Continuous Linear Operators Research Articles

An Application of the Browder-Minty Theorem in a Problem of Partial Differential Equations

In this work, we will show existence of weak solution for a semilinear elliptic problem using as main tool the Browder-Minty Theorem. First, we will make a brief introduction about basic theory of the Sobolev Spaces to support our study and provide sufficient tools for the development of our work. Then we will take a quick approach on the Browder-Minty Theorem and use this result to show the existence of at least one weak solution to an elliptic Partial Differential Equations (PDE) problem whose nonlinearity, denoted by f, is a known function. For this, in addition to the already mentioned results, we will also use as study tools: Embedding Sobolev Theorems, Linear Continuous Operators Theory, Poincaré Inequality and Hölder Inequality.

Reiterative Distributional Chaos on Banach Spaces

If we change the upper and lower densities in the definition of distributional chaos of a continuous linear operator on a Banach space [Formula: see text] by the Banach upper and Banach lower densities, respectively, we obtain Li–Yorke chaos. Motivated by this, we introduce the notions of reiterative distributional chaos of types [Formula: see text], [Formula: see text] and [Formula: see text] for continuous linear operators on Banach spaces, which are characterized in terms of the existence of an irregular vector with additional properties. Moreover, we study its relations with other dynamical properties and present the conditions for the existence of a vector subspace [Formula: see text] of [Formula: see text], such that every nonzero vector in [Formula: see text] is both irregular for [Formula: see text] and distributionally near zero for [Formula: see text].

Closed ideals of operators acting on some families of sequence spaces

We study the lattice of closed ideals in the algebra of continuous linear operators acting on pth Tandori and \(p'\)th Cesàro sequence spaces, \(1\leqslant p<\infty \), which we show are isomorphic to the classical sequence spaces \((\oplus _{n=1}^\infty \ell _\infty ^n)_p\) and \((\oplus _{n=1}^\infty \ell _1^n)_{p'}\), respectively. We also show that Tandori sequence spaces are complemented in certain Lorentz sequence spaces, and that the lattice of closed ideals for certain other Lorentz and Garling sequence spaces has infinite cardinality.

Sparsity of solutions for variational inverse problems with finite-dimensional data

In this paper we characterize sparse solutions for variational problems of the form min _{uin X} phi (u) + F(mathcal {A}u), where X is a locally convex space, mathcal {A} is a linear continuous operator that maps into a finite dimensional Hilbert space and phi is a seminorm. More precisely, we prove that there exists a minimizer that is “sparse” in the sense that it is represented as a linear combination of the extremal points of the unit ball associated with the regularizer phi (possibly translated by an element in the null space of phi ). We apply this result to relevant regularizers such as the total variation seminorm and the Radon norm of a scalar linear differential operator. In the first example, we provide a theoretical justification of the so-called staircase effect and in the second one, we recover the result in Unser et al. (SIAM Rev 59(4):769–793, 2017) under weaker hypotheses.

Convex Stone-Weierstrass theorems and invariant convex sets

A \textit {convex polynomial} is a convex combination of the monomials $\{1, x, x^2, \ldots \}$. This paper establishes that the convex polynomials on $\mathbb R$ are dense in $L^p(\mu )$ and weak$^*$ dense in $L^\infty (\mu )$ whenever $\mu $ is a compactly supported regular Borel measure on $\mathbb {R}$ and $\mu ([-1,\infty )) = 0$. It is also shown that the convex polynomials are norm dense in $C(K)$ precisely when $K \cap [-1, \infty ) = \emptyset $, where $K$ is a compact subset of the real line. Moreover, the closure of the convex polynomials on $[-1,b]$ is shown to be the functions that have a convex power series representation. A continuous linear operator $T$ on a locally convex space $X$ is \textit {convex-cyclic} if there is a vector $x \in X$ such that the convex hull of the orbit of $x$ is dense in $X$. The previous results are used to characterize which multiplication operators on various real Banach spaces are convex-cyclic. Also, it is shown for certain multiplication operators that every nonempty closed invariant convex set is a closed invariant subspace.

Some Expansion of Fuzzy Paranormal Operators

Let be a Fuzzy Hilbert space over the fields of and FB( ) is the set of all fuzzy continuous linear operator on .In this paper we introduce the expansion of different fuzzy paranormal operators like n- fuzzy paranormal operator, *- fuzzy paranormal operator and nth -fuzzy paranormal operator, which all are developed from paranormal operators and their characteristics. The study resulted the properties of an n- fuzzy paranormal operator, * fuzzy paranormal operator and nth -fuzzy paranormal operator and their relationship between them. To investigate the nature of these operators, all it needs the nature of the n- fuzzy paranormal operator.

