Compact symetric bilinear forms

Transcription

1 Compact symetric bilinear forms Mihai Mathematics Department UC Santa Barbara IWOTA 2006

2 IWOTA 2006 Compact forms [1] joint work with: J. Danciger (Stanford) S. Garcia (Pomona College) B. Gustafsson (KTH Stockholm) E. Prodan (Princeton) V. Prokhorov (U. So. Alabama) H.S. Shapiro (KTH Stockholm)

3 IWOTA 2006 Compact forms [2] Sources Takagi, T., On an algebraic problem related to an analytic theorem of Caratheodory and Fejer and on an allied theorem of Landau, Japan J. Math. 1 (1925), Friedrichs, K., On certain inequalities for analytic functions and for functions of two variables, Trans. Amer. Math. Soc. 41 (1937), Glazman, I. M., An analogue of the extension theory of hermitian operators and a non-symmetric one-dimensional boundary-value problem on a half-axis, Dokl. Akad. Nauk SSSR 115(1957), and numerous more recent ramifications

4 IWOTA 2006 Compact forms [3] Main problems Spectral analysis of the Friedrichs form [f, g] = fgda, G f, g L 2 a(g); Asymptotics of the singular numbers of Hankel operators on multiply connected domains; Green function estimates for certain Schrödinger operators

5 IWOTA 2006 Compact forms [4] Tools Hilbert space with a bilinear symmetric (compact) form [x, y] complex symmetric operators T w.r. to [x, y] a refined polar decomposition for T an abstract AAK theorem for compact T s some analysis of complex variables and potential theory

6 IWOTA 2006 Compact forms [5] Friedrichs operator G bounded planar domain da Area measure L 2 a(g) Bergman space w.r. to da P : L 2 (G) L 2 a(g) Bergman projection The form [f, g] = defines Friedrichs operator G fgda = f, F g, F g = P (g). f, g L 2 a(g)

7 IWOTA 2006 Compact forms [6] Angle operator F is anti-linear, but S = F 2 is complex linear, s.a., f, Sf = F f 2, and Sf = P CP Cf, where C is complex conjugation in L 2. S is the angle operator between L 2 a(g) and CL 2 a(g). Side remark: for h H (G), denote by T h the Toeplitz operator on L 2 a(g). Then F T h = T hf.

8 IWOTA 2006 Compact forms [7] Compactness Assume that G has real analytic, smooth boundary (not necessarily simply connected). One can write z = S(z) with S analytic in a neighborhood of G. Thus where suppµ G. = 2i[f, g] = G Therefore S = F 2 is compact. G f(z)g(z)dz dz = f(z)g(z)s(z)dz = fgsdµ,

9 IWOTA 2006 Compact forms [8] Corners Assume that G has piece-wise smooth boundary, with finitely many corners, of interior angles 0 < α k 2π. Friedrichs: For every k, sin α k 2 σ ess (S). α k Moreover, there exists a constant c(g) < 1: for every f L 2 a(g), G fda = 0, G f 2 da 2 c(g) G f 2 da.

10 IWOTA 2006 Compact forms [9] Inverse spectral problem P.-Shapiro: There exists a continuous family of planar domains with unitarily equivalent Friedrichs operators, and such that no two domains are related by an affine transform. The measure µ in the example has three atoms.

11 IWOTA 2006 Compact forms [10] Asymptotics P.-Prokhorov: Assume G real analytic and smooth, and let λ k = (λk (S)) λ k+1 be the eigenvalues of F. Then and lim sup(λ 0 λ 1... λ n ) 1/n2 exp( 1/C( G, suppµ)), n lim sup n λ 1/n n exp( 1/C( G, suppµ)) lim inf n λ1/n n exp( 2/C( G, suppµ)), where C(.,.) is the (Green) capacitor of the two sets.

12 IWOTA 2006 Compact forms [11] Boundary forms Assume that Γ = G is real analytic, smooth. with Smirov class projections P ±. For a C(Γ) the Hankel operator L 2 (Γ) = E 2 (G) E 2 (G) H a f = P (af), is well defined.

13 IWOTA 2006 Compact forms [12] Approximation theory questions lead to with suppµ compact in G. a(z) = 1 2πi dµ(w) z w, Let R : E 2 (G) L 2 (µ) be the restriction operator. Then H a 2 = (R CR) 2 and asymptotics (as in the planar case) can be derived with a similar proof.

14 IWOTA 2006 Compact forms [13] Hilbert space part in the proof Double orthogonal system of vectors (u n ): [u k, u n ] = λ n δ kn, u k, u n = δ kn, where one can choose a positive spectrum: λ 0 λ Weyl-Horn inequality: For every system of vectors g 0,..., g n one has det([g k, g n ]) λ 0...λ n det( g k, g n ).

15 IWOTA 2006 Compact forms [14] Min-Max Danciger: Assume [.,.] is a compact bilinear symmetric form in a complex Hilbert space H. Let σ 0 σ 1 σ 2 0 be the singular values, repeated according to multiplicity. Then min max R[x, x] = σ 2n codimv =n x V x =1 σ n = 2 min codimv =n max R[x, y] (x,y) V (x,y) =1 for all n 0. Here V denotes a C-linear subspace of H H.

16 IWOTA 2006 Compact forms [15] Abstract Friedrichs Inequality Danciger-Garcia-P.: Same conditions: [.,.] compact, with spectrum σ k and eigenvalues u k. Then [x, x] σ 2 x 2 whenever x is orthogonal to the vector σ 1 u 0 + i σ 0 u 1. Furthermore, the constant σ 2 is the best possible for x restricted to a subspace of H of codimension one.

