Shift invariant subspaces of large index in the Bloch space
Nikiforos Biehler








§ ABSTRACT 
We consider the shift operator M_z, defined on the Bloch space and the little Bloch space and we study the corresponding lattice of invariant subspaces. The index of a closed invariant subspace E is defined as ind(E) = (E/M_zE).  We construct closed, shift invariant subspaces in the Bloch space that can have index as large as the cardinality of the unit interval [0,1]. Next we focus on the little Bloch space, providing a construction of closed, shift invariant subspaces that have arbitrary large index. Finally we establish several results on the index for the weak-star topology of a Banach space and prove a stability theorem for the index when passing from (norm closed) invariant subspaces of a Banach space to their weak-star closure in its second dual. This is then applied to prove the existence of weak-star closed invariant subspaces of arbitrary index in the Bloch space.



Meshless quadrature formulas 

arising from numerical differentiation
    
Oleg DavydovDepartment of Mathematics,
	University of Giessen, Arndtstrasse 2, 35392 Giessen, Germany,
	<oleg.davydov@math.uni-giessen.de>, 
Bruno Degli EspostiDepartment of Mathematics
	and Computer Science, University of Florence,
	Viale Morgagni 67, 50134 Firenze, Italy,
	<bruno.degliesposti@unifi.it>

    September 9, 2024
=======================================================================================================================================================================================================================================================================================================================






§ INTRODUCTION & MAIN RESULTS



Consider the unit disc = { z ∈  :   |z| < 1 } of the complex plane. The Bloch space  is defined as the set of the functions f, analytic in , that satisfy sup_|z|<1(1-|z|^2)|f'(z)| < +∞. The quantity 


    f_ = |f(0)| + sup_|z|<1(1-|z|^2)|f'(z)|


is a norm on the Bloch space, which makes it into a Banach space. The closure of analytic polynomials with respect to that norm is a subspace of the Bloch space, called the little Bloch space, and is denoted by _0. An equivalent way of defining the little Bloch space, is as the subspace of functions in the Bloch space that satisfy lim_|z| 1^-(1-|z|^2)|f'(z)| = 0. Functions in the Bloch space enjoy several nice properties. For a function f in the Bloch space, the quantity sup_|z|<1(1-|z|^2)|f'(z)| is Möbius invariant, meaning it remains unchanged after composing f on the right by any automorphism of the unit disc. Another well known fact is that an analytic function belongs to the Bloch space if and only if it is Lipschitz with respect to the hyperbolic metric on the unit disc. The Bloch space has been thoroughly studied (in <cit.>,<cit.> for example) as it is linked with many topics in analytic function theory. 




We consider the operator of multiplication M_z :, M_zf(z) = zf(z), also called the Shift operator, and we are interested in studying the lattice of closed invariant subspaces of M_z. Let E be a closed shift invariant subspace.




2.5cm0.4pt

2020 Mathematics Subject Classification: 30H30, 30B10, 47A15, 47B91. 


Keywords and phrases: Bloch space, little Bloch space, invariant subspaces, shift operator, index, lacunary Taylor series.





It follows from general properties of the shift operator that zE is closed (see Lemma 1.2), allowing us to define the index of E to be the quantity


    ind(E) := (E/zE).





Our goal is to show that, for the spaces under consideration, there exist invariant subspaces for which the index can be as large as possible, that is to say, as large as the space permits it to be. 




Our motivation comes from a series of papers, starting of course from the celebrated Beurling Theorem, which characterizes the invariant subspaces of the shift operator in the classical Hardy space H^2. A fact following from that characterisation is that every invariant subspace has the so called codimension one (also called index one) property, i.e. every non-trivial invariant subspace has index equal to one. A classic reference for properties of the index, in this context, and the codimension one property is Richter's article <cit.>.




In a 1985 paper C. Apostol, H. Bercovici, C. Foias and C. Pearcy (<cit.>) proved that what was previously true for the Hardy space, is no longer true for the classical Bergman space A^2 of functions analytic in the unit disc and square integrable with respect to planar Lebesgue measure. In particular they proved the existence of invariant subspaces E_k such that ind(E_k) = k, for k =1,2, …, +∞. Numerous results have been published since then, in hope to understand the lattice of invariant subspaces of the shift operator. In one direction, we have several results proving the codimension one property, such as in the classical Dirichlet space (<cit.>) of analytic functions in the unit disc whose derivative belongs to A^2, or in the space ℓ^1_A of Taylor series in the unit disc with summable coefficients (<cit.>). In the other direction we have plenty of constructions, utilising different properties of each space, to prove the existence of invariant subspaces of arbitrary index. The first concrete example of an invariant subspace of index 2 in A^2 has been given by H. Hedenmalm in <cit.>, using results on sampling and interpolation in the Bergman space. Later lacunary series were used by A. Borichev for a wide range of spaces including the classical Bergman spaces, a variety of mixed norm spaces, growth spaces and some weighted sequence spaces (<cit.>). In <cit.> E. Abakumov and A. Borichev proved the existence of invariant subspaces of arbitrary index for a variety of weighted sequence spaces using solution sets of convolution equations. In particular they show that the space ℓ^p_A, of Taylor series in the unit disc, with p-summable Taylor coefficients, contains invariant subspaces of arbitrary index as long as p>2. The case 1<p<2 is still an open problem. Special families of inner functions  have been used for H^∞ (<cit.>,<cit.>). 




In the recent years the Bloch space has attracted a lot of attention. In a recent paper (<cit.>), A. Limani and A. Nicolau answer several open questions in the Bloch space, related to invariant subspaces and cyclicity. In particular, they prove a Beurling-type theorem for singly generated, weak-star invariant subspaces in the Bloch space. This gives more motivation to study the index of the shift invariant subspaces in the Bloch space.




In this paper, we exploit ideas developed by A. Borichev in <cit.> and construct lacunary series, with almost maximal growth, in order to prove the existence of closed, shift invariant subspaces of arbitrary large index in each of the spaces  and _0 as well as weak-star closed invariant subspaces in , which under this topology is not a Banach space. The maximal growth for Bloch functions can be obtained by integrating the derivative of a function and using the definition of the norm. 



In particular for every f ∈ we have:


    |f(z)| ≤1/2f_log1 + |z|/1- |z|≲f_log1/1- |z|.


A more accurate result concerning the growth of Bloch functions comes from Makarov's Law of Iterated Logarithm (<cit.>). Inequality (1) also means that integrating a Bloch function, results in a function in the little Bloch space, which will be used with little mention throughout the text. 




By lacunary series we will always mean some function of the form:


    f(z) = ∑_n=0^∞ a_n z^s_n  ,  z ∈  ,


where {a_n}_n=0^∞ and {s_n}_n=0^∞, and such that the exponents s_n grow sufficiently fast. The rate at which the sequence s_n should grow is described by the Ostrowski-Hadamard gap theorem (see also Fabry's gap theorem), which asserts that if s_n+1/s_n≥ q > 1 for some q>1 and all n ∈, and if the Taylor coefficients are such that the radius of convergence of the above series is equal to 1, then that function has the unit circle as a natural boundary. Such sequences {s_n} will simply be called lacunary sequences. Even though series of this type are notoriously badly behaved, they can be useful to construct functions with one's desired properties. Lacunary functions are neatly characterised in the Bloch space:


Let f(z) = ∑_n=0^∞ a_n z^s_n be a lacunary series. Then f belongs to the Bloch space (little Bloch space resp.) if and only if (a_n) is bounded ( a_n  0 resp.)

 
Proof of this can be found in <cit.>, Theorem 1.14. A key tool in proving a given invariant subspace has the desired index is the following lemma on summation of indices.


Let X be a Banach space of analytic functions on the unit disc, satisfying the division property(<cit.>), i.e. 




  * X is a Banach space contained in Hol(), the space of analytic functions on .


  * Evaluation functionals k_λ are bounded on X, for all λ∈.


  * zf belongs to X whenever f ∈ X.


  * If f ∈ X and f(λ) = 0, then there exists a function g∈ X such that (z - λ)g(z) = f(z).



Let N ∈{1,2, …, +∞}, and for each 0 ≤ n < N let E_n  X be an invariant subspaces of index one, and f_n ∈ E_n with f_n(0) ≠ 0. Suppose moreover that for each 0 ≤ n < N there exists c_n > 0 such that


    c_n|g(0)| ≤g + h _X   ,     g ∈ E_n   , h ∈⋁_k ≠ nE_k.


Then ind(⋁_0 ≤ n < N E_n) = N.



A proof of the above lemma is given in <cit.>. In the original paper of S. Richter <cit.> it is proven that when the shift operator M_z is defined on such a space X, then M_z is bounded from above and below, meaning also that zE is closed whenever E is closed. As a consequence the index is well defined for every closed invariant subspace. In our case the spaces E_n will always be cyclic invariant subspaces [f_n] := span{ pf_n   :   p   polynomial} with f_n(0) ≠ 0, and it will be enough to check condition (2) for polynomial multiples of the generating functions. A proof of the fact that the Bloch space verifies the requirements of Lemma 1.2. will be provided in the Section 2. 




