Triple Variational Principles for Operator Functions Matthias - PowerPoint PPT Presentation

Triple Variational Principles for Operator Functions Matthias Langer University of Strathclyde, Glasgow

Operator functions Let T be an operator function in a Hilbert space H defined on Ω ⊂ C : T ( λ ) is a closed operator in H for every λ ∈ Ω . For example, T ( λ ) = λ 2 A + λB + C with bounded operators A, B, C , or T ( λ ) = A − λI with a closed operator A . ❊✐❣❡♥✈❛❧✉❡s✳ ✕ ✵ ✷ ✡ ✐s ❝❛❧❧❡❞ ❡✐❣❡♥✈❛❧✉❡ ♦❢ ❚ ✭ ✮ ✾ ① ✵ ✷ ❞♦♠✭ ❚ ✭ ✕ ✵ ✮✮ ♥ ❢ ✵ ❣ ✿ ❚ ✭ ✕ ✵ ✮ ① ✵ ❂ ✵ ✭ ✮ ✵ ✐s ❡✐❣❡♥✈❛❧✉❡ ♦❢ ❚ ✭ ✕ ✵ ✮ ❙♣❡❝tr✉♠✳ ✕ ✵ ✷ ✛ ✭ ❚ ✮ ✭ ✮ ✵ ✷ ✛ ✭ ❚ ✭ ✕ ✵ ✮✮ ✕ ✵ ✷ ✛ ❡ss ✭ ❚ ✮ ✭ ✮ ✵ ✷ ✛ ❡ss ✭ ❚ ✭ ✕ ✵ ✮✮

Operator functions Let T be an operator function in a Hilbert space H defined on Ω ⊂ C : T ( λ ) is a closed operator in H for every λ ∈ Ω . For example, T ( λ ) = λ 2 A + λB + C with bounded operators A, B, C , or T ( λ ) = A − λI with a closed operator A . Eigenvalues. λ 0 ∈ Ω is called eigenvalue of T ⇐ ⇒ ∃ x 0 ∈ dom( T ( λ 0 )) \ { 0 } : T ( λ 0 ) x 0 = 0 ⇐ ⇒ 0 is eigenvalue of T ( λ 0 ) ❙♣❡❝tr✉♠✳ ✕ ✵ ✷ ✛ ✭ ❚ ✮ ✭ ✮ ✵ ✷ ✛ ✭ ❚ ✭ ✕ ✵ ✮✮ ✕ ✵ ✷ ✛ ❡ss ✭ ❚ ✮ ✭ ✮ ✵ ✷ ✛ ❡ss ✭ ❚ ✭ ✕ ✵ ✮✮

Operator functions Let T be an operator function in a Hilbert space H defined on Ω ⊂ C : T ( λ ) is a closed operator in H for every λ ∈ Ω . For example, T ( λ ) = λ 2 A + λB + C with bounded operators A, B, C , or T ( λ ) = A − λI with a closed operator A . Eigenvalues. λ 0 ∈ Ω is called eigenvalue of T ⇐ ⇒ ∃ x 0 ∈ dom( T ( λ 0 )) \ { 0 } : T ( λ 0 ) x 0 = 0 ⇐ ⇒ 0 is eigenvalue of T ( λ 0 ) Spectrum. λ 0 ∈ σ ( T ) ⇐ ⇒ 0 ∈ σ ( T ( λ 0 )) λ 0 ∈ σ ess ( T ) ⇐ ⇒ 0 ∈ σ ess ( T ( λ 0 ))

