A Weyl Matrix Perspective on Unbounded Non-Self-Adjoint Jacobi Matrices.

Area of Science:

• Spectral theory
• Operator theory
• Linear algebra

Background:

• The Pushnitski-Štampach method encodes bounded non-self-adjoint Jacobi matrices using spectral measures and phase functions.
• Existing methods often rely on moment-based approaches, which can be complex.
• Generalizing these encoding techniques to unbounded Jacobi matrices remains a significant challenge.

Purpose of the Study:

• To develop an alternative perspective on encoding non-self-adjoint Jacobi matrices.
• To generalize the Pushnitski-Štampach correspondence to the unbounded case.
• To establish a bijection between a specific class of Jacobi matrices and spectral data.

Main Methods:

• Utilized Weyl functions as an alternative to spectral moments for matrix encoding.
• Developed a novel approach to handle unbounded non-self-adjoint Jacobi matrices.
• Proved the continuity of the established mapping and the injectivity using a local Borg-Marchenko theorem.

Main Results:

• Established a simplified and generalized correspondence for encoding non-self-adjoint Jacobi matrices, applicable to unbounded cases.
• Demonstrated a bijection between proper Jacobi matrices with positive off-diagonal elements and a specific class of spectral data.
• Proved the continuity of this bijection and established a local Borg-Marchenko theorem for unbounded non-self-adjoint Jacobi matrices.

Conclusions:

• The Weyl function-based approach offers a more streamlined method for encoding non-self-adjoint Jacobi matrices.
• The study successfully extends the encoding framework to unbounded Jacobi matrices, broadening its applicability.
• The developed local Borg-Marchenko theorem provides a valuable tool for analyzing unbounded non-self-adjoint Jacobi matrices.