 With all the preliminary information established, we present the results contained in this article. 



For every N ∈{ 1 , 2, … , + ∞} there exists an invariant subspace E_N such that ind(E_N) = N.


Since the Bloch space is non-separable with that norm we are able to produce an example of an invariant subspace with uncountable index.


There exists an invariant subspace E of  such that ind(E) = card([0,1]).


The above two theorems utilise elementary properties of the Bloch space and its norm, and the vectors generating the cyclic subspaces are constructed inductively. If one wants to pass from the Bloch space to its “little-oh” analogue, one will be forced to use different techniques. Since functions in the little Bloch space satisfy (1-|z|^2)|f'(z)|  0 as |z|  1^-, the norm on the space is unable to capture the behaviour of the function by looking close to the boundary. This impediment proves fatal to the argument used in Theorems 1.2 and 1.3, but gave us motivation to develop a new argument, which requires the construction of lacunary functions with several special properties. The growth of the functions, the most crucial part of their behaviour, is studied with the aid of a classical theorem of R. Salem and A. Zygmund about the distribution function of trigonometric series. In this approach we construct functions which are almost maximally large on sufficiently massive subsets of the unit disc, and use an iterative argument to prove good bounds on the L^p means of polynomials on the aforementioned sets.


For every N ∈{ 1 , 2, … , + ∞} there exists an invariant subspace E_N _0 such that ind(E_N) = N.


Finally, we turn to a more abstract setting and extend several results from S. Richter's paper <cit.> for the weak-star topology. In particular, let X_0, X, Y be Banach spaces of analytic functions satisfying the division property, and that satisfy the following dualities: X_0^∗≅ Y and  Y^∗≅ X. The space X_0 can always be continuously embedded into X. Given an invariant subspace E  X_0, write E^w^∗ for its weak-star closure in X. Under some natural assumptions on the spaces we prove the following theorem: 


Let E be an invariant subspace of X_0 such that ind(E)= N for some N ∈{1, 2, … ,+∞}. Then the subspace E^w^∗ X is invariant, weak-star closed and ind(E^w^∗) = N.


We may then equip the Bloch space with the weak-star topology, inherited from its pre-dual, which can be identified as the Bergman space A^1(). This weaker topology makes the Bloch space separable, and can be viewed as a more natural choice for topology when studying certain problems, such as existence of cyclic vectors of the shift operator. With this set-up, it makes sense to look for invariant subspaces that are weak-star closed. Since the space becomes separable with the weak-star topology we cannot expect the index to be as large as we demonstrate in Theorem 1.3, but at most countable. Theorem 1.6 can be combined with Theorem 1.4 to obtain weak-star closed, invariant subspaces of arbitrary large, index for the Bloch space.


For every N ∈{ 1 , 2, … , + ∞} there exists a weak-star closed invariant subspace E_N such that ind(E_N) = N.


Theorem 1.7 provides an interesting antithesis to a phenomenon previously known in H^∞. There, the invariant subspaces behave quite differently when we pass from a strong to a weaker topology. On one hand, there are plenty of examples demonstrating that norm-closed invariant subspaces may have arbitrary large index. On the other hand, once the space is equipped with the weak-star topology, it a Beurling-type theorem will hold, as in the classical case. Theorem 1.7 demonstrates that it is no longer the case in the Bloch space.


    In each one of the Theorems 1.3, 1.5, 1.7 it is actually proven that there is a sequence of functions f_n, 1 ≤ n < ∞ in the appropriate space (either  or _0) such that for every 1 ≤ n_1 < ⋯ < n_N < ∞, the invariant subspace ⋁_1 ≤ k ≤ N[f_n_k] has index equal to N. 


The paper is organized as follows. Section 2 is dedicated to the proof of Theorems 1.3 and 1.4. In section 3 we provide the proof as well as the necessary backgroung for Theorem 1.5. Finally section 4 is left to prove Theorem 1.6 and Corollary 1.7, as well as to extend the required results from Richter's article.









§ INVARIANT SUBSPACES IN THE BLOCH SPACE


We begin by verifying that the Bloch space satisfies the hypotheses of Lemma 1.2, i.e, that it is indeed a Banach space of analytic functions satisfying the division property. The case of the little Bloch space is almost identical; one simply needs to verify that dividing out the zero of a given function in the little Bloch space, results in a function which is still in the little Bloch space.  Property (i) is obvious, and inequality (1) guarantees (ii) and (iii) are satisfied. It remains to check (iv), a.k.a the division property. Let E_0 = { f ∈  |   f(0) = 0 }, and R_0 : E_0 the operator that maps f ↦ f(z) / z. We will prove that R_0 is bounded, and hence well-defined as well. The general case λ∈ follows from the Möbius invariance of the Bloch space.



Consider f ∈ with f(0) = 0 and fix 0 < s < 1. We can find an analytic function g ∈ Hol() such that f(z) = zg(z). We have:



    ‖ R_0(f) ‖_ = |g(0)| + sup_|z|<1 (1-|z|^2)|g'(z)|.


Obviously we have that g(0) = f'(0) so |g(0)| ≤‖ f ‖_. For the rest we may write :


    sup_|z|<1 (1-|z|^2)|g'(z)| ≤max{sup_|z|≤ s (1-|z|^2)|g'(z)| , sup_s<|z|<1 (1-|z|^2)|g'(z)|  }


Let 0<  < 1-s. and define γ to be the anti-clockwise oriented circle centered at the origin and of radius s +. Then, when |z|< s + we get by Cauchy's formula:


    g'(z) = 1/2π i∫_γg'(ζ)/ζ - z dζ =1/2π i∫_γf'(ζ)-g(ζ)/ζ(ζ - z) dζ = 
    
    = 1/2π i∫_γf'(ζ)/ζ(ζ - z) dζ -1/2π i∫_γζ g(ζ)/ζ^2(ζ - z) dζ = 
    
    = 1/2π i∫_γf'(ζ)(1-|ζ|^2)/ζ(ζ - z)(1-|ζ|^2) dζ - 1/2π i∫_γζ g(ζ)(1-|ζ|^2)/ζ^2(ζ - z)(1-|ζ|^2) dζ.


From this we get the estimate:


    sup_|z| ≤ s(1-|z|^2)|g'(z)|    ≤sup_|z| ≤ s+(1-|z|^2)|f'(z)|·1/s1/1-(s+)^2
        +    sup_|z| ≤ s+|(1-|z|^2)|f(z)|·1/s(s+)1/1-(s+)^2.


This is true for every >0 small enough, hence the above inequality, along with the fact that the integration operator is bounded on the Bloch space (inequality (1)), we get that there is a constant C(s)>0, independent of f such that sup_|z|<s (1-|z|^2)|g'(z)| ≤ C(s) f_. On the other hand : 


    sup_s <|z|<1 (1-|z|^2)|g'(z)| ≤1/ssup_s <|z|<1 (1-|z|^2)|zg'(z)| = 1/ssup_s <|z|<1 (1-|z|^2)|f'(z) - g(z)| ≤
    ≤1/ssup_s <|z|<1 (1-|z|^2)|f'(z)| + 1/ssup_ <|z|<1 (1-|z|^2)|g(z)| ≤
    ≤1/ssup_s <|z|<1 (1-|z|^2)|f'(z)| + 1/s^2sup_s <|z|<1 (1-|z|^2)|f(z)|.


Similarly we obtain a constant C'(s)>0 such that sup_s< |z|<1 (1-|z|^2)|g'(z)| ≤ C'(s) f_. Overall g_≤max{1, C(s), C'(s)}f_, and the proof is finished. 




We may pass to the proof of Theorems 1.3 and 1.4. For every n∈ we introduce the auxiliary function U_n(r) = (1-r^2)nr^n-1 , 0 ≤ r < 1. Note that sup_0<r<1U_n(r) = z^n_. We also consider the sequence of radii r_n = (1-1/n)^1/2. The radius r_n is sufficiently close to the maximizing point of the function U_n, and we see that U_n(r_n) 1/√(e). We begin by stating a lemma that describes the construction of several lacunary sequences with some additional properties. 


There exists a sequence { s(n,i)}_1 ≤ i ≤ n < ∞ such that:



  * 1 < s(1,1) < s(2,1) < s(2,2) < s(3,1) < s(3,2) < s(3,3) < s(4,1) < s(4,2) < ⋯





  * For every 1 ≤ i ≤ n, we have s(n+1,i)/s(n,i)≥ 2, for all n ∈.





  * For every (n,i) ≠ (n',i') we have U_s(n,i)(r_s(n',i')) < 1/2^n+n'.




The proof of this lemma will be given at the end of this section. We may proceed to prove Theorem 1.3. 










Proof of Theorem 1.3.




