Class Of Linear Maps Research Articles

On linear maps preserving complex symmetry

An operator T on a complex Hilbert space H is said to be complex symmetric if there exists a conjugate-linear, isometric involution C:H→H so that CTC=T⁎. This paper is devoted to describing which linear maps leave the class of complex symmetric operators invariant. Complete characterizations are obtained for several classes of linear maps, including similarity transformations, surjective linear isometries, multiplication operators and certain completely positive maps.

Dynamical properties of 2-torus parabolic maps

In this paper, a class of linear maps on the 2-torus are discussed. Discussions are focused on the case that the maps are parabolic. It is shown that the maximal invariant set for a 2-torus parabolic map is indeed invariant, and is almost closed, and the Lebesgue measure restricted to a maximal invariant set is invariant. Under this invariant measure, all Lyapunov exponents of a parabolic map are zero. In certain simple cases, the Lebesgue measure of the maximal invariant sets are computed and estimated. For the case the maps are invertible, it is shown that the inverse of a non-horocyclic parabolic map is no longer a parabolic map. Interesting properties of the conjugation of invertible parabolic maps by automorphisms of the torus are characterized, and a conjugation invariant for such maps are obtained. And it is proven that all these maps can be reduced to a family of one parameter rigid rotations.

Remarks on orthogonality preserving mappings in normed spaces and some stability problems

We consider the Birkhoff-James orthogonality in normed spaces and classes of linear mappings exactly and approximately preserving this relation. Some related stability problems are posed.

Linear mappings approximately preserving orthogonality

We consider the class of linear mappings, between real or complex inner product spaces, such that for each two orthogonal vectors in the domain their values are ɛ-orthogonal in the target space. By ɛ-orthogonality ( 0 ⩽ ɛ < 1 ) we mean the relation u ⊥ ɛ v ⇔ | 〈 u | v 〉 | ⩽ ɛ ‖ u ‖ ‖ v ‖ .

Extensions of Type G and Marginal Infinite Divisibility

We say that a random variate on a Euclidean space is marginal infinitely divisible with respect to a class of linear mappings on that space if each of these mappings results in an infinitely divisible random variate. Special cases are applied in a multivariate extension of the concept of type G probability laws. Random nonnegative matrices play a central role.

Concurrent forms of signal processing algorithms

Forms of algorithms that facilitate rapid processing on affiliated systolic arrays are examined. Classes of linear maps A: E/sup /n to E/sup /n that can be computed on p*p arrays at speed. O(p) where p= square root n are identified. The array architectures which provide the requisite computational support are proposed. The expansion of arbitrary linear maps in terms of the fast maps is considered. The results include a definitive method for minimal expansions and for best approximations of an a priori order. A detailed comparative example which illustrates the principles in question is also included. >

On a Class of Linear Maps for Data Compression

A method for data compression with linear maps has been developed which is found to produce further reduction in overhead storage requirement, compression/decompression time, and clustering overhead as compared to the affine map method in certain cases. Algorithms have been developed for cluster minimization, cluster identification, and compression matrix calculation that may be applied with advantage in both the methods.

Equivalence classes of linear mappings with applications to algebraic cryptography, II

Equivalence classes of linear mappings with applications to algebraic cryptography, II

Equivalence classes of linear mappings with applications to algebraic cryptography, I

Equivalence classes of linear mappings with applications to algebraic cryptography, I

Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. I. Mapping Theorems and Ordering of Functions of Noncommuting Operators

A new calculus for functions of noncommuting operators is developed, based on the notion of mapping of functions of operators onto $c$-number functions. The class of linear mappings, each member of which is characterized by an entire analytic function of two complex variables, is studied in detail. Closed-form solutions for such mappings and for the inverse mappings are obtained and various properties of these mappings are studied. It is shown that the most commonly occurring rules of association between operators and $c$-numbers (the Weyl, the normal, the antinormal, the standard, and the antistandard rules) belong to this class and are, in fact, the simplest ones in a clearly defined sense. It is shown further that the problem of expressing an operator in an ordered form according to some prescribed rule is equivalent to an appropriate mapping of the operator on a $c$-number space. The theory provides a systematic technique for the solution of numerous quantum-mechanical problems that were treated in the past by ad hoc methods, and it furnishes a new approach to many others. This is illustrated by a number of examples relating to mappings and ordering of operators.

On two classes of linear mappings, acting between Banach spaces and Banach lattices

On two classes of linear mappings, acting between Banach spaces and Banach lattices

Inductive and projective limits of normed spaces

Let {Ui, Uij} be an inductive system of normed linear spaces Ui and continuous linear maps uij; Uj → Ui. (We write j ≺ i if uij: Uj → Ui.) An inductive limit of the system with respect to a class of spaces A in and maps f in is a space Uu in Uu and a system ui → Uu of maps in such that (i) whenever j ≺ i, and that (ii) if A is any space in and fi: Ui → A is any system of maps in for which then there is a unique map f: Uu → A in such that fi = fo ui for each i. If is the class of all vector spaces and is the class of linear maps, we obtain the algebraic inductive limit, which we denote simply by U. The usual choice is to take to be the class of locally convex spaces and the class of continuous linear maps; the inductive limit UL then always exists [1, § 16 C]. If is again the continuous linear mappings but contains only normed spaces, the corresponding inductive limit UN may not always exist. However, if in addition we require that contains just contractions (norm-decreasing linear mappings), then an inductive limit Uc will exist if every uij is a contraction [2]. We shall give a condition under which these limits coincide (as far as possible), and consider the corresponding condition for projective limits.