The point spectrum of some non-selfadjoint Jacobi matrices
by
M. Malejki
Institute of Mathematics of the Polish Academy, Krakow
Coauthors: J. Janas, Y. Mykytyuk

In the presentation I am going to speak about the point spectrum of some non-selfadjoint Jacobi matrices given by

Jf=\alphan-1fn-1+qnfn+ \betan fn+1,

for f={f_n}_n=1^\infty in l², f₀=0, lim \alpha_n=\alpha, and lim \beta_n=\beta.
We reduce this situation to J that is a compact perturbation of J₀=S+\rhoS^*, where S is an unilateral shift operator in l² and \rho in (0, 1) (i.e. \alpha = 1,  \beta = \rho).
Denote

E={\rhoz+1/z in C|  |z|=1},   \Omega = {\rhoz+1/z in C|  1 < |z| < 1/\rho}.

It is known that \sigma(J₀)=[`(\Omega)], \sigma<sub>ess</sub>(J<sub>0</sub>)=E and \sigma<sub>p</sub>(J<sub>0</sub>)=\phi.
By the Perron and Kreuser theorems (see ) we have that for any compact perturbation J of J<sub>0</sub> we have \sigma<sub>p</sub> (J) \cap \Omega = \phi.
In the case when J-J<sub>0</sub> is in the trace class it is possible to show similarity of J to an operator T=J<sub>0</sub>+(., p)e<sub>1</sub>, where p={p<sub>n</sub>} in l<sup>2</sup> and [`lim]p_n <= \rho^1/2. Using this fact we prove that \sigma_p(J) \cap [`(\Omega)]=\phi and \sigma_p(J) consists of only finite number of eigenvalues.
In the case J-J₀ is not in the trace class we use the trasfer matrix approach (see ) to find the suffucient conditions for \sigma_p (J) \cap E=\phi.
Similar problems of the point spectrum for non-selfadjoint Jacobi matrices we can also find for example in or .

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J. Janas and S.N. Naboko, On the point spectrum some Jacobi matrices, J. Operator Theory, 40(1998), 113-132.

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Date received: June 6, 2000