Shift operators, Cauchy integrals and approximations

We will study spaces of analytic functions corresponding to Borel measures of the form

where $dA$ and $dm$ are the area and arc-length measures on, respectively, the unit disk $\mathbb{D}:=\{z\in\mathbb{C}:|z|<1\}$ and its boundary circle $\mathbb{T}:=\{z\in\mathbb{C}:|z|=1\}$. The radial weight $G(1-|z|)$ living on $\mathbb{D}$ is defined in terms of a continuous, increasing and positive function $G$, and the weight $w$ living on $\mathbb{T}$ is a general Borel measurable non-negative integrable function. Given such a measure, we may construct first the Lebesgue space $\mathcal{L}^{2}(\mu)$ of (equivalence classes of) Borel measurable functions living on the carrier of $\mu$, and next consider its subspace $\mathcal{P}^{2}(\mu)$, by which we denote the smallest closed subspace of $\mathcal{L}^{2}(\mu)$ which contains the set $\mathcal{P}$ of analytic polynomials. The space $\mathcal{P}^{2}(\mu)$ will be the setting for the first part of our study.

The shift operator $M_{z}:\mathcal{P}^{2}(\mu)\to\mathcal{P}^{2}(\mu)$, which takes a function $f(z)$ to $zf(z)$, is a subnormal operator, in the sense that it is the restriction of a normal operator, namely $M_{z}:\mathcal{L}^{2}(\mu)\to\mathcal{L}^{2}(\mu)$, to an invariant subspace. From the point of view of an operator theorist, the significance of the pair $(\mathcal{P}^{2}(\mu),M_{z})$ lies in the fact the study of subnormal operators can essentially be reduced to the study of the operator $M_{z}:\mathcal{P}^{2}(\mu)\to\mathcal{P}^{2}(\mu)$ for some measure $\mu$ which is compactly supported in the plane. The monograph [6] by Conway is an excellent source of information on this topic.

For measures such as (1.1), the space $\mathcal{P}^{2}(\mu)$ is, like $\mathcal{L}^{2}(\mu)$, a space of Borel measurable functions on the closed disk $\overline{\mathbb{D}}=\mathbb{D}\cup\mathbb{T}$. In certain cases it is even a space of analytic functions on $\mathbb{D}$. In such a case, each element $f\in\mathcal{P}^{2}(\mu)$, a priori interpreted as a function on $\overline{\mathbb{D}}$, has a unique restriction $f_{\mathbb{D}}$ to the disk $\mathbb{D}$. The restriction $f_{\mathbb{D}}$ must be an analytic function by the virtue of it being a locally uniform limit of analytic polynomials. We will below use the term irreducible for such a space which is in this sense ”analytic”. It is a difficult problem (and in general open) to determine which weight pairs $(G,w)$ as in (1.1) produce an irreducible space. Khrushchev in the article [16] solved certain special cases of the problem. For instance, his results apply to $G(t)=t^{n}$ for some $n>0$, and $w=1_{E}$ being a characteristic function of a set $E\subset\mathbb{T}$ in a certain class (defined in terms of Beurling-Carleson conditions). Already these results have fascinating applications to function theory, of which there are plenty in [16]. The article [25] builds on Khrushchev’s work, explains the structure of $\mathcal{P}^{2}(\mu)$ when $w=1_{E}$ and $E$ is a general subset of $\mathbb{T}$, and showcases further applications to the theory of the Cauchy integral operator and de Branges-Rovnyak spaces.

Recently, the author found in [24] an exact condition for irreducibility of $\mathcal{P}^{2}(\mu)$ in the case when $G(t)$ decays at least exponentially as $t\to 0^{+}$, thus confirming a conjecture by Kriete and MacCluer from [20]. Roughly speaking, if $G(t)$ is smaller than the weight $\exp(-ct^{-1})$ for some $c>0$, or more precisely if

but large enough to satisfy

for some $d>0$, then the space $\mathcal{P}^{2}(\mu)$ is irreducible if and only if the carrier set of the measure $d\mu_{\mathbb{T}}=w\,d\textit{m}$ on $\mathbb{T}$ can be covered by intervals $I$ satisfying the condition

In order to properly state the result we will need to define the following concept of core sets. For our purposes this concept is critical, and it will appear frequently throughout the article.

