HANKEL OPERATORS AND INVARIANT SUBSPACES OF THE DIRICHLET SPACE SHUAIBING LUO AND STEFAN RICHTER Abstract. The Dirichlet space D is the space of all analytic func- tions f on the open unit disc D such that f 0 is square integrable with respect to two-dimensional Lebesgue measure. In this paper we prove that the invariant subspaces of the Dirichlet shift are in 1-1 correspondence with the kernels of the Dirichlet-Hankel oper- ators. We then apply this result to obtain information about the invariant subspace lattice of the weak product D D and to some questions about approximation of invariant subspaces of D. Our main results hold in the context of superharmonically weighted Dirichlet spaces. 1. Introduction If f 2 Hol(D) denotes an analytic function on the open unit disc D, then we use f^(n) for its nth Taylor coefficient. We write H2 2 for the Hardy space of the unit disc, it has norm given by kfkH2 = R 2 jdzj P1 ^ 2 jzj=1 jf(z)j 2π = n=0 jf(n)j . In this paper we will consider weighted Dirichlet spaces of the form Z 0 2 H = ff 2 Hol(D): jf (z)j U(z)dA(z) < 1g; D where U is a non-negative superharmonic function on D and dA denotes 2-dimensional Lebesgue measure. Particular examples of such weights are U(z) = (1 − jzj2)1−α for 0 < α ≤ 1. By the representation theorem for superharmonic functions (see [12], page 109) such weights can be represented by use of a finite Borel measure µ on the closed unit disc. We write Z Z 2 1 − wz dµ(w) 1 − jzj Uµ(z) = log 2 + 2 dµ(w); z 2 D; jwj<1 z − w 1 − jwj jwj=1 j1 − wzj Date: January 7, 2015. 2000 Mathematics Subject Classification. Primary 47B32, 47B35; Secondary 30H99. Work of the second author was supported by the National Science Foundation, grant DMS-0901642. Part of this work appeared in the first author's dissertation. 1 2 LUO AND RICHTER then the correspondence µ ! Uµ is a bijection of the collection of finite positive measures on D onto the set of positive superharmonic functions on D, and it is a theorem of Aleman, [1], Theorem IV.1.9 that for all f 2 H2 we have Z Z 0 2 (1.1) jf (z)j Uµ(z)dA(z) = Dw(f)dµ(w); D D R jf(z)−f(w)j2 jdzj where Dw(f) = jzj=1 jz−wj2 2π is called the local Dirichlet integral of f at w 2 D. Here we follow the convention that Dw(f) = 1, if f does not have a nontangential limit at w 2 @D. We define Z 2 D(µ) = ff 2 H (D): Dw(f)dµ(w) < 1g D 2 2 R with norm kfk = kfk 2 + Dw(f)dµ(w). In the special case where H D µ = m is normalized linear Lebesgue measure on the unit circle we obtain Um ≡ 1 and D(m) = D, the classical Dirichlet space. In this case we have 1 Z Z dA X kfk2 = jfj2dm + jf 0j2 = (n + 1)jf^(n)j2; π @D D n=0 Note also that D(µ) = H2 when µ = 0. For measures µ that are supported in @D these spaces arose in [16]. The general case was con- sidered in [1], [22], and [19], where all the basic results about D(µ) can be found. For any space of analytic functions B on the open unit disc D we de- note by (Mz; B) the linear transformation defined by (Mzf)(z) = zf(z) and we use Lat(Mz; B) to denote the collection of invariant subspaces of (Mz; B). We write M(B) for the set of multipliers of B, M(B) = f' 2 Hol(D): 'f 2 B for all f 2 Bg and for ' 2 M(B) we use M' 2 B(B) for the corresponding mul- tiplication operator, f ! 'f. Furthermore for ' 2 M(B) we write k'kM = kM'k for the multiplier norm. 2 2 2 Let H = H ;D(µ), or La = L (dA) \ Hol(D), the Bergman space. 