Hardy spaces of slit domains

Transcription

1 Lund, Washington and Lee, Richmond

2 Set up: Ω is a bounded domain in C H 2 (Ω) is the Hardy space on Ω S : H 2 (Ω) H 2 (Ω), (Sf )(z) = zf (z) Problem: Describe the S-invariant subspaces of H 2 (Ω).

3 Ω = D: Theorem (Beurling) If M is a subspace (closed linear manifold) of H 2 (D) with SM M, then M = ΘH 2 (D), where Θ is a D-inner function. Θ is D-inner means Θ H (D) and Θ = 1 a.e. Similar result when Ω = ins(γ), where γ is a closed smooth (enough) curve

4 Ω = A = {r < z < R} Lat(S, H 2 (A)) described by Sarason, Royden, Hitt Ω = ins(γ) \ n j=1 ins(γ j) Lat(S, H 2 (Ω)) described by Yakubovich, Aleman-Richter Ω is a crescent domain Lat(S, H 2 (Ω)) described by Aleman and Olin

5 Aleman-Feldman-R consider the slit disk G := D \ [0, 1].

6 For f H 2 (G), f + (x) := lim f (x + iy), a.e. x [0, 1] y 0 + f (x) := lim f (x + iy), a.e. x [0, 1] y 0 f (ζ) := lim r 1 f (rζ), a.e. ζ D

7 If ψ : G D, then f 2 H 2 (G) = 1 0 ( f f 2) ψ dx 2π 2π + f 2 ψ dθ 0 2π ψ (ξ) ξ 1/2 ξ 1, ξ G

8 Lemma Suppose M is a H (G)-invariant subspace of H 2 (G). Then M = ΘH 2 (G), where Θ is a G-inner function. Θ is G-inner means Θ φ is D-inner, where φ : D G.

9 There are many other S-invariant subspaces of H 2 (G). Ex: ρ : [0, 1] C be measurable E a measurable subset of [0, 1]. M(ρ, E) := {f H 2 (G) : f + = ρf a.e. on E} M(ρ, E) is closed (since norm is an integral on G) and is S-invariant. M(ρ, E) is not always ΘH 2 (G)

10 Perhaps every S-invariant subspace of H 2 (G) is M = ΘM(ρ, E) ρ : [0, 1] C, E [0, 1], Θ G-inner M(ρ, E) := {f H 2 (G) : f + = ρf a.e. on E}

11 Lemma For f H 2 (G) TFAE: f M(1, [0, 1]) f has an AC across [0, 1] Lemma ball(m(1, [0, 1])) is a normal family of analytic functions on D.

12 Ex: Let M 0 := {f M(1, [0, 1]) : f (0) = 0}. M 0 is closed and S-invariant M 0 ΘM(ρ, E) Θ can not account for the zero at z = 0 since 0 G M 0 = {f H 2 (G) : f z H2 (G), f + = 1f a.e. on [0, 1]} F (z) = z is G-outer

13 Fix ɛ (0, 1) and let G ɛ := D \ [ ɛ, 1]. Theorem Let M be a non-trivial invariant subspace of H 2 (G) with greatest common G-inner divisor Θ M. Then there exists a a measurable set E [0, 1], a measurable function ρ : [0, 1] C a G ɛ -outer function F ɛ, such that M = Θ M { f H 2 (G) : } f H 2 (G ɛ ), f + = ρf a.e. on E F ɛ

14 { M = Θ M f H 2 (G) : } f H 2 (G ɛ ), f + = ρf a.e. on E F ɛ RHS is not necc closed for randomly chosen F ɛ F ɛ is not unique Θ M, ρ, E are (essentially) unique

15 M Lat(S, H 2 (G)) is cyclic if there is an f so that [f ] = {z n f : z N 0 } = M.

16 Is every M cyclic? Theorem If { M = Θ M f H 2 (G) : } f H 2 (G ɛ ), f + = ρf a.e. on E, F ɛ and then M is not cyclic. Converse is not true. m 1 ([0, 1] \ E) > 0

17 What is [f ]? Theorem If f and 1/f H 2 (G), then where ρ = f + /f. [f ] = {f H 2 (G) : f + = ρf a.e. on [0, 1]},

18 Is every M 2-cyclic? Theorem If M Lat(S, H 2 (G)), then there f, g H 2 (G) such that M = [f, g] := {z n f, z m g : m, n N 0 }. f, g can be chosen to be solutions to two extremal problems (just like in Beurling s theorem).

19 When is [f, g] = H 2 (G)? Theorem If f, g H 2 (G) \ {0}, then [f, g] = H 2 (G) if and only if f and g have no non-trivial common G-inner factor and the set { x [0, 1) : f + (x) f (x) = g + } (x) g (x) has Lebesgue measure zero.

20 VOTCAM Virginia Operator Theory and Complex Analysis Meeting Saturday - November 10, University of Richmond Vern Paulsen (Houston) Warren Wogen (Chapel Hill) David Sherman (Virginia) Leiba Rodman (William and Mary)