Inner Functions of Numerical Contractions Hwa-Long Gau Department - - PowerPoint PPT Presentation

Inner Functions of Numerical Contractions

Hwa-Long Gau

Department of Mathematics, National Central University, Chung-Li 320, Taiwan (jointly with Pei Yuan Wu)

August 10, 2010

Hwa-Long Gau Inner Functions of Numerical Contractions 1/29

Numerical Ranges

H : a complex Hilbert space B(H) = { all bounded linear operators on H}, A ∈ B(H) Definition (numerical range of A) W (A) = {Ax, x : x ∈ H, x = 1} Basic properties: (1) W (A) ⊂ C is always bounded and convex. (|Ax, x| ≤ Axx ≤ A) (2) If dim H < ∞, then W (A) is closed. (3) W (U∗AU) = W (A), where U is unitary.

• pf. U∗AUx, x = A(Ux), (Ux) and U({x : x = 1}) = {x : x = 1}

(4) A = λI, λ ∈ C ⇔ W (A) = {λ}

• pf. (A − λI)x, x = 0, ∀x ⇔ A = λI

(5) σ(A) ⊂ W (A).

• pf. If (A − λI)xn → 0, xn = 1 ⇒ Axn, xn → λ

Hwa-Long Gau Inner Functions of Numerical Contractions 2/29

Numerical Ranges

H : a complex Hilbert space B(H) = { all bounded linear operators on H}, A ∈ B(H) Definition (numerical range of A) W (A) = {Ax, x : x ∈ H, x = 1} Basic properties: (1) W (A) ⊂ C is always bounded and convex. (|Ax, x| ≤ Axx ≤ A) (2) If dim H < ∞, then W (A) is closed. (3) W (U∗AU) = W (A), where U is unitary.

• pf. U∗AUx, x = A(Ux), (Ux) and U({x : x = 1}) = {x : x = 1}

(4) A = λI, λ ∈ C ⇔ W (A) = {λ}

• pf. (A − λI)x, x = 0, ∀x ⇔ A = λI

(5) σ(A) ⊂ W (A).

• pf. If (A − λI)xn → 0, xn = 1 ⇒ Axn, xn → λ

Hwa-Long Gau Inner Functions of Numerical Contractions 2/29

Numerical Ranges

H : a complex Hilbert space B(H) = { all bounded linear operators on H}, A ∈ B(H) Definition (numerical range of A) W (A) = {Ax, x : x ∈ H, x = 1} Basic properties: (1) W (A) ⊂ C is always bounded and convex. (|Ax, x| ≤ Axx ≤ A) (2) If dim H < ∞, then W (A) is closed. (3) W (U∗AU) = W (A), where U is unitary.

• pf. U∗AUx, x = A(Ux), (Ux) and U({x : x = 1}) = {x : x = 1}

(4) A = λI, λ ∈ C ⇔ W (A) = {λ}

• pf. (A − λI)x, x = 0, ∀x ⇔ A = λI

(5) σ(A) ⊂ W (A).

• pf. If (A − λI)xn → 0, xn = 1 ⇒ Axn, xn → λ

Hwa-Long Gau Inner Functions of Numerical Contractions 2/29

Numerical Ranges

H : a complex Hilbert space B(H) = { all bounded linear operators on H}, A ∈ B(H) Definition (numerical range of A) W (A) = {Ax, x : x ∈ H, x = 1} Basic properties: (1) W (A) ⊂ C is always bounded and convex. (|Ax, x| ≤ Axx ≤ A) (2) If dim H < ∞, then W (A) is closed. (3) W (U∗AU) = W (A), where U is unitary.

• pf. U∗AUx, x = A(Ux), (Ux) and U({x : x = 1}) = {x : x = 1}

(4) A = λI, λ ∈ C ⇔ W (A) = {λ}

• pf. (A − λI)x, x = 0, ∀x ⇔ A = λI

(5) σ(A) ⊂ W (A).

• pf. If (A − λI)xn → 0, xn = 1 ⇒ Axn, xn → λ

Hwa-Long Gau Inner Functions of Numerical Contractions 2/29

Numerical Ranges

H : a complex Hilbert space B(H) = { all bounded linear operators on H}, A ∈ B(H) Definition (numerical range of A) W (A) = {Ax, x : x ∈ H, x = 1} Basic properties: (1) W (A) ⊂ C is always bounded and convex. (|Ax, x| ≤ Axx ≤ A) (2) If dim H < ∞, then W (A) is closed. (3) W (U∗AU) = W (A), where U is unitary.

• pf. U∗AUx, x = A(Ux), (Ux) and U({x : x = 1}) = {x : x = 1}

(4) A = λI, λ ∈ C ⇔ W (A) = {λ}

• pf. (A − λI)x, x = 0, ∀x ⇔ A = λI

(5) σ(A) ⊂ W (A).

