To the Spectral Theory of Discrete Hausdorff Operators

Abstract

We show that under an arithmetic condition the spectrum of a bounded multidimensional discrete Hausdorff operator in the Lebesgue space is an annulus (or a disc) centered at the origin, provided the perturbation matrices commute and are either positive or negative definite. Conditions for a point spectrum of such an operator to be empty are given and its norm is computed.

On the spectra of multidimensional normal discrete Hausdorff operators

“…) with respect to dist H . It is known that dist H is a metric on compact subsets (see, e.g., [35,§21,VII]). Thus, (3.5) shows that 𝜁𝜎( c,A ) = 𝜎( c,A ), and (i) follows.…”

“…The structure of the spectrum of discrete normal Hausdorff operators in L 2 (R d ) was investigated in [21] for the case of positive or negative definite perturbation matrices. In this paper, the general case of normal discrete Hausdorff operators in L 2 (R d ) is considered.…”

“…The structure of the spectrum of discrete normal Hausdorff operators in $$$ {L}^2\left({\mathrm{\mathbb{R}}}^d\right) $$$ was investigated in [21] for the case of positive or negative definite perturbation matrices. In this paper, the general case of normal discrete Hausdorff operators in $$$ {L}^2\left({\mathrm{\mathbb{R}}}^d\right) $$$ is considered.…”

“…□ Corollary 3.9 (cf. [21]). Let, in addition to the assumptions of Theorem 3.4 (i), all matrices (A(k)) k∈Z be negative definite and  c,A ≠ O.…”