Continuous Linear Transformations Research Articles

Partially continuous pretopological and topological operators for intuitionistic fuzzy sets

In this paper, pretopological and topological operators are introduced based on partially continuous linear transformations of the membership and non-membership functions for intuitionistic fuzzy sets. They turn out to be a generalization of the topological operators for intuitionistic fuzzy sets.On the other hand it is a generalization of the fuzzy set pretopological operators introduced by Wenzhong and Kimfung.

Continuous linear transformations on generalized fuzzy normed spaces

Continuous linear transformations on generalized fuzzy normed spaces

Euclidean-valued group cohomology is always reduced

Let [Formula: see text] be a topological field and [Formula: see text] a countable discrete group. Then for any linear [Formula: see text]-action on a finite-dimensional vector space over [Formula: see text], the groups of coboundaries in the inhomogeneous bar resolution are closed in all degrees, and hence the cohomology is reduced in all degrees. This can be deduced from a more general automatic-closure theorem for continuous linear transformations between inverse limits of finite-dimensional vector spaces.

Weak Multiplier Bimonoids

Based on the novel notion of `weakly counital fusion morphism', regular weak multiplier bimonoids in braided monoidal categories are introduced. They generalize weak multiplier bialgebras over fields and multiplier bimonoids in braided monoidal categories. Under some assumptions the so-called base object of a regular weak multiplier bimonoid is shown to carry a coseparable comonoid structure; hence to possess a monoidal category of bicomodules. In this case, appropriately defined modules over a regular weak multiplier bimonoid are proven to constitute a monoidal category with a strict monoidal forgetful type functor to the category of bicomodules over the base object. Braided monoidal categories considered include various categories of modules or graded modules, the category of complete bornological spaces, and the category of complex Hilbert spaces and continuous linear transformations.

SEMI-ASYMPTOTIC NON-EXPANSIVE ACTIONS OF SEMI-TOPOLOGICAL SEMIGROUPS

Abstract. In this paper we extend Takahashi’s fixed point theorem ondiscrete semigroups to general semi-topological semigroups. Next we de-fine the semi-asymptotic non-expansive action of semi-topological semi-groups to give a partial affirmative answer to an open problem raised byA.T-M. Lau. 1. IntroductionA (not necessarily linear) self-mapping T : E → E on a Banach space Eis called non-expansive if kT(x) − T(y)k ≤ kx − yk for all x,y ∈ E. In 1963R. DeMarr proved the following common fixed point theorem for commutingfamilies of non-expansive self-mappings [6].Theorem 1.1. For any non-empty compact convex subset K of a Banach spaceE each commuting family of non-expansive self mappings on K has a commonfixed point in K.This generalizes the celebrated Markov-Kakutani fixed point theorem (forthe case of Banach spaces) on commuting families of continuous linear trans-formations on Hausdorff topological vector spaces leaving certain nonemptycompact convex subset invariant. DeMarr’s theorem has been generalized inseveral directions by Belluce and Kirk [2,3], Takahashi [16], Mitchell [14], Lauand Holmes [7,8].DeMarr theorem suggests that the action of certain commutative semigrouphas a fixed point. It is then natural to seek the same type offixed point propertyfor the actions of more general semigroups.Let S be a semi-topological semigroup, that is a semigroup with a Hausdorfftopology with separately continuous multiplication. We say that S is rightreversible if it has finite intersection property for closed left ideals. An actionof S on a topological space E is a mapping (s,x) → s(x) from S × E into

Spaces of entire functions represented by vector valued Dirichlet series of two complex variables

Let Y be the space of all entire functions f : ℂ2 → E defined by the vector valued Dirichlet series, where E is a complex Banach algebra with the unit element. We study various topologies defined on the space Y and characterize continuous linear transformations on Y.

How and Why to Solve the Operator Equation AX −XB = Y

The entities A, B, X, Y in the title are operators, by which we mean either linear transformations on a finite-dimensional vector space (matrices) or bounded (= continuous) linear transformations on a Banach space. (All scalars will be complex numbers.) The definitions and statements below are valid in both the finite-dimensional and the infinite-dimensional cases, unless the contrary is stated. 1991 Mathematics Subject Classification 15A24, 47A10, 47A62, 47B47, 47B49, 65F15, 65F30.

