在Cd中，由函数g（z）= ∑n=0∞anzn（an≥0）生成的解析Hilbert空间Hdg（Bd√R）是酉不变的再生核Hilbert空间. 本文证明了，当d≥ 2时，若sup{anRn}<+∞，则有球代数A（Bd√R）中的函数f∉M，即Hdg（Bd√R）上的乘子代数M是H∞（Bd√R）的真子集. 由此可知，若存在M > 0，使得0 ≤ a0 ≤ a1 ≤ … ≤ M，n = 0，1，2，…，则Hdg（Bd√R）不是次正规的.因而不存在Cd中的正测度μ，使得对任何f ∈Hdg（Bd√R）||f||Hdg2 = ∫Cd |f|(z)|2dμ(z)，而且在Hdg（Bd√R）上的von Neumann不等式不成立.

Let B_d be the unit ball ofd-dimensional complex Euclid space C^d, H_d^g (B_d^√R) the reproducing kernel Hilbert space with U-invariant reproducing kernel K(z,w) = g(<z,w>), where g(z) = ∑_n=0^∞ a_nz_n (a_n ≥ 0) is the generating function of the space H_d^g (B_d^√R). In this paper, we show that the multiplier algebra of the space H_d^g (B_d^√R) is a proper subset of H^∞(B_d^√R), and there exists a holomorphic self-mapping ø from B_d^√Rinto B_d^√Rsuch that the multiplication operator M_øis not bounded when the sequence {a_n}_n^∞=0 is bounded andd ≥ 2. Furthermore, we prove that if the coefficients sequence {a_n}_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 H_d^g (B_d^√R) is not subnormal, in other words, there is not any positive measure μ on C^d such that ||f|| H_d^g2 =∫ _C^d |f(z)|²dμ(z), for each f∈H_d^g (B_d^√R), and then von Neumann's inequality does not hold on the space H_d^g (B_d^√R).