Numerical ranges and essential numerical ranges Spectral approximation Multiplier tricks for spectral convergence Sabine B¨ ogli (Imperial College London) Re � 15 � 10 � 5 5 10 15 Quantissima III Venice, 19 August 2019 Based on joint work with M. Marletta (Cardiff): Essential numerical ranges for linear operator pencils . To appear in IMA Journal of Numerical Analysis . Sabine B¨ ogli (Imperial) Multiplier tricks
Numerical ranges and essential numerical ranges Spectral approximation Motivation Let { e k : k ∈ N } be standard ONB of l 2 ( N ) . Consider T with matrix representation �� � � − n 0 : n ∈ N = diag( − 1 , 1 , − 2 , 2 , − 3 , 3 , . . . ) . diag 0 n Re � 15 � 10 � 5 5 10 15 Sabine B¨ ogli (Imperial) Multiplier tricks
Numerical ranges and essential numerical ranges Spectral approximation Motivation Let { e k : k ∈ N } be standard ONB of l 2 ( N ) . Consider T with matrix representation �� � � − n 0 : n ∈ N = diag( − 1 , 1 , − 2 , 2 , − 3 , 3 , . . . ) . diag 0 n Re � 15 � 10 � 5 5 10 15 Compress to V n := span { e k : k = 1 , . . . , 2 n } , n ∈ N � spectral convergence. Sabine B¨ ogli (Imperial) Multiplier tricks
Numerical ranges and essential numerical ranges Spectral approximation Motivation Let { e k : k ∈ N } be standard ONB of l 2 ( N ) . Consider T with matrix representation �� � � − n 0 : n ∈ N = diag( − 1 , 1 , − 2 , 2 , − 3 , 3 , . . . ) . diag 0 n Re � 15 � 10 � 5 5 10 15 Compress to V n := span { e k : k = 1 , . . . , 2 n } , n ∈ N � spectral convergence. Let λ ∈ R \ σ ( T ) . Change ONB: span { e 2 n − 1 , e 2 n } = span { f n , g n } , f n := cos( θ n ) e 2 n − 1 + sin( θ n ) e 2 n , θ n := λ 2 n − π with 4 . g n := − sin( θ n ) e 2 n − 1 + cos( θ n ) e 2 n , Sabine B¨ ogli (Imperial) Multiplier tricks
Numerical ranges and essential numerical ranges Spectral approximation Motivation Let { e k : k ∈ N } be standard ONB of l 2 ( N ) . Consider T with matrix representation �� � � − n 0 : n ∈ N = diag( − 1 , 1 , − 2 , 2 , − 3 , 3 , . . . ) . diag 0 n Re � 15 � 10 � 5 5 10 15 Compress to V n := span { e k : k = 1 , . . . , 2 n } , n ∈ N � spectral convergence. Let λ ∈ R \ σ ( T ) . Change ONB: span { e 2 n − 1 , e 2 n } = span { f n , g n } , f n := cos( θ n ) e 2 n − 1 + sin( θ n ) e 2 n , θ n := λ 2 n − π with 4 . g n := − sin( θ n ) e 2 n − 1 + cos( θ n ) e 2 n , Then compression to H n := V n − 1 ⊕ span { f n } has eigenvalue λ n := � Tf n , f n � = n sin( λ/n ) → λ / ∈ σ ( T ) . So λ is a spurious eigenvalue (point of spectral pollution). Sabine B¨ ogli (Imperial) Multiplier tricks
Numerical ranges and essential numerical ranges Spectral approximation Motivation Let �� � � �� � � n 0 − 1 0 : n ∈ N : n ∈ N A := BT = diag , B := diag . 0 n 0 1 � ( T − λ ) f = 0 ⇐ ⇒ ( A − λB ) f = 0 (eigenvalue problem for linear pencil). Sabine B¨ ogli (Imperial) Multiplier tricks
Numerical ranges and essential numerical ranges Spectral approximation Motivation Let �� � � �� � � n 0 − 1 0 : n ∈ N : n ∈ N A := BT = diag , B := diag . 0 n 0 1 � ( T − λ ) f = 0 ⇐ ⇒ ( A − λB ) f = 0 (eigenvalue problem for linear pencil). ◮ Introduce essential numerical range W e ( T ) � contains all possible spurious eigenvalues obtained by projection methods. Here: W e ( T ) = R . Sabine B¨ ogli (Imperial) Multiplier tricks
Numerical ranges and essential numerical ranges Spectral approximation Motivation Let �� � � �� � � n 0 − 1 0 : n ∈ N : n ∈ N A := BT = diag , B := diag . 0 n 0 1 � ( T − λ ) f = 0 ⇐ ⇒ ( A − λB ) f = 0 (eigenvalue problem for linear pencil). ◮ Introduce essential numerical range W e ( T ) � contains all possible spurious eigenvalues obtained by projection methods. Here: W e ( T ) = R . ◮ Introduce essential numerical range W e ( A, B ) � contains all possible spurious eigenvalues obtained by projection methods. Here: W e ( A, B ) = ∅ . Sabine B¨ ogli (Imperial) Multiplier tricks
Numerical ranges and essential numerical ranges Spectral approximation Motivation Let �� � � �� � � n 0 − 1 0 : n ∈ N : n ∈ N A := BT = diag , B := diag . 0 n 0 1 � ( T − λ ) f = 0 ⇐ ⇒ ( A − λB ) f = 0 (eigenvalue problem for linear pencil). ◮ Introduce essential numerical range W e ( T ) � contains all possible spurious eigenvalues obtained by projection methods. Here: W e ( T ) = R . ◮ Introduce essential numerical range W e ( A, B ) � contains all possible spurious eigenvalues obtained by projection methods. Here: W e ( A, B ) = ∅ . In example: no spectral pollution occurs if we calculate eigenvalues λ n given by ( A H n − λ n B H n ) f n = 0 . Sabine B¨ ogli (Imperial) Multiplier tricks