Decompositions of Differentiable Semigroups

A differentiable semigroup is a topological semigroup $(S, * )$ in which $S$ is a differentiable manifold based on a Banach space and the associative multiplication function * is continuously differentiable. If $e$ is an idempotent element of such a semigroup we show that there is an open set $U$ containing $e$ so that there is a ${C^1}$ retraction $\Phi$ of $U$ into the set of idempotents of $S$ so that $\Phi (x)\Phi (y) = \Phi (x)$ for $x$ and $y$ in $U$ and $x\Phi (x)$ is in the maximal subgroup of $S$ determined by $\Phi (x)$ for each $x$ in $U$. This leads to a natural decomposition of $S$ near $e$ into the union of a collection of mutually disjoint and mutually homeomorphic local differentiable subsemigroups whose intersections with $U$ are the point inverses under $\Phi$. In case $S$ is the semigroup under composition of continuous linear transformations on a Banach space, in the case of a nontrivial idempotent $e$, the existence of $\Phi$ implies that operators near an $e$ have nontrivial invariant subspaces. A dual right handed result holds.

Decompositions of differentiable semigroups

A differentiable semigroup is a topological semigroup ( S , ∗ ) (S, * ) in which S S is a differentiable manifold based on a Banach space and the associative multiplication function * is continuously differentiable. If e e is an idempotent element of such a semigroup we show that there is an open set U U containing e e so that there is a C 1 {C^1} retraction Φ \Phi of U U into the set of idempotents of S S so that Φ ( x ) Φ ( y ) = Φ ( x ) \Phi (x)\Phi (y) = \Phi (x) for x x and y y in U U and x Φ ( x ) x\Phi (x) is in the maximal subgroup of S S determined by Φ ( x ) \Phi (x) for each x x in U U . This leads to a natural decomposition of S S near e e into the union of a collection of mutually disjoint and mutually homeomorphic local differentiable subsemigroups whose intersections with U U are the point inverses under Φ \Phi . In case S S is the semigroup under composition of continuous linear transformations on a Banach space, in the case of a nontrivial idempotent e e , the existence of Φ \Phi implies that operators near an e e have nontrivial invariant subspaces. A dual right handed result holds.

Non‐linear principal component analysis of maximum expiratory flow‐volume curves

AbstractThe ability to classify maximum expiratory flow‐volume (MEFV) curves has been explored for several types of principal component analysis (PCA). The PCAs differed in their methods of transformation of the data. These included discrete non‐linear transformation with or without additional restrictions, or continuous linear (classical PCA) and piecewise linear transformation using first degree B‐splines.Two hundred and fifty curves of males between 21 and 30 years of age were involved in these analyses. Although no real pathology occurs among these males, still a clearly differentiated classification of curve types was found. The results were passively gauged with respect to smoking habits and respiratory symptoms. The analysis using piecewise linear transformations differentiated best between people with and without respiratory symptoms, and between smokers and non‐smokers.In general pathology is rare and to be found at the lower tail of a lung function measurement. As the piecewise linear transformations resulted in a more pronounced negative skewness of the relevant variables, the gain of this transformation with respect to the discrete analyses is to be found in the ability to use the information content within a category and with respect to the continuous linear analysis the ability to pronounce this information.

Continuous Linear Transformations of n-Convex Functions

Continuous Linear Transformations of n-Convex Functions

An Operator Without Invariant Subspaces on a Nuclear Frechet Space

Let X be a locally convex topological complex vector space. Throughout this paper, the term operator on X will mean a continuous linear transformation mapping X into itself, and the term invariant subspace for an operator T on X will mean a closed subspace M of X such that M F {O}, M P X and TM C M. It is still an open problem whether or not every operator on an infinite dimensional separable Hilbert space has an invariant subspace. On the other hand P. Enflo [8] constructed an operator without invariant subspaces on a (nonreflexive) Banach space. In this paper we construct an operator T on an infinite dimensional nuclear Frechet space F, such that T has no invariant subspaces, and T2 has an invariant subspace. Moreover, the space F is a nuclear countably Hilbert space, in the sense of Gelfand and Vilenkin [12]. (That is, F is the intersection of a decreasing sequence of Hilbert spaces Hn n = 1, 2,.. ., such that for every n, the imbedding of Hn+ 1 into Hn is a nuclear mapping, and the topology on F is defined by the sequence of Hn norms.) The elements of F are complex sequences c = (ci)'j=0 and the operator T is the associated backward shift; that is

