Banach Algebra Of Continuous Functions Research Articles

On generalization of Petryshyn's fixed point theorem and its application to the product of $ n $-nonlinear integral equations

<abstract><p>Regarding the Hausdorff measure of noncompactness, we provide and demonstrate a generalization of Petryshyn's fixed point theorem in Banach algebras. Comparing this theorem to Schauder and Darbo's fixed point theorems, we can skip demonstrating closed, convex and compactness properties of the investigated operators. We employ our fixed point theorem to provide the existence findings for the product of $ n $-nonlinear integral equations in the Banach algebra of continuous functions $ C(I_a) $, which is a generalization of various types of integral equations in the literature. Lastly, a few specific instances and informative examples are provided. Our findings can successfully be extended to several Banach algebras, including $ AC, C^1 $ or $ BV $-spaces.</p></abstract>

2-Local isometries between Banach algebras of continuous functions with involution

We give a description of 2-local real isometries between $$C(X,\tau )$$ and $$C(Y,\eta )$$ where X and Y are compact Hausdorff spaces, X is also first countable and $$\tau $$ and $$\eta $$ are topological involutions on X and Y, respectively. In particular, we show that every 2-local real isometry T from $$C(X,\tau )$$ to $$ C(Y,\eta )$$ is a surjective real linear isometry whenever X is also separable.

Applications of Measure of Noncompactness and Darbo’s Fixed Point Theorem to Nonlinear Integral Equations in Banach Spaces

ABSTRACTWe prove a theorem on the existence of solutions of some nonlinear functional integral equations in the Banach algebra of continuous functions on the interval [0,a]. Then we consider a nonlinear integral equation of fractional order and give some sufficient conditions for existence of solutions of this equation. We use fixed point theorems associated with the measure of noncompactness as the main tool. Our existence results include several results obtained in previous studies. Finally we present some examples which show that our results are applicable.

On the Solutions of a Class of Nonlinear Integral Equations in the Banach Algebra of the Continuous Functions and Some Examples

AbstractIn this paper, we study the existence of the solutions of a class of functional integral equations which contain a lot of classical nonlinear integral equations as special cases. We consider the solvability of the equations in the Banach algebra of continuous functions on a closed and bounded interval. The main tools here are the measure of noncompactness and the suitable fixed point theorem for the product of two operators in the Banach algebra.

Disjointness preserving linear operators between Banach algebras of vector-valued functions

We present vector-valued versions of two theorems due to A. Jimenez-Vargas, by showing that, if B(X,E) and B(Y,F) are certain vector- valued Banach algebras of continuous functions and T : B(X,E) ! B(Y,F) is a separating linear operator, then b T :

Univalent functions in the Banach algebra of continuous functions

In this paper, we investigate several interesting properties of a composition operator defined on the open unit ball B 0 of the Banach algebra C(T). We also consider the Noshiro-Warschawski theorem in the Banach algebra of continuous functions.MSC:30C45, 46J10.

Multiplicative spectrum of ultrametric Banach algebras of continuous functions

Let K be an ultrametric complete field and let E be an ultrametric space. Let A be the Banach K-algebra of bounded continuous functions from E to K and let B be the Banach K-algebra of bounded uniformly continuous functions from E to K. Maximal ideals and continuous multiplicative semi-norms on A (resp. on B) are studied by defining relations of stickiness and contiguousness on ultrafilters that are equivalence relations. So, the maximal spectrum of A (resp. of B) is in bijection with the set of equivalence classes with respect to stickiness (resp. to contiguousness). Every prime ideal of A or B is included in a unique maximal ideal and every prime closed ideal of A (resp. of B) is a maximal ideal, hence every continuous multiplicative semi-norms on A (resp. on B) has a kernel that is a maximal ideal. If K is locally compact, every maximal ideal of A (resp. of B) is of codimension 1. Every maximal ideal of A or B is the kernel of a unique continuous multiplicative semi-norm and every continuous multiplicative semi-norm is defined as the limit along an ultrafilter on E. Consequently, on A as on B the set of continuous multiplicative semi-norms defined by points of E is dense in the whole set of all continuous multiplicative semi-norms. Ultrafilters show bijections between the set of continuous multiplicative semi-norms of A, Max ( A ) and the Banaschewski compactification of E which is homeomorphic to the topological space of continuous multiplicative semi-norms. The Shilov boundary of A (resp. B) is equal to the whole set of continuous multiplicative semi-norms.

