arXiv subjects
Publications and source records attributed to A. R. Mirotin.
The extension of Hille-Phillips functional calculus of semigroup generators which leads to unbounded operators is given. Connections of this calculus to Bochner-Phillips functional calculus are indicated, and several examples are considered.
The main purpose of this work is to ascertain when arithmetic operations with periodic functions whose domains may not coincide with the whole real line preserve periodicity.
Toeplitz operators on spaces $H^p(G)\ (1< p<\infty)$ associated with compact connected Abelian group $G$ with ordered dual are considered and the generalization of the classical Gohberg-Krein theorem on the Fredholm index of such operators with continuous symbols is proved. Applications to spectral theory of Toeplitz operators are given and examples of evident computation of index have been considered.
Let a function f with real poles be expanded in a Wolff-Denjoy series with positive coefficients. The main result of the note states that if we subtract its linear part from the function 1/f, then the remaining fractional part of this function will also expand into Wolff-Denjoy series (its poles are also real, and the coefficients of the series are negative). In other words, for Wolff-Denjoy series of the indicated form, an analogue of the well-known Wiener lemma in the theory of Fourier series is true up to a linear term. Applications of the result to operator theory are given.
Let $ψ$ be a Bernstein function. A.~Carasso and T.~Kato obtained necessary and sufficient conditions for $ψ$ to have a property that $ψ(A)$ generates a quasibounded holomorphic semigroup for every generator $A$ of a bounded $C_0$-semigroup in a Banach space, in terms of some convolution semigroup of measures associated with $ψ$. We give an alternative to Carasso-Kato's criterium, and derive several sufficient conditions for $ψ$ to have the above-mentioned property.
The theorem is proved that generalizes the Gelfand generalization of the Paley-Wiener tauberian theorem to general abelian topological semigroups with invariant measure. Several corollaries of this theorem are given.
Let $A$ be a densely defined closed operator in a complex Banach space $X.$ Conditions for left invertibility of operators of the form $\sum_{j=1}^\infty a_j (α_j -A)^{-1}$ are given. Several examples are considered.
The main goal of this work is to examine the structure of normal Hausdorff operators on $\mathbb{R}^n$. We show that normal Hausdorff operator in $L^2(\mathbb{R}^n)$ is unitary equivalent to the operator of multiplication by some matrix-function (its matrix symbol) in the space $L^2(\mathbb{R}^n; \mathbb{C}^{2^n}).$ Several corollaries that show that properties of a Hausdorff operator are closely related to the properties of its symbol are considered. In particular, the norm and the spectrum of such operators are described in terms of the symbol.
A multidimensional generalization of the Bernstein class of functions and the properties of functions of the introduced class are examined. In particular, a new proof of the integral representation of Bernstein functions of many variables is given. Examples are considered.
Sufficient conditions for a semigroup measure algebra to have contractible Gelfand spectrum are given and it is shown that for a wide class of semigroups these conditions are also necessary.
Let $S$ be a discrete abelian semigroup with unit and concellations and $\widehat {S}$ the semigroup of semicharacters of $S$. We shaw that the strong boundary and the Shilov boundary of the algebra of generalized analytic functions defined on $\widehat {S}$ are unions of some maximal subgroups of $\widehat {S}$. We shaw also that the both boundaries coincide with the character group of $S$ if $S$ does not contain nontrivial simple ideals. In the last case the Gelfand spectrum of the algebra under consideration is calculated.
Some assertions in harmonic analysis on the infinite dimensional torus are stated and their equivalence to Riemann hypothesis is proved.
The properties of the compactness of interpolation sets of algebras of generalized analytic functions are investigated and convenient sufficient conditions for interpolation are given.
The functional calculus of semigroup generators, based on the class of Bernstein functions in several variables is developed, the condition for holomorphy of semigroups, generated by operators which arisen in the calculus is given, and in the one-dimensional case the moment inequality for such operators is proved.
In this paper several joint spectra for a finite commuting family of closed operators in Banach space are considered, some new relations between these spectra established (earlier only the inclusion of the Taylor spectrum in the commutant one was known), and in the case of semigroup generators spectral mapping theorems for such spectra are proved. Several of this theorems are generalizations of preceding results due to the author. Applications to stability of multiparametric semigroups are given.
The multidimensional functional calculus of semigroup generators, based on the class of Bernstein functions in several variables is developed, the spectral mapping theorems for joint spectra have been stated, the condition for holomorphy of semigroups, generated by operators which arises in the calculus is given, and the moment inequality for such operators is proved.
Generalization of functions of bounded mean oscillation and Hankel operators to the case of compact abelian groups with linearly ordered dual is considered. Spaces of functions of bounded mean oscillation and of bounded mean oscillation of analytic type on such groups are described in terms of boundedness of corresponding Hankel operators under the assumption that the dual group contains a minimal positive element.
In the recent paper by Mark C. Ho (2014) the notion of a $λ$-Toeplitz operator on the Hardy space $H^2(\mathbb{T})$ over the one-dimensional torus $\mathbb{T}$ was introduced and it was shown (under the supplementary condition) that for $λ\in \mathbb{T}$ the essential spectrum of such an operator is invariant with respect to the rotation $z\mapsto λz$; if in addition $λ$ is not of finite order the essential spectrum is circular. In this paper, we generalize these results to the case when $\mathbb{T}$ is replaced by an arbitrary compact Abelian group whose dual is totally ordered.