Assumptions I • Ω ⊆ C domain; ∆ ⊆ Ω ∩ R interval • T ( λ ) is m-sectorial for each λ ∈ Ω • T ( λ ) is self-adjoint for λ ∈ ∆ • Let t ( λ ) be the closed quadratic form: t ( λ )[ x ] = ( T ( λ ) x, x ); assume that D := dom( t ( λ )) is independent of λ • for each x ∈ D : λ �→ t ( λ )[ x ] is analytic on Ω ❊✐❣❡♥✈❛❧✉❡s ❛♥❞ t❤❡ ❢✉♥❝t✐♦♥ t ( · ) ▲❡t ✕ ✵ ❜❡ ❛♥ ❡✐❣❡♥✈❛❧✉❡ ♦❢ ❚ ✇✐t❤ ❡✐❣❡♥✈❡❝t♦r ① ✵ ✳ ❚❤❡♥ t ✭ ✕ ✵ ✮❬ ① ✵ ❪ ❂ ✭ ❚ ✭ ✕ ✵ ✮ ① ✵ ❀ ① ✵ ✮ ❂ ✵ ❀ ✐✳❡✳ t❤❡ ❢✉♥❝t✐♦♥ ✕ ✼✦ t ✭ ✕ ✮❬ ① ✵ ❪ ❤❛s ❛ ③❡r♦ ❛t ✕ ✵ ✳

Assumptions I • Ω ⊆ C domain; ∆ ⊆ Ω ∩ R interval • T ( λ ) is m-sectorial for each λ ∈ Ω • T ( λ ) is self-adjoint for λ ∈ ∆ • Let t ( λ ) be the closed quadratic form: t ( λ )[ x ] = ( T ( λ ) x, x ); assume that D := dom( t ( λ )) is independent of λ • for each x ∈ D : λ �→ t ( λ )[ x ] is analytic on Ω Eigenvalues and the function t ( · ) Let λ 0 be an eigenvalue of T with eigenvector x 0 . Then t ( λ 0 )[ x 0 ] = ( T ( λ 0 ) x 0 , x 0 ) = 0 , i.e. the function λ �→ t ( λ )[ x 0 ] has a zero at λ 0 .

‘Hyperbolic’ or ‘overdamped’ case Let ∆ = [ α, β ) and assume that for each x ∈ D \ { 0 } : t ( α )[ x ] > 0 , t ( · )[ x ] has exactly one zero in ( α, β ) . Denote this unique zero by p ( x ). The mapping x �→ p ( x ) is called generalised Rayleigh functional. ■❢ ✕ ✵ ✐s ❛♥ ❡✐❣❡♥✈❛❧✉❡ ♦❢ ❚ ✇✐t❤ ❡✐❣❡♥✈❡❝t♦r ① ✵ ✱ t❤❡♥ ♣ ✭ ① ✵ ✮ ❂ ✕ ✵ ✿ ■♥ t❤❡ ❝❛s❡ ✇❤❡♥ ❚ ✭ ✕ ✮ ❂ ❆ � ✕■ ❢♦r ❛ s❡❧❢✲❛❞❥♦✐♥t ♦♣❡r❛t♦r ❆ ✇✐t❤ q✉❛❞r❛t✐❝ ❢♦r♠ a ✇❡ ❤❛✈❡ t ✭ ✕ ✮❬ ① ❪ ❂ a ❬ ① ❪ � ✕ ❦ ① ❦ ✷ ❛♥❞ ❤❡♥❝❡ ♣ ✭ ① ✮ ❂ a ❬ ① ❪ ❦ ① ❦ ✷

‘Hyperbolic’ or ‘overdamped’ case Let ∆ = [ α, β ) and assume that for each x ∈ D \ { 0 } : t ( α )[ x ] > 0 , t ( · )[ x ] has exactly one zero in ( α, β ) . Denote this unique zero by p ( x ). The mapping x �→ p ( x ) is called generalised Rayleigh functional. If λ 0 is an eigenvalue of T with eigenvector x 0 , then p ( x 0 ) = λ 0 . ■♥ t❤❡ ❝❛s❡ ✇❤❡♥ ❚ ✭ ✕ ✮ ❂ ❆ � ✕■ ❢♦r ❛ s❡❧❢✲❛❞❥♦✐♥t ♦♣❡r❛t♦r ❆ ✇✐t❤ q✉❛❞r❛t✐❝ ❢♦r♠ a ✇❡ ❤❛✈❡ t ✭ ✕ ✮❬ ① ❪ ❂ a ❬ ① ❪ � ✕ ❦ ① ❦ ✷ ❛♥❞ ❤❡♥❝❡ ♣ ✭ ① ✮ ❂ a ❬ ① ❪ ❦ ① ❦ ✷