• pf. If (A − λI)xn → 0, xn = 1 ⇒ Axn, xn → λ

Hwa-Long Gau Inner Functions of Numerical Contractions 2/29

Numerical Ranges

H : a complex Hilbert space B(H) = { all bounded linear operators on H}, A ∈ B(H) Definition (numerical range of A) W (A) = {Ax, x : x ∈ H, x = 1} Basic properties: (1) W (A) ⊂ C is always bounded and convex. (|Ax, x| ≤ Axx ≤ A) (2) If dim H < ∞, then W (A) is closed. (3) W (U∗AU) = W (A), where U is unitary.

• pf. U∗AUx, x = A(Ux), (Ux) and U({x : x = 1}) = {x : x = 1}

(4) A = λI, λ ∈ C ⇔ W (A) = {λ}

• pf. (A − λI)x, x = 0, ∀x ⇔ A = λI

(5) σ(A) ⊂ W (A).

• pf. If (A − λI)xn → 0, xn = 1 ⇒ Axn, xn → λ

Hwa-Long Gau Inner Functions of Numerical Contractions 2/29

Numerical Ranges

H : a complex Hilbert space B(H) = { all bounded linear operators on H}, A ∈ B(H) Definition (numerical range of A) W (A) = {Ax, x : x ∈ H, x = 1} Basic properties: (1) W (A) ⊂ C is always bounded and convex. (|Ax, x| ≤ Axx ≤ A) (2) If dim H < ∞, then W (A) is closed. (3) W (U∗AU) = W (A), where U is unitary.

• pf. U∗AUx, x = A(Ux), (Ux) and U({x : x = 1}) = {x : x = 1}

(4) A = λI, λ ∈ C ⇔ W (A) = {λ}

• pf. (A − λI)x, x = 0, ∀x ⇔ A = λI

(5) σ(A) ⊂ W (A).

• pf. If (A − λI)xn → 0, xn = 1 ⇒ Axn, xn → λ

Hwa-Long Gau Inner Functions of Numerical Contractions 2/29

Numerical Ranges

H : a complex Hilbert space B(H) = { all bounded linear operators on H}, A ∈ B(H) Definition (numerical range of A) W (A) = {Ax, x : x ∈ H, x = 1} Basic properties: (1) W (A) ⊂ C is always bounded and convex. (|Ax, x| ≤ Axx ≤ A) (2) If dim H < ∞, then W (A) is closed. (3) W (U∗AU) = W (A), where U is unitary.

• pf. U∗AUx, x = A(Ux), (Ux) and U({x : x = 1}) = {x : x = 1}

(4) A = λI, λ ∈ C ⇔ W (A) = {λ}

• pf. (A − λI)x, x = 0, ∀x ⇔ A = λI

(5) σ(A) ⊂ W (A).

• pf. If (A − λI)xn → 0, xn = 1 ⇒ Axn, xn → λ

Hwa-Long Gau Inner Functions of Numerical Contractions 2/29

Examples

(6) If A is normal, then W (A) = convex hull(σ(A)). Examples: (1) N =    a1 ... an    ⇒ W (N) = convex hull (σ(N))

a1 a2 a3 an

W(N) pf.    a1 ... an       x1 . . . xn    ,    x1 . . . xn    = a1|x1|2 + · · · + an|xn|2 ∈RHS (2) J2 = 1

• ⇒ W (J2) = {z ∈ C : |z| ≤ 1

2}

pf. 1 x1 x2

• ,

x1 x2

• = x2x1 = |x1x2|eiθ =
• t(1 − t)eiθ

⇒ W (J2) = {

• t(1 − t)eiθ : 0 ≤ t ≤ 1, θ ∈ R} = {z ∈ C : |z| ≤ 1

2} Hwa-Long Gau Inner Functions of Numerical Contractions 3/29

Examples

(6) If A is normal, then W (A) = convex hull(σ(A)). Examples: (1) N =    a1 ... an    ⇒ W (N) = convex hull (σ(N))

a1 a2 a3 an

W(N) pf.    a1 ... an       x1 . . . xn    ,    x1 . . . xn    = a1|x1|2 + · · · + an|xn|2 ∈RHS (2) J2 = 1

• ⇒ W (J2) = {z ∈ C : |z| ≤ 1

2}

pf. 1 x1 x2

• ,

x1 x2

• = x2x1 = |x1x2|eiθ =
• t(1 − t)eiθ

⇒ W (J2) = {

• t(1 − t)eiθ : 0 ≤ t ≤ 1, θ ∈ R} = {z ∈ C : |z| ≤ 1

2} Hwa-Long Gau Inner Functions of Numerical Contractions 3/29

Examples

(6) If A is normal, then W (A) = convex hull(σ(A)). Examples: (1) N =    a1 ... an    ⇒ W (N) = convex hull (σ(N))

a1 a2 a3 an

W(N) pf.    a1 ... an       x1 . . . xn    ,    x1 . . . xn    = a1|x1|2 + · · · + an|xn|2 ∈RHS (2) J2 = 1

