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Linear Differential Equations and Functions of Operators

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Linear Differential Equations and Functions of Operators Linear differential equations and functions of operators Andreas Ros´en (formerly Axelsson) Link¨opingUniversity February 2011 Andreas Ros´en (Link¨opingUniversity) Diff. equations & functions of operators February 2011 1 / 26 The simplest example 7 −9 Given initial data x(0) 2 C2 and coefficient matrix A := , solve 3 −5 the linear system of first order ODEs x0(t) + Ax(t) = 0 for x : R ! C2: 3 1 Coordinates in eigenbasis y(t) := V −1x(t), where V := , give 1 1 decoupled equations 4 0 y 0(t) + y(t) = 0 for y : R ! C2: 0 −2 e−4t 0 Solution to the initial ODE is x(t) = V V −1x(0): 0 e2t Definition 4 0 e−4t 0 A = V V −1 gives e−tA := V V −1. 0 −2 0 e2t Andreas Ros´en (Link¨opingUniversity) Diff. equations & functions of operators February 2011 2 / 26 Functional calculus through diagonalization of matrices A generic matrix A 2 Cn×n is diagonalizable, −1 A = V diag(λ1; : : : ; λn) V ; for some invertible \change-of-basis" matrix V 2 Cn×n. Definition For a function φ : σ(A) = fλ1; : : : ; λng ! C defined on the spectrum of A, define the matrix −1 φ(A) := V diag φ(λ1); : : : ; φ(λn) V For fixed A, the map φ 7! φ(A) from symbol φ : σ(A) ! C to matrix φ(A) 2 Cn×n is an algebra-homomorphism: (φψ)(A) = φ(A) (A): −tA −tA −sA −(t+s)A Example: fe gt2R is a group of matrices, e e = e . Andreas Ros´en (Link¨opingUniversity) Diff. equations & functions of operators February 2011 3 / 26 1 Functions of matrices 2 Functions of linear Hilbert space operators 3 Spectral projections and the Kato conjecture 4 Operational calculus and maximal regularity Andreas Ros´en (Link¨opingUniversity) Diff. equations & functions of operators February 2011 4 / 26 Functional calculus for general matrices For general A 2 Cn×n there are natural definitions of matrices A2; A3; A4;:::; (λI − A)−1; λ2 = σ(A): We want to define in a natural way a matrix φ(A) for any φ 2 H(σ(A)) := fφ :Ω ! C holomorphic ;Ω ⊃ σ(A) openg: Definition For a function φ 2 H(σ(A)), define the matrix 1 Z φ(A) := φ(λ)(λI − A)−1dλ, 2πi γ where γ is a closed curve in Ω counter clockwise around σ(A). The holomorphic functional calculus of A is the map H(σ(A)) 3 φ 7! φ(A) 2 Cn×n. Andreas Ros´en (Link¨opingUniversity) Diff. equations & functions of operators February 2011 5 / 26 Properties of the holomorphic functional calculus 1 The holomorphic functional calculus of A is an algebra-homomorphism and its range is a commutative subalgebra of Cn×n: φ(A) (A) = (φψ)(A) = ( φ)(A) = (A)φ(A): 2 For polynomials we have (zk )(A) = Ak ; for k = 0; 1; 2;::: 3 If H(σ(A)) 3 φk ! φ uniformly on compact subsets of Ω, then φk (A) ! φ(A). 4 For diagonal matrices we have φ diag(λ1; : : : ; λn) = diag(φ(λ1); : : : ; φ(λn)): 5 For any invertible \change-of-basis" matrix V , we have φ(VAV −1) = V φ(A)V −1. Andreas Ros´en (Link¨opingUniversity) Diff. equations & functions of operators February 2011 6 / 26 Example: a Jordan block Fix a 2 C and consider A = aI + N, where N is nilpotent: Nm = 0 for some m. Spectrum is σ(A) = fag and m−1 1 1 1 X Nk = = : λI − (aI + N) λ − a I − N=(λ − a) (λ − a)k+1 k=0 Cauchy's formula for derivatives gives m−1 m−1 X 1 Z φ(λ) X φ(k)(a) φ(A) = dλ Nk = Nk : 2πi (λ − a)k+1 k! k=0 γ k=0 For example 02 31 2 0 1 00 1 (3) 3 3100 φ(3) φ (3) 2 φ (3) 6 φ (3) B60 310 7C 6 0 φ(3) φ0(3) 1 φ00(3) 7 φ B6 7C = 6 2 7 : @40 0 31 5A 4 0 0 φ(3) φ0(3) 5 0 0 0 3 0 0 0 φ(3) Andreas Ros´en (Link¨opingUniversity) Diff. equations & functions of operators February 2011 7 / 26 Generalization to bounded operators on Hilbert spaces Letting n ! 