‘Hyperbolic’ or ‘overdamped’ case Let ∆ = [ α, β ) and assume that for each x ∈ D \ { 0 } : t ( α )[ x ] > 0 , t ( · )[ x ] has exactly one zero in ( α, β ) . Denote this unique zero by p ( x ). The mapping x �→ p ( x ) is called generalised Rayleigh functional. If λ 0 is an eigenvalue of T with eigenvector x 0 , then p ( x 0 ) = λ 0 . In the case when T ( λ ) = A − λI for a self-adjoint operator A with quadratic form a we have t ( λ )[ x ] = a [ x ] − λ � x � 2 and hence p ( x ) = a [ x ] � x � 2

Variational principle for eigenvalues In situations as above (or more general situations) variational principles were proved by Duffin 1955 Rogers 1964 Turner 1967 Hadeler 1968 Werner 1971 Barston 1974 Binding, Eschw´ e, H. Langer 2000 Eschw´ e, M.L. 2004 Voss 2015 Jacob, M.L., Trunk 2016 . . .

Theorem. Consider the situation as above and assume that σ ess ( T ) ∩ [ α, β ) = ∅ , α ∈ ρ ( T ) . Then the spectrum of T in ( α, β ) consists of eigenvalues: λ 1 ≤ λ 2 ≤ · · · and λ n = min x ∈ L \{ 0 } p ( x ) = max max inf p ( x ) . L ⊂D L ⊂H x ∈D\{ 0 } dim L = n dim L = n − 1 x ⊥ L

Comparison of two operator functions Let T 1 and T 2 be two operator functions as above and assume that D 1 ⊇ D 2 and t 1 ( λ )[ x ] ≤ t 2 ( λ )[ x ] , x ∈ D 2 , λ ∈ ∆ . ❚❤❡♥ ♣ ✶ ✭ ① ✮ ✔ ♣ ✷ ✭ ① ✮ ❀ ① ✷ ❉ ✷ ❀ ❛♥❞ ❤❡♥❝❡ ✕ ✭✶✮ ❂ ♠✐♥ ① ✷ ▲ ♥❢ ✵ ❣ ♣ ✶ ✭ ① ✮ ✔ ♠❛① ♠✐♥ ① ✷ ▲ ♥❢ ✵ ❣ ♣ ✶ ✭ ① ✮ ♠❛① ♥ ▲ ✚❉ ✶ ▲ ✚❉ ✷ ❞✐♠ ▲ ❂ ♥ ❞✐♠ ▲ ❂ ♥ ① ✷ ▲ ♥❢ ✵ ❣ ♣ ✷ ✭ ① ✮ ❂ ✕ ✭✷✮ ✔ ♠✐♥ ♠❛① ✿ ♥ ▲ ✚❉ ✷ ❞✐♠ ▲ ❂ ♥

Comparison of two operator functions Let T 1 and T 2 be two operator functions as above and assume that D 1 ⊇ D 2 and t 1 ( λ )[ x ] ≤ t 2 ( λ )[ x ] , x ∈ D 2 , λ ∈ ∆ . Then p 1 ( x ) ≤ p 2 ( x ) , x ∈ D 2 , and hence λ (1) = min x ∈ L \{ 0 } p 1 ( x ) ≤ max min x ∈ L \{ 0 } p 1 ( x ) max n L ⊂D 1 L ⊂D 2 dim L = n dim L = n x ∈ L \{ 0 } p 2 ( x ) = λ (2) min max ≤ . n L ⊂D 2 dim L = n