2 By Beurling's theorem we have a precise knowledge of Lat(Mz;H ), 2 the Beurling lattice. Indeed, if (0) 6= M 2 Lat(Mz;H ), then M = 'H2 for some inner function ', i.e. ' is in the unit ball of H1 and satisfies j'(eit)j = 1 a.e., [5]. Less is known about the invari- ant subspaces of the Dirichlet and Bergman shifts, but it is well- established that the Bergman lattice differs in structure substantially HANKEL OPERATORS AND INVARIANT SUBSPACES 3 from the Beurling lattice, [2], while there are a number of similari- 2 ties between Lat(Mz;H ) and Lat(Mz;D(µ)), [15, 18, 1]. We refer the reader to [9] for a nice overview of what is currently known about the Dirichlet lattice Lat(Mz;D). The analogues of inner functions play an important role in the invariant subspace theory of both the Bergman and Dirichlet spaces. For (0) 6= M 2 Lat(Mz; H) we let n = inffk : 9f 2 M with f (k)(0) 6= 0g. Then the extremal problem supfRef (n)(0) : f 2 M; kfk ≤ 1g has a unique solution, which will be called the extremal function for M. In the case H = H2 the extremal function for M is the inner function satisfying M = 'H2. It is easy to check that up to a multiplicative constant of modulus one extremal n functions ' in H are characterized by hz '; 'i = δ0n, where δ0n = 1 for n = 0 and = 0 for n > 0. The main result that we want to bring to the attention of the reader is a new analogy between the Beurling and Dirichlet lattices that is based on the theory of Hankel operators H! H. Here we have written H = ff : f 2 Hg for the space of complex conjugates of H. This is a Hilbert space with inner product hf; giH = hg; fiH, f; g 2 H. As in [4] or [19] we define X (H) = fb 2 H : 9 C > 0 jh' ; bij ≤ Ck'kk k; 8'; 2 Hol(D)g: Note that for every b 2 X (H) the map ('; ) ! h' ; bi extends to be a bounded sesquilinear form on H × H. Thus with each b 2 X (H) we may associate the Hankel operator Hb 2 B(H; H), hHb'; iH = h' ; biH; '; 2 Hol(D): If H = H2 our definition of Hankel operator differs by a rank 1 operator from the common definition as operator H2 ! H2? ⊆ L2(@D). For 2 H = La our definition coincides with what is typically referred to as "little Hankel operator". Carleson measures can be used to describe X (H) in the cases where 2 2 H = H ;D, or La. Recall that a positive measure µ on D is called a Carleson measure for H, if there is a C > 0 such that R jpj2dµ ≤ D Ckpk2 for all polynomials p. Then it is well-known that X (H2) = BMOA = fb 2 H2 : jb0j2(1 − jzj2)dA is a Carleson measure for H2g, 2 [8]. Similarly, X (La) is the Bloch space, and also 2 0 2 2 2 2 X (La) = fb 2 D : jb j (1 − jzj ) dA is a Carleson measure for Lag; see e.g. [24]. Furthermore, in [4] it was shown that X (D) = fb 2 D : jb0j2dA is a Carleson measure for Dg: 4 LUO AND RICHTER One checks that hHb(zf); iH = hHbf; z iH for all f 2 H and 2 Hol(D). This implies that for each b 2 X (H) we have ker Hb 2 Lat(Mz; H). Theorem 1.1. Let µ be a nonnegative finite Borel measure supported in D and let M 2 Lat(Mz;D(µ)). Then there is a b 2 X (D(µ)) such that M = ker Hb: ∗ If M 6= (0), if ' is the extremal function for M, then b = Mz ' 2 X (D(µ)) and M = ker Hb. For µ = 0 one obtains H2. Of course, in this case the result is well- known. For the Bergman space the direct analogue of Theorem 1.1 is false, see [23]. We will prove Theorem 1.1 in Section 2. In the later sections of the paper we will apply the theorem to obtain further results about these 2 2 spaces. If H = H is the Hardy space, or H = La is the Bergman space, p p then it is natural to view H as part of the family of H - or La-spaces, and investigate how properties of functions and operators on these spaces change as the parameter p changes. However, if H = D(µ), or any abstract reproducing kernel Hilbert space, then it is unclear what the most natural class of related spaces should be. In Section 3 of this paper we investigate the weak product D(µ) D(µ), which we consider 1 1 to be a natural analogue of the spaces H and La in the Hardy and Bergman theories. The weak product of H is denoted by H H and it is defined to be the collection of all functions h 2 Hol(D) such that there are se- P1 quences ffigi≥1; fgigi≥1 ⊆ H with i=1 kfikkgik < 1 and h(z) = P1 P1 i=1 fi(z)gi(z) for all z 2 D. Note that whenever i=1 kfikkgik < 1, then 1 1 X 2 X jfi(z)jjgi(z)j ≤ kkzk kfikkgik < 1; i=1 i=1 thus the series will converge locally uniformly to the analytic function h.