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 �| ≤ � Ax �� x � ≤ � 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 = λ I , λ ∈ C ⇔ W ( A ) = { λ } pf. � ( A − λ I ) x , x � = 0 , ∀ x ⇔ A = λ I (5) σ ( A ) ⊂ W ( A ). pf. If ( A − λ I ) x n → 0 , � x n � = 1 ⇒ � Ax n , x n � → 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 �| ≤ � Ax �� x � ≤ � 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 = λ I , λ ∈ C ⇔ W ( A ) = { λ } pf. � ( A − λ I ) x , x � = 0 , ∀ x ⇔ A = λ I (5) σ ( A ) ⊂ W ( A ). pf. If ( A − λ I ) x n → 0 , � x n � = 1 ⇒ � Ax n , x n � → 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 �| ≤ � Ax �� x � ≤ � 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 = λ I , λ ∈ C ⇔ W ( A ) = { λ } pf. � ( A − λ I ) x , x � = 0 , ∀ x ⇔ A = λ I (5) σ ( A ) ⊂ W ( A ). pf. If ( A − λ I ) x n → 0 , � x n � = 1 ⇒ � Ax n , x n � → 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 �| ≤ � Ax �� x � ≤ � 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 = λ I , λ ∈ C ⇔ W ( A ) = { λ } pf. � ( A − λ I ) x , x � = 0 , ∀ x ⇔ A = λ I (5) σ ( A ) ⊂ W ( A ). pf. If ( A − λ I ) x n → 0 , � x n � = 1 ⇒ � Ax n , x n � → 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 �| ≤ � Ax �� x � ≤ � 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 = λ I , λ ∈ C ⇔ W ( A ) = { λ } pf. � ( A − λ I ) x , x � = 0 , ∀ x ⇔ A = λ I (5) σ ( A ) ⊂ W ( A ). pf. If ( A − λ I ) x n → 0 , � x n � = 1 ⇒ � Ax n , x n � → 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 �| ≤ � Ax �� x � ≤ � 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 = λ I , λ ∈ C ⇔ W ( A ) = { λ } pf. � ( A − λ I ) x , x � = 0 , ∀ x ⇔ A = λ I (5) σ ( A ) ⊂ W ( A ). pf. If ( A − λ I ) x n → 0 , � x n � = 1 ⇒ � Ax n , x n � → 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 �| ≤ � Ax �� x � ≤ � 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 = λ I , λ ∈ C ⇔ W ( A ) = { λ } pf. � ( A − λ I ) x , x � = 0 , ∀ x ⇔ A = λ I (5) σ ( A ) ⊂ W ( A ). pf. If ( A − λ I ) x n → 0 , � x n � = 1 ⇒ � Ax n , x n � → 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 �| ≤ � Ax �� x � ≤ � 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 = λ I , λ ∈ C ⇔ W ( A ) = { λ } pf. � ( A − λ I ) x , x � = 0 , ∀ x ⇔ A = λ I (5) σ ( A ) ⊂ W ( A ). pf. If ( A − λ I ) x n → 0 , � x n � = 1 ⇒ � Ax n , x n � → 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: a 1   0 a 1 a 2   ... (1) N =  ⇒ W ( N ) = convex hull ( σ ( N ))  W(N) a n a 3 0 a n       a 1 0 x 1 x 1    .   .  ...  � = a 1 | x 1 | 2 + · · · + a n | x n | 2 ∈ RHS . . pf. �  ,     . . 0 a n x n x n � 0 � 1 ⇒ W ( J 2 ) = { z ∈ C : | z | ≤ 1 (2) J 2 = 2 } 0 0 � 0 � � x 1 � � x 1 � � 1 � = x 2 x 1 = | x 1 x 2 | e i θ = t (1 − t ) e i θ pf. � , 0 0 x 2 x 2 � t (1 − t ) e i θ : 0 ≤ t ≤ 1 , θ ∈ R } = { z ∈ C : | z | ≤ 1 ⇒ W ( J 2 ) = { 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: a 1   0 a 1 a 2   ... (1) N =  ⇒ W ( N ) = convex hull ( σ ( N ))  W(N) a n a 3 0 a n       a 1 0 x 1 x 1    .   .  ...  � = a 1 | x 1 | 2 + · · · + a n | x n | 2 ∈ RHS . . pf. �  ,     . . 0 a n x n x n � 0 � 1 ⇒ W ( J 2 ) = { z ∈ C : | z | ≤ 1 (2) J 2 = 2 } 0 0 � 0 � � x 1 � � x 1 � � 1 � = x 2 x 1 = | x 1 x 2 | e i θ = t (1 − t ) e i θ pf. � , 0 0 x 2 x 2 � t (1 − t ) e i θ : 0 ≤ t ≤ 1 , θ ∈ R } = { z ∈ C : | z | ≤ 1 ⇒ W ( J 2 ) = { 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: a 1   0 a 1 a 2   ... (1) N =  ⇒ W ( N ) = convex hull ( σ ( N ))  W(N) a n a 3 0 a n       a 1 0 x 1 x 1    .   .  ...  � = a 1 | x 1 | 2 + · · · + a n | x n | 2 ∈ RHS . . pf. �  ,     . . 0 a n x n x n � 0 � 1 ⇒ W ( J 2 ) = { z ∈ C : | z | ≤ 1 (2) J 2 = 2 } 0 0 � 0 � � x 1 � � x 1 � � 1 � = x 2 x 1 = | x 1 x 2 | e i θ = t (1 − t ) e i θ pf. � , 0 0 x 2 x 2 � t (1 − t ) e i θ : 0 ≤ t ≤ 1 , θ ∈ R } = { z ∈ C : | z | ≤ 1 ⇒ W ( J 2 ) = { 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: a 1   0 a 1 a 2   ... (1) N =  ⇒ W ( N ) = convex hull ( σ ( N ))  W(N) a n a 3 0 a n       a 1 0 x 1 x 1    .   .  ...  � = a 1 | x 1 | 2 + · · · + a n | x n | 2 ∈ RHS . . pf. �  ,     . . 0 a n x n x n � 0 � 1 ⇒ W ( J 2 ) = { z ∈ C : | z | ≤ 1 (2) J 2 = 2 } 0 0 � 0 � � x 1 � � x 1 � � 1 � = x 2 x 1 = | x 1 x 2 | e i θ = t (1 − t ) e i θ pf. � , 0 0 x 2 x 2 � t (1 − t ) e i θ : 0 ≤ t ≤ 1 , θ ∈ R } = { z ∈ C : | z | ≤ 1 ⇒ W ( J 2 ) = { 2 } Hwa-Long Gau Inner Functions of Numerical Contractions 3/29