11,794,918 research outputs found
Functions of perturbed normal operators
In \cite{Pe1}, \cite{Pe2}, \cite{AP1}, \cite{AP2}, and \cite{AP3} sharp
estimates for were obtained for self-adjoint operators and
and for various classes of functions on the real line . In this note we
extend those results to the case of functions of normal operators. We show that
if belongs to the H\"older class \L_\a(\R^2), 0<\a<1, of functions of
two variables, and and are normal operators, then
\|f(N_1)-f(N_2)\|\le\const\|f\|_{\L_\a}\|N_1-N_2\|^\a. We obtain a more
general result for functions in the space \L_\o(\R^2)=\big\{f:
|f(\z_1)-f(\z_2)|\le\const\o(|\z_1-\z_2|)\big\} for an arbitrary modulus of
continuity \o. We prove that if belongs to the Besov class
B_{\be1}^1(\R^2), then it is operator Lipschitz, i.e.,
\|f(N_1)-f(N_2)\|\le\const\|f\|_{B_{\be1}^1}\|N_1-N_2\|. We also study
properties of in the case when f\in\L_\a(\R^2) and
belongs to the Schatten-von Neuman class \bS_p.Comment: 6 page
Indices of fixed points not accumulated by periodic points
We prove that for every integer sequence satisfying Dold relations there
exists a map , , such that
, where denotes the origin, and
.Comment: 11 pages, 2 figures. Final version to appear in Topol. Methods
Nonlinear Ana
Functions of normal operators under perturbations
In \cite{Pe1}, \cite{Pe2}, \cite{AP1}, \cite{AP2}, and \cite{AP3} sharp
estimates for were obtained for self-adjoint operators and
and for various classes of functions on the real line . In this paper
we extend those results to the case of functions of normal operators. We show
that if a function belongs to the H\"older class \L_\a(\R^2), 0<\a<1,
of functions of two variables, and and are normal operators, then
\|f(N_1)-f(N_2)\|\le\const\|f\|_{\L_\a}\|N_1-N_2\|^\a. We obtain a more
general result for functions in the space
\L_\o(\R^2)=\big\{f:~|f(\z_1)-f(\z_2)|\le\const\o(|\z_1-\z_2|)\big\} for an
arbitrary modulus of continuity \o. We prove that if belongs to the Besov
class B_{\be1}^1(\R^2), then it is operator Lipschitz, i.e.,
\|f(N_1)-f(N_2)\|\le\const\|f\|_{B_{\be1}^1}\|N_1-N_2\|. We also study
properties of in the case when f\in\L_\a(\R^2) and
belongs to the Schatten-von Neuman class \bS_p.Comment: 32 page
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