Embry truncated complex moment problem

✍ Scribed by Il Bong Jung; Eungil Ko; Chunji Li; Sang Soo Park

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 261 KB
• Volume: 375
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/s0024-3795(03)00617-7

✦ Synopsis

Let T be a cyclic subnormal operator on a Hilbert space H with cyclic vector x 0 and let γ ij := (T * i T j x 0 , x 0 ), for any i, j ∈ N ∪ {0}. The Bram-Halmos' characterization for subnormality of T involved a moment matrix M(n). In a parallel approach, we construct a moment matrix E(n) corresponding to Embry's characterization for subnormality of T . We discuss the relationship between M(n) and E(n) via the full moment problem. Next, given a collection of complex numbers γ ≡ {γ ij } (0 i + j 2n, |ij | n) with γ 00 > 0 and γ ji = γij , we consider the truncated complex moment problem for γ ; this entails finding a positive Borel measure µ supported in the complex plane C such that γ ij = zi z j dµ(z). We show that this moment problem can be solved when E(n) 0 and E(n) admits a flat extension E(n + k), where k = 1 when n is odd and k = 2 when n is even.