I give a characterization of the pseudoconvex Hartogs domains $A$ in $C^2$ that satisfy the equation $H^2(A,C)=0$, where $H^2(A,C)$ is the second cohomology group of $A$ with coefficients in the constant sheaf $C$.

Let $ℐ$ be a coherent subsheaf of a locally free sheaf $𝒪\left(E_0\right)$ and suppose that $ℱ=𝒪\left(E_0\right)/ℐ$ has pure codimension. Starting with a residue current $R$ obtained from a locally free resolution of $ℱ$ we construct a vector-valued Coleff-Herrera current $\mu$ with support on the variety associated to $ℱ$ such that $\phi$ is in $ℐ$ if and only if $\mu \phi =0$. Such a current $\mu$ can also be derived algebraically from a fundamental theorem of Roos about the bidualizing functor, and the relation between these two approaches is discussed. By a construction...

We study a certain operator of multiplication by monomials in the weighted Bergman space both in the unit disk of the complex plane and in the polydisk of the $n$-dimensional complex plane. Characterization of the commutant of such operators is given.

The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foiaş. Just as a contraction is related to the Szegö kernel $k_S(z,w)={(1-zw̅)}^{-1}$ for |z|,|w| < 1, by means of $(1/k_S)(T,T*)\ge 0$, we consider an arbitrary open connected domain Ω in ℂⁿ, a complete Pick kernel k on Ω and a tuple T = (T₁, ..., Tₙ) of commuting bounded operators on a complex separable Hilbert space ℋ such that (1/k)(T,T*) ≥ 0. For a complete Pick kernel the 1/k functional calculus makes sense in a beautiful...

We give a concrete description of complex symmetric monomial Toeplitz operators $T_{z^p{\overline{z}}^q}$ on the weighted Bergman space $A^2\left(\Omega \right)$, where $\Omega$ denotes the unit ball or the unit polydisk. We provide a necessary condition for $T_{z^p{\overline{z}}^q}$ to be complex symmetric. When $p,q\in {\mathbb{N}}^2$, we prove that $T_{z^p{\overline{z}}^q}$ is complex symmetric on $A^2\left(\Omega \right)$ if and only if $p_1=q_2$ and $p_2=q_1$. Moreover, we completely characterize when monomial Toeplitz operators $T_{z^p{\overline{z}}^q}$ on $A^2\left({𝔻}_n\right)$ are $J_U$-symmetric with the $n\times n$ symmetric unitary matrix $U$.