• ⇒ W (J2) = {z ∈ C : |z| ≤ 1

2}

pf. 1 x1 x2

• ,

x1 x2

• = x2x1 = |x1x2|eiθ =
• t(1 − t)eiθ

⇒ W (J2) = {

• t(1 − t)eiθ : 0 ≤ t ≤ 1, θ ∈ R} = {z ∈ C : |z| ≤ 1

2} Hwa-Long Gau Inner Functions of Numerical Contractions 3/29

Examples

(6) If A is normal, then W (A) = convex hull(σ(A)). Examples: (1) N =    a1 ... an    ⇒ W (N) = convex hull (σ(N))

a1 a2 a3 an

W(N) pf.    a1 ... an       x1 . . . xn    ,    x1 . . . xn    = a1|x1|2 + · · · + an|xn|2 ∈RHS (2) J2 = 1

• ⇒ W (J2) = {z ∈ C : |z| ≤ 1

2}

pf. 1 x1 x2

• ,

x1 x2

• = x2x1 = |x1x2|eiθ =
• t(1 − t)eiθ

⇒ W (J2) = {

• t(1 − t)eiθ : 0 ≤ t ≤ 1, θ ∈ R} = {z ∈ C : |z| ≤ 1

2} Hwa-Long Gau Inner Functions of Numerical Contractions 3/29

Examples

(6) If A is normal, then W (A) = convex hull(σ(A)). Examples: (1) N =    a1 ... an    ⇒ W (N) = convex hull (σ(N))

a1 a2 a3 an

W(N) pf.    a1 ... an       x1 . . . xn    ,    x1 . . . xn    = a1|x1|2 + · · · + an|xn|2 ∈RHS (2) J2 = 1

• ⇒ W (J2) = {z ∈ C : |z| ≤ 1

2}

pf. 1 x1 x2

• ,

x1 x2

• = x2x1 = |x1x2|eiθ =
• t(1 − t)eiθ

⇒ W (J2) = {

• t(1 − t)eiθ : 0 ≤ t ≤ 1, θ ∈ R} = {z ∈ C : |z| ≤ 1

2} Hwa-Long Gau Inner Functions of Numerical Contractions 3/29

Examples

(6) If A is normal, then W (A) = convex hull(σ(A)). Examples: (1) N =    a1 ... an    ⇒ W (N) = convex hull (σ(N))

a1 a2 a3 an

W(N) pf.    a1 ... an       x1 . . . xn    ,    x1 . . . xn    = a1|x1|2 + · · · + an|xn|2 ∈RHS (2) J2 = 1

• ⇒ W (J2) = {z ∈ C : |z| ≤ 1

2}

pf. 1 x1 x2

• ,

x1 x2

• = x2x1 = |x1x2|eiθ =
• t(1 − t)eiθ

⇒ W (J2) = {

• t(1 − t)eiθ : 0 ≤ t ≤ 1, θ ∈ R} = {z ∈ C : |z| ≤ 1

2} Hwa-Long Gau Inner Functions of Numerical Contractions 3/29

Examples

(6) If A is normal, then W (A) = convex hull(σ(A)). Examples: (1) N =    a1 ... an    ⇒ W (N) = convex hull (σ(N))

a1 a2 a3 an

W(N) pf.    a1 ... an       x1 . . . xn    ,    x1 . . . xn    = a1|x1|2 + · · · + an|xn|2 ∈RHS (2) J2 = 1

• ⇒ W (J2) = {z ∈ C : |z| ≤ 1

2}

pf. 1 x1 x2

• ,

x1 x2

• = x2x1 = |x1x2|eiθ =
• t(1 − t)eiθ

⇒ W (J2) = {

• t(1 − t)eiθ : 0 ≤ t ≤ 1, θ ∈ R} = {z ∈ C : |z| ≤ 1

2} Hwa-Long Gau Inner Functions of Numerical Contractions 3/29

Numerical Ranges of 2 × 2 matrices and Jn

(3) A = a b c

• ⇒

W(A) = |b| a c

(4) Jn =       1 ... ... 1       ⇒ W (Jn) = {z ∈ C : |z| ≤ cos

π n+1}

W(Jn) cos

π n+1 Hwa-Long Gau Inner Functions of Numerical Contractions 4/29

Numerical Ranges of 2 × 2 matrices and Jn

(3) A = a b c

• ⇒

W(A) = |b| a c

(4) Jn =       1 ... ... 1       ⇒ W (Jn) = {z ∈ C : |z| ≤ cos

π n+1}

W(Jn) cos

π n+1 Hwa-Long Gau Inner Functions of Numerical Contractions 4/29

Compressions of the shift

H2 : Hardy space S : unilateral shift on H2 (i.e. (Sf )(z) = zf (z), f ∈ H2) φ ∈ H∞, an inner function (i.e. |φ| = 1 a.e. on ∂D) H(φ) = H2 ⊖ φH2 S(φ) is the compression of S to H(φ). i.e. S = ∗ ∗ S(φ)

