Scholarlink

In Chap. 14 we present the notion of the characteristic operator function of an unbounded almost solvable extension \(A_B\) of a symmetric operator A and express the characteristic function via the Weyl function \(M(\cdot )\) and the boundary operator B. In the special case where A is a trivial operator, this formula reduces to the classical Livšic–Brodskiı̆ formula. In Sects. 14.2 and 14.3 we present some analogues of classical results on the characteristic function of bounded operators: multiplication and realization theorems. In Sect. 14.4 we show that the \(\Pi {\mathfrak L}\) -resolvent matrix of a symmetric operator A coincides with the characteristic function of the operator \(S^* =A^*\upharpoonright {\mathfrak L}^{\perp }\) . We use this result to illustrate the connection between the Schur algorithm for the moment problem and the factorization of the resolvent matrix.

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Characteristic Functions and Resolvent Matrices

  • 章节

  • 发表日期：

    2025年

查看全文
  • Volodymyr Derkach,
  • Mark Malamud

摘要

In Chap. 14 we present the notion of the characteristic operator function of an unbounded almost solvable extension \(A_B\) of a symmetric operator A and express the characteristic function via the Weyl function \(M(\cdot )\) and the boundary operator B. In the special case where A is a trivial operator, this formula reduces to the classical Livšic–Brodskiı̆ formula. In Sects. 14.2 and 14.3 we present some analogues of classical results on the characteristic function of bounded operators: multiplication and realization theorems. In Sect. 14.4 we show that the \(\Pi {\mathfrak L}\) -resolvent matrix of a symmetric operator A coincides with the characteristic function of the operator \(S^* =A^*\upharpoonright {\mathfrak L}^{\perp }\) . We use this result to illustrate the connection between the Schur algorithm for the moment problem and the factorization of the resolvent matrix.