Operators In Finite-dimensional Spaces Research Articles

$C$--normal operators

A new class of operators, larger than C - symmetric operators and different than normal one named C - normal operatorsis introduced. Basic properties are given. Characterizations of this operators in finite dimensional spaces using relations with conjugate normal metrices are presented. Characterizations of Toeplitz operators and composition operators as C - normal operators are given. Bunches of examples are presented.

Multiparameter System of Operators with Two Parameters in Finite Dimensional Spaces

The authors have proved the existence of the multiple basis on the eigen and associated elements of the two parameter system of operators in finite dimensional spaces. The proof uses the notion of the abstract analog of resultant of two operator pencils, acting, generally speaking, in different Hilbert spaces. In this paper necessary and sufficient conditions of the existence of multiple completeness of the eigen and associated vectors of two parameter system of operators in finite dimensional Hilbert space is given.

QUADRATIC EQUATIONS IN HILBERTIAN OPERATORS AND APPLICATIONS

In this paper bounded linear operators in Hilbert space satisfying general quadratic equations are characterized. Necessary and sufficient conditions for sets of operators satisfying two such equations to compare relative to a weak ordering are presented. In addition, averaging operators in finite dimensional spaces are determined, and in this case it is shown that they are unitary models for all projections. It is pointed out, by an example, that the latter result does not hold in infinite dimensions. A key application to certain second order random fields of Karhunen type is given. The main purpose is to present the structure of bounded non-self adjoint operators solving quadratic equations, and indicate their use.

Pairs of quaternionic selfadjoint operators with numerical range in a halfplane

The property is studied that two selfadjoint operators on a quaternionic Hilbert space have the joint numerical range in a halfplane bounded by a line passing through the origin. This property is expressed in various ways, in particular, in terms of compressions to two dimensional subpaces, and in terms of linear dependence over the reals. The canonical form for two selfadjoint quaternionic operators in finite dimensional spaces is the main technical tool.

The Lagrange principle in the problem of optimal inversion of linear operators in finite-dimensional spaces with a priori information about its solution

The problem of solving a system of linear algebraic equations is examined. An application of the Lagrange principle to the optimal recovery in this problem is described. New optimal methods that use available information about the errors in the data and a priori information about the solution are proposed for solving such systems.

On the orbit of invariant subspaces of linear operators in finite-dimensional spaces (new proof of a Halmos's result)

P.R. Halmos proved that for a linear operator A over a finite-dimensional complex vector space E, every A-invariant subspace of E is the range of a commutant of A. His proof was based on a generalization of the concept of eigenvector. In this note, we give an invariant proof of this Halmos's theorem.

On the Uniqueness Problem for Dirichlet Operators

Strong uniqueness inL2and inL1for Dirichlet operators in finite dimensional spaces is studied.

A relationship between mixed discriminants and the joint spectrum of a family of commuting operators in finite-dimensional space

We study the properties of the polynomial operator pencil $$L(\lambda ) = \sum\limits_{i = 0}^n {\lambda ^{n - i} M_i } , M_i :\mathcal{H} \to \mathcal{H}, i = 0, \ldots , n$$ , where\(\mathcal{H}\) is ak-dimensional Hilbert space, and prove that the mixed discriminants {d j } j=0 nk , defined as the coefficients of the polynomial $$det L(\lambda ) = \sum\limits_{j = 0}^{nk} {d_j } \lambda ^{nk - j} $$ , are completely determined by the joint spectrum of the family {M i } i=0 n . A generalization of Gershgorin's well-known theorem on the position of the eigenvalues of a matrix to the case of a polynomial matrix pencil is obtained.

The boundary spectrum of linear operators in finite-dimensional spaces

1. The basic concepts. Let X be an n-dimensional real or complex vector space, n < ∞. Let A : X → X be a linear operator. It is called power bounded if the semigroup Π(A) = {A}0 of its natural powers is bounded. (This property can be defined in terms of any norm: sup{‖A‖ : k ∈ N} < ∞. The choice of the norm does not matter.) An operator A is called double power bounded if A is invertible and A,A are both power bounded. (Thus, sup{‖A‖ : k ∈ Z} < ∞.) We only consider the power bounded operators. Since in this case the spectral radius r(A) does not exceed 1, the spectrum σ(A) consists of two parts, the boundary (or peripheral) spectrum

Discrete Coherent States and Probability Distributions in Finite-Dimensional Spaces

Operator bases are discussed in connection with the construction of phase space representatives of operators in finite-dimensional spaces, and their properties are presented. It is also shown how these operator bases allow for the construction of a finite harmonic oscillator-like coherent state. Creation and annihilation operators for the Fock finite-dimensional space are discussed and their expressions in terms of the operator bases are explicitly written. The relevant finite-dimensional probability distributions are obtained and their limiting behavior for an infinite-dimensional space are calculated which agree with the well known results

On extreme operators in finite-dimensional spaces

On extreme operators in finite-dimensional spaces

Composition of operators in finite dimensional spaces with indefinite metrics

Composition of operators in finite dimensional spaces with indefinite metrics

Projections, generalized inverses, and quadratic forms

The purpose of this paper is to present a unified treatment of the apparently unrelated topics, namely projections and generalized inverses, with some applications to the distribution theory of quadratic forms in Gaussian random variables. Other applications will also be mentioned. Only finite dimensional euclidean spaces are considered, but some of the theory can be extended to the infinite dimensional case using, for instance, the methods of [l]. However, the treatment then will no longer be elementary. It is customary to consider linear operators in finite dimensional spaces as matrices, and we shall do so here. For this we choose a fixed but arbitrary basis, and all considerations are related to it. Thus “projection” is synonymous with “idempotent matrix”. The properties of such matrices play a key role in what follows. The importance of projections has been noticed in the past (e.g., refs. 2-4, to name a few). However, they limited themselves to the cases of orthogonal projections, or symmetric idempotent matrices. Unless otherwise stated, the idempotent matrices are not symmetric in this paper. In Section I, several properties of such matrices are proved. It turns out that, by our methods, the results of the above authors can be simplified. In Section II, we deduce the existence (and uniqueness) of the so-called generalized inverse of a matrix [5], a computational formula [6], as well as a “spectral theorem” for such matrices using the results of Section I. Finally in Section III, we apply this theory in deriving the distributions of some quadratic forms in