A unified flow approach to smooth, even LpL_p-Minkowski problems

We study long-time existence and asymptotic behaviour for a class of anisotropic, expanding curvature flows. For this we adapt new curvature estimates, which were developed by Guan, Ren and Wang to treat some stationary prescribed curvature problems. As an application we give a unified flow approach to the existence of smooth, even $L_p$-Minkowski problems in $\mathbb{R}^{n+1}$ for $p>-n-1.$Comment: 21 pages. Comments are welcom

Estimates of operator moduli of continuity

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}.Comment: 50 page

Diagonal Riccati Stability and Applications

We consider the question of diagonal Riccati stability for a pair of real matrices A, B. A necessary and sufficient condition for diagonal Riccati stability is derived and applications of this to two distinct cases are presented. We also describe some motivations for this question arising in the theory of generalised Lotka-Volterra systems

Almost commuting functions of almost commuting self-adjoint operators

Let $A$ and $B$ be almost commuting (i.e, AB-BA\in\bS_1) self-adjoint operators. We construct a functional calculus \f\mapsto\f(A,B) for \f in the Besov class B_{\be,1}^1(\R^2). This functional calculus is linear, the operators \f(A,B) and $\psi(A,B)$ almost commute for \f,\,\psi\in B_{\be,1}^1(\R^2), \f(A,B)=u(A)v(B) whenever \f(s,t)=u(s)v(t), and the Helton--Howe trace formula holds. The main tool is triple operator integrals.Comment: 6 page

Symbols of truncated Toeplitz operators

We consider three problems connected with coinvariant subspaces of the backward shift operator in Hardy spaces $H^p$: 1) properties of truncated Toeplitz operators; 2) Carleson-type embedding theorems for the coinvariant subspaces; 3) factorizations of pseudocontinuable functions from $H^1$. These problems turn out to be closely connected and even, in a sense, equivalent. The new approach based on the factorizations allows us to answer a number of challenging questions about truncated Toeplitz operators posed by Donald Sarason.Comment: 19 page