Xian Min XU
Let Bd be the unit ball ofd-dimensional complex Euclid space Cd, Hdg (Bd√R) the reproducing kernel Hilbert space with U-invariant reproducing kernel K(z,w) = g(<z,w>), where g(z) = ∑n=0∞ anzn (an ≥ 0) is the generating function of the space Hdg (Bd√R). In this paper, we show that the multiplier algebra of the space Hdg (Bd√R) is a proper subset of H∞(Bd√R), and there exists a holomorphic self-mapping ø from Bd√Rinto Bd√Rsuch that the multiplication operator Møis not bounded when the sequence {an}n∞=0 is bounded andd ≥ 2. Furthermore, we prove that if the coefficients sequence {an}n∞=0 of the generating function g is a bounded, non-decreasing sequence, i.e. there exists a positive number M such that 0 ≤ a0 ≤ a1 ≤ … ≤ M, n = 0, 1, 2, …, then the space Hdg (Bd√R) is not subnormal, in other words, there is not any positive measure μ on Cd such that ||f|| Hdg2 =∫ Cd |f(z)|2dμ(z), for each f∈Hdg (Bd√R), and then von Neumann's inequality does not hold on the space Hdg (Bd√R).