Continuous Linear Map Research Articles

A Kakutani–Mackey-like theorem

We give a partial extension of a Kakutani–Mackey theorem for quasi-complemented vector spaces. This can be applied in the representation theory of certain complemented (non-normed) topological algebras. The existence of continuous linear maps, in the context of quasi-complemented vector spaces, is a very important issue in their study. Relative to this, we prove that every Hausdorff quasi-complemented locally convex space has continuous linear maps, under which a certain quasi-complemented locally convex space turns to be pre-Hilbert.

RETRACTED ARTICLE: New Riesz representations of linear maps associated with certain boundary value problems and their applications

In this paper, we obtain new Riesz representations of continuous linear maps associated with certain boundary value problems in the set of all closed bounded convex non-empty subsets of any Banach space. As applications, the Riesz integral representation results are also given.

Linear maps on C⁎-algebras which are derivations or triple derivations at a point

We prove new results about generalized derivations on C⁎-algebras. By considering the triple product {a,b,c}=2−1(ab⁎c+cb⁎a), we introduce the study of linear maps which are triple derivations or triple homomorphisms at a point. We prove that a continuous linear map T on a unital C⁎-algebra is a generalized derivation whenever it is a triple derivation at the unit element. If we additionally assume T(1)=0, then T is a ⁎-derivation and a triple derivation. Furthermore, a continuous linear map on a unital C⁎-algebra which is a triple derivation at the unit element is a triple derivation. Similar conclusions are obtained for continuous linear maps which are derivations or triple derivations at zero.We also give an automatic continuity result, that is, we show that generalized derivations on a von Neumann algebra and linear maps on a von Neumann algebra which are derivations or triple derivations at zero are all continuous even if not assumed a priori to be so.

RETRACTED ARTICLE: A sharp Trudinger type inequality for harmonic functions and its applications

The present paper introduces a sharp Trudinger type inequality for harmonic functions based on the Cauchy-Riesz kernel function, which includes modified Poisson type kernel in a half plane considered by Xu et al. (Bound. Value Probl. 2013:262, 2013). As applications, we not only obtain Morrey representations of continuous linear maps for harmonic functions in the set of all closed bounded convex nonempty subsets of any Banach space, but also deduce the representation for set-valued maps and for scalar-valued maps of Dunford-Schwartz.

Almost n-Multiplicative Maps between Frechet Algebras

For the Fr'{e}chet algebras $(A, (p_k))$ and $(B, (q_k))$ and $n in mathbb{N}$, $ngeq 2$, a linear map $T:A rightarrow B$ is called textit{almost $n$-multiplicative}, with respect to $(p_k)$ and $(q_k)$, if there exists $varepsilongeq 0$ such that$$q_k(Ta_1a_2cdots a_n-Ta_1Ta_2cdots Ta_n)leq varepsilon p_k(a_1) p_k(a_2)cdots p_k(a_n),$$for each $kin mathbb{N}$ and $a_1, a_2, ldots, a_nin A$. The linear map $T$ is called textit{weakly almost $n$-multiplicative}, if there exists $varepsilongeq 0$ such that for every $kin mathbb{N}$ there exists $n(k)in mathbb{N}$ with$$q_k(Ta_1a_2cdots a_n-Ta_1Ta_2cdots Ta_n)leq varepsilon p_{n(k)}(a_1) p_{n(k)}(a_2)cdots p_{n(k)}(a_n),$$for each $k in mathbb{N}$ and $a_1, a_2, ldots, a_nin A$.The linear map $T$ is called $n$-multiplicative if$$Ta_{1}a_{2} cdots a_{n} = Ta_{1} Ta_{2} cdots Ta_{n},$$for every $a_{1}, a_{2},ldots, a_{n} in A$.In this paper, we investigate automatic continuity of (weakly) almost $n$-multiplicative maps between certain classes of Fr'{e}chet algebras, including Banach algebras. We show that if $(A, (p_k))$ is a Fr'{e}chet algebra and $T: A rightarrow mathbb{C}$ is a weakly almost $n$-multiplicative linear functional, then either $T$ is $n$-multiplicative, or it is continuous. Moreover, if $(A, (p_k))$ and $(B, (q_k))$ are Fr'{e}chet algebras and $T:A rightarrow B$ is a continuous linear map, then under certain conditions $T$ is weakly almost $n$-multiplicative for each $ngeq 2$. In particular, every continuous linear functional on $A$ is weakly almost $n$-multiplicative for each $ngeq 2$.