 We consider a sequence  { s(n,i)}_1 ≤ i ≤ n < ∞, as constructed using Lemma 2.1. We define a collection of functions as follows:


    f_i(z) = 1 + ∑_n=i^∞z^s(n,i)  , z∈,   1 ≤ i < ∞.


Note that condition (ii) of Lemma 2.1 states that for each i, the function f_i is lacunary. Moreover, Proposition 1.1 guarantees that these functions all belong to the Bloch space as their Taylor coefficients are bounded. For each N ∈ we define the invariant subspace E_N = ⋁_1 ≤ i ≤ N [f_i]. For N = ∞ we define E_∞ =⋁_1 ≤ i < ∞ [f_i] and our goal is to show that ind(E_N) = N. According to Lemma 1.2, it suffices to show that for every 0 ≤ M < ∞ and for every 1 ≤ i_0 < ∞ there exists some c_i_0 > 0 such that for every 1 ≤ i_1 < i_2 < ⋯ < i_M, with i_j ≠ i_0 for j ≠ 0, and for every polynomials p_0, p_1, …, p_M, we have:


    c_i_0|p_0(0)| = c_i_0|f_i_0(0)p_0(0)| ≤‖ f_i_0p_0 + ∑_j=1^M f_i_jp_j ‖_.


To that end, fix 0 ≤ M < ∞, and consider 1 ≤ i_0 < ∞ and 1 ≤ i_1 < i_2 < ⋯ < i_M satisfying i_j ≠ i_0 for j ≠ 0 as well as polynomials p_0,p_1,…,p_M. To simplify notation we denote U_n,i := U_s(n,i) and r_n,i := r_s(n,i). For n ≥ i_0 we have that:


    ‖ f_i_0p_0 + ∑_j=1^Mf_i_jp_j ‖_   ≥sup_|z| = r_n,i_0(1-|z|^2)|(f_i_0p_0 + ∑_j=1^Mf_i_jp_j)'| ≥
       sup_|z|=r_n,i_0(1-|z|^2)|p_0(z)s(n,i_0)z^s(n,i_0)-1|  
       - sup_|z|=r_n,i_0(1-|z|^2)|p_0(z) ∑_k ≥ i_0k ≠ ns(k,i_0)z^s(k,i_0)-1| 
       - sup_|z|=r_n,i_0(1-|z|^2)|p'_0(z)f_i_0(z)| 
       - ∑_j=1^Msup_|z|=r_n,i_0(1-|z|^2)|p_j(z)f'_i_j(z)| 
       - ∑_j=1^Msup_|z|=r_n,i_0(1-|z|^2)|p'_j(z)f_i_j(z)|.


For (5) we have that:


    sup_|z|=r_n,i_0(1-|z|^2)|p_0(z)s(n,i_0)z^s(n,i_0)-1| = U_n,i_0(r_n,i_0) ·sup_|z| = r_n,i_0|p_0(z)|.


Note that U_n,i_0(r_n,i_0) 1/√(e), as n ∞. The maximum principle guarantees  that 
 sup_|z| = r_n,i_0|p_0(z)| p_0_∞, as n ∞. For (6) we have that:


    sup_|z|=r_n,i_0(1-|z|^2)|p_0(z) ∑_k ≠ ns(k,i_0)z^s(k,i_0)-1| ≤p_0_∞·∑_k ≠ n U_k,i_0(r_n,i_0) ≤p_0_∞·∑_k ≠ n1/2^k+n,


where we used property (iii) of Lemma 2.1. The quantities in (7) and (9) can be treated alike. If 0 ≤ j ≤ M, then:


    sup_|z|=r_n,i_0(1-|z|^2)|p'_j(z)f_i_j(z)| ≤p'_j_∞·sup_|z|=r_n,i_0(1-|z|^2)|f_i_j(z)|.


Since functions in the Bloch space grow at most logarithmically, we obtain that 
 sup_|z|=r_n,i_0(1-|z|^2)|f_i_j(z)| n ∞⟶ 0. 



Finally, for (8) we have that:


    ∑_j=1^Msup_|z|=r_n,i_0(1-|z|^2)|p_j(z)f'_i_j(z)| ≤∑_j=1^M p_j_∞·sup_|z|=r_n,i_0(1-|z|^2)|f'_i_j(z)|  
    ≤∑_j=1^M p_j_∞·∑_k = i_j^∞U_k,i_j(r_n,i_0) ≤∑_j=1^M p_j_∞·∑_k = i_j^∞1/2^k+n.


where we used Lemma 2.1 once again. By substituting (10)-(13) into (5)-(9), we get that:


    U_n,i_0(r_n,i_0)  ·sup_|z| = r_n,i_0|p_0(z)| ≤‖ f_i_0p_0 + ∑_j=1^Mf_i_jp_j ‖_ + p_0_∞·∑_k ≠ n1/2^k+n
     + ∑_j=0^M p'_j_∞·sup_|z|=r_n,i_0(1-|z|^2)|f_i_j(z)| + ∑_j=1^M p_j_∞·∑_k = i_j^∞1/2^k+n
     ≤‖ f_i_0p_0 + ∑_j=1^Mf_i_jp_j ‖_ + p_0_∞·1/2^n + ∑_j=0^M p'_j_∞·sup_|z|=r_n,i_0(1-|z|^2)|f_i_j(z)| + 1/2^n·∑_j=1^M p_j_∞.


By letting n ∞ we get that:


    1/√(e)·p_0_∞≤‖ f_i_0p_0 + ∑_j=1^Mf_i_jp_j ‖_ ,


and hence:


    1/√(e)· |p_0(0)| ≤1/√(e)·p_0_∞≤‖ f_i_0p_0 + ∑_j=1^Mf_i_jp_j ‖_.


And we obtained inequality (4) with c_i_0 = 1/√(e).                                              ▪












The above proof may serve as a model to prove Theorem 1.4. The challenge that arises is to define a continuum of functions with the properties described above, instead of countably many. 











Proof of Theorem 1.4.




We apply Lemma 2.1 and set s_n = s(n,1), n ≥ 1. The properties of the sequence may be summarized as follows:



  * 1 < s_1 < s_2 < ⋯.





  * s_n+1/s_n≥ 2, for all n ∈.





  * For every n ≠ n' we have U_s_n(r_s_n') < 1/2^n+n'.


To define an invariant subspace, we are in need of a lemma that we will use without proof, as a detailed proof can be found in Lemma 3.2, <cit.>.


There exists a family {N_α  :  α∈ [0,1] } such that for every M ∈ and every finitely many indices α_0,α_1, …, α_M ∈ [0,1] with α_0 ≠α_i for 1 ≤ i  ≤ M we have:


    card( N_α_0\⋃_i=1^M N_α_i) = ∞.




We consider the family {N_α}_α∈ [0,1] and we define for each α∈ [0,1] a function:


    f_α(z) = 1 + ∑_n∈ N_αz^s_n  , z∈.


Once again, these functions are lacunary, and they belong to the Bloch space. The invariant subspace is defined similarly, E = ⋁_α∈ [0,1][f_α] and we aim to show that ind(E) = (E/zE) = card([0,1]). By definition, it suffices to show that for any α_0 ∈ [0,1] we have:


    f_α_0 + zE ∉span{ f_α + zE   : α∈ [0,1]   , α≠α_0 },


where the closure is in the quotient topology of E/zE, and f_α+ zE denotes the equivalence class of f_α in the quotient space. Let α_0 ∈ [0,1]. It is sufficient to find some constant c >0 such that:


    f_α_0 + zE + u _E/zE≥ c   ,
    for every   u ∈span{ f_α + zE   :  α∈ [0,1]   , α≠α_0 }


To that end, consider any finite number of indices α_1, …, α_M ∈ [0,1] with α_i ≠α_0, and any polynomials q,p_1,…,p_M. Then we need to show that 


    f_α_0(1 + z · q) + ∑_i=1^M p_i f_α_i_≥ c_0.


Set p_0 = 1 + zq. From Lemma 2.2 we have that there exists some increasing sequence k_n  N_α_0 such that k_n ∉⋃_i=1^M N_α_i. Starting once again as in Theorem 1.3 we obtain:


    ‖ f_α_0p_0 + ∑_j=1^M f_α_jp_j ‖_   ≥sup_|z| = r_k_n(1-|z|^2)|(f_α_0p_0 + ∑_j=1^M f_α_jp_j)'| ≥
       sup_|z|=r_k_n(1-|z|^2)|p_0(z)s_k_nz^s_k_n-1|  
       - sup_|z|=r_k_n(1-|z|^2)|p_0(z) ∑_m ≠ k_ns_mz^s_m-1| 
       - sup_|z|=r_k_n(1-|z|^2)|p'_0(z)f_α_0(z)| 
       - ∑_j=1^Msup_|z|=r_k_n(1-|z|^2)|p_j(z)f'_α_j(z)| 
       - ∑_j=1^Msup_|z|=r_k_n(1-|z|^2)|p'_j(z)f_α_j(z)|.