1, we extend the above holomorphic functional calculus from matrices A 2 Cn×n; to bounded linear operators T : H!H on Hilbert space H (or more generally to Banach space operators). The Dunford (or Riesz{Dunford or Dunford{Taylor) integral 1 Z φ(T ) := φ(λ)(λI − T )−1dλ 2πi γ defines a bounded linear operator φ(T ): H!H. Spectrum σ(T ) ⊂ C is compact (but typically not finite). Symbol φ :Ω ! C is holomorphic on open neighbourhood Ω ⊃ σ(T ). Closed curve γ ⊂ Ω encircles σ(T ) counter clockwise. Andreas Ros´en (Link¨opingUniversity) Diff. equations & functions of operators February 2011 8 / 26 Example: self-adjoint operators Let T : H!H be self-adjoint: hTf ; gi = hf ; Tgi, f ; g 2 H. The spectral theorem shows −1 T = VMλV ; where V : L2(X ; dµ) !H is a bijective isometry, with a Borel measure dµ on a σ-compact space X , and Mλ : L2(X ; dµ) ! L2(X ; dµ) is multiplication by a real-valued function λ 2 L1(X ; dµ): (Mλf )(x) := λ(x)f (x) for f 2 L2(X ; dµ) and a.e. x 2 X . Then 1 the spectrum σ(T ) ⊂ R is the essential range of λ : X ! R, and 2 φ(T ) is similar to multiplication by φ(λ(x)), or more precisely −1 φ(T ) = VMφ◦λV : Andreas Ros´en (Link¨opingUniversity) Diff. equations & functions of operators February 2011 9 / 26 A boundedness conjecture Even though φ is assumed to be holomorphic on a neighbourhood of σ(T ) in the definition of φ(T ), it is natural to ask whether φ(T ) only depends on φjσ(T ) ? Do we have the estimate kφ(T )kH!H ≤ sup jφ(λ)j ? λ2σ(T ) For self-adjoint operators we have kT kH!H = kMφ◦λkL2!L2 = sup jφ(λ)j: λ2σ(T ) Andreas Ros´en (Link¨opingUniversity) Diff. equations & functions of operators February 2011 10 / 26 The conjecture is false! The estimate kφ(T )kH!H ≤ sup jφ(λ)j λ2σ(T ) cannot be true for general non-selfadjoint operators, since changing to an equivalent norm on H may change the RHS, but not the LHS (not σ(T )!). Example 0 1 A simple counter example is A = : We have A2 = 0, so 0 0 σ(A) = f0g. However, φ(0) φ0(0) A = : 0 φ(0) We cannot bound A (φ(0) and φ0(0)) by φ(0), uniformly in φ. Andreas Ros´en (Link¨opingUniversity) Diff. equations & functions of operators February 2011 11 / 26 A classical positive result The estimate kφ(T )kH!H ≤ sup jφ(λ)j λ2σ(T ) holds if we replace the spectrum σ(T ) in the RHS by the disk fjλj ≤ kT kg⊃ σ(T ): The following is a classical result by J. von Neumann (Eine Spektraltheorie f¨urallgemeine Operatoren eines unit¨arenRaumes. Math. Nachr., 1951). Theorem (von Neumann) Let T : H!H be a bounded linear operator on a Hilbert space H. Then kφ(T )kH!H ≤ sup jφ(λ)j jλ|≤kT k holds for all φ holomorphic on some neighbourhood of jλj ≤ kT k. Andreas Ros´en (Link¨opingUniversity) Diff. equations & functions of operators February 2011 12 / 26 A sharpening of von Neumann's theorem M. Crouzeix (Numerical range and functional calculus in Hilbert space. J. Funct. Anal., 2007) has proved the following. Theorem (Crouzeix) Let W (T ) := fhTf ; f i ; kf k = 1g be the numerical range of a linear Hilbert space operator T : H!H. Then kφ(T )kH!H ≤ 11:08 supλ2W (T ) jφ(λ)j holds uniformly for all φ holomorphic on some neighbourhood of W (T ). Hausdorff–Toeplitz theorem: W (T ) is a convex set. σ(T ) ⊂ W (T ) ⊂ fjλj ≤ kT kg The second inclusion quite sharp: kT k ≤ 2 supfjλj ; λ 2 W (T )g. For a normal( A∗A = AA∗) matrix A, W (A) is the convex hull of σ(A). Andreas Ros´en (Link¨opingUniversity) Diff. equations & functions of operators February 2011 13 / 26 Spectral projections: a simple example Often one needs to apply φ which are not holomorphic on a convex neigbourhood of σ(T ) (like W (T )). Example 23 0 03 Let A := 40 7 15, so that σ(A) = f3; 7g (algebraic multiplicity 2 for 0 0 7 ( 20 0 03 0; jλ − 3j < 1; λ = 7). For φ(λ) = we have φ(A) := 4010 5 being 1; jλ − 7j < 1; 0 01 the spectral projection onto the two-dimensional generalized eigenspace at λ = 7. Von Neumann's and Crouzeix's theorems do not apply, but still kφ(A)k ≤ C supfjφ(λ)j ; jλ − 3j < 1 or jλ − 7j < 1g is clear from the Dunford integral! Andreas Ros´en (Link¨opingUniversity) Diff. equations & functions of operators February 2011 14 / 26 Partial differential equations (PDEs) Applications to PDEs typically involves functional calculus of unbounded n differential operators D on a space like H = L2(R ) (rather than bounded operators T ). Pn 2 Consider the positive Laplace operator −∆ = − 1 @k ; with spectrum σ(−∆) = R+. The heat equation @t ft − ∆ft = 0 has solution −t(−∆) ft = e f0; t > 0: 2 The wave equation @t ft − ∆ft = 0 has solution p p sin(t −∆) ft = cos(t −∆)f0 + p (@t f )0; t 2 R: −∆ n For functions ft (x) = f (t; x) we write x 2 R for the space variable and t for the time/evolution variable.
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