Cyclic vectors in Dirichlet-type spaces

Transcription

1 Cyclic vectors in Dirichlet-type spaces Constanze Liaw (Baylor University) at TeXAMP 2013

2 This presentation is based on joint work with C. Bénéteau, A. Condori, D. Seco, A. Sola. Thanks to NSF for their support.

3 Broader Impacts of the problem of cyclicity Invariant subspace problem and cyclic vectors: Does every bounded operator T on a Hilbert space H have a non-trivial closed invariant subspace (i.e. T (W ) W )? NO, IF one can find an operator T such that every 0 ϕ H is cyclic (i.e. H = clos span{t n ϕ : n N}). Structure (basic building blocks) of a function space determined by its cyclic vectors Brown Shields conjecture For physicists, the cyclicity of an operator means that the spectrum has multiplicity one

4 One complex variable

5 Dirichlet-type spaces and cyclic vectors Consider the Dirichlet-type spaces D α, i.e. bounded analytic functions on the unit disk D C with norm f 2 D α = k=0 (k + 1)α a k 2 <, where f(z) = k=0 a kz k Bergman A 2 = D 1 ; Hardy H 2 = D 0 ; and Dirichlet D = D 1 A vector f is cyclic (under the forward shift) for D α if D α = span{z k f(z) : k N {0}} The constant function 1 is cyclic for D α f D α cyclic, implies f(z) 0 for z D The fewer zeros the easier is cyclicity.

6 Optimality Note f is cyclic in D α iff N n (f, α) := inf p n p n f 1 2 D α 0 as n If f(z) = 1 z, then p n = (order n Taylor poly. of 1/f) yields p n f 1 2 D α = n + 2 Two types of results: Optimal sequence of polynomials p n The optimal rate of decay of these norms N n (f, α) as n

7 Example of explicit optimal approximants For f(z) = 1 z, optimal for H 2 : C n (z) = D : R n (z) = A 2 : S n (z) = n k=0 n k=0 n k=0 ( 1 k n + 1 ( 1 H k+1 ( 1 H n+2 ) z k, ) z k, H n = k(k + 3) (n + 1)(n + 4) ) z k. n k=2 1 k,

8 Polynomials that have no zeros in D are cyclic in D α for α 1. Rate of decay Let H n = n k=2 1 k and note that H n log n for large n. Definition For α < 1, we set ϕ α (n) = n α 1, n N. For α = 1, we use ϕ 1 (n) = 1/H n, n N. Theorem (Bénéteau Condori L. Seco Sola, J. d A. accepted) Suppose f D α, α 1, can be extended analytically to some strictly bigger disk. Suppose also that f does not vanish in D. Then there exists a constant C 0 so that the optimal norm satisfies N n (f, α) C 0 ϕ α (n + 1). Moreover, for polynomial f with zero on T, and α = 1, 0, 1, there is a constant C 1 so that C 1 ϕ α (n + 1) N n (f, α).

9 Partial result on the Brown Shields conjecture

10 Outer Vectors in H 2 are cyclic iff they are outer For α 0: If f cyclic in D α, then f outer Logarithmic capacity Non-tangentially f (ζ) = lim z ζ T f(z) For f D, f exists outside a set of logarithmic capacity zero Zero set Z(f) = {ζ T : f (ζ) = 0} Brown Shields: If f D is cyclic, then Z(f) has capacity zero Brown Shields Conjecture (1984) A vector f D is cyclic iff it is outer and has Z(f) capacity zero. Brown Cohn: For any closed set of logarithmic capacity zero E T, there exists a cyclic function f in D with Z(f) = E.

11 Two weak versions of the Brown Shields conjecture: Theorem (Hedenmalm Shields 1990, Richter Sundberg 1994) A vector f D is cyclic, if it is outer and Z(f) is countable. Theorem (El-Fallah Kellay Ransford 2006) The condition countable can be replaced by one which is closer to capacity zero, but VERY complicated.

12 Theorem (Bénéteau Condori L. Seco Sola, J. d A. accepted) Suppose f D and log f D. Then f is cyclic in D. Theorem (Bénéteau Condori L. Seco Sola, J. d A. accepted) Let f H and q = log f D α, α 1. Suppose there exist polynomials q n of degree n that approach q in D α norm with sup Re(q(z) q n (z)) + log q n q C z D for some constant C > 0. Then f is cyclic in D α. Brown Cohn s examples satisfy above assumptions.

13 Two complex variables

14 Dirichlet-type space on the bidisk Bidisk D 2 = {(z 1, z 2 ) C 2 : z 1 < 1, z 2 < 1} Holomorphic f : D 2 C belongs to the Dirichlet-type space D α if its power series f(z 1, z 2 ) = k=0 l=0 a k,lz k 1 zl 2 satisfies f 2 α = k=0 l=0 Function f D α is cyclic, if (k + 1) α (l + 1) α a k,l 2 < D α := span{z1 kzl 2f : k = 0, 1,... ; l = 0, 1,...} Let P n, n N, be the polynomials of the form p n = n k=0 l=0 n c k,l z1 k z2 l f is cyclic iff N n (f, α) := inf pn P n p n f 1 2 D α n 0

15 Reductions to functions of one variable

16 Reduction to functions of one variable Consider J α,m,n := { f D α : f = e.g. f(z 1, z 2 ) = 1 z 1 z 2 J α,1,1 Consider the mappings k=0 a k z1 Mk z2 Nk L M,N : D 2α D α via L M,N (F )(z 1, z 2 ) = F (z M 1 zn 2 ), R M,N : J α,m,n D 2α via R M,N (f)(z) = f(z 1/M, 1) If f J α,m,n, there exist constants such that }, c 2 R(f) D2α f α c 1 R(f) D2α Note the change from D α for bidisk to D 2α for disk!

17 Theorem (Bénéteau Condori L. Seco Sola, submitted 2013) Let f J α,m,n have the property that R(f) = f(z 1/M, 1) is a function that admits an analytic continuation to the closed unit disk, whose zeros lie in C \ D. Then f is cyclic in D α, and there exists a constant C = C(α, f, M, N) such that N n (f, α) Cϕ 2α (n + 1). This result is sharp in the sense that, if R(f) has at least one zero on T, then there exists c = c(α, f, M, N) such that for large n: cϕ 2α (n + 1) N n (f, α). { } n 2α 1 for 2α < 1 Here ϕ 2α (n) = 1/ n k=2 1 increases if α > 1/2. k for 2α = 1

18 Examples Functions like f(z 1, z 2 ) = 1 z 1, f(z 1, z 2 ) = (1 z 1 z 2 ) N, N N, and f(z 1, z 2 ) = z 2 1 z2 2 2(cos θ)z 1z 2 + 1, θ R, satisfy the assumptions of the theorem Polynomial g(z 1, z 2 ) = 1 z 1 z 2 is not cyclic in D α for α > 1/2, although it is only zero for z 1 = z 2 = 1 Notice that g is outer, but its zero set {z 1 = z 2 = 1} has non-zero logarithmic capacity

19 Open problems The Brown-Shields conjecture for functions on the bidisk: Is the condition that f D is outer and the zero set of f (on the boundary) has logarithmic capacity 0 sufficient for f to be cyclic? Sub-problem: Characterize the cyclic polynomials f D α for each α 1.