RETRACTED ARTICLE: New applications of the existence of solutions for equilibriumequations with Neumann type boundary condition

Using the existence of solutions for equilibrium equations with aNeumann type boundary condition as developed by Shi and Liao (J. Inequal. Appl.2015:363, 2015), we obtain the Rieszintegral representation for continuous linear maps associated with additiveset-valued maps with values in the set of all closed bounded convex non-emptysubsets of any Banach space, which are generalizations of integral representationsfor harmonic functions proved by Leng, Xu and Zhao (Comput. Math. Appl. 66:1-18,2013). We also deduce the Rieszintegral representation for set-valued maps, for the vector-valued maps ofDiestel-Uhl and for the scalar-valued maps of Dunford-Schwartz.

Continuous extension of functions from countable sets

We give a characterization of countable discrete subspace A of a topological space X such that there exists a (linear) continuous mapping φ:Cp⁎(A)→Cp(X) with φ(y)|A=y for every y∈Cp⁎(A). Using this characterization we answer two questions of A. Arhangel'skii. Moreover, we introduce the notion of well-covered subset of a topological space and prove that for well-covered functionally closed subset A of a topological space X there exists a linear continuous mapping φ:Cp(A)→Cp(X) with φ(y)|A=y for every y∈Cp(A).

ON THE EVENTUAL PERIODICITY OF PIECEWISE LINEAR CHAOTIC MAPS

We present a family of continuous piecewise linear maps of the unit interval into itself that are all chaotic in the sense of Li and Yorke [‘Period three implies chaos’, Amer. Math. Monthly82 (1975), 985–992] and for which almost every point (in the sense of Lebesgue) in the unit interval is an eventually periodic point of period $p,p\geq 3$, for a member of the family.

Spaces ℓ-dominated by I or [formula omitted

If X is compact metrizable and has finite fd-height then the unit interval, I, ℓ-dominates X, in other words, there is a continuous linear map of Cp(I) onto Cp(X). Related results are given for spaces ℓ-dominated by the reals.

Derivations on FCIN algebras

Let $\mathcal{L}$ be an algebra generated by the commuting independent nests, $\mathcal{M}$ is an ultra-weakly closed subalgebra of $\mathbf{B(H)}$ which contains $alg\mathcal{L}$ and $\phi$ is a norm continuous linear mapping from $alg\mathcal{L}$ into $\mathcal{M}$. In this paper we will show that a norm continuous linear derivable mapping at zero point from $Alg\mathcal{L}$ to $\mathcal{M}$ is a derivation.

Orthogonally additive polynomials on Banach function algebras

For a Banach function algebra A, we consider the problem of representing a continuous d-homogeneous polynomial P:A→X, where X is an arbitrary Banach space, that satisfies the property P(f+g)=P(f)+P(g) whenever f,g∈A are such that supp(f)∩supp(g)=∅. We show that such a polynomial can be represented as P(f)=T(fd)(f∈A) for some continuous linear map T:A→X for a variety of Banach function algebras such as the algebra of continuous functions C0(Ω) for any locally compact Hausdorff space Ω, the algebra of Lipschitz functions lipα(K) for any compact metric space K and α∈]0,1[, the Figà–Talamanca–Herz algebra Ap(G) for some locally compact groups G and p∈]1,+∞[, the algebras AC([a,b]) and BVC([a,b]) of absolutely continuous functions and of continuous functions of bounded variation on the interval [a,b]. In the case where A=Cn([a,b]), P can be represented as P(f)=∑T(n1,…,nd)(f(n1)⋯f(nd)), where the sum is taken over (n1,…,nd)∈Zd with 0≤n1≤…≤nd≤n, for appropriate continuous linear maps T(n1,…,nd):Cn−nd([a,b])→X.

