Vol 211, No 9 (2020)

Under the assumption that the rational closure of a group algebra of a left-ordered group in the ring of operators of the module of formal Malcev series is a division ring, we find a canonical form of nonsingular matrices of this division ring. Bibliography: 10 titles.

A natural counterpart of Vitushkin's criterion is obtained in the problem of uniform approximation of functions by solutions of second-order homogeneous elliptic equations with constant complex coefficient on compact subsets of $\mathbb R^d$, $d\ge3$. It is stated in terms of a single (scalar) capacity connected with the leading coefficient of the Laurent series. The scheme of approximation uses methods in the theory of singular integrals and, in particular, constructions of certain special Lipschitz surfaces and Carleson measures. Bibliography: 23 titles.

We consider a family of equivalent norms (called operator $E$-norms) on the algebra $\mathfrak B(\mathscr H)$ of all bounded operators on a separable Hilbert space $\mathscr H$ induced by a positive densely defined operator $G$ on $\mathscr H$. By choosing different generating operators $G$ we can obtain the operator $E$-norms producing different topologies, in particular,the strong operator topology on bounded subsets of $\mathfrak B(\mathscr H)$.We obtain a generalised version of the Kretschmann-Schlingemann-Werner theorem, which shows that the Stinespring representation of completely positive linear maps is continuous with respect to the energy-constrained norm of complete boundedness on the set of completely positive linear maps and the operator $E$-norm on the set of Stinespring operators.The operator $E$-norms induced by a positive operator $G$ are well defined for linear operators relatively bounded with respect to the operator $\sqrt G$, and the linear space of such operators equipped with any of these norms is a Banach space. We obtain explicit relations between operator $E$-norms and the standard characteristics of $\sqrt G$-bounded operators. Operator $E$-norms allow us to obtain simple upper bounds and continuity bounds for some functions depending on $\sqrt G$-bounded operators used in applications.Bibliography: 29 titles.