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Deficiency numbers of operators generated by infinite Jacobi matrices
- Publication Year :
- 2015
-
Abstract
- Let $A_j,B_j$ $(j=0,1,\ldots)$ be $m \times m$ matrices, whose elements are complex numbers, $A_j$ are selfadjoint matrices and $B_j^{-1}$ exist. We study the deficiency index problem for minimal closed symmetric operator $L$ with domain $D_L$, generated by the Jacobi matrix $\textbf{J}$ with entries $A_j,B_j$ in the Hilbert space $l_m^2$ of sequences $ u=(u_0,u_1, \ldots), u_j \in C^m$ by mapping $u \rightarrow \textbf{J}u$, i.e. by the formula $Lu=lu$ for $u \in D_L$, where $lu=((lu)_0,(lu)_1, \ldots)$ and $$ (lu)_0:=A_0u_0+B_0u_1, \quad (lu)_j:=B^*_{j-1}u_{j-1}+A_ju_j+B_ju_{j+1}, \;\; j=1,2, \ldots $$ It is well known that the case of the minimal deficiency numbers of the operator $L$ corresponds to the determinate case, and the case of the maximal deficiency numbers of this operator corresponds to the completely indeterminate case of the matrix power moment problem. In this paper we obtain new conditions of the minimal, maximal and not maximal deficiency numbers of the operator $L$ in terms of the entries of the matrix $\textbf{J}$. The special attention is paid to the case $m=1$, i.e. we present some conditions on the elements of the numerical tridiagonal Jacobi matrix, which ensure the realization of the determinate case of the classical power moment problem.<br />9 pages, in Russian
- Subjects :
- Pure mathematics
Tridiagonal matrix
General Mathematics
010102 general mathematics
Hilbert space
010103 numerical & computational mathematics
01 natural sciences
Power (physics)
Moment problem
Mathematics - Spectral Theory
symbols.namesake
Operator (computer programming)
Jacobian matrix and determinant
FOS: Mathematics
symbols
0101 mathematics
Spectral Theory (math.SP)
M-matrix
Mathematics
Subjects
Details
- Language :
- Russian
- Database :
- OpenAIRE
- Accession number :
- edsair.doi.dedup.....b39bd7b47a0d86063529c83d4efcc1a9