Search

In this paper, unitary symmetrizers are considered. It is well known that using Newton operatoralgorithm, similar to the usual Newton algorithm, for extracting the square root, one can provethat for every Hermitian operator T 0, there exists a unique Hermitian operator S 0 suchthat T = S2. Moreover, S commutes with every bounded operator R with which commutes T. Theoperator S is called a square root of the operator T and is denoted by T1=2. The existence of thesquare root allows one to determine the absolute value jTj = (TT)1=2 of the bounded operator T.For every bounded linear operator T : H ! H there exists a unique partially isometric operatorU : H ! H such that T = UjTj, KerU = KerT. Such an equality is called a polar expansionof the operator T. The Schmidt operator is understood as the unitary multiplier of the polarexpansion of a compact inverse operator, with the help of which E. Schmidt was the rst to obtainthe expansion of a compact and not-self-adjoint operator and introduced so-called s-numbers.This paper shows that the unitary symmetrizer of an operator diers only in sign from the adjointSchmidt operator. The main result of the paper: if A is an invertible and compact operator, andS is a unitary operator such that the operator SA is self-adjoint, then the operator AS is alsoself-adjoint and the formula S = U holds, where U is the Schmidt operator.

In this paper we consider inverse problems for a fractional heat equation, where the fractional time derivative is taken into account in Riemann-Liouville sense. For the solution of this equation, we have to find an unknown right - hand side depending only on a spatial variable. The problem modeling the process of determining the temperature and density of sources in the process of fractional heat conductivity with respect to given initial and final temperatures is considered. Problems with general boundary conditions with respect to the spatial variable that are not strongly regular are investigated. The existence and uniqueness of classical solution to the problem are proved. The problem is considered independent from a corresponding spectral problem for an operator of multiple differentiation with not strongly regular boundary conditions has the basis property of root functions.

We study the nature of the spectrum of the periodic problem for the heat equation with a lower-order term and with a deviating argument. A significant influence of the lower-order term on the correct solvability of this problem is found. We obtain a criterion for the strong solvability of the above-mentioned problem.