This article is semi-review and is also methodological in nature. The article discusses generalizations of the Jacobson-Chevalley density theorem, on the basis of which a number of results from linear algebra are presented, related to centralizers of locally algebraic linear operators. Also, based on Jacobson’s approach to constructing Galois theory based on the density theorem, a proof of Hilbert’s theorem 90 and some of its well-known generalizations are given.

Let a von Neumann algebra M of operators act on a Hilbert space H, I be the unit of M, τ be a faithful semifinite normal trace on M. Let S(M, τ ) be the ^∗-algebra of all τ -measurable operators and L₁(M, τ ) be the Banach space of all τ -integrable operators, P, Q ∈ S(M, τ ) be idempotents. If P − Q ∈ L₁(M, τ ) then τ (P − Q) ∈ R. In particular, if A  = A³  ∈ L₁(M, τ ), then τ (A) ∈ R. If P −Q ∈ L₁(M, τ ) and PQ ∈ M, then for all n ∈ N we have (P −Q)²ⁿ⁺¹ ∈ L₁(M, τ ) and τ ((P − Q)²ⁿ⁺¹) = τ (P − Q) ∈ R. If A ∈ L₂(M, τ ) and U ∈ M is an isometry, then ∥UA − A∥²₂ ≤ 2∥(I − U )AA^∗∥₁.

The problems on the location of the matrix spectrum inside or outside domains bounded by ellipses or parabolas are studied. Special Lyapunov-type equations are connected with these problems. Theorems about the unique solvability of such equations are proved. Conditions for perturbations of matrix entries are obtained, which guarantee that the spectra of the perturbed matrices belong to the specified domains as well.

We study the classes of Turing degrees of selector functions in which a rigid computable structure is relatively computably categorical. It is proved that for some structures such classes of degrees can be represented as the unions of upper cones of c.e. degrees. In addition we show that there are non-c.e. upper cones realized as the degrees in which some computable structure is relatively computably categorical.

We give a description of involutions in formal matrix ring K_s(R) over an UFD. Some results on equivalency of involutions were also obtained.

The Beurling–Malliavin Theorem on the multiplier, considered in a subharmonic framework in the first part of our work, already in its original classical version within the framework of entire functions of exponential type, allowed in the 1960s to completely solve the problem of the radius of completeness of exponential system in the form of a remarkable criterion and exclusively in geometric terms for the exponents of this exponential system without any additional restrictions on the relative position of these exponents. The exact formulations of the Beurling– Malliavin Theorem on the radius of completeness in the introduction are somewhat adapted as a problem of the possible minimum growth along the real axis R of subharmonic functions with given constraints on their Riesz distributions of masses. In this largely overview part of the paper, we discuss the Beurling–Malliavin Theorem on the radius of completeness, along with its somewhat more general subharmonic manifestations. Thus, our results from 2014-16 allow us to obtain significantly more subtle results with respect to the Beurling – Malliavin Theorem itself on the radius of completeness with a defect excess of no more than 1 or 2 for exponents in classical rigid Banach spaces of continuous functions on a segment of fixed length or functions with integrable module in the p-th degree on such segments.