In this paper we obtain some equivalent characterizations of Bloch functions on general bounded strongly pseudoconvex domains with smooth boundary, which extends the known results in [1, 9, 10].

Nous donnons des résultats théoriques sur l’idéal de Macaev et la trace de Dixmier. Ensuite, nous caractérisons les symboles antiholomorphes $\overline{f}$ tels que l’opérateur de Hankel $H_{\overline{f}}$ sur l’espace de Bergman à poids soit dans l’idéal de Macaev et nous donnons la trace de Dixmier. Pour cela, nous regardons le comportement des normes de Schatten ${𝒮}^p$ quand $p$ tend vers $1$ et nous nous appuyons sur le résultat de Engliš et Rochberg sur l’espace de Bergman. Nous parlons aussi des puissances de tels opérateurs. Abstract....

Let $\varphi$ and $\psi$ be holomorphic self-maps of the unit disk, and denote by $C_{\varphi }$, $C_{\psi }$ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators $C_{\varphi }-C_{\psi }$ from Bloch spaces to Bloch spaces in the unit disk. Compactness of the difference is also characterized.

Let $K$ be a compact set in an open set $D$ on a Stein manifold $\Omega$ of dimension $n$. We denote by $H^{\infty }\left(D\right)$ the Banach space of all bounded and analytic in $D$ functions endowed with the uniform norm and by $A_K^D$ a compact subset of the space $C\left(K\right)$ consisted of all restrictions of functions from the unit ball ${𝔹}_{H^{\infty }\left(D\right)}$. In 1950ies Kolmogorov posed a problem: does$${\mathscr{H}}_{\epsilon }\left(A_K^D\right)\sim \tau {\left(ln\frac{1}{\epsilon }\right)}^{n+1},\epsilon \to 0,$$where ${\mathscr{H}}_{\epsilon }\left(A_K^D\right)$ is the $\epsilon$-entropy of the compact $A_K^D$. We give here a survey of results concerned with this problem and a related problem on the strict asymptotics of Kolmogorov diameters...