• n H2 = φH2 ⊕ H(φ).
• r

(S(φ)g)(z) = PH(φ)(zg(z)), ∀ g ∈ H(φ), where PH(φ) is the orthogonal projection of H2 onto H(φ). Note: rank (I − S(φ)∗S(φ)) = 1, dim ker S(φ) ≤ 1 and φ(S(φ)) = 0.

Hwa-Long Gau Inner Functions of Numerical Contractions 5/29

Examples of S(φ)

Examples. (1) If φ(z) = zn ⇒ H(φ) = H2 ⊖ znH2 = {1, z, . . . , zn−1} ∀ g ∈ H(φ) ⇒ g(z) = n−1

j=0 ajzj

(S(φ)g)(z) = PH(φ)(zg(z)) = n−1

j=1 aj−1zj ⇒ S(φ) = Jn (Jordan block)

(2) If φ(z) = n

j=1 z−aj 1−aj z , |aj| < 1 ⇒ dim H(φ) = n.

S(φ) ∼ =          

a1 ... ai · · · aij ... . . . aj ... an

         

n×n

, aij = (1 − |ai|2)(1 − |aj|2), if j = i + 1

• (1 − |ai|2)(1 − |aj|2)(−ai+1) · · · (−aj−1),

if j > i + 1

Hwa-Long Gau Inner Functions of Numerical Contractions 6/29

Examples of S(φ)

Examples. (1) If φ(z) = zn ⇒ H(φ) = H2 ⊖ znH2 = {1, z, . . . , zn−1} ∀ g ∈ H(φ) ⇒ g(z) = n−1

j=0 ajzj

(S(φ)g)(z) = PH(φ)(zg(z)) = n−1

j=1 aj−1zj ⇒ S(φ) = Jn (Jordan block)

(2) If φ(z) = n

j=1 z−aj 1−aj z , |aj| < 1 ⇒ dim H(φ) = n.

S(φ) ∼ =          

a1 ... ai · · · aij ... . . . aj ... an

         

n×n

, aij = (1 − |ai|2)(1 − |aj|2), if j = i + 1

• (1 − |ai|2)(1 − |aj|2)(−ai+1) · · · (−aj−1),

if j > i + 1

Hwa-Long Gau Inner Functions of Numerical Contractions 6/29

Numerical ranges of S(φ)

Theorem (Gau & Wu, 1997) A ∼ = S(φ) with dim H(φ) = n, λ ∈ ∂D Then ∃ a unique (n + 1)-gon P such that (1) P is inscribed in ∂D; (2) P is circumscribed about W (A); (3) P has λ as a vertex; (4) P = W (U) where U is an (n + 1) × (n + 1) unitary dilation of A.

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

W(A) W(Uθ) D

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

W(A) W(Uθ) D

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

W(A) W(Uθ) D

Hwa-Long Gau Inner Functions of Numerical Contractions 7/29

Unitary dilations of S(φ)

Example: Let φ(z) = 3

j=1 z−aj 1−aj z , where aj = ei2jπ/3/2, j = 1, 2, 3.

S(φ) ∼ =   1 1 1/8  , Uθ =     1 1 1/8 √ 63/8 eiθ√ 63/8 −eiθ/8     ⇒ ∂W (Uθ) is inscribed in unit circle and circumscribed about W (S(φ))

−1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1

W(A) W(Uθ) D

Hwa-Long Gau Inner Functions of Numerical Contractions 8/29

Lucas’ theorem and Siebeck theorem

Lucas’ Theorem p(z) = (z − a1)(z − a2) · · · (z − an), p′(z) = n(z − b1)(z − b2) · · · (z − bn−1) ⇒ bj ∈ convex hull ({a1, a2, . . . , an}) for all j = 1, . . . n − 1 Siebeck Theorem p(z) = (z − a1)(z − a2)(z − a3), p′(z) = 3(z − b1)(z − b2) ⇒ ∃ an ellipse E with foci b1, b2 s.t. E is inscribed in △a1a2a3, and, E ∩ ajaj+1 = (aj + aj+1)/2, for j = 1, 2, 3 (a4 = a1).

a1 a2 a3 b1 b2 a1 a2 a3 an b1 b2 bn-1 a4 b3

Question What is the analogue for n-gons?