Strongly compatible generators of groups on Fréchet spaces

We consider the linear Cauchy problem(1){ut=a(D)u,t∈Ru(0)=u0, where a(D):X→X is a continuous linear operator on a Fréchet space X. By imposing a condition (which is neither stronger nor weaker than the equicontinuity of the powers of a(D)), we present the necessary and sufficient conditions for the generation of a uniformly continuous group on X, which provides the unique solution of (1). In addition, for every pseudodifferential operator a(D) with constant coefficients defined on FLloc2, which is a Fréchet space of distributions, we also provide the necessary and sufficient conditions such that the restriction {eta(D)}t⩾0 is a well defined semigroup on L2 and E′. We conclude that the heat equation solution on FLloc2 for all t∈R extends the standard solution on Hilbert spaces for t⩾0.

Continuous linear operators on Orlicz-Bochner spaces

Abstract Let (Ω, Σ, μ) be a complete σ-finite measure space, φ a Young function and X and Y be Banach spaces. Let Lφ(X) denote the corresponding Orlicz-Bochner space and $\begin{array}{} \displaystyle \mathcal T^\wedge_\varphi \end{array}$ denote the finest Lebesgue topology on Lφ(X). We examine different classes of ( $\begin{array}{} \displaystyle \mathcal T^\wedge_\varphi \end{array}$, ∥ ⋅ ∥Y)-continuous linear operators T : Lφ(X) → Y: weakly compact operators, order-weakly compact operators, weakly completely continuous operators, completely continuous operators and compact operators. The relationships among these classes of operators are established.

The invariant subspace problem for the space of smooth functions on the real line

We construct a continuous linear operator acting on the space of smooth functions on the real line without non-trivial invariant subspaces. This is a first example of such an operator acting on a Fréchet space without a continuous norm. The construction is based on the ideas due to C. Read who constructed a continuous operator without non-trivial invariant subspaces on the Banach space ℓ1.

A remark on a paper of P. B. Djakov and M. S. Ramanujan

Let l be a Banach sequence space with a monotone norm in which the canonical system (e_{n}) is an unconditional basis. We show that if there exists a continuous linear unbounded operator between l-K\{o}the spaces, then there exists a continuous unbounded quasi-diagonal operator between them. Using this result, we study in terms of corresponding K\{o}the matrices when every continuous linear operator between l-K\{o}the spaces is bounded. As an application, we observe that the existence of an unbounded operator between l-K\{o}the spaces, under a splitting condition, causes the existence of a common basic subspace.

On coincidence results for summing multilinear operators: interpolation, $$\ell _1$$-spaces and cotype

Grothendieck’s theorem asserts that every continuous linear operator from $$\ell _1$$ to $$\ell _2$$ is absolutely (1, 1)-summing. This kind of result is commonly called coincidence result. In this paper we investigate coincidence results in the multilinear setting, showing how the cotype of the spaces involved affect such results. The special role played by $$\ell _1$$ spaces is also investigated with relation to interpolation of tensor products. In particular, an open problem on the interpolation of m injective tensor products is solved.

A Note on Supercyclic Operators in Locally Convex Spaces

We treat some questions related to supercyclicity of continuous linear operators when acting in locally convex spaces. We extend results of Ansari and Bourdon and consider doubly power bounded operators in this general setting. Some examples are given.

Order spectrum of the Cesàro operator in Banach lattice sequence spaces

The discrete Cesaro operator C acts continuously in various classical Banach sequence spaces within $$ {\mathbb {C}}^{{\mathbb {N}}}.$$ For the coordinatewise order, many such sequence spaces X are also complex Banach lattices [eg. $$c_0, \ell ^p $$ for $$ 1 < p \le \infty , $$ and $$ {{\text {ces}}}(p)$$ for $$ p \in \{ 0 \} \cup ( 1, \infty )$$]. In such Banach lattice sequence spaces, C is always a positive operator. Hence, its order spectrum is well defined within the Banach algebra of all regular operators on X. The purpose of this note is to show, for every X belonging to the above list of Banach lattice sequence spaces, that the order spectrum $$ \sigma _\mathrm{o} (C)$$ of Ccoincides with its usual spectrum $$ \sigma ( C)$$ when C is considered as a continuous linear operator on the Banach space X.

Continuity of pseudodifferential operators on mixed-norm Lebesgue spaces

Mixed-norm Lebesgue spaces found their place in the study of some questions in the theory of partial differential equations, as it can be seen from recent interest in continuity of certain classes of pseudodifferential operators on these spaces. We present a general framework for dealing with continuity of linear operators on these spaces. This allows us to prove the boundedness of a large class of pseudodifferential operators and also the boundedness of integral operators on mixed-norm Lebesgue spaces. In some cases, the generalisations to mixed-norm Sobolev spaces are obtained as well, together with applications to some interpolation and compactness results.