Since k_n ≠ m for all indices m appearing in the sums above, condition (iii') is satisfied, and hence we may replicate the argument of Theorem 1.3. Therefore by letting k_n ∞ we obtain:


    1/√(e)·p_0_∞≤‖ f_α_0p_0 + ∑_j=1^M f_α_jp_j ‖_,


and since p_0_∞ = 1+zq_∞≥ 1 we  arrive at:


    1/√(e)≤‖ f_α_0p_0 + ∑_j=1^M f_α_jp_j ‖_,


which is inequality (17) with c=1/√(e).                                                     ▪












We finish this section by proving Lemma 2.1.









Proof of Lemma 2.1.




We will construct the sequence s(n,i) inductively. We may define s(1,1) > 1 as we like. To define the integer s(2,1), we take it to be s(2,1) ≥ 2 s(1,1) so that it satisfies conditions (i) and  (ii). For condition (iii), we notice that 


    For every   n∈,    U_n(r)  0,   as  r  1^-   and
    for every   r ∈ (0,1),    U_n(r)  0,   as   n ∞.


This means we can choose s = s(2,1) large enough so that:


    U_2,1(r_1,1) < 1/2^2+1  and  
    U_1,1(r_2,1) < 1/2^2+1.


To continue, assume that we have already defined all terms up to s(n,i) for some (n,i). The next term has the form s(n+1,1), if n=i, and has the form s(n,i+1) if n>i. Without loss of generality, we may assume the first case. First choose s(n+1,1) large enough so that s(n+1,1) > s(n,n) and s(n+1,1) ≥ 2 s(n,1). That way conditions (i) and (ii) are taken care of. For the last one we take s(n+1,1) additionally as large as to have:


    U_n+1,1(r_m,i) < 1/2^n+1+m  for all   1 ≤ m ≤ n   and   1 ≤ i ≤ m 
    
    U_m,i(r_n+1,1) < 1/2^n+1+m  for all   1 ≤ m ≤ n   and   1 ≤ i ≤ m.


This is possible because of (25), (26) and because we have only finitely many predefined terms for which we need to verify the inequalities. By the inductive hypothesis we can construct the whole sequence s(n,i) satisfying all properties (i),(ii)
and (iii). The proof is complete.▪










Lemma 2.1 may be adapted for the spaces _α, 0<α < 1, which are Bloch type spaces with norm |f(0)|+sup_|z|<1(1-|z|^2)^α|f'(z)|. One can then prove Theorem 1.3 and 1.4 for those spaces.





§ INVARIANT SUBSPACES IN THE LITTLE BLOCH SPACE


In this section we will construct invariant subspaces in the little Bloch space of arbitrary, but countable, index. The section is split into two parts. We begin by stating some preliminary results, that will be used in the construction. Then we pass to the construction of the functions generating the invariant subspaces and prove that it has the correct index. The proof of the desired index is achieved with the aid of Lemma 1.2. We denote by m the normalized Lebesgue measure of [0,2π] or = {z ∈  :   |z| = 1}.




 §.§ Preliminary results


The first result is known as Makarov's inequality and it will be useful to us since it permits us to pass from estimates involving Bloch norms, to estimates involving L^p-means . Following that, is an exponential version of it, which follows by a simple calculation.

(Makarov's Inequality)
Let g ∈. Then for every 0 ≤ r < 1 and every n ∈ the following inequality holds:


    ( |g(rζ)|^2n)^1/2n≤g_( 1 + (n!)^1/2n√(log1/1-r)).




A proof of the above can be found in Makarov's original paper <cit.>. Theorem 8.9 in <cit.> provides a refined version involving the above numerical constants.

(Makarov's Inequality; exponential form)
Let g ∈, with g_≤ 1. Then for every 1-1/e≤ r<1 we have:


    ∫_rexp{|g(ζ)|^2/8log1/1-r} dm(ζ) ≤ 2




From Makarov's inequality for the function g we obtain for every n ∈:


    ∫_r|g|^2n dm ≤g_^2n( 1 + (n!)^1/2n√(log1/1-r))^2n


When r ≥ 1 - 1/e we get 1 + (n!)^1/2n√(log1/1-r)≤ 2(n!)^1/2n√(log1/1-r), and so:


    ∫_r|g|^2n dm ≤ 4^n n! log^n 1/1-r


Hence


    ∫_r( |g|^2/8log1/1-r)^n 1/n! dm ≤1/2^n


Using the monotone convergence theorem and summing up all the integrals, we get the result.



Theorem 3.3 is a classic result of Salem and Zygmund, as in their original paper <cit.>. 

(Salem-Zygmund)
Let F(z) = ∑_k=1^∞c_kz^n_k, z ∈, be a lacunary power series, i.e. n_k+1/n_k≥ q > 1 for some q >1 and all k ∈. Let C_N = 1/√(2)(|c_1|^2 + ⋯ + |c_N|^2)^1/2 and let R_N,I_N be the real and imaginary parts of the N-th partial sum respectively. If C_N  +∞ and c_N/C_N 0, then:


    m( { z∈  :  R_N(z)/C_N≤ x , I_N(z)/C_N≤ y }) 1/2π∫_-∞^x ∫_-∞^y e^-1/2(t^2+s^2) dt ds,    x,y ∈.




We will need to work with the modulus of a complex function instead of the real and imaginary part. We thus compute the asymptotic distribution of it.


Let F_N be the partial sum of the series F(z) = ∑_k=1^∞c_kz^n_k, z ∈. Under the hypotheses of Theorem 3.3 we have that:


    m( { z ∈  :  |F_N(z)|/C_N≤ x })  1 - e^-x^2/2,    x ≥ 0.
 













Let U_N = R_N/C_N and V_N = I_N/C_N. We will compute the distribution of √(|U_N|^2+|V_N|^2). First of all, for x ≥ 0 we have |U_N| ≤ x  |U_N|^2 ≤ x^2, which means that:


    m(|U_N|^2 ≤ x^2) 1/√(2 π)∫_-x^x e^- 1/2 t^2 dt = 2/√(2 π)∫_0^x e^- 1/2 t^2 dt


And thus by a change of variables:


    m(|U_N|^2 ≤ x) 1/√(2 π)∫_0^x 1/√(t) e^- t/2 dt =
    1/√(2 π)∫_-∞^x 1/√(t) e^- t/2_{t ≥ 0}  dt


The same can be said about the distribution of |V_N|^2. Since the joint distribution of U_N and V_N is asymptotically Gaussian, the variables are asymptotically independent. That means that we can use the continuous mapping theorem, to compute the asymptotic distribution of |U_N|^2 + |V_N|^2 by convoluting the density functions of each of  the summands. Therefore the density of |U_N|^2 + |V_N|^2 will be given by the formula:


    ϱ(x)    =  ∫_-∞^∞1/2π1/√(x-y)1/√(y)_{y ≥ 0 }_{x ≥ y } e^-(x-y)/2 e^-y/2 dy 
       = e^-x/2/2π∫_0^x 1/√(x-y)1/√(y) dy ,    x ≥ 0.


We substitute √(y) = u to obtain:


    ϱ(x) = e^-x/2/2π∫_0^√(x)2/√(x)1/√(1 - (u/√(x))^2) du,    x ≥ 0.


Then we substitute u/√(x) = sin t to get:


    ϱ(x) = e^-x/2/2π∫_0^π/22/√(x)√(x)cos t/cos t  dt = e^-x/2/2  ,     x ≥ 0.


What we computed means that:


    m( |U_N|^2 + |V_N|^2 ≤ x	) ∫_0^x 1/2 e^-t/2 dt ,    x ≥ 0.


Hence a final change of variables gives us that:


    m ( |F_N|/C_N≤ x ) ∫_0^x t e^-1/2t^2 dt = 1-e^-x^2/2 ,    x ≥ 0.













 We introduce a family of lacunary polynomials which will be the basic building blocks to construct functions that generate invariant subspaces of high index. Let f_s(z) = ∑_m=s^2s z^3^m, s ∈ be a lacunary polynomial. We associate to every f_s a radius r_s = 1 - 1/3^2s, and a function X_s(r) = 1/√(log1/1-r) f_s(r), 0<r<1. 


Let f_s and r_s be as above. There exists a constant C>0 such that:


    1/C· s ≤‖ f_s(r_s ·) ‖_L^2()^2 ≤ C · s    , s ≥ 1.















By Parseval's formula we have:


    ‖ f_s(r_s ·) ‖_L^2^2 = ∑_m=s^2sr_s^2 · 3^m≤ s+1


On the other hand,


    ∑_m=s^2sr_s^2 · 3^m≥∑_m=s^2sr_s^2 · 3^2s = ∑_m=s^2s(1 - 1/3^2s)^2 · 3^2s = (1 - 1/3^2s)^2 · 3^2s (2s - s +1).


Since (1 - 1/n)^2n converges to 1/e^2 as n  + ∞ we get the reverse inequality.