Hwa-Long Gau Inner Functions of Numerical Contractions 9/29

Theorem

Theorem (Gau & Wu, 1998) p(z) = (z − a1)(z − a2) · · · (z − an+1), where aj ∈ ∂D, ∀j, and aj’s are distinct p′(z) = (n + 1)(z − b1)(z − b2) · · · (z − bn) Let U = diag (a1, . . . , an+1) and φ(z) = n

j=1 z−bj 1−bj z . Then 1

σ(S(φ)) = {b1, b2, . . . bn};

2

U is a unitary dilation of S(φ);

3

W (S(φ)) ∩ ajaj+1 = {(aj + aj+1)/2}, j = 1, . . . , n + 1.

−1 −0.5 0.5 1 −1 −0.5 0.5 1

∂D a5 a4 a3 a2 a1 b1 b4 b2 b3

−1 −0.5 0.5 1 −1 −0.5 0.5 1

∂D a5 a4 a3 a2 a1 b1 b4 b2 b3 W(S(φ))

Hwa-Long Gau Inner Functions of Numerical Contractions 10/29

Projective geometry theorem

Recall: If two triangles △ABC and △A′B′C ′ in the plane have their six (distinct) vertices on a conic alternatively, then there is another conic which is tangent to their six edges.

A A' B' C' C B C' A A' B' C' C B

Question What is the analogue for n-gons?

Hwa-Long Gau Inner Functions of Numerical Contractions 11/29

Theorem

Theorem (Gau & Wu, 2004) If a1, a′

1, a2, a′ 2, . . . , an+1, a′ n+1 (in this order) are 2n + 2 distinct points on ∂D,

then ∃ unique S(φ) with dim H(φ) = n s.t. W (S(φ)) is circumscribed by the two (n + 1)-gons a1a2 . . . an+1 and a′

1a′ 2 . . . a′ n+1.

−1.5 −1 −0.5 0.5 1 1.5 −1.5 −1 −0.5 0.5 1 1.5

∂D a′

1

a′

2

a′

3

a4 a′

4

a5 a′

5

a1 a2 a3

−1.5 −1 −0.5 0.5 1 1.5 −1.5 −1 −0.5 0.5 1 1.5

a′

1

a′

2

a′

3

a′

4

a′

5

a1 a2 a3 a4 a5 W(S(φ)) ∂D

Hwa-Long Gau Inner Functions of Numerical Contractions 12/29

Numerical radius

A ∈ B(H) Definition w(A) = sup{|z| : z ∈ W (A)} (numerical radius of A) r(A) = sup{|z| : z ∈ σ(A)} (spectral radius of A) Basic properties: (1) r(A) ≤ w(A) ≤ A. (2) If A is normal, then w(A) = A. (3) A/2 ≤ w(A) ≤ A.

• pf. A ≤ Re A + Im A = w(Re A) + w(Im A) ≤ w(A) + w(A)

Examples: W (

• 2
• ) = D ⇒ w(
• 2
• ) = 1 and
• 2
• = 2

Recall: A is a numerical contraction if w(A) ≤ 1.

Hwa-Long Gau Inner Functions of Numerical Contractions 13/29

Numerical radius

A ∈ B(H) Definition w(A) = sup{|z| : z ∈ W (A)} (numerical radius of A) r(A) = sup{|z| : z ∈ σ(A)} (spectral radius of A) Basic properties: (1) r(A) ≤ w(A) ≤ A. (2) If A is normal, then w(A) = A. (3) A/2 ≤ w(A) ≤ A.

• pf. A ≤ Re A + Im A = w(Re A) + w(Im A) ≤ w(A) + w(A)

Examples: W (

• 2
• ) = D ⇒ w(
• 2
• ) = 1 and
• 2
• = 2

Recall: A is a numerical contraction if w(A) ≤ 1.

Hwa-Long Gau Inner Functions of Numerical Contractions 13/29

Numerical radius

A ∈ B(H) Definition w(A) = sup{|z| : z ∈ W (A)} (numerical radius of A) r(A) = sup{|z| : z ∈ σ(A)} (spectral radius of A) Basic properties: (1) r(A) ≤ w(A) ≤ A. (2) If A is normal, then w(A) = A. (3) A/2 ≤ w(A) ≤ A.

• pf. A ≤ Re A + Im A = w(Re A) + w(Im A) ≤ w(A) + w(A)

Examples: W (

• 2
• ) = D ⇒ w(
• 2
• ) = 1 and
• 2
• = 2

Recall: A is a numerical contraction if w(A) ≤ 1.

