This note corrects an error in the definition of the rate function in Jacquier, Pakkanen, and Stone (2018) and slightly simplifies some proofs.

In this note, we give a new property of de Branges–Rovnyak kernels. As the main theorem, it is shown that the exponential of de Branges–Rovnyak kernel is strictly positive definite if the corresponding Schur class function is nontrivial.

Introduced recently in mathematical finance by Bayer et al. (2016), the rough Bergomi model has proved particularly efficient to calibrate option markets. We investigate some of its probabilistic properties, in particular proving a pathwise large deviations principle for a small-noise version of the model. The exponential function (continuous but superlinear) as well as the drift appearing in the volatility process fall beyond the scope of existing results, and a dedicated analysis is needed.

We discuss the concept of inner function in reproducing kernelHilbert spaces with an orthogonal basis of monomials and examine connections between inner functions and optimal polynomial approximants to ${1}/{f}\;$, where $f$ is a function in the space. We revisit some classical examples from this perspective, and show how a construction of Shapiro and Shields can be modiûed to produce inner functions.

Let $\left( X,\,B,\,\mu \right)$ be a $\sigma $-finite measure space and let $H\,\subset \,{{L}^{2}}\left( X,\,\mu \right)$ be a separable reproducing kernel Hilbert space on $X$. We show that the multiplier algebra of $H$ has property $\left( {{A}_{1}}\left( 1 \right) \right)$.

Bohr's inequality says that if $f(z) = \sum^{\infty}_{n = 0} a_n z^n$ is a bounded analytic function on the closed unit disc, then $\sum^{\infty}_{n = 0} \lvert a_n\rvert r^n \leq \Vert f\Vert_{\infty}$ for $0 \leq r \leq 1/3$ and that $1/3$ is sharp. In this paper we give an operator-theoretic proof of Bohr's inequality that is based on von Neumann's inequality. Since our proof is operator-theoretic, our methods extend to several complex variables and to non-commutative situations.

We obtain Bohr type inequalities for the algebras of bounded analytic functions and the multiplier algebras of reproducing kernel Hilbert spaces on various higher-dimensional domains, for the non-commutative disc algebra ${\mathcal A}_n$, and for the reduced (respectively full) group C*-algebra of the free group on $n$ generators.

We also include an application to Banach algebras. We prove that every Banach algebra has an equivalent norm in which it satisfies a non-unital version of von Neumann's inequality.

2000 Mathematical Subject Classification: 47A20, 47A56.