Let f_s and r_s be as above. There exists a constant c > 0 such that for every >0 and every M > 0 there exists arbitrary large s ∈ with the property that for every 0 ≤ x ≤ M we have


    m ( {ζ∈  :   |f_s(r_s ζ)| > x √(log1/1-r_s)}) ≥ e^-cx^2 - .













We wish to apply Theorem 3.4. First we notice that:


    √(log1/1-r_s) = √(2s log3)≍‖ f_s(r_s ·)‖_L^2 + ∞.


Following the notation of Theorem 3.4, the above means that:


    C_2s = 1/√(2)(|c_1|^2 + |c_2|^2 + ⋯ + |c_2s|^2)^1/2 = 1/√(2)‖ f_s(r_s ·)‖_L^2 + ∞


Moreover it is clear that all non-zero Taylor coefficients are bounded. This means that both conditions of Theorem 3.4 are satisfied. The fact that we can apply Theorem 3.4 to a lacunary polynomial of this form comes from the fact that the convergence in Salem and Zygmund's Theorem is guaranteed only by the “length” s+1 of the block, and the fact that the lacunary gap on the exponents is at least 3. Finally, the convergence is uniform in x whenever it belongs to a fixed bounded interval. 










We finish this section by observing the following property of the functions X_s(r): 


    For every   s ∈,    X_s(r)  0 ,   as   r  1^-   and
    for every   r ∈ (0,1),    X_s(r)  0,   as   s ∞.


These properties are analogous to (18) and (19). Property (20) is straightforward. For property (21), fix r ∈ (0,1). Then:


    X_s(r) = 1/√(log1/1-r)∑_m=s^2sr^3^m≤1/√(log1/1-r)∑_m=s^2sr^3^s = 1/√(log1/1-r) (s+1) r^3^s


And (s+1)r^3^s converges to zero as s ∞.





 §.§ Invariant subspaces in _0 of arbitrary index - Construction


We consider the following functions:







    f_i(z) = 1 + ∑_j=i^∞δ_j f_i,j(z)    , z ∈,    1 ≤ i < + ∞ ,


where f_i,j are the blocks


    f_i,j(z) = ∑_m = s(i,j)^2s(i,j) z^3^m   , z∈,



and δ_j = 2^9 ·√(c/j), with c the constant appearing in Lemma 3.6. Since every function is lacunary and the coefficients δ_j tend to zero, these functions belong to _0. To each block are assigned its associated parameters:




  * i ≥ 1   ,   j ≥ i set

    r(i,j) = 1 - 1/3^2s(i,j)






  * i ≥ 1   ,  j ≥ i   ,    r ∈ (0,1) set

    X_i,j(r) = 1/√(log1/1-r)∑_m=s(i,j)^2s(i,j)r^3^m






There exists a sequence {s(i,j)}_1 ≤ i ≤ j < ∞, such that:




  (1.i.j) For all i ∈ and j ≥ i we have s(i,j) > max{ 2^4j+4 , s(i',j')   :   i' < i ,   j' < j  }






  (2.i.j) For all i ∈ and j ≥ i and (i',j') ≠ (i,j) we have 

    X_i,j(r(i',j')) ≤1/δ_j·1/2^i+i'+j+2j'+2






  (3.i.j) For all i ∈ and j ≥ i we have 


    m( {ζ∈ : |f_i,j(r(i,j) ζ)| > √(1/c · 2^j+6)·√(log1/1-r(i,j))}) ≥  1 -  1/2^j+5,


where c is the constant appearing in Lemma 3.6.





Set E_i,j = {ζ∈  :  |f_i,j(r(i,j)ζ)| ≥√(1/c · 2^j+6)·√(log1/1-r(i,j))} and F_i,j = \ E_i,j


We define the sequence inductively, respecting the lexicographic order. It is clear that to obtain the above conditions it just suffices to make sure that s(i,j) is large enough at each step. In particular, for condition (3.i.j) we apply Lemma 3.6 with x = √(1/c · 2^j+6), = 1/2^j+6 and apply the inequality e^-x > 1 - x which holds for x<1.  












Proof of Theorem 1.5.




As in the proof of Theorem 1.3, we define again E_N = ⋁_1 ≤ i ≤ N[f_i] _0 and E_∞ =⋁_1 ≤ i < ∞[f_i] _0 where f_i, 1 ≤ i < ∞ are the functions described in the beginning of section 3.2 and the implicit sequence { s(i,j)}_1 ≤ i≤ j < ∞ satisfies the conditions of Proposition 3.7. Following a similar argument as in Theorem 1.3, we fix 1 ≤ M < ∞, 1 ≤ i_1 < i_2 < ⋯ < i_M ∈ and p_1,p_2, …, p_M be polynomials, such that ∑_m=1^M p_m f_i_m_≤ 1. We wish to apply Lemma 2.1 to the function ∑_m=1^M p_m f_i_m in order to bound the values p_m(0), 1 ≤ m ≤ M, by some constants independent of the polynomials. 



Define for every 1 ≤ m ≤ M and every j ≥ i_m the set:


    U_i_m,j = {ζ∈ E_i_m,j  :   |p_m(r(i_m,j)ζ)| ≥ 2^j-1}.


We will implement the following scheme. We assume that for some J > i_M we have the following two inequalities:


    m(U_i_m,J) ≤J/2^J+5   ,   1 ≤ m ≤ M,  
    p_m_L^2^J+1(r(i_m,J+1))≤ 2^J+1   ,   1 ≤ m ≤ M,


and we will prove that :


    m(U_i_m,J-1) ≤J-1/2^J+4   ,   1 ≤ m ≤ M,  
    p_m_L^2^J(r(i_m,J))≤ 2^J   ,   1 ≤ m ≤ M.


The fact that we can assume (22) and (23) for all J large enough comes from the fact that the polynomials p_m are bounded on the closed unit disc. Give the success of this argument, we may iterate it, starting from some radius close enough to one until we obtain:


    p_m_L^2^i_M(r(i_m,i_M))≤ 2^i_M   ,   1 ≤ m ≤ M,


which means that |p_m(0)| ≤ 2^i_M for every 1 ≤ m ≤ M. The constant does not depend on the polynomials p_m, so using Lemma 2.1 we will have proven Theorem 1.5.



For the rest of the proof, fix some 1 ≤ n ≤ M and some J > i_M. First we demonstrate how to obtain inequality (25). Let A = {ζ∈  :   |p_n(r(i_n,J)ζ)| ≥ 2^J-1}. Then: 


    m(A)     = m(A ∩ E_i_n,J) + m(A ∩ F_i_n,J) 
       ≤ m(U_i_n,J) + m(F_i_n,J)  
       ≤J/2^J+5 + 1/2^J+5
       ≤J/2^J·1/2^5 + 1/2^5
       ≤ 2 ·1/2^5 = 1/16.


We may now use Minkowski's inequality to write:


    p_n_L^2^J(r(i_n,J))  =  ( ∫_\ A|p_n(r(i_n,J)ζ)|^2^J dm(ζ) + ∫_A|p_n(r(i_n,J)ζ)|^2^J dm(ζ) )^1/2^J
    ≤( ∫_\ A|p_n(r(i_n,J)ζ)|^2^J dm(ζ) )^1/2^J + ( ∫_A|p_n(r(i_n,J)ζ)|^2^J dm(ζ) )^1/2^J .


Applying Cauchy-Schwartz inequality to the second integral we obtain:


    p_n_L^2^J(r(i_n,J))≤( ∫_\ A|p_n(r(i_n,J)ζ)|^2^J dm(ζ) )^1/2^J + √(m(A))·p_n_L^2^J+1(r(i_n,J)).


Using the bound for the polynomial on the set \ A, our hypothesis (23) and the fact that r(i_n,J) < r(i_n,J+1), we get that


    p_n_L^2^J(r(i_n,J))≤  2^J-1 + 1/4· 2^J+1 = 2^J.


Inequality (25) is proven. We proceed to prove inequality (24). By (1.i.j) we see that s(i,j) > s(1,1) ≥ 3 and hence all radii r(i,j) satisfy r(i,j) ≥ 1 - 1/e. We may thus apply Proposition 3.2 for the function  ∑_m=1^M p_m f_i_m, on the radius r(i_n,J-1) and get:



    ∫_r(i_n,J-1)exp{|∑_m=1^M p_m f_i_m|^2/8log1/1-r(i_n,J-1)} dm ≤ 2.


Applying Jensen's inequality for the exponential function yields:


    exp{∫_r(i_n,J-1)|∑_m=1^M p_m f_i_m|^2/8log1/1-r(i_n,J-1) dm }≤ 2.


