Let be a complex Banach space, with the unit ball . We study the spectrum of a bounded weighted composition operator on determined by an analytic symbol with a fixed point in such that is a relatively compact subset of , where is an analytic function on .
Let A stand for a Toeplitz operator with a continuous symbol on the Bergman space of the polydisk or on the Segal-Bargmann space over . Even in the case N = 1, the spectrum Λ(A) of A is available only in a few very special situations. One approach to gaining information about this spectrum is based on replacing A by a large “finite section”, that is, by the compression of A to the linear span of the monomials . Unfortunately, in general the spectrum of does not mimic the spectrum of A as...
We establish -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.
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 of the polydisc onto the mixed norm space of the unit disc for any 0 < p < ∞ and 0 < q ≤ ∞.
This paper characterizes the boundedness and compactness of weighted composition operators between a weighted-type space and the Hardy space on the unit ball of ℂⁿ.
Let H() denote the space of all holomorphic functions on the unit ball ⊂ ℂⁿ. Let φ be a holomorphic self-map of and u∈ H(). The weighted composition operator on H() is defined by . We investigate the boundedness and compactness of induced by u and φ acting from Zygmund spaces to Bloch (or little Bloch) spaces in the unit ball.