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1

Asymptotics of the discrete spectrum for complex Jacobi matrices

Malejki M.

Opuscula Mathematica

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2014

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Vol. 34, no. 1

139--160

EN

The spectral properties and the asymptotic behaviour of the discrete spectrum for a special class of infinite tridiagonal matrices are given. We derive the asymptotic formulae for eigenvalues of unbounded complex Jacobi matrices acting in l2(N).

2

Asymptotic behaviour and approximation of eigenvalues for unbounded block Jacobi matrices

Malejki M.

Opuscula Mathematica

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2010

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Vol. 30, no. 3

311-330

EN

The research included in the paper concerns a class of symmetric block Jacobi matrices. The problem of the approximation of eigenvalues for a class of a self-adjoint unbounded operators is considered. We estimate the joint error of approximation for the eigenvalues, numbered from 1 to N, for a Jacobi matrix J by the eigenvalues of the finite submatrix J(n) of order pn x pn, where N = max{k ∈ N : k ≤ rpn} and r ∈ (0, 1) is suitably chosen. We apply this result to obtain the asymptotics of the eigenvalues of J in the case p = 3.

3

Approximation of eigenvalues of some unbounded self-adjoint discrete Jacobi matrices by eigenvaules of finite submatrices

Malejki M.

Opuscula Mathematica

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2007

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Vol. 27, no. 1

37-49

EN

We investigate the problem of approximation of eigenvalues of some self-adjoint operator in the Hilbert space l2(N) by eigenvalues of suitably chosen principal finite submatrices of an infinite Jacobi matrix that defines the operator considered. We assume the Jacobi operator is bounded from below with compact resolvent. In our research we estimate the asymptotics (with n → ∞) of the joint error of approximation for the first n eigenvalues and eigenvectors of the operator by the eigenvalues and eigenvectors of the finite submatrix of order n x n. The method applied in our research is based on the Rayleigh-Ritz method and Volkmer's results included in [7]. We extend the method to cover a class of infinite symmetric Jacobi matrices with three diagonals satisfying some polynomial growth estimates.

4

Operators in Banach modules and the Weyl functional calculus for generators of n-parameter C0-groups

Malejki M.

Opuscula Mathematica

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2001

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Vol. 21

65-92

EN

In the paper, I am going to work with unbounded operators in special Banach modules over real Clifford algebras. I propose the definition of n-spectrum for these operators. The mentioned definition is similar to one considered by A. Mclntosh and A. Pryde for bounded operators. The main aim of this work is to describe the support of the Weyl functional calculus T(A) for several operators (formulas for T(A) is according to Anderson) using the Clifford algebras methods. A = (A1,... , Am) is a tuple of operators in a Banach space. The operators Aj are not necessarily bounded but it will be assumed that the tuple A is a generator of the m-parameter C0-group, which satisfies a polynomial growth condition. The final result is the formula suppT(A) = σ(n)(A) ∩ Rm, where σ(n)(A) is the n-spectrum for an operator associated to the tuple A and n ≤ m is an integer.

PL

W tej pracy zajmuję się badaniem nieograniczonych operatorów działających w pewnych modułach Banacha nad algebrami Clifforda. Proponuję tu definicję pewnego rodzaju widma dla takich operatorów, zwanego n-widmem. Definicja ta jest wzorowana na pracy A. Mclntosha i A. Pryde'a, którzy postawili ją dla operatorów ograniczonych. Głównym celem tej pracy jest jednak opisanie nośnika dla rachunku funkcyjnego Weyla T(A) (dla abstrakcyjnych operatorów - wprowadzonego przez W. O. Andersona) dla układów operatorów. Udaje się to wykonać dzięki wykorzystaniu analizy cliffordowskiej. Układ A = (A1,... , Am) jest generatorem m-parametrowej silnie ciągłej grupy operatorów w przestrzeni Banacha. Będę zakładać, że wzrost tej grupy jest ograniczony wielomianowo. Głównym rezultatem tej pracy jest wykazanie równości suppT(A) = σ(n)(A) ∩Rm, gdzie σ(n)(A) jest wspomnianym n-widmem dla pewnego operatora związanego z układem A, natomiast n ≤ m jest pewną liczbą naturalną.