Dropping the assumption that T is hyperbolic We relax the conditions that t ( α )[ x ] > 0 and that t ( · )[ x ] has exactly one zero. Assume that for each x ∈ D \ { 0 } and λ ∈ ∆: t ′ ( λ )[ x ] < 0 . t ( λ )[ x ] = 0 ⇒ ❆ ♠❛♣♣✐♥❣ ♣ ✿ ❉ ♥ ❢ ✵ ❣ ✦ R ❬ ❢✝✶❣ ✐s ❝❛❧❧❡❞ ❣❡♥❡r❛❧✐s❡❞ ❘❛②❧❡✐❣❤ ❢✉♥❝t✐♦♥❛❧ ❢♦r ❚ ✐❢ ❢♦r ❛❧❧ ① ✷ ❉ ♥ ❢ ✵ ❣ ✿ ✐❢ t ✭ ✕ ✵ ✮❬ ① ❪ ❂ ✵ ❀ ♣ ✭ ① ✮ ❂ ✕ ✵ ✐❢ t ✭ ✕ ✮❬ ① ❪ ❁ ✵ ♦♥ ✁ ❀ ♣ ✭ ① ✮ ❁ ☛ ✐❢ t ✭ ✕ ✮❬ ① ❪ ❃ ✵ ♦♥ ✁ ✿ ♣ ✭ ① ✮ ✕ ☞

Dropping the assumption that T is hyperbolic We relax the conditions that t ( α )[ x ] > 0 and that t ( · )[ x ] has exactly one zero. Assume that for each x ∈ D \ { 0 } and λ ∈ ∆: t ′ ( λ )[ x ] < 0 . t ( λ )[ x ] = 0 ⇒ A mapping p : D \ { 0 } → R ∪ {±∞} is called generalised Rayleigh functional for T if for all x ∈ D \ { 0 } : p ( x ) = λ 0 if t ( λ 0 )[ x ] = 0 , p ( x ) < α if t ( λ )[ x ] < 0 on ∆ , p ( x ) ≥ β if t ( λ )[ x ] > 0 on ∆ .

Triple variational principle A subspace M ⊂ D is called t ( λ )-non-negative if t ( λ )[ x ] ≥ 0 for all x ∈ M ; it is called maximal t ( λ )-non-negative if it is maximal with this property. Denote by M + α the set of all maximal t ( α )-non-negative subspaces of D . ❚❤❡♦r❡♠✳ ❬▼✳▲✳✱ ❙tr❛✉ss ✷✵✶✻❪ ❆ss✉♠❡ t❤❛t ❬ ☛❀ ☞ ✮ ❭ ✛ ❡ss ✭ ❚ ✮ ❂ ∅ ❀ ☛ ✷ ✚ ✭ ❚ ✮ ✿ ❚❤❡♥ t❤❡ s♣❡❝tr✉♠ ♦❢ ❚ ✐♥ ✭ ☛❀ ☞ ✮ ❝♦♥s✐sts ♦❢ ❡✐❣❡♥✈❛❧✉❡s✿ ✕ ✶ ✔ ✕ ✷ ✔ ✁ ✁ ✁ ❛♥❞ ✕ ♥ ❂ s✉♣ s✉♣ ✐♥❢ ♣ ✭ ① ✮ ✿ ▼✷ M ✰ ① ✷▼♥❢ ✵ ❣ ▲ ✚▼ ☛ ❞✐♠ ▲ ❂ ♥ � ✶ ① ❄ ▲ ✭❚❤❡ ✐♥❡q✉❛❧✐t② ❵ ✕ ✬ ✇❛s ♣r♦✈❡❞ ✐♥ ❬❊s❝❤✇✓ ❡✱ ❍✳ ▲❛♥❣❡r ✷✵✵✷❪ ✇❤❡♥ ❚ ✭ ✕ ✮ ❛r❡ ❜♦✉♥❞❡❞✳✮