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It is well known that for any inner function
$\theta $
defined in the unit disk
$\mathbb {D}$
, the following two conditions: (i) there exists a sequence of polynomials
$\{p_n\}_n$
such that
$\lim _{n \to \infty } \theta (z) p_n(z) = 1$
for all
$z \in \mathbb {D}$
and (ii)
$\sup _n \| \theta p_n \|_\infty < \infty $
, are incompatible, i.e., cannot be satisfied simultaneously. However, it is also known that if we relax the second condition to allow for arbitrarily slow growth of the sequence
$\{ \theta (z) p_n(z)\}_n$
as
$|z| \to 1$
, then condition (i) can be met for some singular inner function. We discuss certain consequences of this fact which are related to the rate of decay of Taylor coefficients and moduli of continuity of functions in model spaces
$K_\theta $
. In particular, we establish a variant of a result of Khavinson and Dyakonov on nonexistence of functions with certain smoothness properties in
$K_\theta $
, and we show that the classical Aleksandrov theorem on density of continuous functions in
$K_\theta $
is essentially optimal. We consider also the same questions in the context of de Branges–Rovnyak spaces
$\mathcal {H}(b)$
and show that the corresponding approximation result also is optimal.
Given a holomorphic self-map
$\varphi $
of
$\mathbb {D}$
(the open unit disc in
$\mathbb {C}$
), the composition operator
$C_{\varphi } f = f \circ \varphi $
,
$f \in H^2(\mathbb {\mathbb {D}})$
, defines a bounded linear operator on the Hardy space
$H^2(\mathbb {\mathbb {D}})$
. The model spaces are the backward shift-invariant closed subspaces of
$H^2(\mathbb {\mathbb {D}})$
, which are canonically associated with inner functions. In this paper, we study model spaces that are invariant under composition operators. Emphasis is put on finite-dimensional model spaces, affine transformations, and linear fractional transformations.
We show that any weakly separated Bessel system of model spaces in the Hardy space on the unit disc is a Riesz system and we highlight some applications to interpolating sequences of matrices. This will be done without using the recent solution of the Feichtinger conjecture, whose natural generalization to multidimensional model subspaces of
${\mathrm {H}}^2$
turns out to be false.
Let
$\mathcal {N}$
be the Nevanlinna class, and let B be a Blaschke product. It is shown that the natural invertibility criterion in the quotient algebra
$\mathcal {N} / B \mathcal {N}$
, that is,
$|f| \ge e^{-H} $
on the set
$B^{-1}\{0\}$
for some positive harmonic function H, holds if and only if the function
$- \log |B|$
has a harmonic majorant on the set
$\{z\in \mathbb {D}:\rho (z,\Lambda )\geq e^{-H(z)}\}$
, at least for large enough functions H. We also study the corresponding class of positive harmonic functions H on the unit disc such that the latter condition holds. We also discuss the analogous invertibility problem in quotients of the Smirnov class.
Let $(z_k)$ be a sequence of distinct points in the unit disc $\mathbb {D}$ without limit points there. We are looking for a function $a(z)$ analytic in $\mathbb {D}$ and such that possesses a solution having zeros precisely at the points $z_k$, and the resulting function $a(z)$ has ‘minimal’ growth. We focus on the case of non-separated sequences $(z_k)$ in terms of the pseudohyperbolic distance when the coefficient $a(z)$ is of zero order, but $\sup _{z\in {\mathbb D}}(1-|z|)^p|a(z)| = + \infty$ for any $p > 0$. We established a new estimate for the maximum modulus of $a(z)$ in terms of the functions $n_z(t)=\sum \nolimits _{|z_k-z|\le t} 1$ and $N_z(r) = \int_0^r {{(n_z(t)-1)}^ + } /t{\rm d}t.$ The estimate is sharp in some sense. The main result relies on a new interpolation theorem.
This note characterizes, in terms of interpolating Blaschke products, the symbols of Hankel operators essentially commuting with all quasicontinuous Toeplitz operators on the Hardy space of the unit circle. It also shows that such symbols do not contain the complex conjugate of any nonconstant singular inner function.
It is known that if
$S(z)$
is a non-constant singular inner function defined on the unit disk, then
$\min _{|z|\le r}|S(z)|\to 0$
as
$r\to 1^-$
. We show that the convergence can be arbitrarily slow.
For an inner function u, we discuss the dual operator for the compressed shift $P_u S|_{{\mathcal {K}}_u}$, where ${\mathcal {K}}_u$ is the model space for u. We describe the unitary equivalence/similarity classes for these duals as well as their invariant subspaces.
The relationship between the distribution of zeros of an infinite Blaschke product $B$ and the inclusion in weighted Bergman spaces $A_{\unicode[STIX]{x1D6FC}}^{p}$ of the derivative of $B$ or the derivative of functions in its model space $H^{2}\ominus \mathit{BH}^{2}$ is investigated.
In this paper we discuss the range of a co-analytic Toeplitz operator. These range spaces are closely related to de Branges–Rovnyak spaces (in some cases they are equal as sets). In order to understand its structure, we explore when the range space decomposes into the range of an associated analytic Toeplitz operator and an identifiable orthogonal complement. For certain cases, we compute this orthogonal complement in terms of the kernel of a certain Toeplitz operator on the Hardy space, where we focus on when this kernel is a model space (backward shift invariant subspace). In the spirit of Ahern–Clark, we also discuss the non-tangential boundary behavior in these range spaces. These results give us further insight into the description of the range of a co-analytic Toeplitz operator as well as its orthogonal decomposition. Our Ahern–Clark type results, which are stated in a general abstract setting, will also have applications to related sub-Hardy Hilbert spaces of analytic functions such as the de Branges–Rovnyak spaces and the harmonically weighted Dirichlet spaces.
Let ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$ be Dirichlet spaces with superharmonic weights induced by positive Borel measures $\unicode[STIX]{x1D707}$ on the open unit disk. Denote by $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ Möbius invariant function spaces generated by ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$. In this paper, we investigate the relation among ${\mathcal{D}}_{\unicode[STIX]{x1D707}}$, $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ and some Möbius invariant function spaces, such as the space $BMOA$ of analytic functions on the open unit disk with boundary values of bounded mean oscillation and the Dirichlet space. Applying the relation between $BMOA$ and $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$, under the assumption that the weight function $K$ is concave, we characterize the function $K$ such that ${\mathcal{Q}}_{K}=BMOA$. We also describe inner functions in $M({\mathcal{D}}_{\unicode[STIX]{x1D707}})$ spaces.
If A(z) belongs to the Bergman space , then the differential equation f″+A(z)f=0 is Blaschke-oscillatory, meaning that the zero sequence of every nontrivial solution satisfies the Blaschke condition. Conversely, if A(z) is analytic in the unit disc such that the differential equation is Blaschke-oscillatory, then A(z) almost belongs to . It is demonstrated that certain “nice” Blaschke sequences can be zero sequences of solutions in both cases when A ∈ or A ∉ . In addition, no condition regarding only the number of zeros of solutions is sufficient to guarantee that A ∈ .
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