Estimating Norms of Commutators
We find estimates on the norm of a commutator of the form [f(x), y] in terms of the norm of [x, y], assuming that x and y are bounded linear operators on Hilbert space, with x normal and with spectrum within the domain of f. In particular, we discuss ‖[x2, y]‖ and ‖[x1/2, y]‖ for 0 ≤ x ≤ 1. For larger values of δ = ‖[x, y]‖, we can rigorously calculate the best possible upper bound ‖[f(x), y]‖ ≤ ηf(δ) for many f. In other cases, we have conducted numerical experiments that strongly suggest that we have in many cases found the correct formula for the best upper bound.
Operator H\"older--Zygmund functions
It is well known that a Lipschitz function on the real line does not have to be operator Lipschitz. We show that the situation changes dramatically if we pass to H\"older classes. Namely, we prove that if $f$ belongs to the H\"older class $\L_\a(\R)$ with $0 0$. Then we find a sharp estimate for $\|f(A)-f(B)\|$ for functions $f$ of class $\L_\o\df\{f: \o_f(\d)\le\const\o(\d)\}$ for an arbitrary modulus of continuity $\o$. In particular, we study moduly of continuity, for which $\|f(A)-f(B)\|\le\const\o(\|A-B\|)$ for self-adjoint $A$ and $B$, and for an arbitrary function $f$ in $\L_\o$. We obtain similar estimates for commutators $f(A)Q-Qf(A)$ and quasicommutators $f(A)Q-Qf(B)$. Finally, we estimate the norms of finite differences $\sum\limits_{j=0}^m(-1)^{m-j}(m j)f\big(A+jK\big)$ for $f$ in the class $\L_{\o,m}$ that is defined in terms of finite differences and a modulus continuity $\o$ of order $m$. We also obtaine similar results for unitary operators and for contractions.
Net Spaces in Categorical Topology a
In the paper we introduce the notion of S‐nets, S a nonempty construct, as a generalization of the usual nets. S‐net spaces and continuous maps between them are then defined in a natural way. For S‐net spaces we introduce a number of axioms and study the categories obtained. We find some topological and Cartesian closed categories of S‐net spaces and describe their relations to certain known topological categories.