The set $\text{core}(w)$ is open, and it does not depend on the particular representative of $w$ in the space $\mathcal{L}^{1}(\mathbb{T})$ of equivalence classes of functions which are Lebesgue integrable on $\mathbb{T}$ with respect to $d\textit{m}$.

Carriers are obviously not unique. The set

is a carrier for $w$. If $w$ is only defined up to a set of $m$-measure zero, then we may take as a carrier for $w$ any set differing from (1.4) by a set of $m$-measure zero. Since $\log 0=-\infty$, it is obvious from (1.2) that $\text{core}(w)$ is essentially contained in any carrier of $w$.

Irreducibility of $\mathcal{P}^{2}(\mu)$-spaces of the form (1.1) with $G$ satisfying (ExpDec) and (LogLogInt) can be characterized in terms of core sets. The next theorem, fundamental to our study, follows from [24, Theorem A], with the non-trivial part being the equivalence of the third condition and the other two.

In particular, the following measures $\mu$ correspond to irreducible $\mathcal{P}^{2}(\mu)$:

and

If the $\text{core}(w)$ is not a carrier of $w$, then the space $\mathcal{P}^{2}(\mu)$ will contain a full Lebesgue space $\mathcal{L}^{2}(w_{r}d\textit{m})$, members of which live only on $\mathbb{T}$. Here $w_{r}$ denotes a certain residual weight. The residuals play no role in the statements of our main results, but will be important in the proofs. Their definition is postponed to coming sections.

The reader might wonder what happens in the case $\beta<1$ in ($T1$). Then (ExpDec) is violated, and condition $(iii)$ in \threfIrrDef implies $(ii)$, but the converse is false. This can be inferred from work of Khruschev in [16], and this idea is further elaborated on in [25]. Also one might ask what happens if $\alpha\geq 1$ in ($T2$), which means that (LogInt) is violated. This case is less interesting: Volberg’s theorem in [30] implies that $\mathcal{P}^{2}(\mu)$ is then either a close cousin of the Hardy space $H^{2}$ (this happens when $\int_{\mathbb{T}}\log w\,d\textit{m}>-\infty$) or it is not a space of analytic functions at all (if $\int_{\mathbb{T}}\log w\,d\textit{m}=-\infty$). See also the introductory section to [24] for a more detailed account.

Having established fairly sharp conditions for irreducibility, a way opens to an operator and function theoretic study of this class of spaces. Motivated by certain applications which will soon be detailed, in the first part of the article we study the structure of $M_{z}$-invariant subspaces of $\mathcal{P}^{2}(\mu)$ generated by functions in $H^{\infty}$, the algebra of bounded analytic functions in $\mathbb{D}$. This question readily reduces to the study of invariant subspaces generated by singular inner functions

where $\nu$ is a finite positive singular Borel measure on $\mathbb{T}$. For $h\in H^{\infty}$, we will denote by $[h]$ the smallest $M_{z}$-invariant subspace containing $h$. It is well-known that any singular inner function generates a non-trivial invariant subspace in the classical Hardy space $H^{2}$ of square-summable Taylor series, and it is almost as well-known that in order for $S_{\nu}$ to generate a non-trivial invariant subspace in the standard weighted Bergman spaces (which are $\mathcal{P}^{2}(\mu)$-spaces of the kind (1.1) themselves, with $G(t)=t^{n}$ for some $n>-1$, and $w\equiv 0$) we must have $\nu(A)>0$ for some Beurling-Carleson set $A$ (see [18], [19], [27]).

Our first main result characterizes the cyclic singular inner functions in the considered class of $\mathcal{P}^{2}(\mu)$-spaces. By cyclicity we mean that $[S_{\nu}]=\mathcal{P}^{2}(\mu)$. It is not hard to see that the minimal considered rate of decay (ExpDec) of the part of $\mu$ living on $\mathbb{D}$ makes every non-vanishing bounded function be cyclic in $\mathcal{P}^{2}(\mu)$ in the case that $w=0$. Thus only properties of $w$ can stop $S_{\nu}$ from being cyclic.

Note that $\text{core}(w)$ is open, and hence Borel measurable, so $\nu\big{(}\text{core}(w)\big{)}$ makes perfect sense.