A double commutant theorem for conjugate selfadjoint operators

Let A be a bounded linear transformation on the complex separable Hilbert space H. If there is a conjugation Q on H such that A = QA*Q, we say that A is conjugate selfadjoint. In this note we examine commutativity properties of conjugate selfadjoint operators which possess cyclic vectors. 1. Preliminaries. Let H be a complex Hilbert space with a countably infinite basis, and let (f, g) denote the inner product of two vectors in H. By H ff H we mean the Hilbert space of vectorsf E3 g having inner product (fl D gI,f2 E 92) = (fl,f2) + (g1, g2). A linear manifold is a subset which is closed under vector addition and under multiplication by complex numbers. A subspace is a linear manifold which is closed in the norm topology induced by the inner product. The smallest subspace containing the set U ??of fn) will be denoted by V{fn). If F is a subset of Hilbert space, clos F will denote the closure of F in the norm topology and F' = { gl( g, f) = 0, f E F). Whenever A is a continuous linear transformation on H, its graph F(A) = { fD Af If E H) is a subspace of H ff H. One can easily verify that F(A)= { A*(-f) Ef fIf E H), where A* is the adjoint of The germinal idea of representing a linear transformation through its associated graph subspace originated in the work of J. von Neumann [1]. Hereafter we shall refer to a continuous (or bounded) linear transformation as an operator. If A and B are operators on H, we define (A ff B)(f E g) = Af ff Bg. The set of all operators on H that commute with A is called the commutant of This algebra will be denoted by (A)'. The double commutant of A, designated {A}, is the algebra of all operators on H that commute with every member of (A }'. It is self evident that (A)' is abelian if and only if (A)' = (A A. A transformation Q on H is said to be a conjugation if Q2 = I and (Qf, Qg) = (g, f) for every f and g in H. Intuitively speaking, Q replaces each element of H by its conjugate with respect to the real subspace consisting of all fixed points of Q [3, p. 357]. If there is a conjugation Q such that A = QA*Q, we shall call A conjugate selfadjoint. The well-known Hankel operators [4] belong to this class. Received by the editors July 18, 1980 and, in revised form, January 12, 1981. 1980 Mathematics Subject Classification. Primary 47A05; Secondary 46J99.

Commutation properties of automorphism groups and mappings of von Neumann algebras

We prove a commutation theorem for point ultraweakly continuous oneparameter groups of automorphisms of von Neumann algebras. If α t , is such a group in Aut( R ) for a von Neumann algebra R , we show the equivalence of the following three conditions on an ultraweakly continuous linear transformation μ: R → R : (a) μ commutes weakly with the infinitesimal generator for α t ; (b) μ ° α t = α t ° μ, t ∈ R; and ( c) μ leaves invariant each of the spectral subspaces associated with α t . A simple condition which is applicable when μ is an automorphism is pointed out.

Baer Endomorphism Rings and Closure Operators

A Baer ring is a ring in which every right (and left) annihilator ideal is generated by an idempotent. Generalizing quite naturally from the fact that the endomorphism ring of a vector space is a Baer ring, Wolfson [5; 6] investigated questions such as when the endomorphism ring of a free module is a Baer ring, and when the ring of continuous linear transformations on a pair of dual vector spaces is a Baer ring. A further generalization was made in [7], where the question of when the endomorphism ring of a torsion-free module over a semiprime left Goldie ring is a Baer ring was treated.

Norm Convergence of Tn

Throughout this paper X will denote a complex Banach space and all operators T will be assumed to be continuous linear transformations from X into X. If T is an operator then ┘(T), γ(T), and R(T) will denote the spectrum of T, the spectral radius of T, and range of T, respectively. This paper contains necessary and sufficient conditions for the (norm) convergence of {Tn} when T is an operator on X.

