General Solution of the Stieltjes Truncated Matrix Moment Problem

Abstract

Abstract. The description of all solutions of the truncated Stieltjes matrix moment problem consisting in finding all s × s matrix measures dσ (t) on [0, ∞) with given first 2n + 1 power s × s matrix moments (Cj ) n j=0 is obtained in a general case, when the block Hankel matrix Γn := (C j+k ) n j,k=0 may be non-invertible. Special attention is paid to the description of canonical solutions for which dσ (t) is a sum of at most sn + s point matrix "masses" with the minimal sum of their ranks. Mathematics Subject Classification (2000). Primary 30E05, 30E10; Secondary 82C70, 82D10.

On Distinguished Solutions of Truncated Matricial Hamburger Moment Problems

“…The approach of M. G. Kreȋn is characterized by the use of methods of operator theory. A modern refreshment of M. G. Kreȋn's method can be found in recent work of V. M. Adamyan, I. M. Tkachenko, and M. Urrea (see [3][4][5][6]). …”

On Truncated Matricial Stieltjes Type Moment Problems

“…For a comprehensive survey on the state of affairs concerning matricial versions of moment problems we refer the reader to [18] and the references cited therein. Against to the background of this paper we will here particularly mention the following works which deal with matricial generalizations of Stieltjes type moment problems: Adamyan and Tkachenko [1,2], Andô [4], Bolotnikov [5][6][7], Bolotnikov and Sakhnovich [8], Chen and Hu [9], Chen and Li [10], Dyukarev [13,14], Dyukarev and Katsnel'son [19,20], Hu and Chen [23]. Most of the topics and methods of this paper are strongly influenced by our former work [18].…”

Local moment problem

“…The quest of dσ ⊥ Λ (t) consists in the search of some special solutions d σ(t) of the truncated Hamburger moment problem, which satisfy the additional restriction σ(Λ − 0) − σ(+0) = 0 for the set of moments(2)We call the latter moment problem the truncated Hamburger moment problem with the gap [0, Λ]. Using as in [3][4][5][6] the approach based on the extension theory of Hermitian operators we obtain first the solvability criterium of the truncated Hausdorff problem, which is treated as a version of the Hamburger problem [1], where the sought σ(t) should be constant out of (0, Λ).Theorem 0.1 A system of real numbers b 0 , ..., b 2m are power moments of non-negative measure dσ Λ (t) on [0, Λ] if and only if a) the Hankel matrix Γ m := (b k+j ) m k,j=0 is non-negative; b) for any set of complex numbers ξ 0 , . .…”

“…We call the latter moment problem the truncated Hamburger moment problem with the gap [0, Λ]. Using as in [3][4][5][6] the approach based on the extension theory of Hermitian operators we obtain first the solvability criterium of the truncated Hausdorff problem, which is treated as a version of the Hamburger problem [1], where the sought σ(t) should be constant out of (0, Λ).…”