Non-self-adjoint Jacobi matrices with a rank-one imaginary part

• Published in: Journal of functional analysis Vol. 241; no. 2; pp. 383 - 438
• Main Authors: Arlinskiĭ, Yury, Tsekanovskiĭ, Eduard
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15-12-2006
• Subjects: Inverse problem; Jacobi operators; Non-self-adjoint problems; Jacobi operators; Non-self-adjoint problems; Inverse problem
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/j.jfa.2006.05.002

We develop direct and inverse spectral analysis for finite and semi-infinite non-self-adjoint Jacobi matrices with a rank-one imaginary part. It is shown that given a set of
n not necessarily distinct nonreal numbers in the open upper (lower) half-plane uniquely determines an
n
×
n
Jacobi matrix with a rank-one imaginary part having those numbers as its eigenvalues counting algebraic multiplicity. Algorithms of reconstruction for such finite Jacobi matrices are presented. A new model complementing the well-known Livsic triangular model for bounded linear operators with a rank-one imaginary part is obtained. It turns out that the model operator is a non-self-adjoint Jacobi matrix. We show that any bounded, prime, non-self-adjoint linear operator with a rank-one imaginary part acting on some finite-dimensional (respectively separable infinite-dimensional Hilbert space) is unitarily equivalent to a finite (respectively semi-infinite) non-self-adjoint Jacobi matrix. This obtained theorem strengthens a classical result of Stone established for self-adjoint operators with simple spectrum. We establish the non-self-adjoint analogs of the Hochstadt and Gesztesy–Simon uniqueness theorems for finite Jacobi matrices with nonreal eigenvalues as well as an extension and refinement of these theorems for finite non-self-adjoint tri-diagonal matrices to the case of mixed eigenvalues, real and nonreal. A unique Jacobi matrix, unitarily equivalent to the operator of integration
(
F
f
)
(
x
)
=
2
i
∫
x
l
f
(
t
)
d
t
in the Hilbert space
L
2
[
0
,
l
]
is found as well as spectral properties of its perturbations and connections with the well-known Bernoulli numbers. We also give the analytic characterization of the Weyl functions of dissipative Jacobi matrices with a rank-one imaginary part.