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Deficiency indices of block Jacobi matrices and Miura transformation. (English) Zbl 1507.47075

Summary: We study the infinite Jacobi block matrices under the discrete Miura-type transformations which relate matrix Volterra and Toda lattice systems to each other and the situations when the deficiency indices of the corresponding operators are the same. A special attention is paid to the completely indeterminate case (i.e., then the deficiency indices of the corresponding block Jacobi operators are maximal). It is shown that there exists a Miura transformation which retains the complete indeterminacy of Jacobi block matrices appearing in the Lax representation for such systems, namely, if the Lax matrix of Volterra system is completely indeterminate, then so is the Lax matrix of the corresponding Toda system, and vice versa. We consider an implication of the obtained results to the study of matrix orthogonal polynomials as well as to the analysis of self-adjointness of scalar Jacobi operators.

MSC:

47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
44A60 Moment problems
37K15 Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
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