17 IWOTA 2006 Compact forms [16] The ellipse Ω t - the interior of the ellipse where t > 0 is a parameter. Quadrature identity x 2 cosh 2 t + Ω t f(z) da(z) = (sinh 2t) y2 sinh 2 t < 1, 1 1 f(x) 1 x 2 dx

18 IWOTA 2006 Compact forms [17] hence fgda = (sinh 2t) Ω t 1 1 f(x)g(x) 1 x 2 dx. Singular values σ n (t) and normalized (in L 2 a(ω t )) singular vectors e n : σ n (t) = e n (z) = (n + 1) sinh 2t sinh[2(n + 1)t] 2n + 2 π sinh[2(n + 1)t] U n(z) where U n denotes the nth Chebyshev polynomial of the second kind.

19 IWOTA 2006 Compact forms [18] Since U 0 = 1, U 1 = 2z, and σ 2 = 3 sinh 2t sinh 6t, we obtain σ1 e 0 (z) i σ 0 e 1 (z) = 2 π sinh 4t (1 2iz), then the inequality ( ) f 2 da 3 sinh 2t f 2 da Ω t sinh 6t Ω t holds whenever f (1 2iz). Furthermore, the preceding inequality is the best possible that can hold on a subspace of L 2 a(ω t ) of codimension one.

20 IWOTA 2006 Compact forms [19] Takagi s work Original approach to the Carathéodory-Fejér problem; leads to the form B(f, g) = (ufg)(n) (0), f, g H 2 (T). n! where u(z) = c 0 + c 1 z + + c n z n is a prescribed Taylor polynomial at the origin.

21 IWOTA 2006 Compact forms [20] Extremal problem There exists an analytic function F in the unit disk such that and F M if and only if F (z) = c 0 + c 1 z + + c n z n + O(z n+1 ) max f 2 =1 1 n! (uf 2 ) (n) (0) M, where f is a polynomial of degree n and f 2 denotes the l 2 -norm of its coefficients.

22 IWOTA 2006 Compact forms [21] C-symmetric operators Let H be a Hilbert space with an anti-linear conjugation C, which is isometric: Cx = x, C 2 = I. An operator T is called C-symmetric, if T C = CT, i.e. T is symmetric w.r. to the form {x, y} = x, Cy. Our examples were of the form [x, y] = T x, Cy = {T x, y}.

23 IWOTA 2006 Compact forms [22] Polar decomposition Garcia-P.: T L(H) is C-symmetric if and only if T = CJ T where J is another isometric anti-linear conjugation which commutes with T. Refines f. dim. decompositions of Takagi and Schur, and infinite dim. ones of Godic-Lucenko.

24 IWOTA 2006 Compact forms [23] Compact C-sym. operators admits J-invariant solutions, hence T u k = σ k u k J T u k = σ k u k, and T u k = CJ T u k = σ k Cu k. T = max{λ 0; there exists x 0, T x = λcx}.

25 IWOTA 2006 Compact forms [24] C-symmetric approximants If T = CJ T is compact C-symmetric, then choose F n 0, F n T = T F n, of rank n + 1, so that T F n σ n+1 = σ n+1 ( T ). Then T n = CJF n is C-symmetric, and T T n σ n+1.

26 IWOTA 2006 Compact forms [25] Unbounded operators H with isometric conjugation C T : D(T ) H closed graph, densely defined is C-symmetric, if D(T ) CD(T ) and CT C T. For instance Schrödinger operators with complex potentials, or certain PDE s with non-symmetric boundary values are C-symmetric.

27 IWOTA 2006 Compact forms [26] Example Let q(x) be a real valued, continuous, even function on [ 1, 1] and let α be a nonzero complex number satisfying α < 1. For a small parameter ɛ > 0, we define the operator with domain [T α f](x) = if (x) + ɛq(x)f(x), D(T α ) = {f L 2 [ 1, 1] : f L 2 [ 1, 1], f(1) = αf( 1)}.

28 IWOTA 2006 Compact forms [27] If C denotes the conjugation operator [Cu](x) = u( x) on L 2 [ 1, 1], then it follows that that nonselfadjoint operator T α satisfies T α = CT 1/α C and T α = T 1/α and hence T α is a C-selfadjoint operator. Takagi s anti-linear equation (T α λ)u n = σ n Cu n, will give, for n = 0 the norm of the resolvent of T α.

29 IWOTA 2006 Compact forms [28] Application 2 D Laplace operator with zero (Dirichlet) boundary conditions over a finite domain (with smooth boundary) Ω R d. v(x) 0 be a scalar potential, which is 2 D-relatively bounded, with relative bound less than one. H : D( 2 D) L 2 (Ω); H = 2 D + v(x), the associated selfadjoint Hamiltonian with compact resolvent.

30 IWOTA 2006 Compact forms [29] Assumption on H: its energy spectrum σ consists of two parts, σ [0, E ] [E +, ), which are separated by a gap G E + E > 0. Let E (E, E + ) and G E = (H E) 1 average Ḡ E (x 1, x 2 ) 1 ωɛ 2 dx be the resolvent and take the dy G E (x, y), x x 1 ɛ y x 2 ɛ where ω ɛ is the volume of a sphere of radius ɛ in R d.

31 IWOTA 2006 Compact forms [30] Green function estimate Garcia-Prodan-P.: For q smaller than a critical value q c (E), there exists a constant C q,e, independent of Ω, such that: C q,e is given by C q,e = F (q, E) = ḠE(x 1, x 2 ) C q,e e q x 1 x 2. ωɛ 1 e 2qɛ min E ± E q q/f (q,e) with (E + E q 2 )(E E + q 2 ) 4E. The critical value q c (E) is the positive solution of the equation q = F (q, E).