Topological Conjugate Piecewise Linear Unimodal Mappings of an Interval Into Itself

Let f, g : [0, 1] → [0, 1] be a pair of continuous piecewise linear unimodal mappings and let f be defined as follows: f(x) = 2x for x ≤ 1/2 and f(x) = 2 − 2x for x > 1/2. Also let h : [0, 1] → [0, 1] be a piecewise continuously differentiable homeomorphism such that fh = hg. Then h is piecewise linear and the mapping f, together with the ascending (or descending) monotone part of g, completely determines g and h.

An Interesting Infinite Series and Its Implications to Operator Theory

The following is a discussion regarding a specific class of operators acting on the space of entire functions, denoted $H(\mathbb{C})$. A diagonal operator $D$ on $H(\mathbb{C})$ is defined to be a continuous linear map, sending $H(\mathbb{C})$ into $H(\mathbb{C})$, that has the monomials $z^n$ as its eigenvectors and $\{\lambda_n\}$ as the corresponding eigenvalues. A closed subspace $M$ is invariant for $D$ if $Df\in M$ for all $f\in M$. The study of invariant subspaces is a popular topic in modern operator theory. We observe that the closed linear span of the orbit, which we write $\overline{\mbox{span}}\{D^kf:k\geq0\}=\overline{\mbox{span}}\{\sum^{\infty}_{n=0}a_n\lambda_n^kz^n:k\geq0\}$, is the smallest closed invariant subspace for $D$ containing $f$. If every invariant subspace for a diagonal operator $D$ on $H(\mathbb{C})$ can be expressed as a closed linear span of some subset of the eigenvectors of $D$, we say that $D$ admits spectral synthesis on $H(\mathbb{C})$. Until recently, it was not known whether or not every diagonal operator on $H(\mathbb{C})$ admitted spectral synthesis. This article focuses on using techniques from calculus and linear algebra to construct a class of operators which fail spectral synthesis on $H(\mathbb{C})$. If the reader is not familiar with the operator theory definitions provided in the background, he or she can still appreciate the construction of an interesting infinite series relying on properties of logarithms, various convergence tests, and Cramer's Rule.

English

Let $\mathcal {H}$ be a complex Hilbert space, $A$ a positive operator with closed range in $\mathscr {B}(\mathcal {H})$ and $\mathscr {B}_{A}(\mathcal {H})$ the sub-algebra of $\mathscr {B}(\mathcal {H})$ of all \mbox {$A$-self}-adjoint operators. Assume $\phi \colon \mathscr {B}_{A}(\mathcal {H})$ onto itself is a linear continuous map. This paper shows that if $\phi $ preserves \mbox {$A$-unitary} operators such that $\phi (I)=P$ then $\psi $ defined by $\psi (T)=P\phi (PT)$ is a homomorphism or an anti-homomorphism and $\psi (T^{\sharp })=\psi (T)^{\sharp }$ for all $T \in \mathscr {B}_{A}(\mathcal {H})$, where $P=A^{+}A$ and $A^{+}$ is the Moore-Penrose inverse of $A$. A similar result is also true if $\phi $ preserves \mbox {$A$-quasi}-unitary operators in both directions such that there exists an operator $T$ satisfying $P\phi (T)=P$.

Pseudo-differential operator involving generalized Hankel–Clifford transformation

Pseudo-differential operators (p.d.os) involving generalized Hankel–Clifford transformation associated with the symbol [Formula: see text] whose derivatives satisfy certain growth condition are defined and the Zemanian type function spaces [Formula: see text] and [Formula: see text] are introduced. It is shown that p.d.o’s are continuous linear map of the space [Formula: see text] and [Formula: see text] into itself. Also an Integral representation of p.d.o is obtained.

