Let $M=G/K$ be a Hermitian symmetric space of the noncompact type and let $\pi$ be a discrete series representation of $G$ holomorphically induced from a unitary character of $K$. Following an idea of Figueroa, Gracia-Bondìa and Vàrilly, we construct a Stratonovich-Weyl correspondence for the triple $(G,\pi ,M)$ by a suitable modification of the Berezin calculus on $M$. We extend the corresponding Berezin transform to a class of functions on $M$ which contains the Berezin symbol of $d\pi \left(X\right)$ for $X$ in the Lie algebra $𝔤$ of $G$. This allows...

We construct and study a Stratonovich-Weyl correspondence for the holomorphic representations of the Jacobi group.

We prove Strichartz's conjecture regarding a characterization of Hardy-Sobolev spaces.

Let HE∞ be the space of all bounded holomorphic functions on the unit ball of the Banach space E. In this note we study the algebra homomorphisms on HE∞ which are strict continuous.

Let $X$ be a completely regular Hausdorff space, $E$ a real Banach space, and let $C_b(X,E)$ be the space of all $E$-valued bounded continuous functions on $X$. We study linear operators from $C_b(X,E)$ endowed with the strict topologies ${\beta }_z$$(z=\sigma ,...$

Let $X$ be a completely regular Hausdorff space, $C_b\left(X\right)$ the space of all scalar-valued bounded continuous functions on $X$ with strict topologies. We prove that these are locally convex topological algebras with jointly continuous multiplication. Also we find the necessary and sufficient conditions for these algebras to be locally $m$-convex.

We study strict u-ideals in Banach spaces. A Banach space X is a strict u-ideal in its bidual when the canonical decomposition $X***=X*\oplus X^{\perp }$ is unconditional. We characterize Banach spaces which are strict u-ideals in their bidual and show that if X is a strict u-ideal in a Banach space Y then X contains c₀. We also show that ${\ell }_{\infty }$ is not a u-ideal.

Let ${\left\{\beta \left(n\right)\right\}}_{n=0}^{\infty }$ be a sequence of positive numbers and $1\le p<\infty$. We consider the space $H^p\left(\beta \right)$ of all power series $f\left(z\right)={\sum }_{n=0}^{\infty }\widehat{f}\left(n\right)z^n$ such that ${\sum }_{n=0}^{\infty }{\left|\widehat{f}\left(n\right)\right|}^p\beta {\left(n\right)}^p<\infty$. We investigate strict cyclicity of $H_p^{\infty }\left(\beta \right)$, the weakly closed algebra generated by the operator of multiplication by $z$ acting on $H^p\left(\beta \right)$, and determine the maximal ideal space, the dual space and the reflexivity of the algebra $H_p^{\infty }\left(\beta \right)$. We also give a necessary condition for a composition operator to be bounded on $H^p\left(\beta \right)$ when $H_p^{\infty }\left(\beta \right)$ is strictly cyclic.

If G is the closure of $L_{\infty }$ in exp L₂, it is proved that the inclusion between rearrangement invariant spaces E ⊂ F is strictly singular if and only if it is disjointly strictly singular and E ⊊ G. For any Marcinkiewicz space M(φ) ⊂ G such that M(φ) is not an interpolation space between $L_{\infty }$ and G it is proved that there exists another Marcinkiewicz space M(ψ) ⊊ M(φ) with the property that the M(ψ) and M(φ) norms are equivalent on the Rademacher subspace. Applications are given and a question of Milman...