Opial type inequalities for cosine and sine operator functions

Various L _p form Opial type inequalities are given for cosine and sine operator functions with applications.     

Topological mixing and Hypercyclicity Criterion for sequences of operators

For a sequence of continuous linear operators on a separable Fréchet space , we discuss necessary conditions and sufficient conditions for to be topologically mixing, and the relations between topological mixing and the Hypercyclicity Criterion. Among them are: 1) topological mixing is equivalent to being hereditarily densely hypercyclic; 2) the Hypercyclicity Criterion with respect to the full sequence implies topological mixing; 3) topological mixing implies the Hypercyclicity Criterion with respect to some sequence that cannot be syndetic in general, and also implies condition (b) of the Hypercyclicity Criterion with respect to the full sequence. Applications to two examples of operators on the Fréchet space of entire functions are also discussed.

     

Poincaré like inequalities for semigroups, cosine and sine operator functions

Here we present Poincaré type general L _p inequalities regarding semigroups, cosine and sine operator functions.     

Interpolation by hypercyclic functions for differential operators

We prove that, given a sequence of points in a complex domain Ω without accumulation points, there are functions having prescribed values at the points of the sequence and, simultaneously, having dense orbit in the space of holomorphic functions on Ω. The orbit is taken with respect to any fixed nonscalar differential operator generated by an entire function of subexponential type, thereby extending a recent result about MacLane-hypercyclicity due to Costakis, Vlachou and Niess.     

Second order differential operators with Feller-Wentzell type boundary conditions

In this paper, we study a basic generation problem concerning the second order differential operator in the space C[0,1] of complex continuous functions equipped with Feller-Wentzell type boundary conditions, which originates from the work of Feller [W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2) 55 (1952) 468-519]. We prove successfully that the operator, under suitable assumptions, generates a strongly continuous cosine function on C[0,1] (or on a subspace of C[0,1]), by means of an operator matrix analysis combined with perturbation, approximation, and similarity techniques.     

Hereditarily hypercyclic operators and mixing

We show that hereditary transitivity (respectively strongly hereditary transitivity) is equivalent to weak mixing (respectively strong mixing) in a discrete dynamical system with Polish phase space. We also study the connection between local orbit structure and hypercyclicity, and obtain a “local hypercyclicity criterion.”     

Some necessary and sufficient conditions for Hypercyclicity Criterion

We give necessary and sufficient conditions for an operator on a separable Hilbert space to satisfy the hypercyclicity criterion. This paper is a part of the second author’s Doctoral thesis, written at Shiraz University under the direction of the first author.     

STRONGLY CONTINUOUS INTEGRATED COSINE OPERATOR FUNCTIONS WITH GROWTH ω

For a continuous increasing function ω ： [0, ∞） → （0, ∞） of finite exponetial type, we establish a Hille-Yosida type theorem for strongly continuous integrated cosine operator functions with O（ω）. It includes the well-known polynomially bounded and exponentially bounded cases.     

Semigroups of operators,cosine operator functions,and linear differential equations

This survey presents a systematic exposition of the elements of the theory of operator semigroups (OS's) in Banach space from Hille-Yosida to the end of 1989. There is a parallel exposition of the theory of cosine operator functions (COF's). The paper contains the following divisions: Linear differential equations in Banach space, reduction of the Cauchy problem for second order equations to the Cauchy problem for first order equations, one-parameter OS's and COF's. differentiable OS's. analytic OS's. Fredholm OS's. positive OS's. stable OS's. spectral properties of OS's and COF's. compactness properties of OS's and COF's. uniformly continuous OS's and COF's. almost periodic OS's and COF's. uniformly bounded OS's and COF's. the theory of perturbations for OS's and COF's. ad joint OS's and COF's.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 28, pp. 87–202, 1990.     

Linear differential operators and operator matrices of the second order

Linear differential operators (equations) of the second order in Banach spaces of vector functions defined on the entire real axis are studied. Conditions of their invertibility are given. The main results are based on putting a differential operator in correspondence with a second-order operator matrix and further use of the theory of first-order differential operators that are defined by the operator matrix. A general scheme is presented for studying the solvability conditions for different classes of second-order equations using second-order operator matrices. The scheme includes the studied problem as a special case.     

