TY - JOUR

T1 - Embry truncated complex moment problem

AU - Jung, Il Bong

AU - Ko, Eungil

AU - Li, Chunji

AU - Park, Sang Soo

N1 - Funding Information:
The authors would like to express their gratitude to the referee for valuable comments. This work was supported by Korea Research Foundation Grant (KRF-2002-070-C00006).

PY - 2003/12/1

Y1 - 2003/12/1

N2 - Let T be a cyclic subnormal operator on a Hilbert space ℋ with cyclic vector x0 and let γij:=(T*iT jx0,x0), 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≡{γij} (0 ≤ i + j ≤ 2n, |i-j| ≤ n) with γ00 > 0 and γ ji = γ̄ij, 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 γij = ∫z̄izjdμ(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.

AB - Let T be a cyclic subnormal operator on a Hilbert space ℋ with cyclic vector x0 and let γij:=(T*iT jx0,x0), 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≡{γij} (0 ≤ i + j ≤ 2n, |i-j| ≤ n) with γ00 > 0 and γ ji = γ̄ij, 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 γij = ∫z̄izjdμ(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.

KW - Cyclic subnormal operators

KW - Truncated complex moment problem

KW - k-Hyponormal operator

UR - http://www.scopus.com/inward/record.url?scp=0142062893&partnerID=8YFLogxK

U2 - 10.1016/S0024-3795(03)00617-7

DO - 10.1016/S0024-3795(03)00617-7

M3 - Article

AN - SCOPUS:0142062893

SN - 0024-3795

VL - 375

SP - 95

EP - 114

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 1-3

ER -