Analytic Operator Functions Research Articles

Nijenhuis geometry

This work is the first, and main, of a series of papers in progress dedicated to Nijenhuis operators, i.e., fields of endomorphisms with vanishing Nijenhuis torsion. It serves as an introduction to Nijenhuis Geometry that should be understood in much wider context than before: from local description at generic points to singularities and global analysis. The goal of the present paper is to introduce terminology, develop new important techniques (e.g., analytic functions of Nijenhuis operators, splitting theorem and linearisation), summarise and generalise basic facts (some of which are already known but we give new self-contained proofs), and more importantly, to demonstrate that the research programme proposed in the paper is realistic by proving a series of new, not at all obvious, results.

Approximation by complex Chlodowsky-Szasz-Durrmeyer operators in compact disks

This paper presents a study on the approximation properties of the operators constructed by the composition of Chlodowsky operators and Szasz-Durrmeyer operators. We give the approximation properties and obtain a Voronovskaya-type result for these operators for analytic functions of exponential growth on compact disks. Furthermore, a numerical example with an illustrative graphic is given to compare for the error estimates of the operators.

Method for Localizing the Zeros of Analytic Functions Based on the Krawczyk Operator

The problem of localizing the zeros of analytic functions using the Krawczyk operator is considered. Formulas are obtained for computing the image of the Krawczyk operator for analytic functions. Based on this, a new algorithm for localizing the zeros is proposed. The use of the method is demonstrated by numerical examples including the search for the zeros of the Riemann zeta function.

Inverses of generators of integrated fractional resolvent operator functions

Abstract This paper is devoted to the inverse generator problem in the setting of generators of integrated resolvent operator functions. It is shown that if the operator A is the generator of a tempered β-times integrated α-resolvent operator function ((α, β)-ROF) and it is injective, then the inverse operator A −1 is the generator of a tempered (α, γ)-ROF for all γ > β + 1/2, by means of an explicit representation of the integrated resolvent operator function based in Bessel functions of first kind. Analytic resolvent operator functions are also considered, showing that A −1 is in addition the generator of a tempered (δ, 0)-ROF for all δ < α. Moreover, the optimal decay rate of (α, β)-ROFs as t → ∞ is given. These result are applied to fractional Cauchy problem unsolved in the fractional derivative.

Accumulation of complex eigenvalues of a class of analytic operator functions

For analytic operator functions, we prove accumulation of branches of complex eigenvalues to the essential spectrum. Moreover, we show minimality and completeness of the corresponding system of eigenvectors and associated vectors. These results are used to prove sufficient conditions for eigenvalue accumulation to the poles and to infinity of rational operator functions. Finally, an application of electromagnetic field theory is given.

The block numerical range of analytic operator functions

We introduce the block numerical range Wn(L) of an operator function L with respect to a decomposition H = H1⊕. . .⊕Hn of the underlying Hilbert space. Our main results include the spectral inclusion property and estimates of the norm of the resolvent for analytic L . They generalise, and improve, the corresponding results for the numerical range (which is the case n = 1) since the block numerical range is contained in, and may be much smaller than, the usual numerical range. We show that refinements of the decomposition entail inclusions between the corresponding block numerical ranges and that the block numerical range of the operator matrix function L contains those of its principal subminors. For the special case of operator polynomials, we investigate the boundedness of Wn(L) and we prove a Perron-Frobenius type result for the block numerical radius of monic operator polynomials with coefficients that are positive in Hilbert lattice sense.

Geometric properties of the complex Baskakov-Stancu operators in the unit disk

In this article, we determine certain conditions under which the partial sums involving thecomplex Baskakov-Stancu operators of analytic univalent functions of bounded turning are also of boundedturning. Moreover, we consider some geometric properties such as starlikeness and convexity for these partialsums. The lower bound of the partial sums of univalent functions is computed using the lower bound of thecomplex Baskakov-Stancu operators of analytic functions

A Note on Random Perturbations of a Multiple Eigenvalue of a Hermitian Operator

The problem of a random Hermitian perturbation of a multiple isolated eigenvalue of a Hermitian operator is considered. It is shown that the combined multiplicities of the perturbed eigenvalues converge in probability to the multiplicity of the eigenvalue of the target operator. Also the asymptotic distribution of a certain average of these eigenvalues, centered at the target, is obtained. As a tool differentiation of analytic functions of operators is employed in conjunction with an ensuing “delta-method”. The result is of a probabilistic rather than statistical nature.

Variational characterizations for eigenfunctions of analytic self-adjoint operator functions

In this paper we consider Rellich's diagonalization theorem for analytic self-adjoint operator functions and investigate variational principles for their eigenfunctions and interlacing statements. As an application, we present a characterization for the eigenvalues of hyperbolic operator polynomials.

Some Asymptotic Theory for Functional Regression and Classification

Exploiting an expansion for analytic functions of operators, the asymptotic distribution of an estimator of the functional regression parameter is obtained in a rather simple way; the result is applied to testing linear hypotheses. The expansion is also used to obtain a quick proof for the asymptotic optimality of a functional classification rule, given Gaussian populations.

Classes of analytic functions with fractional powers defined by means of a certain linear operator

Motivated by the success of the familiar Dziok–Srivastava convolution operator, we introduce here a closely-related linear operator for analytic functions with fractional powers. By means of this linear operator, we then define and investigate a class of analytic functions. Finally, we determine certain conditions under which the partial sums of the linear operator of bounded turning are also of bounded turning. We also illustrate an application of a fractional integral operator.

