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 -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 for |z|,|w| < 1, by means of , 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 on the weighted Bergman space , where denotes the unit ball or the unit polydisk. We provide a necessary condition for to be complex symmetric. When , we prove that is complex symmetric on if and only if and . Moreover, we completely characterize when monomial Toeplitz operators on are -symmetric with the symmetric unitary matrix .
Let Ω be a C∞-domain in Cn. It is well known that a holomorphic function on Ω behaves twice as well in complex tangential directions (see [GS] and [Kr] for instance). It follows from well known results (see [H], [RS]) that some converse is true for any kind of regular functions when Ω satisfies(P) The real tangent space is generated by the Lie brackets of real and imaginary parts of complex tangent vectorsIn this paper we are interested in the behavior of holomorphic Hardy-Sobolev functions in...