Rate of growth of hypercyclic entire functions

Let

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be a weighted shift operator on the space of entire functions. Suppose that |a_a| → ∞ monotonically, a particular such operator being the differentiation operator Df = f′.l It is known that T possesses hypercyclic functions, that is, functions f for which is dense in . In this paper a sharp result on the permissible rates of growth to T-hypercyclic entire functions is obtained, and it is shown that for every permissible growth rate there is a dense subspace of T-hypercyclic functions satisfying this growth condition. These results generalise earlier results by the author and by Armitage on the differentiation operator.     

The umbral calculus on logarithmic algebras

D. E. Loeb and G.-C. Rota, using the operator of differentiation D, constructed the logarithmic algebra that is the generalization of the algebra of formal Laurent series. They also introduced Appell graded logarithmic sequences and binomial (basic) graded logarithmic sequences as sequences of elements of the logarithmic algebra and extended the main results of the classical umbral calculus on such sequences. We construct an algebra by an operator d that is defined by the formula (1.1). This algebra is an analog of the logarithmic algebra. Then we define sequences analogous to Boas-Buck polynomial sequences and extend the main results of the nonclassical umbral calculus on such sequences. The basic logarithmic algebra constructed by the operator of q-differentiation is considered. The analog of the q-Stirling formula is obtained.     

Weighted differentiation composition operator from logarithmic Bloch spaces to Bloch‐type spaces

In this paper, we give a new characterization for the boundedness of weighted differentiation composition operator from logarithmic Bloch spaces to Bloch‐type spaces and calculate its essential norm in terms of the n‐th power of induced analytic self‐map on the unit disk. From which a sufficient and necessary condition of compactness of this operator follows immediately.     

On one class of differential operators and their application

We use a generalized differentiation operator to construct a generalized shift operator, which makes it possible to define a generalized convolution operator in the space H(?). Next, we consider the characteristic function of this operator and introduce a generalized Laplace transform. We study the homogeneous equation of the generalized convolution operator, investigate its solvability, and consider the multipoint Vallée Poussin problem.     

On One Class of Almost Bounded Perturbations of Smooth Restrictions of a Closed Operator

We investigate one class of perturbations of a closed densely-defined operator in a Hilbert space. These perturbations change the domain of definition of the operator. We prove that the perturbed operator S is closed and densely defined. We construct the adjoint operator S*.     

Problems for equations with special parabolic operator of fractional differentiation

We establish the well-posedness of the Cauchy problem and the two-point boundary-value problem for an equation with an operator of fractional differentiation that corresponds to the singular parabolic Beltrami – Laplace operator on a surface of the Dini class.     

Best approximation of a differentiation operator in L2-space

This paper contains the magnitude of the best approximation in the L₂-sense of a k-th order differentiation operator of a bounded linear operator A(f) which acts on the class of functions which are differentiable n times.Translated from Matematicheskie Zametki, Vol. 3, No. 2, pp. 157–164, February, 1968.     

A model for operators with cyclic adjoint

It is proved that every bounded linear operator on a complex Banach space whose adjoint has aw ^*-cyclic vector, is similar to the differentiation operator on a Banach space of entire functions of finite exponential type. The relation of this model to the existence of non-trivial invariant subspaces is discussed.     

Galerkin method for a nonstationary equation with a monotone operator

In the present paper, we consider the Galerkin method for a quasilinear differentialoperator equation with a leading self-adjoint operator A(t) and a subordinate monotone operator K. For the projection subspaces we take linear spans of eigenelements of an operator similar to the leading operator A(t). We obtain new estimates for the Galerkin method and consider applications to an initial-boundary value problem for a parabolic equation of higher order.     

On a Class of Analytic Operator Functions and Their Linearizations

We consider an operator function T in a Krein space which can formally be written as (0.1)but the last term on the right of (0.1) is replaced by a relatively form‐compact perturbation of a similar form. We study relations between the operator function T, a selfadjoint operator M in some Krein space, associated with T, and an operator which can be constructed with the help of the operator function –T^–1. The results are applied to a Sturm‐Liouville problem with a coefficient depending rationally on the eigenvalue parameter.     