Having settled the cyclicity question, we turn our attention to the invariant subspace $[S_{\nu}]$ generated by a singular inner function corresponding to a measure $\nu$ which places all its mass on the core: $\nu(\mathbb{T})=\nu\big{(}\text{core}(w)\big{)}$. In other words, $\text{core}(w)$ is a carrier for $\nu$. A problem which arises in the theory of normalized Cauchy integrals and de Branges-Rovnyak spaces $\mathcal{H}(b)$ (to be discussed below) is to determine which functions are contained in the intersection $H^{2}\cap[S_{\nu}]$, or sometimes in $\mathcal{N}^{+}\cap[S_{\nu}]$, where $\mathcal{N}^{+}$ is the Smirnov class of the disk $\mathbb{D}$ (see [13] for precise definitions):

In this context, we have the following result.

In other words, if $f\in\mathcal{N}^{+}$ can be approximated by polynomial multiples of $S_{\nu}$ in the norm of $\mathcal{P}^{2}(\mu)$, and $\nu$ places all of its mass on $\text{core}(w)$, then $S_{\nu}$ appears in the inner-outer factorization of $f$. Under the additional assumption that $w$ is bounded, a simple argument will show that in fact $[S_{\nu}]\cap H^{2}=S_{\nu}H^{2}$. In [21] and [23], the feature of $S_{\nu}$ appearing in \threfPermanenceMainTheorem is called its permanence property. It is obvious that a singular inner function satisfying the permanence property cannot be cyclic.

For the considered class of spaces, \threfCyclicityMainTheorem and \threfPermanenceMainTheorem completely determine the structure of $M_{z}$-invariant subspaces generated by bounded analytic functions. Indeed, it follows that if $h=BS_{\nu}U\in H^{\infty}$ is the inner-outer factorization of $h$ into a Blaschke product $B$, singular inner function $S_{\nu}$ and outer function $U$, then

where $\nu_{w}$ is the restriction of the singular measure $\nu$ to the set $\text{core}(w)$.

Irreducible spaces find applications in the theory of Cauchy integrals.

Trivial examples of functions $f$ satisfying (RSD) are the analytic polynomials, and functions which extend analytically to a larger disk $r\mathbb{D}=\{z\in\mathbb{C}:|z|<r\}$, $r>1$. In those cases, the limit in (RSD) is zero even when the term $\exp\big{(}c\sqrt{n}\big{)}$ in (RSD) is replaced by $\exp\big{(}cn^{\alpha}\big{)}$ for $\alpha<1$. Conversely, if $f$ has an analytic extension to a disk around the origin of radius larger than 1, then $|f_{n}|=\mathcal{O}\big{(}\exp(-cn)\big{)}$ for some $c>0$.

Let us assume that $\nu$ is a finite Borel measure for which the Cauchy integral

is a function satisfying (RSD). Can we say something about the nature of the measure $\nu$? The Cauchy integral $\mathcal{C}_{\nu}$ has a representation of the form

where $\{\nu_{n}\}_{n\geq 0}$ is the sequence of Fourier coefficients of $\nu$ indexed by non-negative integers. The rest of the coefficients are annihilated under $\mathcal{C}$, and the condition (RSD) gives us no information about $\nu_{n}$ for $n<0$. However, the following statement is a consequence of the irreducibility of spaces corresponding to measures of the form ($T1$).

The function $\log|g|$ is, in general, not integrable on the entire open set $U$ appearing in \threfUncertThmRSD.

In a way, \threfUncertThmRSD is similar to the classical theorem of brothers Riesz on structure of measures $\nu$ on $\mathbb{T}$ with vanishing positive Fourier coefficients. In our setting, the vanishing of the coefficients is replaced by a weaker condition of their rapid decay forced by the condition (RSD). It should be noted that if we were to replace in (RSD) the term $\exp\big{(}c\sqrt{n}\big{)}$ by $\exp\big{(}cn^{\alpha}\big{)}$ for any $\alpha<1/2$, and thus consider the weaker unilateral spectral decay condition

then a structural result for $\nu$ as in \threfUncertThmRSD does not hold: $d\nu=g\,d\textit{m}$ will still be absolutely continuous, but examples show that $g$ can be chosen so that the set in (1.8) is closed and contains no interval. This follows from a related work of Khrushchev in [16]. There should be room for a slight improvement of the result (see the discussion in Section 1.6.4 below). We ought to mention also that Volberg in [30] found spectral decay conditions making the set in (1.8) fill up the whole circle $\mathbb{T}$. We will return to both these works below.