Using the inequality |x + y|^2 = |x|^2 + |y|^2 + 2Re(xy) ≥ |x|^2 + |y|^2 - 2|x||y| and applying it for x = p_nδ_J-1 f_i_n,J-1 and y = ∑_m=1^M p_m f_i_m - p_nδ_J-1 f_i_n,J-1 we obtain:


    | ∑_m=1^M p_m f_i_m|^2 ≥ 
    |p_nδ_J-1 f_i_n,J-1|^2 -  2 |p_nδ_J-1 f_i_n,J-1| ·| p_n ( ∑_j ≥ i_nj ≠ J-1δ_j f_i_n,j) + ∑_1 ≤ m ≤ Mm ≠ n p_m ( ∑_ j ≥ i_m δ_j f_i_m,j) |.


Since we are integrating over the radius r(i_n,J-1) we may use (2.i.j) to get the following bound:


    | p_n ( 1 + ∑_j ≥ i_nj ≠ J-1δ_j f_i_n,j) + ∑_1 ≤ m ≤ Mm ≠ n p_m ( 1 + ∑_ j ≥ i_m δ_j f_i_m,j) | 
    ≤ |p_n| ( 1 +∑_j ≥ i_nj ≠ J-1δ_j X_i_n,j(r(i_n,J-1)) ·√(log1/1-r(i_n,J-1)))  
    
        + ∑_1 ≤ m ≤ Mm ≠ n |p_m| ( 1 + ∑_ j ≥ i_m δ_j X_i_m,j(r(i_n,J-1)) ·√(log1/1-r(i_n,J-1))) 
    ≤ |p_n| ( 1 +∑_j ≥ i_nj ≠ J-11/2^2i_n + j + 2(J-1) + 2 ·√(log1/1-r(i_n,J-1))) 
    
        + ∑_1 ≤ m ≤ Mm ≠ n |p_m| ( 1 + ∑_ j ≥ i_m 1/2^i_n + i_m + j + 2(J-1) + 2 ·√(log1/1-r(i_n,J-1))).


A short calculation then yields: 


    | p_n ( 1 + ∑_j ≥ i_nj ≠ J-1δ_j f_i_n,j) + ∑_1 ≤ m ≤ Mm ≠ n p_m ( 1 + ∑_ j ≥ i_m δ_j f_i_m,j) | ≤∑_m=1^M |p_m|( 1 + 1/2^2J+1√(log1/1-r(i_n,J-1))).


As a result: 


    |∑_m=1^M p_m f_i_m|^2/8log1/1-r(i_n,J-1)≥|p_nδ_J-1 f_i_n,J-1|^2/8log1/1-r(i_n,J-1) - 1/4·|p_n δ_J-1 f_i_n,J-1|/√(log1/1-r(i_n,J-1))·( ∑_m=1^M |p_m|(1/√(log1/1-r(i_n,J-1)) + 1/2^2J+1)  ).


Taking into account condition (1.i.j) gives that 1/√(log1/1-r(i_n,J-1))≤1/2^2J+1. Therefore after integration on the circle of radius r(i_n,J-1) and using Cauchy-Schwartz inequality, we obtain:


    ∫_r(i_n,J-1)|∑_m=1^M p_m f_i_m|^2/8log1/1-r(i_n,J-1) dm  ≥
    ∫_r(i_n,J-1)|p_nδ_J-1 f_i_n,J-1|^2/8log1/1-r(i_n,J-1) dm - 1/4∑_m=1^M 1/2^2J(∫_r(i_n,J-1)|p_nδ_J-1 f_i_n,J-1|^2/log1/1-r(i_n,J-1) dm )^1/2p_m_L^2(r(i_n,J-1)) .




Finally, we notice that because of (1.i.j):


    p_m_L^2(r(i_n,J-1))≤p_m_L^2^J(r(i_n,J-1))≤p_m_L^2^J(r(i_m,J))≤ 2^J.


Moreover J > i_M so M/2^J≤ 1. Therefore we obtain the following inequality:


    2 ≥exp{∫_r(i_n,J-1)|∑_m=1^M p_m f_i_m|^2/8log1/1-r(i_n,J-1) dm }≥exp(X(X-1)),


where X =(∫_r(i_n,J-1)|p_nδ_J-1 f_i_n,J-1|^2/log1/1-r(i_n,J-1) dm )^1/2. As a result it must be that X ≤ 2, which in turn gives:


    ∫_r(i_n,J-1)|p_nδ_J-1 f_i_n,J-1|^2/log1/1-r(i_n,J-1) dm ≤ 32.


Restricting ourselves on the set U_i_n,J-1 and using the growth of the block f_i_n,J-1 and the size of the polynomial p_n yields exactly the desired result, i.e:


    m(U_i_n,J-1) ≤J-1/2^J+4.














§ WEAK STAR CLOSED INVARIANT SUBSPACES & STABILITY OF INDEX


In this last section we discuss the index of a weak-star (w^∗) closed, invariant subspaces of a Banach space. In the first subsection duality in the Bloch spaces is introduced, to provide a concrete example. Following that we extend several of Richter's results from <cit.> for the weak-star topology, and prove Theorem 1.6. Finally we may apply that to the Bloch spaces and obtain Theorem 1.7.




 §.§ Duality in the Bloch spaces


Consider the Bergman space A^1 of integrable functions in the unit disc, as well as the following dual pairings:


    ⟨· , ·⟩ : _0 × A^1 
    ⟨ f , g ⟩ = lim_r  1∫_f(rz) g(rz) dA(z),


and,


    ⟨· , ·⟩ : A^1 ×
    ⟨ g , f ⟩ = lim_r  1∫_g(rz) f(rz) dA(z).


It is proven in <cit.>,<cit.>, <cit.> that these pairings are well defined and realize the dualities ( _0 )^∗≅ A^1 and (A^1)^∗≅. We can therefore endow the space _0 with the weak topology inherited from its dual space, and the space  with the w^∗-topology inherited from its pre-dual. 



The topology in  can be characterized in terms of nets, as is done in <cit.>. If { f_i }_i ∈ I is a net, then:


    f_i w^∗ 0 
    f_i(z)  0    ,    z ∈
    lim sup_if_i_ < + ∞


The above statement remains true if we replace w^∗-convergence by weak convergence and the net { f_i }_i ∈ I belongs to _0. If f ∈_0 then f is norm-cyclic in _0 if and only if f is w^∗-cyclic in  (<cit.>). Since polynomials are norm-dense in _0, the constant function 1 is norm-cyclic in _0 and so 1 must be w^∗-cyclic in , or equivalently, polynomials are w^∗-dense in . In particular, we may consider polynomials belonging in the set { a_0 + a_1z + ⋯ + a_N z^N   :   N ∈, a_i ∈ + i}, thus obtaining a countable, norm-dense set in _0 and hence a countable, w^∗-dense set in . This means that the space (, w^∗) is separable. 

Our aim is to produce w^∗-closed invariant subspaces of arbitrary index in the Bloch space. Since (, w^∗) is separable, the index of an invariant subspace can be at most countable. In <cit.> and <cit.> it is proven that H^∞ with the norm topology contains invariant subspaces of index equal to the cardinality of the interval [0,1]. Theorem 1.4 is the Bloch space equivalent of that. It is also known that if H^∞ is equipped with the w^∗-topology then Beurling's theorem holds, i.e. for every E  H^∞  invariant, then E = ϕ H^∞ for some inner function ϕ (<cit.>). This implies that all invariant subspaces have the index one property, and thus Theorem 1.7 provides a contrasting phenomenon to the situation in H^∞. 


We remind the reader of two properties that hold in the Bloch space, and will be used in what follows. If we have a function f ∈ and F' = f then F ∈_0 and F_≤ C f_ for some constant C independent of f. The second one, called the “division property” states that if f ∈ and f(λ)=0 then f/z-λ∈. In a Banach space of analytic functions (see Lemma 1.2) the division operator R_λ, defined on E_λ = {f ∈  :   f(λ) =0 }, is bounded for any λ∈, which is also equivalent to saying that the operator M_z - λ is bounded below for any λ∈ (<cit.>) . Next is a useful proposition.


Let M_z : be the Shift operator. Then:


  * M_z is w^∗ - w^∗ continuous on 

  * R_λ is w^∗ - w^∗ continuous for every λ∈












To prove continuity of M_z, consider a converging net f_i w^∗ 0. We need to show that zf_i w^∗ 0. Pointwise convergence is obvious. Moreover, 


    zf_i_ = M_zf_i_≤M_zf_i_  so,
    lim sup_izf_i_≤M_zlim sup_i f_i_ < + ∞


The two combined guarantee convergence of the net. For the second claim, consider a converging net f_i w^∗ f in E_λ. There are functions g_i, g ∈ such that (z - λ )g_i = f_i and (z- λ)g = f. This implies that R_λf_i = g_i and R_λf = g, and thus it suffices to show that g_iw^∗g. This can be deduced by the boundedness of R_λ and by proceeding as for the shift operator.

 









 The above proposition has the following consequence: If E is a w^∗-closed subspace of , then M_zE is also w^∗-closed. This means that it is meaningful to consider the quotient E / zE for an invariant subspace which is w^∗-closed, and that will it be a well-defined locally convex space.