Hwa-Long Gau Inner Functions of Numerical Contractions 13/29

Numerical radius

A ∈ B(H) Definition w(A) = sup{|z| : z ∈ W (A)} (numerical radius of A) r(A) = sup{|z| : z ∈ σ(A)} (spectral radius of A) Basic properties: (1) r(A) ≤ w(A) ≤ A. (2) If A is normal, then w(A) = A. (3) A/2 ≤ w(A) ≤ A.

• pf. A ≤ Re A + Im A = w(Re A) + w(Im A) ≤ w(A) + w(A)

Examples: W (

• 2
• ) = D ⇒ w(
• 2
• ) = 1 and
• 2
• = 2

Recall: A is a numerical contraction if w(A) ≤ 1.

Hwa-Long Gau Inner Functions of Numerical Contractions 13/29

Numerical radius

A ∈ B(H) Definition w(A) = sup{|z| : z ∈ W (A)} (numerical radius of A) r(A) = sup{|z| : z ∈ σ(A)} (spectral radius of A) Basic properties: (1) r(A) ≤ w(A) ≤ A. (2) If A is normal, then w(A) = A. (3) A/2 ≤ w(A) ≤ A.

• pf. A ≤ Re A + Im A = w(Re A) + w(Im A) ≤ w(A) + w(A)

Examples: W (

• 2
• ) = D ⇒ w(
• 2
• ) = 1 and
• 2
• = 2

Recall: A is a numerical contraction if w(A) ≤ 1.

Hwa-Long Gau Inner Functions of Numerical Contractions 13/29

Numerical radius

A ∈ B(H) Definition w(A) = sup{|z| : z ∈ W (A)} (numerical radius of A) r(A) = sup{|z| : z ∈ σ(A)} (spectral radius of A) Basic properties: (1) r(A) ≤ w(A) ≤ A. (2) If A is normal, then w(A) = A. (3) A/2 ≤ w(A) ≤ A.

• pf. A ≤ Re A + Im A = w(Re A) + w(Im A) ≤ w(A) + w(A)

Examples: W (

• 2
• ) = D ⇒ w(
• 2
• ) = 1 and
• 2
• = 2

Recall: A is a numerical contraction if w(A) ≤ 1.

Hwa-Long Gau Inner Functions of Numerical Contractions 13/29

Numerical radius

A ∈ B(H) Definition w(A) = sup{|z| : z ∈ W (A)} (numerical radius of A) r(A) = sup{|z| : z ∈ σ(A)} (spectral radius of A) Basic properties: (1) r(A) ≤ w(A) ≤ A. (2) If A is normal, then w(A) = A. (3) A/2 ≤ w(A) ≤ A.

• pf. A ≤ Re A + Im A = w(Re A) + w(Im A) ≤ w(A) + w(A)

Examples: W (

• 2
• ) = D ⇒ w(
• 2
• ) = 1 and
• 2
• = 2

Recall: A is a numerical contraction if w(A) ≤ 1.

Hwa-Long Gau Inner Functions of Numerical Contractions 13/29

Numerical Contractions

D ≡ {z ∈ C : |z| < 1} A(D) ≡ {f ∈ C(D) : f is analytic on D} von Neumann Inequality T ∈ B(H), T ≤ 1, f ∈ A(D). Then f (T) ≤ f ∞. Theorem (Ando, 1973) T ∈ B(H). (1) w(T) ≤ 1 ⇔ T = 2(I − B∗B)1/2B for some contraction B. (2) If w(T) ≤ 1 then ∃ invertible X with X · X −1 ≤ 2 such that X −1TX is a contraction. Note: w(T) ≤ 1 ⇒ T = XCX −1, where C is a contraction. f ∈ A(D) ⇒ f (T) = Xf (C)X −1 ≤ X · f (C) · X −1 ≤ 2f ∞

Hwa-Long Gau Inner Functions of Numerical Contractions 14/29

Drury’s Result

Theorem (Drury, 2008) T ∈ B(H), w(T) ≤ 1, f ∈ A(D), f ∞ ≤ 1. Then f (T) ≤ ν(|f (0)|) ≤ 2 where ν(α) =

• 2 − 3α2 + 2α4 + 2(1 − α2)

√ 1 − α2 + α4 for 0 ≤ α ≤ 1.

• ∞

0.0 0.2 0.4 0.6 0.8 1.0 1.0 1.2 1.4 1.6 1.8 2.0

• Fig. 2. The function α → ν(α).

Note: (1) max{ν(α) : 0 ≤ α ≤ 1} = ν(0) = 2. (2) If w(T) ≤ 1, f ∞ ≤ 1 and f (T) = 2 ⇒ f (0) = 0.

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Drury’s Conjecture

Conjecture (Drury, 2008) f ∈ A(D), f ∞ = 1. T.F.A.E. (1) ∃T ∈ B(H) such that w(T) = 1 and f (T) = 2. (2) f is a finite Blaschke product with f (0) = 0. Firstly, we consider the implication “(2) ⇒ (1)”. That is, f (z) = z

n−1

• j=1

z − λj 1 − λjz , |λj| < 1, ∀ j, n ≥ 1. We want to find an (n + 1)-by-(n + 1) matrix T such that w(T) = 1 and f (T) = 2.