Implicit Function Theorem. Part II

Summary In this article, we formalize differentiability of implicit function theorem in the Mizar system [3], [1]. In the first half section, properties of Lipschitz continuous linear operators are discussed. Some norm properties of a direct sum decomposition of Lipschitz continuous linear operator are mentioned here. In the last half section, differentiability of implicit function in implicit function theorem is formalized. The existence and uniqueness of implicit function in [6] is cited. We referred to [10], [11], and [2] in the formalization.

Quasinormable $$\varvec{C_0}$$-Groups and Translation-Invariant Fréchet Spaces of Type $$\varvec{\mathcal {D}_E}$$

Let E be a locally convex Hausdorff space satisfying the convex compact property and let $$(T_x)_{x \in {\mathbb {R}}^d}$$ be a locally equicontinuous $$C_0$$-group of linear continuous operators on E. In this article, we show that if E is quasinormable, then the space of smooth vectors in E associated to $$(T_x)_{x \in {\mathbb {R}}^d}$$ is also quasinormable. In particular, we obtain that the space of smooth vectors associated to a $$C_0$$-group on a Banach space is always quasinormable. As an application, we show that the translation-invariant Frechet spaces of smooth functions of type $$\mathcal {D}_E$$ (Dimovski et al. in Monatsh Math 177:495–515, 2015) are quasinormable, thereby settling the question posed in [8, Remark 7]. Furthermore, we show that $$\mathcal {D}_E$$ is not Montel if E is a solid translation-invariant Banach space of distributions (Feichtinger and Grochenig in J Funct Anal 86:307–340, 1989). This answers the question posed in [8, Remark 6] for the class of solid translation-invariant Banach spaces of distributions.

Countable tightness and k-property of free topological vector spaces

The free topological vector space V(X) over a Tychonoff space X is a pair consisting of a topological vector space V(X) and a continuous mapping i=iX:X→V(X) such that every continuous mapping f from X to a topological vector space E gives rise to a unique continuous linear operator f‾:V(X)→E with f=f‾∘i. In this paper, the k-property and countable tightness of free topological vector space over some generalized metric spaces are studied. We mainly discuss the characterizations of a space X such that V(X) or the fourth level of V(X) is a k-space or is of countable tightness, respectively.

A set-valued Lagrange theorem based on a process for convex vector programming

ABSTRACTIn this paper, we present a new set-valued Lagrange multiplier theorem for constrained convex set-valued optimization problems. We introduce the novel concept of Lagrange process. This concept is a natural extension of the classical concept of Lagrange multiplier where the conventional notion of linear continuous operator is replaced by the concept of closed convex process, its set-valued analogue. The behaviour of this new Lagrange multiplier based on a process is shown to be particularly appropriate for some types of proper minimal points and, in general, when it has a bounded base.

Extension and integral representation of the finite Hilbert transform in rearrangement invariant spaces

The finite Hilbert transform T is a classical (singular) kernel operator which is continuous in every rearrangement invariant space X over (–1, 1) having non-trivial Boyd indices. For X = Lp , 1 < p < ∞, this operator has been intensively investigated since the 1940’s (also under the guise of the “airfoil equation”). Recently, the extension and inversion of T : X → X for more general X has been studied in [6], where it is shown that there exists a larger space [T, X], optimal in a well defined sense, which contains X continuously and such that T can be extended to a continuous linear operator T : [T, X] → X. The purpose of this paper is to continue this investigation of T via a consideration of the X-valued vector measure mX : A.→ induced by T and its associated integration operator . In particular, we present integral representations of T: X →X based on the L 1-space of mX and other related spaces of integrable functions.

Strong Factorizations of Operators with Applications to Fourier and Cesàro Transforms

Consider two continuous linear operators T:X1(μ)→Y1(ν) and S:X2(μ)→Y2(ν) between Banach function spaces related to different σ-finite measures μ and ν. By means of weighted norm inequalities we characterize when T can be strongly factored through S, that is, when there exist functions g and h such that T(f)=gS(hf) for all f∈X1(μ). For the case of spaces with Schauder basis, our characterization can be improved, as we show when S is, for instance, the Fourier or Cesàro operator. Our aim is to study the case where the map T is besides injective. Then we say that it is a representing operator—in the sense that it allows us to represent each element of the Banach function space X(μ) by a sequence of generalized Fourier coefficients—providing a complete characterization of these maps in terms of weighted norm inequalities. We also provide some examples and applications involving recent results on the Hausdorff–Young and the Hardy–Littlewood inequalities for operators on weighted Banach function spaces.