In most classical Hilbert spaces of analytic functions in the unit disk, the family of functions which extend analytically to a larger disk forms a dense subset of the space. This is not the case in Hilbert spaces of normalized Cauchy integrals. These are the so-called model spaces $K_{\theta}$, where $\theta$ is an inner function, and more generally the de Branges-Rovnyak spaces $\mathcal{H}(b)$, where the symbol $b$ is any analytic self-map of the unit disk. There are several ways to define the space $\mathcal{H}(b)$, the easiest perhaps being by stating that it is the Hilbert space of analytic functions on $\mathbb{D}$ with a reproducing kernel of the form

Alternatively, we may realize it as the space of normalized Cauchy integrals of functions $\tilde{f}\in\mathcal{L}^{2}(\nu_{b})$, given in the special case $b(0)=0$ by the formula

Here $\nu_{b}$ is the Aleksandrov-Clark measure of $b$, these two objects being related by the formula

The normalization refers to multiplication of the Cauchy integral in (1.9) by the factor $1-b(z)$, which ensures that the product lands in $H^{2}$. It is well-known that model spaces $K_{\theta}=\mathcal{H}(\theta)$ correspond to the purely singular measures $\nu_{\theta}$ in (1.10). In fact, every positive finite Borel measure $\nu$ on $\mathbb{T}$ corresponds to a function $b=b_{\nu}$ through the formula (1.10). See [5] for more details.

If $\theta$ is a singular inner function, then $\mathcal{K}_{\theta}$ will contain no functions which extend analytically across $\mathbb{T}$. Moreover, it is a consequence of deep results on cyclicity of singular inner functions of Beurling from [3], and also of more recent results of El-Fallah, Kellay and Seip from [10], that in fact if $\theta$ is singular, then for any non-zero function $f(z)=\sum_{n\geq 0}f_{n}z^{n}\in\mathcal{K}_{\theta}$ and for any $c>0$ it holds that

This fact is not as deep as the two results cited above which imply it, but it is needed in the proof of one of our main results. For this reason, we give an elementary proof in Section 6.1. We mention also that a characterization of density in $\mathcal{K}_{\theta}$ of functions in $\mathcal{A}^{\infty}$, the algebra of functions analytic in $\mathbb{D}$ with all derivatives extending continuously to $\overline{\mathbb{D}}$, has been established [22].

The situation is more interesting, and much more difficult to handle, in the general class of $\mathcal{H}(b)$-spaces. It was proved long ago by Sarason that the set $\mathcal{P}$ of analytic polynomials is contained and dense in $\mathcal{H}(b)$ if and only if $b$ is a non-extreme point of the unit ball of $H^{\infty}$, a condition characterized by

where

In terms of core sets, this result can be stated as follows, and a proof can be found in [28].

Since these conditions are very restrictive, it is tempting to make an effort to capture a larger class of symbols $b$ for which $\mathcal{H}(b)$ contains a dense subset of functions in some nice regularity class which is strictly larger than $\mathcal{P}$. The article [23] connects the approximation problem in $\mathcal{H}(b)$ with the structure of $M_{z}$-invariant subspaces of $\mathcal{P}^{2}(\mu)$, and [21] refines the method to prove the density of $\mathcal{A}^{\infty}\cap\mathcal{H}(b)$ for a large class of symbols $b$. The method from [23] is very general and applies to a wide range of approximation problems in $\mathcal{H}(b)$. In particular, it applies to approximations by functions in the class (RSD). Since our structural results in \threfCyclicityMainTheorem and \threfPermanenceMainTheorem are definitive, we can prove also a definitive result on existence and density of functions $f\in\mathcal{H}(b)$ which satisfy (RSD). In fact, we will prove a much stronger (and optimal) result.

In order to state our result, we will need to quantify the spectral decay of a function $f$ by a condition of the type (RSD) but with $\exp\big{(}c\sqrt{n}\big{)}$ replaced by faster increasing sequences. To this end, we define below in \threfAdmissibleSequenceDef the admissible sequences $M=\{M_{n}\}_{n\geq 0}$. These sequences are logarithmically convex (at least eventually, for large $n$) and are decreasing to zero at least as fast as $\exp\big{(}-c\sqrt{n}\big{)}$, but satisfy a condition of the form

which prohibits, for instance, their exponentially fast decay.

