We consider separately radial (with corresponding group ${𝕋}^n$) and radial (with corresponding group $\mathrm{U}\left(n\right))$ symbols on the projective space ${\mathbb{P}}^n\left(ℂ\right)$, as well as the associated Toeplitz operators on the weighted Bergman spaces. It is known that the $C^{*}$-algebras generated by each family of such Toeplitz operators are commutative (see R. Quiroga-Barranco and A. Sanchez-Nungaray (2011)). We present a new representation theoretic proof of such commutativity. Our method is easier and more enlightening as it shows that the...

Toeplitz operators on strongly pseudoconvex domains in Cn, constructed from the Bergman projection and with symbol equal to a positive power of the distance to the boundary, are considered. The mapping properties of these operators on Lp, as the power of the distance varies, are established.

We establish $L^p$-estimates for the weighted Bergman projection on a nonsingular cone. We apply these results to the weighted Fock space with respect to the minimal norm in ℂⁿ.

We introduce the Bloch space for the minimal ball and we prove that this space can be identified with the dual of a certain analytic space which is strongly related to the Bergman theory on the minimal ball.

For any holomorphic function F in the unit polydisc Uⁿ of ℂⁿ, we consider its restriction to the diagonal, i.e., the function in the unit disc U of ℂ defined by F(z) = F(z,...,z), and prove that the diagonal mapping maps the mixed norm space $H^{p,q,\alpha }\left(Uⁿ\right)$ of the polydisc onto the mixed norm space $H^{p,q,\left|\alpha \right|+(p/q+1)(n-1)}\left(U\right)$ of the unit disc for any 0 < p < ∞ and 0 < q ≤ ∞.

We prove that compactness of the canonical solution operator to $\overline{\partial }$ restricted to $(0,1)$-forms with holomorphic coefficients is equivalent to compactness of the commutator $[𝒫,\overline{M}]$ defined on the whole $L_{(0,1)}^2\left(\Omega \right),$ where $\overline{M}$ is the multiplication by $\overline{z}$ and $𝒫$ is the orthogonal projection of $L_{(0,1)}^2\left(\Omega \right)$ to the subspace of $(0,1)$ forms with holomorphic coefficients. Further we derive a formula for the $\overline{\partial }$-Neumann operator restricted to $(0,1)$ forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplications...

It has been known for a long time that the Szegö projection of tube domains over irreducible symmetric cones is unbounded in $L^p$ for $p\ne 2$. Indeed, this is a consequence of the fact that the characteristic function of a disc is not a Fourier multiplier, a fundamental theorem proved by C. Fefferman in the 70’s. The same problem, related to the Bergman projection, deserves a different approach. In this survey, based on joint work of the author with D. Békollé, G. Garrigós, M. Peloso and F. Ricci, we give...

We study Toeplitz operators $T_a$ with radial symbols in weighted Bergman spaces $A_{\mu }^p$, 1 < p < ∞, on the disc. Using a decomposition of $A_{\mu }^p$ into finite-dimensional subspaces the operator $T_a$ can be considered as a coefficient multiplier. This leads to new results on boundedness of $T_a$ and also shows a connection with Hardy space multipliers. Using another method we also prove a necessary and sufficient condition for the boundedness of $T_a$ for a satisfying an assumption on the positivity of certain indefinite...

We study the limit behavior of weighted Bergman kernels on a sequence of domains in a complex space ${ℂ}^N$, and show that under some conditions on domains and weights, weighed Bergman kernels converge uniformly on compact sets. Then we give a weighted generalization of the theorem given by M. Skwarczyński (1980), highlighting some special property of the domains, on which the weighted Bergman kernels converge uniformly. Moreover, we show that convergence of weighted Bergman kernels implies this property,...

On the Hardy space of the unit disk, we consider operators which are finite sums of products of a Toeplitz operator and a Hankel operator. We then give characterizations for such operators to be zero. Our results extend several known results using completely different arguments.