Publications (5)0 Total impact
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ABSTRACT: We study in this paper properties of functions of perturbed normal operators
and develop earlier results obtained in \cite{APPS2}. We study operator
Lipschitz and commutator Lipschitz functions on closed subsets of the plane.
For such functions we introduce the notions of the operator modulus of
continuity and of various commutator moduli of continuity. Our estimates lead
to estimates of the norms of quasicommutators $f(N_1)R-Rf(N_2)$ in terms of
$\|N_1R- RN_2\|$, where $N_1$ and $N_2$ are normal operator and $R$ is a
bounded linear operator. In particular, we show that if $0<\a<1$ and $f$ is a
H\"older function of order $\a$, then for normal operators $N_1$ and $N_2$, $$
\|f(N_1)R-Rf(N_2)\|\le\const(1-\a)^{-2}\|f\|_{\L_\a}\|N_1R-RN_2\|^\a\|R\|^{1-\a}.
$$ In the last section we obtain lower estimates for constants in operator
H\"older estimates.
08/2011;
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ABSTRACT: In \cite{AP2} we obtained general estimates of the operator moduli of
continuity of functions on the real line. In this paper we improve the
estimates obtained in \cite{AP2} for certain special classes of functions.
In particular, we improve estimates of Kato \cite{Ka} and show that $$
\big\|\,|S|-|T|\,\big\|\le C\|S-T\|\log(2+\log\frac{\|S\|+\|T\|}{\|S-T\|}) $$
for every bounded operators $S$ and $T$ on Hilbert space. Here
$|S|\df(S^*S)^{1/2}$. Moreover, we show that this inequality is sharp.
We prove in this paper that if $f$ is a nondecreasing continuous function on
$\R$ that vanishes on $(-\be,0]$ and is concave on $[0,\be)$, then its operator
modulus of continuity $\O_f$ admits the estimate $$
\O_f(\d)\le\const\int_e^\be\frac{f(\d t)\,dt}{t^2\log t},\quad\d>0. $$
We also study the problem of sharpness of estimates obtained in \cite{AP2}
and \cite{AP4}. We construct a $C^\be$ function $f$ on $\R$ such that
$\|f\|_{L^\be}\le1$, $\|f\|_{\Li}\le1$, and $$
\O_f(\d)\ge\const\,\d\sqrt{\log\frac2\d},\quad\d\in(0,1]. $$
In the last section of the paper we obtain sharp estimates of $\|f(A)-f(B)\|$
in the case when the spectrum of $A$ has $n$ points. Moreover, we obtain a more
general result in terms of the $\e$-entropy of the spectrum that also improves
the estimate of the operator moduli of continuity of Lipschitz functions on
finite intervals, which was obtained in \cite{AP2}.
04/2011;
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ABSTRACT: We generalize our results of \cite{AP2} and \cite{AP3} to the case of maximal dissipative operators. We obtain sharp conditions on a function analytic in the upper half-plane to be operator Lipschitz. We also show that a H\"older function of order $\a$, $0<\a<1$, that is analytic in the upper half-plane must be operator H\"older of order $\a$. Then we generalize these results to higher order operator differences. We obtain sharp conditions for the existence of operator derivatives and express operator derivatives in terms of multiple operator integrals with respect to semi-spectral measures. Finally, we obtain sharp estimates in the case of perturbations of Schatten-von Neumann class $\bS_p$ and obtain analogs of all the results for commutators and quasicommutators. Note that the proofs in the case of dissipative operators are considerably more complicated than the proofs of the corresponding results for self-adjoint operators, unitary operators, and contractions that were obtained earlier in \cite{AP2}, \cite{AP3}, and \cite{Pe6}. Comment: 34 pages
09/2010;
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ABSTRACT: In \cite{Pe1}, \cite{Pe2}, \cite{AP1}, \cite{AP2}, and \cite{AP3} sharp estimates for $f(A)-f(B)$ were obtained for self-adjoint operators $A$ and $B$ and for various classes of functions $f$ on the real line $\R$. In this note we extend those results to the case of functions of normal operators. We show that if $f$ belongs to the H\"older class $\L_\a(\R^2)$, $0<\a<1$, of functions of two variables, and $N_1$ and $N_2$ 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 $f$ 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 $f(N_1)-f(N_2)$ in the case when $f\in\L_\a(\R^2)$ and $N_1-N_2$ belongs to the Schatten-von Neuman class $\bS_p$. Comment: 6 pages
03/2010;
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ABSTRACT: This is a continuation of our papers \cite{AP2} and \cite{AP3}. In those papers we obtained estimates for finite differences $(\D_Kf)(A)=f(A+K)-f(A)$ of the order 1 and $(\D_K^mf)(A)\df\sum\limits_{j=0}^m(-1)^{m-j}(m\j)f\big(A+jK\big)$ of the order $m$ for certain classes of functions $f$, where $A$ and $K$ are bounded self-adjoint operator. In this paper we extend results of \cite{AP2} and \cite{AP3} to the case of unbounded self-adjoint operators $A$. Moreover, we obtain operator Bernstein type inequalities for entire functions of exponential type. This allows us to obtain alternative proofs of the main results of \cite{AP2}. We also obtain operator Bernstein type inequalities for functions of unitary operators. Some results of this paper as well as of the papers \cite{AP2} and \cite{AP3} were announced in \cite{AP1}. Comment: 34 pages
03/2010;