In $(iii)$, the condition of vanishing of $b$ at some $\lambda\in\mathbb{D}$ is the uninteresting case, sice then $\mathcal{H}(b)$ contains a rational function with no poles on $\overline{\mathbb{D}}$. For such a function $(ii)$ holds trivially.

To reach \threfMainTheoremHbExistenceESD we only really need the characterization of irreducibility of $\mathcal{P}^{2}(\mu)$. Proof of the next theorem requires the full strength of the invariant subspace results developed in the first part of this article.

Our proof of \threfMainTheoremHbDensityESD depends crucially on \threfCyclicityMainTheorem and \threfPermanenceMainTheorem, but is otherwise similar to the proofs in [21] and [23]. However, in the present work we obtain new information on which functions in $\mathcal{H}(b)$ fail to be approximable by classes appearing in \threfMainTheoremHbDensityESD. These results are presented in Section 7.

We mentioned earlier that our result is optimal. This is morally true, in the following sense. Assume that $M=\{M_{n}\}_{n\geq 0}$ is a logarithmically convex sequence which is not admissible according to \threfAdmissibleSequenceDef, because we have

For instance, $M$ could be defined by (1.12) for $p=1$. If Volberg or Kriete and MacCluer were interested in approximations in $\mathcal{H}(b)$-spaces, they would have proved the following theorem by a use of their techniques in [20] and [30].

We do not prove the above theorem in the present article. Its proof is completely analogous to the proof of \threfMainTheoremHbExistenceESD. The difference consists merely of a use of theorems and observations of Volberg and Kriete-MacCluer from the above mentioned papers, instead of main theorem of [24] as we do here in the proof of \threfMainTheoremHbExistenceESD.

It follows that the investigation of existence and approximability properties in $\mathcal{H}(b)$ of functions with spectral decay satisfying at least (RSD) is essentially completed in \threfMainTheoremHbExistenceESD and \threfMainTheoremHbDensityESD.

Section 2 deals with construction of special domains which look like wizard hats and which support very large positive harmonic functions. We prove \threfPermanenceMainTheorem and \threfCyclicityMainTheorem in Sections 3 and 4, respectively. Proof of \threfPermanenceMainTheorem relies heavily on results of Section 2. The second part of the article starts in Section 5. There we deal with some preparatory estimates on moments sequences which are needed later. \threfMainTheoremHbExistenceESD and \threfMainTheoremHbDensityESD are proved in Sections 6 and 7. The techniques used in these sections come from [23], but we refine some of the methods and prove auxilliary results of hopefully independent interest. Finally, in Section 8, we prove \threfUncertThmRSD.

For a measure $\mu$ on $\overline{\mathbb{D}}$ we will denote by $\mu_{\mathbb{D}}$ and $\mu_{\mathbb{T}}$ its restriction to $\mathbb{D}$ and $\mathbb{T}$, respectively. In some contexts we will also use the same notations $\mu_{\mathbb{D}}$ and $\mu_{\mathbb{T}}$ to emphasize that the considered measure lives only on $\mathbb{D}$ or $\mathbb{T}$. The area measure $dA$ will always be normalized by the condition $A(\mathbb{D})=1$, and a similar convention will be used also for the arc-length measure on the circle: $m(\mathbb{T})=1$. We let $\log^{+}(x)=\max\big{(}0,\log x\big{)}$.

The symbol $\|\cdot\|_{\mu}$ always denotes the usual $\mathcal{L}^{2}(\mu)$-norm corresponding to the finite positive Borel measure $\mu$. For a set $E\subset\mathbb{T}$, we sometimes use the shorter notation $\mathcal{L}^{2}(E)$ to denote the space of functions on $\mathbb{T}$ which vanish outside of $E$ and are square-integrable with respect to the Lebesgue measure m. The notation $\big{\langle}\cdot,\cdot\big{\rangle}$ denotes different kinds of duality pairings between spaces. By $\big{\langle}\cdot,\cdot\big{\rangle}_{\mathcal{L}^{2}}$ we will denote the standard inner product in $L^{2}(\mathbb{T})$.

The operator $P_{+}:\mathcal{L}^{2}(\mathbb{T})\to H^{2}$ is the orthogonal projection onto the Hardy space $H^{2}$. For a bounded analytic function $h$, the notation $T_{\overline{h}}:H^{2}\to H^{2}$ stands for the co-analytic Toeplitz operator with symbol $h$, this operator being defined by the formula $T_{\overline{h}}f=P_{+}\overline{h}f$.