In 2005, N. Nikolski proved among other things that for any $r\in (0,1)$ and any $K\geq 1$, the condition number $CN(T)=\Vert T\Vert \cdot \Vert T^{-1}\Vert $ of any invertible n-dimensional complex Banach space operators T satisfying the Kreiss condition, with spectrum contained in $\left \{ r\leq |z|<1\right \}$, satisfies the inequality $CN(T)\leq CK(T)\Vert T \Vert n/r^{n}$ where $K(T)$ denotes the Kreiss constant of T and $C>0$ is an absolute constant. He also proved that for $r\ll 1/n,$ the latter bound is asymptotically sharp as $n\rightarrow \infty $. In this note, we prove that this bound is actually achieved by a family of explicit $n\times n$ Toeplitz matrices with arbitrary singleton spectrum $\{\lambda \}\subset \mathbb {D}\setminus \{0\}$ and uniformly bounded Kreiss constant. Independently, we exhibit a sequence of Jordan blocks with Kreiss constants tending to $\infty $ showing that Nikolski’s inequality is still asymptotically sharp as K and n go to $\infty $.

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.

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.

If $B$ is the Blachke product with zeros $\{{{z}_{n}}\},\,\text{then}\,\left| {B}'\left( z \right) \right|\,\le \,{{\Psi }_{B}}\left( z \right)$, where

$${{\Psi }_{B}}\,=\,\sum\limits_{n}{\frac{1-{{\left| {{z}_{n}} \right|}^{2}}}{{{\left| 1-{{\overline{z}}_{n}}z \right|}^{2}}}.}$$

Moreover, it is a well-known fact that, for $0\,<\,p\,<\,\infty $,

$${{M}_{p\left( r,{B}' \right)}}\,=\,{{\left( \frac{1}{2\pi }\,\int_{0}^{2\pi }{{{\left| {B}'\left( \text{r}{{\text{e}}^{i\theta }} \right) \right|}^{p}}d\theta } \right)}^{{1}/{p}\;}},\,0\,\le \,r\,<\,1,$$

is bounded if and only if ${{M}_{p}}\left( r,\,{{\Psi }_{B}} \right)$ is bounded. We find a Blaschke product ${{B}_{0}}$ such that ${{M}_{p}}\left( r,\,{{{{B}'}}_{0}} \right)$ and ${{M}_{p}}\left( r,{{\Psi }_{{{B}_{0}}}} \right)$ are not comparable for any $\frac{1}{2}\,<\,p\,<\,\infty $. In addition, it is shown that, if $0\,<\,p\,<\,\infty$, $B$ is a Carleson-Newman Blaschke product and a weight $\omega $ satisfies a certain regularity condition, then

$${{\int }_{\mathbb{D}}}{{\left| {B}'\left( z \right) \right|}^{p}}\omega \left( z \right)dA\left( z \right)~\asymp {{\int }_{\mathbb{D}}}{{\Psi }_{B}}{{\left( z \right)}^{p}}\omega \left( z \right)dA\left( z \right),$$

where $d\,A\left( z \right)$ is the Lebesgue area measure on the unit disc.

We consider semicrossed products of the disk algebra with respect to endomorphisms defined by finite Blaschke products. We characterize the Jacobson radical of these operator algebras. Furthermore, in the case that the finite Blaschke product is elliptic, we show that the semicrossed product contains no nonzero quasinilpotent elements. However, if the finite Blaschke product is hyperbolic or parabolic with positive hyperbolic step, the Jacobson radical is nonzero and a proper subset of the set of quasinilpotent elements.