 §.§ Extension of Richter's results - Stability of index


We consider the general situation where X_0, X, Y are Banach spaces of analytic functions satisfying the division property, and that satisfy the following dualities: X_0^∗≅ Y and  Y^∗≅ X. We furthermore assume that Proposition 4.1 is true for the space, i.e. M_z and R_λ are continuous with respect to the w^∗-topology in X. We will denote by Lat_X_0(M_z)  the lattice of norm closed, invariant subspaces of X_0 and by Lat_X(M_z,w^∗) the lattice of w^∗-closed, invariant subspaces of X. As mentioned above for given ℳ∈ Lat_X(M_z,w^∗), quotients of the form ℳ/zℳ make sense under the above assumptions, and the projection operator onto the quotient space is always continuous. The dimension of ℳ/zℳ is the same as that of ℳ/(z-λℳ) for λ∈, as follows from general properties of the shift operator and Proposition 4.1. If A is any subset of X, we denote by A^w^∗ the w^∗-closure of A in X. This coincides with all the w^∗-limits of nets in A.  If f ∈ X, we will write [f]_∗ for the  w^∗-closed, invariant subspace generated by f. Moreover, when writing ℳ∨𝒩 for ℳ, 𝒩∈ Lat_X(M_z,w^∗) we will mean the smallest w^∗-closed invariant subspace containing ℳ + 𝒩. Finally, for given ℳ∈ Lat_X(M_z,w^∗) we define 𝒵(ℳ) = {λ∈  |   f(λ) = 0   ,   f ∈ℳ}. 


Let ℳ, 𝒩∈ Lat_X(M_z,w^∗). Then:


  * ind(ℳ∨𝒩) ≤ind(ℳ) + ind(𝒩),

  * If  ind(ℳ) = m ≥ 2, with m finite, and n_1 +n_2 = m then there exist 𝒩_1, 𝒩_2 ∈ Lat_X(M_z,w^∗), 𝒩_1, 𝒩_2 ℳ, such that ind(𝒩_i) = n_i and ind(𝒩_1 ∨𝒩_2) = m.














For the first implication, if either ℳ or 𝒩 have infinite index, then we have nothing to show, so assume the index of both is finite. In that case there exist ℳ_1 ℳ and 𝒩_1 𝒩, finite dimensional subspaces, such that:


    ℳ = zℳ + ℳ_1     ,   𝒩 = z𝒩 + 𝒩_1   and, 
    ind(ℳ) = (ℳ_1)    ,   ind(𝒩) = (𝒩_1).


Then 


    ℳ + 𝒩 = z(ℳ+ 𝒩) + (ℳ_1 + 𝒩_1)  z(ℳ∨𝒩) + (ℳ_1 + 𝒩_1) ℳ∨𝒩.


Since ℳ_1 + 𝒩_1 is finite dimensional, it is w^∗-closed. The space z(ℳ∨𝒩) + (ℳ_1 + 𝒩_1) is then also w^∗-closed. Indeed, consider the natural projection of X onto the quotient X / z(ℳ∨𝒩). Consider a base h_1, …, h_n of ℳ_1 + 𝒩_1. That map is well defined because z(ℳ∨𝒩) is w^∗-closed and continuous. The space spanned by Ph_1, …, Ph_n is finite dimensional, thus closed in the quotient topology which is also locally convex. But looking at the inverse image we see that:


    P^-1(span{Ph_1,…, Ph_n }) = z(ℳ∨𝒩) + (ℳ_1 + 𝒩_1).


By the continuity of P we deduce that this space is w^∗-closed. Since ℳ + 𝒩 is w^∗-dense in ℳ∨𝒩 we get from the above inclusions that


    z(ℳ∨𝒩) + (ℳ_1 + 𝒩_1) = ℳ∨𝒩,


and so,
 

    ind(ℳ∨𝒩) = (ℳ_1 + 𝒩_1) ≤(ℳ_1) + (𝒩_1) = ind(ℳ) + ind(𝒩).

 
To prove the second implication we use a similar argument.
 













Let ℳ∈ Lat_X(M_z,w^∗) and λ∉𝒵(ℳ). The following are equivalent:



  * ind(ℳ)=1,

  * If f∈ℳ such that f(λ)=0 then there exists some h∈ℳ such that (z-λ)h = f,

  * If (z-λ)h = f ∈ℳ for some h ∈ X  then h ∈ℳ.














The equivalence of (2) and (3) is elementary. We will first prove that (1) implies (3). 



Let (z- λ)h ∈ℳ for some h ∈ X, and suppose that h ∉ℳ. The function f:= (z-λ)h ∈ℳ satisfies f(λ)=0 but f ∉ (z-λ)ℳ. Hence the equivalence class f̅∈ℳ/(z-λ)ℳ is non zero. Since λ∉𝒵(ℳ) there exists some function g ∈ℳ such that g(λ) ≠ 0, which also means that g ∉ (z-λ)ℳ so g̅≠ 0. Since ind(ℳ)=1 there exists μ∈\{ 0} such that g̅ = μf̅. That means precisely that g  ∈μ f + (z-λ)ℳ. By evaluating at z=λ we get g(0) = μ f(λ) +0 = 0 which is contradictory. 




To prove that (3) implies (1), suppose that whenever (z-λ)h ∈ℳ for some h ∈ X, we have that h ∈ℳ. Consider f,g ∈ℳ and their respective equivalence classes, f̅,g̅∈ℳ/(z-λ)ℳ. We need to show that they are linearly dependent as vectors and that way conclude that the dimension of the quotient is in fact equal to one. If either f̅ or g̅ are zero then there is nothing to show, so we may assume neither of them are. That in particular means that, thanks to the hypothesis, that f(λ),g(λ)≠ 0. Consider the function g_0(z) = g(z) ·f(λ)/g(λ)∈ℳ. Then f-g_0 ∈ℳ and f(λ) - g_0(λ) = 0. Therefore we can write f-g_0 = (z-λ)h for some h ∈ X, and by the hypothesis that means that h ∈ℳ and thus we conclude that f-g_0 ∈ (z-λ)ℳ. Therefore f-g_0 = 0 f̅-g̅_̅0̅ =0 f̅ = g̅_̅0̅f̅ = f(λ)/g(λ)g̅, and thus the equivalence classes of f and g are linearly dependent.












Let ℳ∈ Lat_X(M_z,w^∗) and λ∉𝒵(ℳ). The following are equivalent:



  * ind(ℳ)=1,

  * There exists a (not necessarily closed) subspace L ℳ such that L^w^∗ = ℳ, with the properties that λ∉𝒵(L) and (z-λ)h ∈ L for some h ∈ X implies h ∈ℳ.













Proving that (1) implies (2) is achieved by simply taking L = ℳ and applying Proposition 4.3. 




To prove the converse, we will verify condition (3) of Proposition 4.3. Let (z-λ)h ∈ℳ for some h ∈ X. Since L^w^∗ = ℳ, there exists a net { f_i}_i∈ I L such that f_i w^∗ (z-λ)h. In particular by continuity of the evaluation functionals we have that f_i(λ) = k_λ(f_i)  k_λ((z-λ)h) = 0. Since λ∉𝒵(L) we can find a g ∈ L such that g(λ)≠ 0. For every i ∈ I we consider the function 


    g_i(z) = f_i(z) - f_i(λ)/g(λ)g(z) ∈ L.


This new net { g_i }_i∈ I has g_i(λ)=0 for all i ∈ I. That means that there is a net { h_i }_i ∈ I X such that g_i = (z-λ)h_i. But by the hypothesis this means that every h_i ∈ℳ for all i ∈ I. Since f_i w^∗(z-λ)h we get that  g_i w^∗(z-λ)h and hence (z-λ)h_i w^∗ (z-λ)h. By continuity of the division operator we get h_i w^∗h. Since h_i ∈ℳ for all i ∈ I, and ℳ is w^∗-closed, we obtain that h ∈ℳ. Condition (3) is therefore satisfied and ind(ℳ)=1.












Let f ∈ X, f ≠ 0. Then ind[f]_∗ = 1.












Since f ≠ 0 there is some λ∈ such that f(λ) ≠ 0. It suffices then to verify condition (2) of Proposition 4.4 by taking L = {pf   :   p   polynomial}.












Let ℳ_1 , ℳ_2 ∈ Lat_X(M_z,w^∗)  have the index one property, and let λ∉𝒵(ℳ_1) ∪𝒵(ℳ_2). The following are equivalent:



  * ind(ℳ_1 ∨ℳ_2)=1,

  * There exist nets {g_i^1 }_i ∈ Iℳ_1,{g_i^2 }_i ∈ Iℳ_2 such that g_i^1(λ) = g_i^2(λ) =1 and g_i^1 - g_i^2 w^∗ 0.