Hwa-Long Gau Inner Functions of Numerical Contractions 16/29

Example 1

Example 1. f (z) = z T = 2

• ⇒ f (T) = T = 2

T = 2 1 1

• = 2(I2 − B∗B)1/2B ⇒ w(T) ≤ 1

Note: (1) In fact, W (T) = D. (2) T = 2

• =

√ 2

1 √ 2

• X

1

• J2
• 1

√ 2

√ 2

• X −1

.

Hwa-Long Gau Inner Functions of Numerical Contractions 17/29

Example 2

Example 2. f (z) = zn, n ≥ 2 T =       

√ 2 1 ... ... 1 √ 2

      

(n+1)×(n+1)

=     

√ 2 1 ... 1

1 √ 2

    

• X

     

1 ... ... 1 1

     

• Jn+1

    

1 √ 2

1 ... 1 √ 2

    

• X −1

. ⇒ f (T) = XJn

n+1X −1

=

• √

2 In−1

1 √ 2

 

1 ...

 

• 1

√ 2

In−1 √ 2

• =

 

2 ...

  ⇒ f (T) = 2

Hwa-Long Gau Inner Functions of Numerical Contractions 18/29

Example 2

T =     

√ 2 1 ... 1

1 √ 2

          

1 ... ... 1 1

          

1 √ 2

1 ... 1 √ 2

     = 2     

1

1 √ 2

...

1 √ 2

    

• (In+1 − B∗B)1/2

     

1 ... ... 1 1

          

1 √ 2

...

1 √ 2

1

    

• B

. ⇒ w(T) ≤ 1. Note: In fact, W (T) = D.

Hwa-Long Gau Inner Functions of Numerical Contractions 19/29

Some Observations

Mn ≡ { all n-by-n complex matrices} Now, f (z) = z

n−1

• j=1

z − λj 1 − λjz , |λj| < 1, ∀ j, n ≥ 1. Let X =  

√ 2 In−1

1 √ 2

  ∈ Mn+1 We need to find a contraction C ∈ Mn+1 such that f (C) =   

1 ...

  . Let T = XCX −1 ⇒ f (T) = Xf (C)X −1 =

• √

2 In−1

1 √ 2

 

1 ...

 

• 1

√ 2

In−1 √ 2

• =

 

2 ...

  ⇒ f (T) = 2

Hwa-Long Gau Inner Functions of Numerical Contractions 20/29

f (XS(φ)X −1) = 2

Now, if f ∈ H∞ is an inner function with f (0) = 0. Let φ(z) = zf (z) and g(z) = f (z)/z. Then S(φ) =

∗ ∗ S(g) ∗

• n H(φ) = ker S(φ) ⊕ H(g) ⊕ ker S(φ)∗

= S(f ) ∗

• n H(φ) = H(f ) ⊕ ker S(φ)∗

f (S(φ)) =

x1 x2

• n H(φ) = ker S(φ) ⊕ H(g) ⊕ ker S(φ)∗

Since 0 = φ(S(φ)) = S(φ)f (S(φ)) ⇒

• x1

x2

• ∈ ker S(φ) ⇒ x2 = 0.

f (S(φ)) = 1 ⇒ |x1| = 1 Let X =  

√ 2 IH(g)

1 √ 2

  on H(φ) = ker S(φ) ⊕ H(g) ⊕ ker S(φ)∗. ⇒ f (XS(φ)X −1) = Xf (S(φ))X −1 = 2.

Hwa-Long Gau Inner Functions of Numerical Contractions 21/29

w(XS(φ)X −1) = 1

∵ rank (I − S(φ)∗S(φ)) = 1 and dim ker S(φ) = 1 ⇒ S(φ)∗S(φ) =

IH(g) 1

• n H(φ) = ker S(φ) ⊕ H(g) ⊕ ker S(φ)∗

⇒ XS(φ)X −1 =  

√ 2 IH(g)

1 √ 2

 

∗ ∗ S(g) ∗

 

1 √ 2

IH(g) √ 2

  = 2  

1

1 √ 2 IH(g)

 

• (I − B∗B)1/2

∗ ∗ S(g) ∗

 

1 √ 2 IH(g)

1

 

• B

⇒ w(XS(φ)X −1) ≤ 1 In fact, W (XS(φ)X −1) = D.

