkd eiθ

(k ≥ 1), we can prove that lim

k→∞Akf , g = A∞f , g

(f , g ∈ L2(Tn)) and A∞Meiθj = Meiθj A∞ for j = 1, . . . n. Hence there exists ϕ in L∞(Tn) such that A∞ = Mϕ. Using the above condition, we also have T = PH2(Dn)A∞|H2(Dn) = PH2(Dn)Mϕ|H2(Dn), that is, T is a Toeplitz operator.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 12 / 18

Results on H2(Dn)

Theorem 8

A bounded linear operator T on H2(Dn) is compact if and only if M∗m

zi TMm zj → 0

in norm for all i, j ∈ {1, ...., n}.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 13 / 18

Results on H2(Dn)

Theorem 8

A bounded linear operator T on H2(Dn) is compact if and only if M∗m

zi TMm zj → 0

in norm for all i, j ∈ {1, ...., n}. Following Feintuch (1989) (and Barr´ ıa and Halmos (1982)) one can now define asymptotic Toeplitz operator as follows:

Definition 9

A bounded linear operator T on H2(Dn) is said to be asymptotic Toeplitz

• perator if there exists A ∈ B(H2(Dn)) such that M∗m

zi TMm zi → A and

M∗m

zi (T − A)Mm zj → 0 as m → ∞ in norm, 1 ≤ i, j ≤ n.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 13 / 18

Results on H2(Dn)

Theorem 8

A bounded linear operator T on H2(Dn) is compact if and only if M∗m

zi TMm zj → 0

in norm for all i, j ∈ {1, ...., n}. Following Feintuch (1989) (and Barr´ ıa and Halmos (1982)) one can now define asymptotic Toeplitz operator as follows:

Definition 9

A bounded linear operator T on H2(Dn) is said to be asymptotic Toeplitz

• perator if there exists A ∈ B(H2(Dn)) such that M∗m

zi TMm zi → A and

M∗m

zi (T − A)Mm zj → 0 as m → ∞ in norm, 1 ≤ i, j ≤ n.

Theorem 10

Let T be a bounded linear operator on H2(Dn). Then T is an asymptotic Toeplitz

• perator if and only if T is a compact perturbation of Toeplitz operator.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 13 / 18

Quotient spaces of H2(Dn)

Let Q be a joint (M∗

z1, . . . , M∗ zn)-invariant subspace of H2(Dn) and

Czi = PQMzi|Q, i = 1, . . . , n.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 14 / 18

Quotient spaces of H2(Dn)

Let Q be a joint (M∗

z1, . . . , M∗ zn)-invariant subspace of H2(Dn) and

Czi = PQMzi|Q, i = 1, . . . , n.

Theorem 11

Let T, A ∈ B(Q), C ∗m

zi TC m zi → A and C ∗m zi (T − A)C m zj → 0 in norm for all

i, j = 1, . . . , n. Then T = A + K, where K ∈ B(Q) is a compact operator and C ∗

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

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 14 / 18

Quotient spaces of H2(Dn)

Proposition 12

Let Θ ∈ H∞(Dn) be an inner function and Q = H2(Dn)/ΘH2(Dn) and A ∈ B(Q). Then C ∗

ziACzi = A for all i = 1, . . . n, if and only if A = 0.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 15 / 18

Quotient spaces of H2(Dn)

Proposition 12

Let Θ ∈ H∞(Dn) be an inner function and Q = H2(Dn)/ΘH2(Dn) and A ∈ B(Q). Then C ∗

ziACzi = A for all i = 1, . . . n, if and only if A = 0.

Summing up the above two results, we have the following generalization of Chalendar and Ross.

Theorem 13

Let Θ ∈ H∞(Dn) be an inner function, and T and A be bounded linear operators

• n Q = H2(Dn)/ΘH2(Dn). Then TFAE:

(i) C ∗m

zi TC m zi → A and C ∗m zi (T − A)C m zj → 0 in norm for all i, j = 1, . . . , n;

(ii) C ∗m

zi TC m zi → 0 in norm for all i = 1, . . . , n;

(iii) T is compact.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 15 / 18

References

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Hardy-Sobolev spaces of the polydisk. J. Operator Theory 61 (2009), 301-312.

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Graduate Texts in Mathematics, 179. Springer-Verlag, New York, 1998.

Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 16 / 18

References

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Amit Maji (SMU, ISIBC) Toeplitz and Asymptotic Toeplitz OTOA, December 13-22, 2016 17 / 18

Thank You