Approximate diagonalization of some Toeplitz operators and matrices. (English) Zbl 1523.47042

Ahmadi Kakavandi, Bijan; Nobari, Elham

Linear Multilinear Algebra 70, No. 21, 7093-7114 (2022).

Two particular self-adjoint Toeplitz operators are considered. The first is \(T=T(\rho)\) with entries \(t_{ij}=\rho^{|i-j|}\), \(\rho\in(0,1)\). Here, the theory of M. Rosenblum and J. Rovnyak [Hardy classes and operator theory. Unabr. and slightly corr. republ. of the 1985 orig. Mineola, NY: Dover (1997; Zbl 0918.47001)] can be applied to give an explicit characterization for the unitary isometry \(V\) such that \(VTV^{-1}=M_x\), where \(M_x\) is the multiplication operator with \(x\) in the spectrum. The second Toeplitz operator has only three nonzero diagonals: \(a_0\in\mathbb{R}\) on the main diagonal and for some fixed \(k\ge2\), \(a_k\in\mathbb{R}\) on the diagonals \(\pm k\). Here, the generic \(T(\cos k\theta)\) for which \(a_0=0\) and \(a_k=1/2\) needs to be analysed. Using Chebyshev polynomials of the second kind, ‘generalized eigenvectors’ \(v_\lambda\) are obtained satisfying \(T(\cos k\theta) v_\lambda = \lambda v_\lambda\), for \(\lambda\) in the spectrum. There are \(k\) generalized eigenvectors for each point in the spectrum.
The eigenvectors of the \(n\times n\) leading Toeplitz matrices are shown to be approximations for the eigenfunctions of the operator with an error depending on \(n\) and on \(\rho\) or \(k\), respectively.
Explicit expressions in the tridiagonal Toeplitz case were also obtained in [S. E. Ekström and S. Serra-Capizzano, Numer. Linear Algebra Appl. 25, No. 5, e2137, 17 p. (2018; Zbl 1513.65095)].

Reviewer: Adhemar Bultheel (Leuven)

MSC:

47B35	Toeplitz operators, Hankel operators, Wiener-Hopf operators
65F15	Numerical computation of eigenvalues and eigenvectors of matrices
62M15	Inference from stochastic processes and spectral analysis

Keywords:

Toeplitz operator; Toeplitz matrix; diagonalisation; dimension reduction; generalised eigenvectors

Citations:

Zbl 0918.47001; Zbl 1513.65095

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