We consider three problems connected with coinvariant subspaces of the backward shift operator in Hardy spaces H p :-properties of truncated Toeplitz operators; -Carleson-type embedding theorems for the coinvariant subspaces; -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.

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Currently, one of the most important problems in Russian refineries is the lack of production of high-octane gasoline that meets current and future environmental requirements. One solution for this problem is blending oxygenates into gasoline, for example, tertiary amyl methyl ether (TAME) and methyl tertiary butyl ether (MTBE). Each of these substances has its advantages and disadvantages. Prospects for obtaining other high-octane components alternative to MTBE and TAME may have good prospects. One choice is dimerization of isobutylene to produce di-isobutylene and the product consists mainly of isooctene isomers. Physico-chemical characteristics of isooctene sample as component of motor gasoline in comparison with MTBE and TAME are investigated. The results showed that the introduction of isooctene, which has low volatility, leads to a decrease in Reid vapor pressure (RVP) of base gasoline at the level of TAME. Furthermore, the antiknock properties of isooctene in various gasoline bases are established. The values of the isooctene blend octane numbers calculated in the ranges of 108–150 by the research methods and 92–136 by the motor methods are calculated according to the obtained results, isooctene has high antiknock detonation efficiency at the MTBE level. Finally, the use of isooctene as gasoline additives gives good prospects to refining companies in light of decreasing the overhead, enhancing the grade of product, as well as decreasing large effects on the environment.

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517.984Under some natural restrictions, we prove that any one-dimensional perturbation of a singular unitary operator on a Hilbert space is unitarily equivalent to a model operator on a space determined (in a certain way) by two functions from the Hardy space H 2. Bibliography: 3 titles.It is well known that for contractions on Hilbert spaces characteristic functions are defined. On the one hand, in their terms a classification for contractions up to the unitary equivalence is obtained. On the other hand, these terms give tools to answer various questions of the spectral analysis. In the noncontractive case, no tool with similar qualities was found, and we think that it does not exist at all. Here we propose a representation of an operator in some function space (function model) in the simplest case. Note that in terms of this model, the characteristic function of an operator can be simply calculated. This gives a possibility to solve classification problems. In addition, many spectral properties of operators can be described in terms of our model. We do not discuss these questions in detail. We restrict our presentation to a theorem on the unitary equivalence of operators from some class to model operators.

In the first part of the paper we discuss a multi-dimensional analogue of the well-known construction by D. Clark that allows one to study families of spectral measures of perturbations of the model contraction.In the second part we present extensions of the relevant results on the boundary behavior of pseudocontinuable functions. We show that, although the most direct analogue of the scalar theorem on the existence of boundary values for pseudocontinuable functions with respect to Clark measures fails in the non-scalar situation, suitable vector-valued versions of such results can be found.