Hwa-Long Gau Inner Functions of Numerical Contractions 22/29

Theorem 1

Theorem Let f ∈ H∞ be an inner function with f (0) = 0. Then W (XS(φ)X −1) = D and f (XS(φ)X −1) = 2, where φ(z) = zf (z) and X =  

√ 2 I

1 √ 2

  on H(φ) = ker S(φ) ⊕ (H(f ) ⊖ ker S(φ)) ⊕ ker S(φ)∗. Check: W (XS(φ)X −1) = D. Let A = XS(φ)X −1. Since w(A) ≤ 1 ⇒ W (A) ⊆ D. Conversely, if eiθ ∈ σ(S(φ)), ⇒ hθ ≡ (I − e−iθS(φ))−1x = ∞

n=0(e−iθS(φ))nx exists.

Let xθ = √ 2hθ + (1 − √ 2)x + (1 − √ 2)hθ, f (S(φ))xf (S(φ))x. Then Axθ, xθ = eiθxθ2 ⇒ eiθ ∈ W (A). If eiθ ∈ σ(S(φ)) ⇒ eiθ ∈ σ(S(φ)) = σ(A) ⊆ W (A). Hence ∂D ⊆ W (A) or W (A) = D.

Hwa-Long Gau Inner Functions of Numerical Contractions 23/29

Review

Conjecture (Drury, 2008) f ∈ A(D), f ∞ = 1. T.F.A.E. (1) ∃T ∈ B(H) such that w(T) = 1 and f (T) = 2. (2) f is a finite Blaschke product with f (0) = 0. We have completed the proof of “(2) ⇒ (1)”. i.e. T = XS(φ)X −1, where φ(z) = zf (z). Now, we consider the implication “(1) ⇒ (2)”.

Hwa-Long Gau Inner Functions of Numerical Contractions 24/29

Theorem 2

Theorem f ∈ H∞, f ∞ = 1, T ∈ B(H), w(T) ≤ 1 If f (T) is well-defined and f (T)x = 2 for some unit vector x ∈ H. Then f is an inner function with f (0) = 0. Proof. 2 = f (T) ≤ ν(|f (0)|) ≤ 2 = ν(0) ⇒ f (0) = 0. Let M = {T nx : n ≥ 0} and A = T|M. By Berger dilation theorem, we have Af (A)x = 0. ⇒ Af (A)Anx = 0, ∀ n ⇒ Af (A) = 0. Choose a positive X such that C ≡ X −1AX is a contraction. Then Cfi(C) = 0, where fi is the inner part of f . Let φ(z) = zfi(z). Then C extends to S(φ) ⊕ S(φ) ⊕ · · · . Since f (C)x = 1, by Sarason’s result, we have C = S(φ) and f = fi.

Hwa-Long Gau Inner Functions of Numerical Contractions 25/29

Crabb’s Result

Theorem (Crabb, 1971) T ∈ B(H). If w(T) ≤ 1 and T nx = 2 for some n ≥ 1 and some unit vector x ∈ H. Then T ∼ = A ⊕ T ′, where A = 2

• ∈ M2 or A =

      

√ 2 1 ... ... 1 √ 2

       ∈ Mn+1 depending on whether n = 1 or n ≥ 2.

Hwa-Long Gau Inner Functions of Numerical Contractions 26/29

Main Result

Theorem f ∈ H∞, f ∞ = 1, T ∈ B(H), w(T) ≤ 1 If f (T) is well-defined and f (T)x = 2 for some unit vector x ∈ H. Then (1) f is an inner function with f (0) = 0; (2) W (T) = D; (3) T ∼ = XS(φ)X −1 ⊕ T ′, where φ(z) = zf (z) and X =  

√ 2 I

1 √ 2

  on H(φ) = ker S(φ)⊕(H(f )⊖ker S(φ))⊕ker S(φ)∗.

Hwa-Long Gau Inner Functions of Numerical Contractions 27/29

Proof

Proof. Let M = {T nx : n ≥ 0} and A = T|M. Choose a positive X such that C ≡ X −1AX is a contraction and X ≤ √ 2. we have shown that C = S(φ) and f is inner with f (0) = 0. 2 = f (A)x = Xf (C)X −1x ≤ X · f (C) · X −1 · x ≤ 2 ⇒ X =

• √

2 X1

1 √ 2

• n

M = ∨{f (C)x} ⊕ (M ⊖ ∨{x, f (C)x} ⊕ ∨{x}. Claim 1. X1 = IM⊖∨{x,f (C)x} Claim 2. M is a reducing subspace for T. Indeed, let y = f (C)x and N = {T ∗ny : n ≥ 0}. ⇒ T ∗|N = XS( φ) X −1, where φ(z) = φ(z). ⇒ T ∗|N ∼ = (T|M)∗ ⇒ M = N

Hwa-Long Gau Inner Functions of Numerical Contractions 28/29

Thanks for your attention!

Hwa-Long Gau Inner Functions of Numerical Contractions 29/29