Suppose that ind(ℳ_1 ∨ℳ_2)=1. Since λ∉𝒵(ℳ_1) ∪𝒵(ℳ_2) there are f_1 ∈ℳ_1, f_2 ∈ℳ_2 such that f_1(λ) = f_2(λ) = 1. We have that f_1 - f_2 ∈ℳ_1 ∨ℳ_2 and (f_1-f_2)(λ) = 0 so f_1 - f_2 = (z-λ)h for some h ∈ X. By Proposition 4.3, h ∈ℳ_1 ∨ℳ_2 and since ℳ_1 + ℳ_2 is dense in ℳ_1 ∨ℳ_2 there exists a net { h_i}_i ∈ Iℳ_1 + ℳ_2 with h_i w^∗h. By the definition of ℳ_1 + ℳ_2 we can find nets {h_i^1 }_i ∈ Iℳ_1,{h_i^2 }_i ∈ Iℳ_2 such that h_i = h_i^1 - h_i^2 w^∗h. Then (z-λ)h_i^1 - (z-λ)h_i^2w^∗ (z-λ)h = f_1 - f_2 by continuity of the shift operator. Define the nets:


    g_i^1 = (z-λ)h_i^1 + f_1 ∈ℳ_1 
    
    g_i^2 = (z-λ)h_i^2 + f_2 ∈ℳ_2  , with 
    
    g_i^1(λ) = 0 + f_1(λ) = 1   and   g_i^2(λ) = 0 + f_2(λ) = 1


and notice that


    g_i^1 - g_i^2 = (z-λ)h_i^1 + f_1 -(z-λ)h_i^2 -f_2 = (z-λ)(h_i^1-h_i^2) +f_1 -f_2 w^∗ (z-λ)h - (z-λ)h = 0,


which gives (2). To prove the contrary we will verify condition (2) of proposition 4.4. To that end, take L = ℳ_1 + ℳ_2 and consider a function h ∈ X with (z-λ)h ∈ℳ_1 + ℳ_2. We will show that h ∈ℳ_1 ∨ℳ_2. We write (z-λ)h ∈ℳ_1 + ℳ_2 as (z-λ)h = f_1 + f_2 with f_i ∈ℳ_i. By the hypothesis there are nets 


    { g_i^1 }_i∈ Iℳ_1 , { g_i^2 }_i∈ Iℳ_2   , with
    
    g_i^1(λ) = g_i^2(λ) = 1   and   g_i^1 - g_i^2 w^∗ 0.


We write:


    f_1(z) = f_1(z) - f_1(λ)g_i^1(z) +  f_1(λ)g_i^1(z), 
    
    f_2(z) = f_2(z) - f_2(λ)g_i^2(z) +  f_2(λ)g_i^2(z).


and notice that:


    f_1 - f_1(λ)g_i^1 ∈ℳ_1   and   f_1(λ) - f_1(λ)g_i^1(λ) = 0    i ∈ I , 
    
    f_2 - f_2(λ)g_i^2 ∈ℳ_2   and   f_2(λ) - f_2(λ)g_i^2(λ) = 0    i ∈ I.


Then there are nets { h_i^1 }_i∈ I,{ h_i^2 }_i∈ I X such that: 


    f_1 - f_1(λ)g_i^1 = (z-λ)h_i^1   and   f_2 - f_2(λ)g_2^1 = (z-λ)h_i^2.


Moreover, by the fact that each of the subspaces is of index one and by Proposition 4.3 we can conclude that in fact h_i^1 ∈ℳ_1 and h_i^2 ∈ℳ_2 for all i ∈ I. Notice as well that f_1(λ)+f_2(λ) = 0  f_1(λ) = -f_2(λ). This permits us to write:


    f_1 + f_2 = (z-λ)h_i^1 + f_1(λ)g_i^1 + (z-λ)h_i^2 + f_2(λ)g_i^2 = (z-λ)(h_i^1 + h_i^2) + f_1(λ)(g_i^1-g_i^2).


Since g_i^1 - g_i^2 w^∗ 0 we have that (z-λ)(h_i^1+h_i^2) w^∗ f_1+f_2 = (z-λ)h. Once again by continuity of the division operator we may conclude that (h_i^1+h_i^2) w^∗ h, which gives that h ∈ℳ_1 ∨ℳ_2.













Let ℳ∈ Lat_X_0(M_z) be an invariant subspace of index 1. Then ℳ^w^∗∈ Lat_X(M_z, w^∗) has index 1.











Let λ∉𝒵(ℳ). Then λ∉𝒵(ℳ^w^∗). We set L = ℳ and then L^w^∗ = ℳ^w^∗. Let h ∈ X with (z - λ)h ∈ L = ℳ X_0. Since ℳ has index 1, we deduce from the analog of Proposition 4.3 for the norm topology case that h ∈ℳ. By Proposition 4.4, ind(ℳ^w^∗) = 1.













Let ℳ∈ Lat_X_0(M_z) be an invariant subspace with ind(ℳ) = M, where M is finite or countably infinite. Then ℳ^w^∗∈ Lat_X(M_z, w^∗) satisfies ind(ℳ^w^∗) ≤ M. 












By (1) of Proposition 4.2, we may write ℳ = ℳ_1 ∨ℳ_2 ∨⋯∨ℳ_M, where each ℳ_n has index 1. Then ℳ^w^∗ = ℳ_1^w^∗∨ℳ_2^w^∗∨⋯∨ℳ_M^w^∗. By combining Propositions 4.2 and 4.7 we obtain the result.
 











    If ℳ X_0, is a norm closed and convex set, then ℳ^w^∗∩ X_0 = ℳ.



    One inclusion is obvious. For the other inclusion consider h ∈ℳ^w^∗∩ X_0. There exists a net {h_i }_i∈ Iℳ X_0 such that h_i w^∗ h in X. Notice that both the net and its limit belong to the space X_0. By the dualities X_0 ≅ Y and Y ≅ X we know that h_i w^∗ h in X is the same as h_i  h weakly in X_0. Hence h belongs to the weak closure of ℳ in X_0. Since ℳ is convex, its weak closure coincides with its norm closure, and as such we deduce that h ∈ℳ, as ℳ is itself norm closed.


We may now prove one of the main theorems of this section.













Proof of Theorem 1.6.




We assume first that M< ∞ and suppose that ind(ℳ^w^∗) < M. Without loss of generality we may assume that 0 ∉𝒵(ℳ).  Let {f_1,f_2,…,f_M } be a set of functions whose equivalence classes in ℳ/zℳ form a base. By our assumption, the set {f_1, f_2, …, f_M }ℳ^w^∗/ zℳ^w^∗ has to be linearly dependent, and therefore there exist λ_1,λ_2, …, λ_M not all zero such that:


    ∑_m=1^Mλ_m f_m = 0   ,   in  ℳ^w^∗/ zℳ^w^∗.


That means that there exists a function h ∈ℳ^w^∗ such that:


    ∑_m=1^Mλ_m f_m =  zh   ,   in   X.


It suffices to prove that h ∈ℳ, because that provides a contradiction to the fact that the equivalence classes of the functions f_m  , 1 ≤ m ≤ M form a base in  ℳ/z ℳ. Notice that (27) actually says that zh ∈ X_0. From the division property on X_0 we deduce that h ∈ X_0. Since h ∈ℳ^w^∗ we get that h ∈ℳ^w^∗∩ X_0. Moreover, ℳ is convex, as it is a linear subspace of X_0. From Lemma 4.9, we obtain that h ∈ℳ. 

In the above argument we essentially demonstrated that, given an ℳ∈ Lat_X_0(M_z) and a set of functions f_1,f_2,…,f_M, linearly independent in the quotient space ℳ/zℳ, the same set of functions forms a linearly independent set in the quotient space ℳ^w^∗/ zℳ^w^∗. In the case where M = ∞ we consider an infinite set { f_1, f_2, …}ℳ that forms a base in ℳ/zℳ. Then for any N ∈ the set {f_1, f_2, … f_N } will form a linearly independent set in ℳ^w^∗/ zℳ^w^∗, proving that for every N ∈, ind(ℳ^w^∗) ≥ N, and hence ind(ℳ^w^∗) = ∞.

▪









Note that the above argument works exactly in the same way if M=1, reproving Proposition 4.7. Lastly, we can apply this to prove Theorem 1.7:




Proof of Theorem 1.7.




Let N ∈{2, …, ∞}. By Theorem 1.5 there exist functions f_n _0, 1 ≤ n < N such that E_N := ∨_n=1^N[f_n] has index N in _0. An application of Theorem 1.6 for for X_0 = _0, Y= A^1 and X = yields the result.

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ACKNOWLEDGEMENTS. 

I would like to thank my advisors Evgeny Abakumov and Alexander Borichev for proposing me the problems and for proof-checking my drafts. I would especially like to thank A. Borichev for sharing with me his rich ideas and valuable techniques.








[heading=bibintoc,title=References] 














NIKIFOROS BIEHLER: 

Univ Gustave Eiffel, Univ Paris Est Creteil, CNRS, LAMA UMR8050, F-77447 Marne-la-Vallée, France
 
 Email address : nikiforos.biehler@univ-eiffel.fr

