The coupling method and operator relations Sanne ter Horst 1 - PowerPoint PPT Presentation

The coupling method and operator relations Sanne ter Horst 1 North-West University IWOTA 2017 Chemnitz, Germany Joint work with M. Messerschmidt, A.C.M. Ran, M. Roelands and M. Wortel 1 This work is based on the research supported in part by the National Research Foundation of South Africa (Grant Numbers 90670, and 93406).

Outline • Application of the Coupling Method • Formalization of the Coupling Method Three Banach space operator relations: MC, EAE and SC • Question 1: Do MC, EAE and SC coincide • Question 2: When are two operators MC/EAE/SC?

The Coupling Method for integral equations Integral operators with semi-separable kernel Define � τ K : L 2 n [0 , τ ] → L 2 ( f ∈ L 2 n [0 , τ ] , ( Kf )( t ) = k ( t , s ) f ( s ) ds , n [0 , τ ]) . 0 Here � C ( t )( I − P ) B ( s ) , s < t ; k ( s , t ) = − C ( t ) PB ( s ) , s > t , with P ∈ Mat n × n a projection and C , B ∈ L 2 n × n [0 , τ ]. C Then K is Hilbert-Schmidt, so I − K is Fredholm. Integral equation: Given g ∈ L 2 n [0 , τ ], find f ∈ L 2 n [0 , τ ] with � τ g = ( I − K ) f , i . e ., g ( t ) = f ( t ) − k ( t , s ) f ( s ) ds . 0

The Coupling Method for integral equations Integral operators with semi-separable kernel Define � τ K : L 2 n [0 , τ ] → L 2 ( f ∈ L 2 n [0 , τ ] , ( Kf )( t ) = k ( t , s ) f ( s ) ds , n [0 , τ ]) . 0 Here � C ( t )( I − P ) B ( s ) , s < t ; k ( s , t ) = − C ( t ) PB ( s ) , s > t , with P ∈ Mat n × n a projection and C , B ∈ L 2 n × n [0 , τ ]. C Then K is Hilbert-Schmidt, so I − K is Fredholm. Integral equation: Given g ∈ L 2 n [0 , τ ], find f ∈ L 2 n [0 , τ ] with � τ g = ( I − K ) f , i . e ., g ( t ) = f ( t ) − k ( t , s ) f ( s ) ds . 0 Associated system With B and C we associate the differential equation: x ( t ) = B ( t ) C ( t ) x ( t ) ˙ ( t ∈ [0 , τ ]) . Write U : [0 , τ ] → Mat n × n for the associated fundamental matrix. C

The Coupling Method for integral equations Bart-Gohberg-Kaashoek ’84 Define S τ = PU ( τ ) P : Im P → Im P

The Coupling Method for integral equations Bart-Gohberg-Kaashoek ’84 Define S τ = PU ( τ ) P : Im P → Im P and � τ H : L 2 n [0 , τ ] → L 2 n [0 , τ ] , ( Hf )( t ) = C ( t ) B ( s ) f ( s ) ds ; 0 � τ Q : L 2 n [0 , τ ] → Im P , Qf = P B ( s ) f ( s ) ds 0 R : Im P → L 2 n [0 , τ ] , ( Qx )( t ) = C ( t ) Px . Then I − H is invertible and � I − K � − 1 � � ( I − H ) − 1 ( I − H ) − 1 R − R = . (MC) Q ( I − H ) − 1 − Q I S τ

The Coupling Method for integral equations Bart-Gohberg-Kaashoek ’84 Define S τ = PU ( τ ) P : Im P → Im P and � τ H : L 2 n [0 , τ ] → L 2 n [0 , τ ] , ( Hf )( t ) = C ( t ) B ( s ) f ( s ) ds ; 0 � τ Q : L 2 n [0 , τ ] → Im P , Qf = P B ( s ) f ( s ) ds 0 R : Im P → L 2 n [0 , τ ] , ( Qx )( t ) = C ( t ) Px . Then I − H is invertible and � I − K � − 1 � � ( I − H ) − 1 ( I − H ) − 1 R − R = . (MC) Q ( I − H ) − 1 − Q I S τ Moreover, there exist invertible operators E and F such that � I − K � � S τ � 0 0 = E F . (EAE) 0 I L 2 0 I Im P n [0 ,τ ] � I � − R The Schur complements of are given by Q I − H S τ = I + Q ( I − H ) − 1 R . I − K = ( I − H ) + RQ and (SC)

The Coupling Method for integral equations Fredholm properties The identity � I − K � � S τ � 0 0 = E F 0 I Im P 0 I L 2 n [0 ,τ ] with E and F invertible yields: I − K (on L 2 n [0 , τ ]) and S τ (on Im P ) have the same ’Fredholm properties.’ And one can show: Ker ( I − K ) = ( I − H ) − 1 R Ker S τ Im ( I − K ) = { f : Q ( I − H ) − 1 f ∈ Im S τ } . and

The Coupling Method for integral equations Fredholm properties The identity � I − K � � S τ � 0 0 = E F 0 I Im P 0 I L 2 n [0 ,τ ] with E and F invertible yields: I − K (on L 2 n [0 , τ ]) and S τ (on Im P ) have the same ’Fredholm properties.’ And one can show: Ker ( I − K ) = ( I − H ) − 1 R Ker S τ Im ( I − K ) = { f : Q ( I − H ) − 1 f ∈ Im S τ } . and Generalized inverse � I � − R Expressing the Moore-Penrose generalized inverse of in terms of its Q I − H Schur complements one can compute the MP generalized inverse of I − K : ( I + K ) + = ( I − H ) − 1 − ( I − H ) − 1 RS + τ Q ( I − H ) − 1 , and solve the integal equation: f = ( I + K ) + g , if g ∈ Im ( I − K ) .