Differential Subordination Defined by New Generalised Derivative Operator for Analytic Functions

A new generalised derivative operator is introduced. This operator generalised many well‐known operators studied earlier by many authors. Using the technique of differential subordination, we will study some of the properties of differential subordination. In addition we investigate several interesting properties of the new generalised derivative operator.

The Quadratic Numerical Range of an Analytic Operator Function

We introduce a new concept of numerical range for analytic operator functions. This so-called quadratic numerical range is induced by a decomposition of the underlying Hilbert space. Like the classical numerical range of an operator function, it is not convex, it has the spectral inclusion property, and it provides resolvent estimates and estimates for the lengths of Jordan chains in boundary points having the exterior cone property. As the quadratic numerical range is contained in the numerical range, it yields tighter enclosures for the eigenvalues and the spectrum of analytic operator functions.

Self-adjoint Analytic Operator Functions: Local Spectral Function and Inner Linearization

In this note we continue the study of spectral properties of a self-adjoint analytic operator function A(z) that was started in [5]. It is shown that if A(z) satisfies the Virozub–Matsaev condition on some interval Δ0 and is boundedly invertible in the endpoints of Δ0, then the ‘embedding’ of the original Hilbert space $${\mathcal{H}}$$ into the Hilbert space $${\mathcal{F}}$$ , where the linearization of A(z) acts, is in fact an isomorphism between a subspace $${\mathcal{H}}(\Delta_{0})$$ of $${\mathcal{H}}$$ and $${\mathcal{F}}$$ . As a consequence, properties of the local spectral function of A(z) on Δ0 and a so-called inner linearization of the operator function A(z) in the subspace $${\mathcal{H}}(\Delta_{0})$$ are established.

The delta method for analytic functions of random operators with application to functional data

In this paper, the asymptotic distributions of estimators for the regularized functional canonical correlation and variates of the population are derived. The method is based on the possibility of expressing these regularized quantities as the maximum eigenvalue and the corresponding eigenfunctions of an associated pair of regularized operators, similar to the Euclidean case. The known weak convergence of the sample covariance operator, coupled with a delta-method for analytic functions of covariance operators, yields the weak convergence of the pair of associated operators. From the latter weak convergence, the limiting distributions of the canonical quantities of interest can be derived with the help of some further perturbation theory.

Approximation by analytic operator functions. Factorizations and very badly approximable functions

This is a continuation of our earlier paper \cite{PT3}. We consider here operator-valued functions (or infinite matrix functions) on the unit circle $\T$ and study the problem of approximation by bounded analytic operator functions. We discuss thematic and canonical factorizations of operator functions and study badly approximable and very badly approximable operator functions. We obtain algebraic and geometric characterizations of badly approximable and very badly approximable operator functions. Note that there is an important difference between the case of finite matrix functions and the case of operator functions. Our criteria for a function to be very badly approximable in the case of finite matrix functions also guarantee that the zero function is the only superoptimal approximant. However in the case of operator functions this is not true.

Self-adjoint analytic operator functions and their local spectral function

For a self-adjoint analytic operator function A ( λ ) , which satisfies on some interval Δ of the real axis the Virozub–Matsaev condition, a local spectral function Q on Δ, the values of which are non-negative operators, is introduced and studied. In the particular case that A ( λ ) = λ I − A with a self-adjoint operator A, it coincides with the orthogonal spectral function of A. An essential tool is a linearization of A ( λ ) by means of a self-adjoint operator in some Krein space and the local spectral function of this linearization. The main results of the paper concern properties of the range of Q ( Δ ) and the description of a natural complement of this range.

Conservative linear systems, unitary colligations and Lax–Phillips scattering: multidimensional generalizations

Connections between conservative linear systems, Lax–Phillips scattering, and operator model theory are well known. A common thread in all the theories is a contractive, analytic, operator-valued function on the unit disc T(z) having a representation of the form T(z) = D + z C (I − zA)−1 B, known as the transfer or frequency-response function in the system-theory community, the scattering function in the mathematical physics community, and the characteristic operator function in the operator theory community. Here we consider analogues of this circle of ideas in the more general setting of multidimensional systems/multi-evolution scattering systems/multivariable function-theoretic operator theory. Three particular extensions are discussed; from the point of view of system theory, these involve (1) a multidimensional linear system with transfer function a contractive analytic operator function on the unit polydisc in complex Euclidean space, (2) a non-commutative multidimensional linear system with evolution along a free semigroup and with transfer function equal to a formal power series in non-commuting indeterminants, and (3) an overdetermined multidimensional linear system with transfer function identified as a bundle mapping between two Hermitian vector bundles over an algebraic curve embedded in complex projective space. This survey updates an earlier survey by the first author appearing in 2000.

Some criteria for univalence of certain integral operators

We derive some criteria for univalence of certain integral operators for analytic functions in the open unit disk.

Some estimates for analytic functions of strip or sectorial operators

Let S be a strip or a sector of the complex plane. Being given a linear operator T on a Hilbert space whose numerical range is contained in S and a holomorphic function f uniformly bounded in S, we provide an estimate of the norm of f(T) with respect to this uniform bound.