Calculation of eigenvalues of a discrete self-adjoint operator perturbed by a bounded operator

We generalize the method of regularized traces which calculates eigenvalues of a perturbed discrete operator for the case of an arbitrary multiplicity of eigenvalues of the unperturbed operator. We obtain a system of equations, enabling one to calculate eigenvalues of the perturbed operator with large ordinal numbers. As an example, we calculate eigenvalues of a perturbed Laplace operator in a rectangle.     

Matrix regular operator space and operator system

We establish a relationship between Schreiner's matrix regular operator space and Werner's (nonunital) operator system. For a matrix ordered operator space V with complete norm, we show that V is completely isomorphic and complete order isomorphic to a matrix regular operator space if and only if both V and its dual space V^∗ are (nonunital) operator systems.     

Hyperelasticity and the potentialness of a nonlinear operator

Summary A quasilinear second order partial differential operator of divergence form, involving N unknown functions and independent variables is transformed into a nonlinear functional operator in a Hilbert space. The constraint of potenlialness imposed on the nonlinear operator is seen to be equivalent in the case N=3 to hyperelasticity in the differential operator of elasiticity. By imposing further conditions on the differential operator, and assuming it is polynomial in the Frechet derivative of the unknown functions it becomes possible to extend to finite hyperelasticity a result proved earlier for infinitesimal hyperelasticity. Entrata in Redazione il 25 gennaio 1971.     

The analyticity of a generalized Ruelle’s operator

In this work we propose a generalization of the concept of Ruelle’s operator for one dimensional lattices used in thermodynamic formalism and ergodic optimization, which we call generalized Ruelle’s operator. Our operator generalizes both the Ruelle operator proposed in [2] and the Perron Frobenius operator defined in [7]. We suppose the alphabet is given by a compact metric space, and consider a general a-priori measure to define the operator. We also consider the case where the set of symbols that can follow a given symbol of the alphabet depends on such symbol, which is an extension of the original concept of transition matrices from the theory of subshifts of finite type. We prove the analyticity of the Ruelle’s operator and present some examples.     

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.     

Perturbation of a periodic operator by a narrow potential

We consider perturbations of a second-order periodic operator on the line; the Schr?dinger operator with a periodic potential is a specific case of such an operator. The perturbation is realized by a potential depending on two small parameters, one of which describes the length of the potential support, and the inverse value of other corresponds to the value of the potential. We obtain sufficient conditions for the perturbing potential to have eigenvalues in the gaps of the continuous spectrum. We also construct their asymptotic expansions and present sufficient conditions for the eigenvalues of the perturbing potential to be absent.     

Some classes of functions of a linear closed operator

A linear closed densely defined operator and some domain Ω lying in the regular set of the operator and containing the negative real semiaxis of the real line are specified in a Banach space. We assume that power estimates for the norm of the resolvent operator are known at zero and infinity. We use the Cauchy integral formula to introduce operator functions generated by scalar functions that are analytic in a certain domain not containing the origin and containing the complement of Ω and satisfy power estimates for their absolute values at zero and infinity. We study some properties of operator functions, which were studied by the authors earlier for the case of an operator whose inverse is bounded; in particular, we study the multiplicative property.     

离散算子的本征投影与本征幂零

本文给出例子，说明离散算子本征投影一致有界推不出其本征幂零一致，有界幂零一致有界也推不出本征幂零的幂一致有界。     

When is the range of a normal operator invariant?

It is proved that if an operator (or a set of operators) leaves the ranges of the spectral projections corresponding to certain subsets of the spectrum of a normal operator, then the operator (respectively, set of operators) leaves the range of the normal operator invariant. This strengthens a result of Peligrad [4].     

The analytic model of a subnormal operator

The analytic model of a pure subnormal opertor and its conjugate subnormal operator is obtained. A mosaic is introduced for subnormal operator. Some results in spectral analysis of subnormal operators are obtained by means of the analytic model and the mosaic. The form of pure subnormal operators with rank two self-commutator is determined.This work is partially supported by a NSF grant no DMS-8502359.     

Multiple operator integrals and higher operator derivatives

In this paper we consider the problem of the existence of higher derivatives of the function t??(A+tK), where ? is a function on the real line, A is a self-adjoint operator, and K is a bounded self-adjoint operator. We improve earlier results by Sten’kin. In order to do this, we give a new approach to multiple operator integrals. This approach improves the earlier approach given by Sten’kin. We also consider a similar problem for unitary operators.     

Generalized Jordan matrix of a linear operator

For any linear operator defined over an arbitrary field k, there is a basis in which this matrix is a generalized Jordan matrix (of the second kind) with elements in the field k. For any linear operator, such a matrix is defined uniquely up to permutation of diagonal blocks.     

Refined operator inequalities for relative operator entropies

Analysis Mathematica - We investigate the relative operator entropies in the more general settings of C*-algebras, real C*-algebras and JC-algebras. We show that all the operator inequalities on...     

The uniform algebra associated with a cyclic subnormal operator

A uniform algebra is associated with each cyclic subnormal operator, and a map is defined from the spectrum of the operator into the maximal ideal space of the uniform algebra. Affirmative answers as to whether this map is continuous are given under reasonable hypothesis on the structure of the spectrum of the operator.     

A hypercyclic finite rank perturbation of a unitary operator

A unitary operator V and a rank 2 operator R acting on a Hilbert space H{\mathcal{H}} are constructed such that V + R is hypercyclic. This answers affirmatively a question of Salas whether a finite rank perturbation of a hyponormal operator can be supercyclic.     

A method for finding the fixed vector of a stochastic operator

The author proposes a new method for finding the fixed vector of a stochastic operator based on the representation of the operator of the linear system as the product of an noninvertible operator of simple form and an invertible operator. A complement to Jentsch’s theorem is obtained.     

To the theory of operator monotone and operator convex functions

We prove that a real function is operator monotone (operator convex) if the corresponding monotonicity (convexity) inequalities are valid for some normal state on the algebra of all bounded operators in an infinite-dimensional Hilbert space. We describe the class of convex operator functions with respect to a given von Neumann algebra in dependence of types of direct summands in this algebra. We prove that if a function from ℝ⁺ into ℝ⁺ is monotone with respect to a von Neumann algebra, then it is also operator monotone in the sense of the natural order on the set of positive self-adjoint operators affiliated with this algebra.     

Matrix analysis and the friedrichs operator of a quadrature domain

The Friedrichs operator of a quadrature domain is a finite rank operator which appears in the joint diagonalization of the Bergman space scalar product and its real part, regarded as a real symmetric form. This note relates some general aspects of the theory of pairs of hermitian or symmetric forms to the specific case of the Friedrichs operator.     

Spectral properties of a nonlocal second-order difference operator

We consider a spectral problem for a nonlocal difference operator of second derivative with variable coefficients and with a complex parameter in the boundary condition. We study the algebraic and geometric multiplicity of the eigenvalues and the sign of their real part. We obtain conditions on the parameter which ensure that the entire spectrum of the operator lies in the right complex half-plane.