Growth order and stability of semigroups and cosine operator functions

Equivalent conditions for a trajectory of a C₀-semigroup T(⋅) (resp. cosine function C(⋅)) of operators to have the growth order O(tα) or o(tα) are expressed in terms of Cesàro and Abel means of the norm of the trajectory. We then deduce characterizations of growth order and stability for T(⋅) and C(⋅). It is also shown that under some Tauberian condition the uniform boundedness (resp. strong convergence) of T(⋅) is equivalent to the uniform boundedness (resp. strong convergence) of its Abel means.     

Infinite order differential operators in spaces of entire functions

Differential operators ?(Δ_θ,ω), where ? is an exponential type entire function of a single complex variable and Δ_θ,ω=(θ+ωz)D+zD², D=∂/∂z, , θ?0, , acting in the spaces of exponential type entire function are studied. It is shown that, for ω?0, such operators preserve the set of Laguerre entire functions provided the function ? also belongs to this set. The latter consists of the polynomials possessing real nonpositive zeros only and of their uniform limits on compact subsets of the complex plane . The operator exp(aΔ_θ,ω), a?0 is studied in more details. In particular, it is shown that it preserves the set of Laguerre entire functions for all . An integral representation of exp(aΔ_θ,ω), a>0 is obtained. These results are used to obtain the solutions to certain Cauchy problems employing Δ_θ,ω.     

Positive linear operators generated by analytic functions

Let φ be a power series with positive Taylor coefficients {a _k } _k=0^∞ and non-zero radius of convergence r ≤ ∞. Let ξ _x , 0 ≤ x < r be a random variable whose values α _k , k = 0, 1, …, are independent of x and taken with probabilities a _k x ^k /φ(x), k = 0, 1, …. The positive linear operator (A _φ f)(x):= E[f(ξ _x )] is studied. It is proved that if E(ξ _x ) = x, E(ξ _x ²) = qx ² + bx + c, q, b, c ∈ R, q > 0, then A _φ reduces to the Szász-Mirakyan operator in the case q = 1, to the limit q-Bernstein operator in the case 0 < q < 1, and to a modification of the Lupaş operator in the case q > 1.     

A weighted identity for partial differential operators of second order and its applications

In this Note, a weighted identity for partial differential operators of second order is established. As its applications, one may deduce all the known controllability/observability results for the parabolic, hyperbolic, Schrödinger and plate equations that are derived via Carleman estimate. Meanwhile, a new controllability/observability result is presented for the parabolic equations with a complex principal part. To cite this article: X. Fu, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Spatial pseudoanalytic functions arising from the factorization of linear second order elliptic operators

Biquaternionic Vekua‐type equations arising from the factorization of linear second order elliptic operators are studied. Some concepts from classical pseudoanalytic function theory are generalized onto the considered spatial case. The derivative and antiderivative of a spatial pseudoanalytic function are introduced and their applications to the second order elliptic equations are considered. Copyright © 2011 John Wiley & Sons, Ltd.     

Hypercyclic and topologically mixing cosine functions on Banach spaces

Our first aim in this paper is to give sufficient conditions for the hypercyclicity and topological mixing of a strongly continuous cosine function. We apply these results to study the cosine function associated to translation groups. We also prove that every separable infinite dimensional complex Banach space admits a topologically mixing uniformly continuous cosine family.

     

Generalization of certain subclasses of multivalent functions with negative coefficients defined by using a differential operator

Making use of a differential operator, we introduce and study a certain class SC_n(j,p,q,α,λ) of p-valently analytic functions with negative coefficients. In this paper, we obtain numerous sharp results including (for example ) coefficient estimates, distortion theorem, radii of close-to convexity, starlikeness and convexity and modified Hadamard products of functions belonging to the class SC_n(j,p,q,α,λ). Finally, several applications investigate an integral operator, and certain fractional calculus operators are also considered.     

Positive self-adjoint operators generated by products of differential expressions

Under the assumption of the product τ:=l⁺l of the regular differential expression l and its adjoint l⁺, being well-formed, a complete characterization of all the positive self-adjoint extensions of the minimal operator generated by τ in terms of boundary conditions is given.