The Coupling Method: Formalization Two Banach space operators U : X → X and V : Y → Y are called matricially coupled (MC), equivalent after extension (EAE) resp. Schur coupled (SC) if: � X � � X � (MC) There exist an invertible operator � U : → such that Y Y � � � V 11 � U U 12 V 12 U − 1 = � � U = and . U 21 U 22 V 21 V (EAE) There exist Banach spaces X 0 and Y 0 and invertible operators E and F s.t. � U � V � � 0 0 = E F . 0 I X 0 0 I Y 0 � X � � X � (SC) There exists an operator matrix S = [ A B C D ] : → with A and D Y Y invertible and U = A − BD − 1 C , V = D − CA − 1 B .

The Coupling Method: Formalization Two Banach space operators U : X → X and V : Y → Y are called matricially coupled (MC), equivalent after extension (EAE) resp. Schur coupled (SC) if: � X � � X � (MC) There exist an invertible operator � U : → such that Y Y � � � V 11 � U U 12 V 12 U − 1 = � � U = and . U 21 U 22 V 21 V (EAE) There exist Banach spaces X 0 and Y 0 and invertible operators E and F s.t. � U � V � � 0 0 = E F . 0 I X 0 0 I Y 0 � X � � X � (SC) There exists an operator matrix S = [ A B C D ] : → with A and D Y Y invertible and U = A − BD − 1 C , V = D − CA − 1 B . In the example I − K and S τ are MC ⇒ I − K and S τ are EAE ⇒ I − K and S τ are SC ⇓ ⇓ Fredholm properties generalized inverse

The Coupling Method: Formalization Two Banach space operators U : X → X and V : Y → Y are called matricially coupled (MC), equivalent after extension (EAE) resp. Schur coupled (SC) if: � X � � X � (MC) There exist an invertible operator � U : → such that Y Y � � � V 11 � U U 12 V 12 U − 1 = � � U = and . U 21 U 22 V 21 V (EAE) There exist Banach spaces X 0 and Y 0 and invertible operators E and F s.t. � U � V � � 0 0 = E F . 0 I X 0 0 I Y 0 � X � � X � (SC) There exists an operator matrix S = [ A B C D ] : → with A and D Y Y invertible and U = A − BD − 1 C , V = D − CA − 1 B . More recent applications • Diffraction theory (Castro, Duduchava, Speck, e.g., 2014) • Truncated Toeplitz operators (Cˆ amara, Partington, 2016) • Connection with Paired Operators approach (Speck, 2017) • Wiener-Hopf factorization (Groenewald, Kaashoek, Ran, 2017)

The Coupling Method: Formalization Two Banach space operators U : X → X and V : Y → Y are called matricially coupled (MC), equivalent after extension (EAE) resp. Schur coupled (SC) if: � X � � X � (MC) There exist an invertible operator � U : → such that Y Y � � � V 11 � U U 12 V 12 U − 1 = � � U = and . U 21 U 22 V 21 V (EAE) There exist Banach spaces X 0 and Y 0 and invertible operators E and F s.t. � U � V � � 0 0 = E F . 0 I X 0 0 I Y 0 � X � � X � (SC) There exists an operator matrix S = [ A B C D ] : → with A and D Y Y invertible and U = A − BD − 1 C , V = D − CA − 1 B . More recent applications • Completeness theorems in dynamical systems (Kaashoek, Verduyn Lunel) • Unbounded operator functions (Engstr¨ om, Torshage, Arxiv)

Do MC, EAE and SC coincide? Question (Bart-Tsekanovskii ’92) Do the operator relations MC, EAE and SC coincide?

Do MC, EAE and SC coincide? Question (Bart-Tsekanovskii ’92) Do the operator relations MC, EAE and SC coincide? ⇐ ⇒ ⇐ MC EAE = SC Early results • Bart-Gohberg-Kaashoek ’84: MC ⇒ EAE • Bart-Tsekanovskii ’92: EAE ⇒ MC (so EAE ⇔ MC) • Bart-Tsekanovskii ’94: SC ⇒ EAE Proof MC = ⇒ EAE � � � � � U � � V � − U 21 I Y 0 0 U 12 U = E F holds with E = and F = and 0 I Y 0 I X U 22 U 21 V 11 U V 12 � V 21 � � − V 12 � V I E − 1 = F − 1 = , . V 11 V 12 U 22 V U 21

Do MC, EAE and SC coincide? Question (Bart-Tsekanovskii ’92) Do the operator relations MC, EAE and SC coincide? ⇐ ⇒ ⇐ MC EAE = SC Early results • Bart-Gohberg-Kaashoek ’84: MC ⇒ EAE • Bart-Tsekanovskii ’92: EAE ⇒ MC (so EAE ⇔ MC) • Bart-Tsekanovskii ’94: SC ⇒ EAE • Remaining implication: Does EAE ⇒ SC hold?

Do MC, EAE and SC coincide? Question (Bart-Tsekanovskii ’92) Do the operator relations MC, EAE and SC coincide? ⇐ ⇒ ⇐ MC EAE = SC Early results • Bart-Gohberg-Kaashoek ’84: MC ⇒ EAE • Bart-Tsekanovskii ’92: EAE ⇒ MC (so EAE ⇔ MC) • Bart-Tsekanovskii ’94: SC ⇒ EAE • Remaining implication: Does EAE ⇒ SC hold? • BT’92: Yes if U and V are Fredholm (Banach space: + index = 0) • BGKR’05: Yes if SC is an equivalence relation (this is true for EAE) (BT=Bart-Tsekanovskii, BGKR=Bart-Gohberg-Kaashoek-Ran)