We consider operators acting in the space C(X) (X is a compact topological space) of the form $Au\left(x\right)=\left({\sum }_{k=1}^N{e}^{{\phi }_k}{T}_{{\alpha }_k}\right)u\left(x\right)={\sum }_{k=1}^N{e}^{{\phi }_k\left(x\right)}u\left({\alpha }_k\left(x\right)\right)$, u ∈ C(X), where ${\phi }_k\in C\left(X\right)$ and ${\alpha }_k:X\to X$ are given continuous mappings (1 ≤ k ≤ N). A new formula on the logarithm of the spectral radius r(A) is obtained. The logarithm of r(A) is defined as a nonlinear functional λ depending on the vector of functions $\phi ={\left({\phi }_k\right)}_{k=1}^N$. We prove that $ln\left(r\left(A\right)\right)=\lambda \left(\phi \right)=ma{x}_{\nu \in Mes}{\sum }_{k=1}^N{\int }_X{\phi }_{k}d{\nu }_k-\lambda *\left(\nu \right)$, where Mes is the set of all probability vectors of measures $\nu ={\left({\nu }_k\right)}_{k=1}^N$ on X × 1,..., N and λ* is some convex lower-semicontinuous functional on $\left(C^N\left(X\right)\right)*$. In other...

This paper studies properties of a large class of algebras of holomorphic functions with bounded growth in several complex variables.The main result is useful in the applications. Using the symbolic calculus of L. Waelbroeck, it gives for instance a theorem of the “Nullstellensatz” type and approximation theorems.

We solve the Kato square root problem for bounded measurable perturbations of subelliptic operators on connected Lie groups. The subelliptic operators are divergence form operators with complex bounded coefficients, which may have lower order terms. In this general setting we deduce inhomogeneous estimates. In case the group is nilpotent and the subelliptic operator is pure second order, we prove stronger homogeneous estimates. Furthermore, we prove Lipschitz stability of the estimates under small...

We study the bases and frames of reproducing kernels in the model subspaces $K_{\Theta }^2=H^2\ominus \Theta H^2$ of the Hardy class $H2$ in the upper half-plane. The main problem under consideration is the stability of a basis of reproducing kernels $k_{{\lambda }_n}\left(z\right)=(1-\overline{\Theta \left({\lambda }_n\right)}\Theta \left(z\right))/(z-{\overline{\lambda }}_n)$ under “small” perturbations of the points ${\lambda }_n$. We propose an approach to this problem based on the recently obtained estimates of derivatives in the spaces $K_{\Theta }^2$ and produce estimates of admissible perturbations generalizing certain results of W.S. Cohn and E. Fricain.

The aim of this paper is to investigate stability of unit ball in Orlicz spaces, endowed with the Luxemburg norm, from the “local” point of view. Firstly, those points of the unit ball are characterized which are stable, i.e., at which the map $z\to \{(x,y):\frac{1}{2}(x+y)=z\}$ is lower-semicontinuous. Then the main theorem is established: An Orlicz space $L^{\varphi }\left(\mu \right)$ has stable unit ball if and only if either $L^{\varphi }\left(\mu \right)$ is finite dimensional or it is isometric to $L^{\infty }\left(\mu \right)$ or $\varphi$ satisfies the condition ${\Delta }_r$ or ${\Delta }_r^0$ (appropriate to the measure $\mu$ and the function...