Extensions of bitriangular operators

We prove that every one-dimensional extension of a bitriangular operator has a cyclic commutant. We also prove that ifT is an extension of a bitriangular operator by an algebraic operator, then the weakly closed algebraW(T) generated byT has a separating vector.This work was partially supported by NSF Grant DMS-9401544.Participant, Workshop in Linear Analysis and Probability, Texas A&M University     

Convergence in Measure and τ-Compactness of τ-Measurable Operators,Affiliated with a Semifinite von Neumann Algebra

Let τ be a faithful normal semifinite trace on a von Neumann algebra. We establish the Leibniz criterion for sign-alternating series of τ-measurable operators and present an analogue of the criterion of series “sandwich” series for τ-measurable operators. We prove a refinement of this criterion for the τ-compact case. In terms of measure convergence topology, the criterion of τ-compactness of an arbitrary τ-measurable operator is established. We also give a sufficient condition of 1) τ-compactness of the commutator of a τ-measurable operator and a projection; 2) convergence of τ-measurable operator and projection commutator sequences to the zero operator in the measure τ.

     

A $$\mathbb{Z}_{2}$$-orbifold model of the symplectic fermionic vertex operator superalgebra

We give an example of an irrational C ₂-cofinite vertex operator algebra whose central charge is −2d for any positive integer d. This vertex operator algebra is given as the even part of the vertex operator superalgebra generated by d pairs of symplectic fermions, and it is just the realization of the c = −2-triplet algebra given by Kausch in the case d = 1. We also classify irreducible modules for this vertex operator algebra and determine its automorphism group. This research is supported in part by a grant from Japan Society for the Promotion of Science.     

On the eigenvectors and eigenvalues of an operator pair

We establish conditions for orthogonality of eigenvectors and the type of eigenvalues (real or purely imaginary) of an operator pair A, B in an equation Au + Bu = 0which permute (or skew-permute) with a given third operator. We consider equations with differential operators A and B (the differential spectral problem), with a differential operator A and an integro-differential operator B, and also with algebraic operators A and B. Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 14–19.     

A Fredholm Determinant Formula for Section Determinants of Bounded Operators

We show that the section determinant of e^A can be expressed, under certain conditions, by the Fredholm determinant of an integral operator. The kernel function of this integral operator is computed explicitly in terms of the operator A. As a simple consequence we derive a Weierstrass type product expansion for the section determinant.     

A criterion for the fundamental principle to hold for invariant subspaces on bounded convex domains in the complex plane

Let D be a bounded convex domain of the complex plane. We study the problem of whether the fundamental principle holds for analytic function spaces on D invariant with respect to the differentiation operator and admitting spectral synthesis. Earlier this problem was solved under a restriction on the multiplicities of the eigenvalues of the differentiation operator. In the present paper, we lift this restriction. Thus, we present a complete solution of the fundamental principle problem for arbitrary nontrivial closed invariant subspaces admitting spectral synthesis on arbitrary bounded convex domains.     

Collapsing and Dirac-Type Operators

We analyze the limit of the spectrum of a geometric Dirac-type operator under a collapse with bounded diameter and bounded sectional curvature. In the case of a smooth limit space B, we show that the limit of the spectrum is given by the spectrum of a certain first-order differential operator on B, which can be constructed using superconnections. In the case of a general limit space X, we express the limit operator in terms of a transversally elliptic operator on a G-manifold X/ with X = X//G. As an application, we give a characterization of manifolds which do not admit uniform upper bounds, in terms of diameter and sectional curvature, on the k-th eigenvalue of the square of a Dirac-type operator. We also give a formula for the essential spectrum of a Dirac-type operator on a finite-volume manifold with pinched negative sectional curvature.     

Approximation of differentiation operator in the space L 2 on semiaxis

We establish an upper bound for the error of the best approximation of the first order differentiation operator by linear bounded operators on the set of twice differentiable functions in the space L ₂ on the half-line. This upper bound is close to a known lower bound and improves the previously known upper bound due to E. E. Berdysheva. We use a specific operator that is introduced and studied in the paper.     

Solution of boundary and eigenvalue problems for second‐order elliptic operators in the plane using pseudoanalytic formal powers

We propose a method for solving boundary value and eigenvalue problems for the elliptic operator D = div p grad + qin the plane using pseudoanalytic function theory and in particular pseudoanalytic formal powers. Under certain conditions on the coefficients p and q with the aid of pseudoanalytic function theory a complete system of null solutions of the operator can be constructed following a simple algorithm consisting in recursive integration. This system of solutions is used for solving boundary value and spectral problems for the operator D in bounded simply connected domains. We study theoretical and numerical aspects of the method. Copyright © 2010 John Wiley & Sons, Ltd.     

On uniform positivity of a symmetrizer

We exhibit ah effective method of constructing a positive definite symmetrizer of a linear operator that is spectrally equivalent to a self-adjoint operator bundle whose operator coefficients satisfy certain conditions. It is proved that these conditions are sufficient for positive definiteness of the symmetrizer.Translated fromMatematicheskie Melody i Fiziko-Mekhanicheskie Polya, Issue 34, 1991, pp. 12–15.