Linear Transformations in the Operational Calculus

A new development of linear transformations in the one-sided operational calculus is presented. The setting for this is a (noncommutative) ring $\mathcal{R}$ of continuous linear transformations on a familiar test function space from distribution theory. Included in $\mathcal{R}$, both algebraically and topologically, are all the right-sided distributions, all the traditional transformations of the operational calculus, such as the exponential shifts, the dilatations and the algebraic derivative, all translations and multiplications by infinitely differentiable functions, and many other new transformations. The development parallels that given by E. Gesztelyi for linear transformations of the Mikusinski operator field, but is cast in a simpler and more flexible setting. The main tool of the investigation is a representation theorem of the type introduced by V. Dolezal and the results are primarily theorems concerning commutativity properties in $\mathcal{R}$. It is shown that a linear transformation (i) commutes with all translations iff it is a distribution (ii) commutes with differentiation iff it is a distribution (iii) commutes with the algebraic derivative if it is a multiplier and (iv) is a distribution and commutes with every dilatation if it is a number. Because of the latter, it becomes reasonable to define a Laplace transform in $\mathcal{R}$ which encompasses (for right-sided distributions) that givenby D. Price. Some results on inversion in $\mathcal{R}$ are given and a number of unsettled problems, perhaps amenable to solutions in this setting, are mentioned.

Reflective and coreflective hulls in the category of locally convex spaces

A new method of characterizing epi-reflective and mono-coreflective hulls in certain subcategories of the category HAUS of Hausdorff spaces and continuous maps is investigated. For comparison, the basic results are first presented briefly for HAUS. The remainder of the paper is concerned with the category LCS of Hausdorff locally convex topological vector spaces and continuous linear transformations. In LCS, reflective means closed under the taking of products and closed subspaces, and coreflective has a similar interpretation involving direct sums and quotients. Theorems analogous to those for HAUS are proved and then applied to study the distribution of reflective and coreflective subcategories in LCS and the hulls of certain classes of spaces. Subcategories of LCS that are both reflective and coreflective are also investigated, and some consequences of results are the representation theorems that every space is a quotient of a complete space, a closed subspace of a barrelled space, and, assuming the nonexistence of a strongly inaccessible cardinal, a closed subspace of a bornological space.

Matrix representations for linear transformations on series analytic in the unit disc

Let S be the space of all complex sequences A such that if £ is a complex number and | z | < 1 then ^ Anzn converges. We present three characterizations of the linear transformations from S to S which have matrix representations. We also characterize the linear transformations from S to the bounded sequences (or to the convergent sequences) which have matrix representations. The characterizations are in terms of natural topologies for the spaces. These results are a blend of Kothe and Toeplitz' much quoted study [3] of complex sequence spaces, Haplanov's beautiful characterization [1] of those matrices which transform S to S, and some rather natural norms for S which have been used by V. Ganapathy Iyer [2] in his study of entire functions. Kothe and Toeplitz study complex sequence spaces, matrices, and linear transformations having a kind of continuity which is independent of norms. A space is said to be normal provided that if x is in the space and | yn | ^ | xn |, n = 0,1, , then y is also in the space. Kothe and Toeplitz show that a continuous linear transformation from a normal space to a normal space has a matrix representation, and conversely, provided that each space contains all the finite sequences. Our space S is normal and the space of bounded sequences is also normal. The continuity criteria used in our theorems (statement (2) in each) are special cases of the continuity condition of Kothe and Toeplitz. It follows from their work that the existence of a matrix for a linear transformation L is necessary and sufficient for L to have analytic continuity (see definition below). Given a matrix transformation from S to S and a norm Nr (0 < r < 1: if A is in S, Nr(A) = Σ*U \AP τp) for S, Haplanov's theorem provides another such norm NB such that the transformation is continuous from the normed linear space {S, Nr} to {S, NR}. Finally, to complete Theorem 1, each linear transformation which is continuous relative to some such pair of norms is represented by a matrix, even though S is complete with respect to neither of the norms. Our second theorem is like the first: the transformations are from S to the bounded sequences (or convergent sequences).

Nonlinear prediction of generalized random processes

Let ( D) be the Schwartz space of infinitely differentiable scalar functions, ϑ, on the real line with compact supports, σ( ϑ), and ( Ω, Σ, P) be a fixed complete probability space. Let X: (D) → L p(Ω, Σ, P) , with 1 < p < ∞, be a generalized random process, i.e., a continuous linear transformation, T = (−∞, t), and β T = σ( X( ϑ) : σ( ϑ) ⊂ T), the σ-field generated by the random variables. Let { ε} i = 1 ∞ ⊂ (D) be such that σ( ε) ⊄ T, i ⩾ 1 . Then relative to the p-th moment of the error as the criterion of optimality, there exists a weak generalized random process (i.e., a closed linear transformation) Y:DY→L p(Ω,β T,P) , where D ( Y) is dense in ( D), such that ‖ Y( ε i) − X( ε i)‖ p is a minimum for each i ⩾ 1.