Lyusternik--Graves Theorems for the Sum of a Lipschitz Function and a Set-valued Mapping

In a paper of 1950 Graves proved that for a function $f$ acting between Banach spaces and an interior point $\bar x$ in its domain, if there exists a continuous linear mapping $A$ which is surjective and the Lipschitz modulus of the difference $f-A$ at $\bar x$ is sufficiently small, then $f$ is (linearly) open at $\bar x$. This is an extension of the Banach open mapping principle from continuous linear mappings to Lipschitz functions. A closely related result was obtained earlier by Lyusternik for smooth functions. In this paper, we obtain Lyusternik--Graves theorems for mappings of the form $f+F$, where $f$ is a Lipschitz continuous function around $\bar x$ and $F$ is a set-valued mapping. Roughly, we give conditions under which the mapping $f+F$ is linearly open at $\bar x$ for $\bar y$ provided that for each element $A$ of a certain set of continuous linear operators the mapping $f(\bar{x}) +A(\cdot - \bar{x}) + F$ is linearly open at $\bar x$ for $\bar y$. In the case when $F$ is the zero mapping, as corollaries we obtain the theorem of Graves as well as open mapping theorems by Pourciau and Páles, and a constrained open mapping theorem by Cibulka and Fabian. From the general result we also obtain a nonsmooth inverse function theorem proved recently by Cibulka and Dontchev. Application to Nemytskii operators and a feasibility mapping in control are presented.

Modeling and Synthesis of Discrete Mappings of Linear Continuous Channel

Linear continuous channel discrete mappings in the form of metric finite-dimensional space operators are formalized. Its coupling with representation in the form of integral transforms on the basis of limiting process from series view is shown. It is shown that semi-digital operators of linear continuous channel discrete mappings on the basis of Pugachev canonical presentation are calculated by force of iterative procedure consistently conditioning basis functions on input and output channel in terms of recurrence input transformation.

Kneading with weights

We generalize Milnor–Thurston's kneading theory to the setting of piecewise continuous and monotone interval maps with a weight associated to each branch. We define a weighted kneading determinant {\cal D}(t) and establish combinatorially two kneading identities, one with the cutting invariant and one with the dynamical zeta function. For the pressure \log \rho_1 of the weighted system, playing the role of entropy, we prove that {\cal D}(t) is non-zero when |t|<1/\rho_1 and has a zero at 1/\rho_1 . Furthermore, our map is semi-conjugate to every map in an analytic family \mathcal S_t, 0<t<1/\rho_1 of piecewise linear maps with slopes proportional to the prescribed weights and defined on a Cantor set. When the original map extends to a continuous map f , the family \mathcal S_t converges as t\rightarrow 1/\rho_1 to a continuous piecewise linear interval map \tilde{f} . Furthermore, f is semi-conjugate to \tilde{f} and the two maps have the same pressure.

A unified existence theory for evolution equations and systems under nonlocal conditions

We investigate the effect of nonlocal conditions expressed by linear continuous mappings over the hypotheses which guarantee the existence of global mild solutions for functional-differential equations in a Banach space. A progressive transition from the Volterra integral operator associated to the Cauchy problem, to Fredholm type operators appears when the support of the nonlocal condition increases from zero to the entire interval of the problem. The results are extended to systems of equations in a such way that the system nonlinearities behave independently as much as possible and the support of the nonlocal condition may differ from one variable to another.

Bifurcation Structures in a Bimodal Piecewise Linear Map: Chaotic Dynamics

In this work, we investigate the bifurcation structure of the parameter space of a generic 1D continuous piecewise linear bimodal map focusing on the regions associated with chaotic attractors (cyclic chaotic intervals). The boundaries of these regions corresponding to chaotic attractors with different number of intervals are identified. The results are obtained analytically using the skew tent map and the map replacement technique.