Draft EE 8235: Lecture 9 1 Lecture 9: Spectral theory for compact normal operators • Resolvent and spectrum of an operator • Compact operators ⋆ Direct extension of matrices • Normal operators ⋆ Commute with its adjoint • Compact normal operators ⋆ Unitarily diagonalizable ⋆ E-functions provide a complete orthonormal basis of H • Riesz-spectral operators
Draft EE 8235: Lecture 9 2 Resolvent • Want to study equations of the form ( λI − A ) ψ = u, {A : H ⊃ D ( A ) − → H ; λ ∈ C ; ψ, u ∈ H } Determine conditions under which A λ = ( λI − A ) is boundedly invertible R λ = ( λI − A ) − 1 exists (1) R λ = ( λI − A ) − 1 is bounded Relevant conditions: (2) The domain of R λ = ( λI − A ) − 1 is dense in H (3) • The resolvent set of A : ρ ( A ) := { λ ∈ C ; (1), (2), (3) hold } • The spectrum of A : σ ( A ) := C \ ρ ( A )
Draft EE 8235: Lecture 9 3 Spectrum R λ = ( λI − A ) − 1 exists (1) R λ = ( λI − A ) − 1 is bounded (2) The domain of R λ = ( λI − A ) − 1 is dense in H (3) • σ ( A ) can be decomposed into σ ( A ) = σ p ( A ) ∪ σ c ( A ) ∪ σ r ( A ) ⋆ Point spectrum σ p ( A ) := { λ ∈ C ; ( λI − A ) is not one-to-one } ⋆ Continuous spectrum σ c ( A ) := { λ ∈ C ; (1) and (3) hold, but (2) doesn’t } ⋆ Residual spectrum σ r ( A ) := { λ ∈ C ; (1) holds but (3) doesn’t }
Draft EE 8235: Lecture 9 4 Examples • Point spectrum { λ ∈ σ p ( A ): e-values ; v ∈ N ( λI − A ): e-functions } • Continuous spectrum multiplication operator on L 2 [ a, b ]: [ M a f ( · )] ( x ) = a ( x ) f ( x ) • Residual spectrum right-shift operator on ℓ 2 ( N ): [ S r f ( · )] ( n ) = f n − 1
Draft EE 8235: Lecture 9 5 Spectral decomposition of compact normal operators • compact, normal operator A on H admits a dyadic decomposition ∞ � [ A v n ] ( x ) = λ n v n ( x ) � ⇒ [ A f ] ( x ) = λ n v n ( x ) � v n , f � for all f ∈ H � v n , v m � = δ nm n = 1 → H , with compact and normal A − 1 A : H ⊃ D ( A ) − � A − 1 v n ∞ ( x ) = λ − 1 � � � n v n ( x ) � λ − 1 A − 1 f � � ⇒ n v n ( x ) � v n , f � , f ∈ H ( x ) = � v n , v m � = δ nm n = 1 ∞ � [ A f ] ( x ) = λ n v n ( x ) � v n , f � , f ∈ D ( A ) n = 1 ∞ � � | λ n | 2 |� v n , f �| 2 < ∞ � D ( A ) = f ∈ H ; n = 1
Draft EE 8235: Lecture 9 6 • compact, normal operator A on H u = u R ( A ) + u N ( A ) � [ A v n ] ( x ) = λ n v n ( x ) , λ n � = 0 ∞ � = v n � v n , u � + u N ( A ) � v n , v m � = δ nm n = 1 • Solutions to ( λI − A ) ψ = u, λ � = 0 1. λ − not an eigenvalue of A ⇒ unique solution ∞ � v n , u � v n + 1 � ψ = λ u N ( A ) λ − λ n n = 1 � λ − eigenvalue of A ⇒ there is a solution iff � v j , u � = 0 for all j ∈ J 2. J − index set s.t. λ j = λ � v j , u � v j + 1 � � ψ = c j v j + λ u N ( A ) λ − λ j j ∈ J j ∈ N \ J
Draft EE 8235: Lecture 9 7 Singular Value Decomposition of compact operators • compact operator A : H 1 − → H 2 admits a Schmidt Decomposition (i.e., an SVD) ∞ � [ A f ] ( x ) = σ n u n ( x ) � v n , f � n = 1 A A † u n σ 2 � � ⇒ { u n } n ∈ N orthonormal basis of H 2 ( x ) = n u n ( x ) A † A v n σ 2 � � ( x ) = n v n ( x ) ⇒ { v n } n ∈ N orthonormal basis of H 1 • matrix M : C n − → C m r r M = U Σ V ∗ = � σ i u i v ∗ � ⇒ σ i u i � v i , f � M f = i i = 1 i = 1 M M ∗ u i σ 2 = i u i M ∗ M v i σ 2 = i v i
Draft EE 8235: Lecture 9 8 Riesz basis • { v n } n ∈ N : Riesz basis of H if ⋆ span { v n } n ∈ N = H ⋆ there are m, M > 0 such that for any N ∈ N and any { α n } , n = 1 , . . . , N N N N | α n | 2 ≤ � α n v n � 2 ≤ M � � � | α n | 2 m n = 1 n = 1 n = 1
Draft EE 8235: Lecture 9 9 • closed A : H ⊃ D ( A ) − → H � { λ n } n ∈ N simple e-values [ A v n ] ( x ) = λ n v n ( x ) { v n } n ∈ N Riesz basis of H A † w n ( x ) = ¯ � � λ n w n ( x ) ⇒ { w n } n ∈ N can be scaled s.t. � w n , v m � = δ nm ⋆ ⋆ every f ∈ H can be represented uniquely by ∞ � v n ( x ) � w n , f � f ( x ) = n = 1 ∞ ∞ | � w n , f � | 2 ≤ � f � 2 ≤ M � � | � w n , f � | 2 m n = 1 n = 1 or by ∞ � w n ( x ) � v n , f � f ( x ) = n = 1 ∞ ∞ 1 1 | � v n , f � | 2 ≤ � f � 2 ≤ � � | � v n , f � | 2 M m n = 1 n = 1
Draft EE 8235: Lecture 9 10 Riesz-spectral operator • closed A : H ⊃ D ( A ) − → H is Riesz-spectral operator if { λ n } n ∈ N simple e-values { v n } n ∈ N [ A v n ] ( x ) = λ n v n ( x ) Riesz basis of H { λ n } n ∈ N totally disconnected A − Riesz-spectral operator with e-pair { ( λ n , v n ) } n ∈ N e-functions of A † s.t. � w n , v m � = δ nm { w n } n ∈ N − � σ ( A ) = { λ n } n ∈ N , ρ ( A ) = { λ n ∈ C ; inf n ∈ N | λ − λ n | > 0 } ∞ 1 ( λI − A ) − 1 f � � � λ ∈ ρ ( A ) ⇒ v n ( x ) � w n , f � ( x ) = λ − λ n n = 1 ∞ ∞ � � | λ n | 2 |� w n , f �| 2 < ∞ � � [ A f ] ( x ) = λ n v n ( x ) � w n , f � , D ( A ) = f ∈ H ; n = 1 n = 1
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