From: BrendaSueCook (brenda@math.missouri.edu)
Date: Wed Mar 14 2007 - 10:54:41 CDT
COLLOQUIUM
University of Missouri-Columbia
Department of Mathematics
Eduard Tsekanovskii
Niagara University
Non-Self-Adjoint Jacobi Matrices
with a Rank-One Imaginary Part
Abstract: 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 a n x
n Jacobi matrix with a rank-one imaginary part having those numbers as
its eigenvalues counting algebraic multiplicity. An algorithm for
reconstruction for such finite Jacobi matrices is 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. This follows
from the fact that any bounded, prime, non-self-adjoint linear operator
with a rank-one imaginary part acting on some finite-dimensional (resp.,
separable infinite-dimensional Hilbert space) is unitarily equivalent to
a finite (resp., semi-infinite) non-self-adjoint Jacobi matrix. This
result strengthens a classical result of Stone established for
self-adjoint operators with simple spectrum. We establish the
non-self-adjoint analogs of various 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. We also
give the analytic characterization of the Weyl functions of dissipative
Jacobi matrices with a rank-one imaginary part.
This talk is based on joint work with Yu. Arlinskii.
3:30 p.m.
Thursday, March 15
114 General Classroom Building
Refreshments will be served at 2:45 p.m. in Room 326 Mathematical
Sciences (Math Lounge).
Please note the change in time for coffee: We are starting early at
2:45p.m. to celebrate Eduard Tsekanovskii's 70th birthday.
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