Numerical ranges and essential numerical ranges Definition and basic properties of numerical ranges Spectral approximation Definition and basic properties of essential numerical ranges Numerical ranges for operators and linear operator pencils Let H be a Hilbert space and let A be a linear operator in H . Recall: The numerical range of A is the convex set W ( A ) = {� Af, f � : f ∈ D ( A ) , � f � = 1 } . Then σ app ( A ) ⊆ W ( A ) with σ app ( A ) := { λ ∈ C : ∃ ( f n ) n ∈ N ⊂ D ( A ) , � f n � = 1 , � ( A − λ ) f n � → 0 } . Sabine B¨ ogli (Imperial) Multiplier tricks
Numerical ranges and essential numerical ranges Definition and basic properties of numerical ranges Spectral approximation Definition and basic properties of essential numerical ranges Numerical ranges for operators and linear operator pencils Let H be a Hilbert space and let A be a linear operator in H . Recall: The numerical range of A is the convex set W ( A ) = {� Af, f � : f ∈ D ( A ) , � f � = 1 } . Then σ app ( A ) ⊆ W ( A ) with σ app ( A ) := { λ ∈ C : ∃ ( f n ) n ∈ N ⊂ D ( A ) , � f n � = 1 , � ( A − λ ) f n � → 0 } . Now study linear pencil λ �→ A − λB with B another linear operator. Define σ ( A, B ) := { λ ∈ C : 0 ∈ σ ( A − λB ) } (and various parts analogously) and � � λ ∈ C : 0 ∈ W ( A − λB ) W ( A, B ) := . Then σ app ( A, B ) ⊆ W ( A, B ) . Sabine B¨ ogli (Imperial) Multiplier tricks
Numerical ranges and essential numerical ranges Definition and basic properties of numerical ranges Spectral approximation Definition and basic properties of essential numerical ranges Essential numerical ranges for operators and linear operator pencils The essential numerical range of A is the convex set � � w W e ( A ) = n →∞ � Af n , f n � : f n ∈ D ( A ) , � f n � = 1 , f n lim → 0 . Then σ e ( A ) ⊆ W e ( A ) ⊆ W ( A ) with w σ e ( A ) := { λ ∈ C : ∃ ( f n ) n ∈ N ⊂ D ( A ) , � f n � = 1 , f n → 0 , � ( A − λ ) f n � → 0 } . Sabine B¨ ogli (Imperial) Multiplier tricks
Numerical ranges and essential numerical ranges Definition and basic properties of numerical ranges Spectral approximation Definition and basic properties of essential numerical ranges Essential numerical ranges for operators and linear operator pencils The essential numerical range of A is the convex set � � w W e ( A ) = n →∞ � Af n , f n � : f n ∈ D ( A ) , � f n � = 1 , f n lim → 0 . Then σ e ( A ) ⊆ W e ( A ) ⊆ W ( A ) with w σ e ( A ) := { λ ∈ C : ∃ ( f n ) n ∈ N ⊂ D ( A ) , � f n � = 1 , f n → 0 , � ( A − λ ) f n � → 0 } . For linear pencils: � � W e ( A, B ) := λ ∈ C : 0 ∈ W e ( A − λB ) . Then σ e ( A, B ) ⊆ W e ( A, B ) ⊆ W ( A, B ) . Sabine B¨ ogli (Imperial) Multiplier tricks
Numerical ranges and essential numerical ranges Definition and basic properties of numerical ranges Spectral approximation Definition and basic properties of essential numerical ranges Essential numerical range FOR OPERATORS ◮ W e ( T ) is closed and convex with W e ( T ) ⊇ conv( σ e ( T )) . Sabine B¨ ogli (Imperial) Multiplier tricks
Numerical ranges and essential numerical ranges Definition and basic properties of numerical ranges Spectral approximation Definition and basic properties of essential numerical ranges Essential numerical range FOR OPERATORS ◮ W e ( T ) is closed and convex with W e ( T ) ⊇ conv( σ e ( T )) . ◮ Salinas (1972): T bounded and hypo-normal ( T ∗ T − TT ∗ ≥ 0 ) = ⇒ W e ( T ) = conv( σ e ( T )) . Sabine B¨ ogli (Imperial) Multiplier tricks
Numerical ranges and essential numerical ranges Definition and basic properties of numerical ranges Spectral approximation Definition and basic properties of essential numerical ranges Essential numerical range FOR OPERATORS ◮ W e ( T ) is closed and convex with W e ( T ) ⊇ conv( σ e ( T )) . ◮ Salinas (1972): T bounded and hypo-normal ( T ∗ T − TT ∗ ≥ 0 ) = ⇒ W e ( T ) = conv( σ e ( T )) . ◮ With Marletta and Tretter: If T is selfadjoint and unbounded, then W e ( T ) = conv( � σ e ( T )) \{±∞} , where the extended essential spectrum � σ e ( T ) is defined as σ e ( T ) with −∞ ( + ∞ ) added if T is unbounded below (above). Sabine B¨ ogli (Imperial) Multiplier tricks
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