Orthonormal q-bases associated to some continuous linear operators on (ℤ p ,K)

Let K be a complete valued field, extension of the p-adic field ℚp. Let q be a unit of ℤp, q not a root of unity and Vq be the closure of the set {qn/n ∈ ℤ} and let Open image in new window (Vq,K) be the Banach algebra of the continuous functions from Vq to K. Let τq be the operator on Open image in new window (Vq,K) defined by τq(f)(x) = f(qx), f ∈ Open image in new window (Vq,K). In her article [10] (see also [11]), A. Verdoodt constructs orthonormal bases associated to specific operators that commute with τq. Let q ∈ K such that |q − 1| < 1, q not a root of unity. Let Open image in new window (ℤp,K) be the Banach algebra of continuous functions from ℤp to K. We give here, as in umbral calculus, a bijective correspondence between a class of orthonormal bases of Open image in new window (ℤp, K) and a class of linear continuous operators which commute with the translation operator τ1: τ1(f)(x) = f(x + 1).

Continuity versus boundedness of the spectral factorization mapping

This paper characterizes the Banach algebras of continuous functions on which the spectral factorization mapping S is continuous or bounded. It is shown that S is continuous if and only if the Riesz projection is bounded on the algebra, and that S is bounded only if the algebra is isomorphic to the algebra of continuous functions. Conse- quently, S can never be both continuous and bounded, on any algebra under consideration. 1. Introduction. The operation by which a functionf on the unit circle is written as f( ) = f+( )f+( ) for all 2 T :=fz2 C :jzj = 1g, with an outer functionf+, is known as spectral factorization. This operation arises in many dierent

Koebe's and Bieberbach's Inequalities in the Banach Algebra of Continuous Functions

Throughout the paper,C(T) denotes the Banach algebra, with sup norm, of continuous complex-valued functions defined on a compact metric spaceT. ByB(f;r) we denote an open ball inC(T) centered atf∈C(T) of radiusr>0, and byU(a;r) the open ball in the complex planeCata∈Cof radiusr>0 (a neighborhood ofa). In compliance with usual notations concerningunivalent functions, bySwe denote the class of functionsf(z)=z+∑∞p=1apzpunivalentin the unit diskE=U(0;1). By means offractional derivative, interesting generalizations of certain inequalities are obtained. The object of the present article is to generalize the classSin the algebraC(T) and to prove a generalization ofKoebe'sandBieberbach's inequalities:Iff∈Sandz∈E,then[formula]

Symbolic Calculus on a Banach Algebra of Continuous Functions

The symbolic calculus on Banach algebras of continuous functions and related spaces is studied. In particular, functions operating on the real part of the algebra are considered. The main tool in this paper is an ultraseparation argument. As a consequence it is shown, for example, that t p on [0, 1) for any p with 0 < p < 1 does not operate on the real part of a Banach function algebra on a compact Hausdorff space unless the algebra contains every continuous function.

Universal measurable extension and the arens extension of a Banach algebra of continuous functions

Universal measurable extension and the arens extension of a Banach algebra of continuous functions

Compact weighted endomorphisms of $C(X)$

A weighted endomorphism of an algebra is defined to be a linear operator which is equal to an endomorphism followed by a multiplier. Thus, if B is an algebra with unit, then T is a weighted endomorphism of B if there is an element u in B and an endomorphism S of B such that T: f u Sf. The problem considered in this note is to characterize compact weighted endomorphisms of C(X), the Banach algebra of continuous functions on a compact Hausdorff space X. If X is a compact Hausdorff space, then it is easy to see that every nonzero endomorphism of C(X) has the form Sf = f o qp for some continuous function 9p from X into X. Consequently, weighted endomorphisms of C(X) have the form f(x) -* u(x)f(qp(x)) for some u E C(X) and continuous function 9p from X into X. This map will be denoted by uCcp and its spectrum by a(uC,). In related papers, the spectra of weighted automorphisms of various algebras were considered by Kitover ([6] and [7]) and in a lattice theoretic setting by Arendt ([1] and [2]) and Schaeffer, Wolff and Arendt [8]. Results on weighted automorphisms and endomorphisms of the disc algebra may be found in [4] and [5]. We begin with some notation and terminology. If B, and B2 are Banach spaces, then a linear map T: B1 -B2 is compact if it maps bounded sets into sequentially compact sets. If X is a set and qp: X -* X, then we let qPn denote the nth iterate of qT, i.e. T0(x) = x and %, (x) = p(P(, 1(x)) for n > 0, x E X. Also, if qp: X -* X, then a point c in X is called a fixed point of 9p of order n if n is a positive integer, Tpn(c) = c and q)k(c) =-c, k = 1, . . . , n1. The principal result of this note will be the following theorem.

Logarithms of moduli of inversible elements of a Banach algebra

The isomorphism of a separable Banach algebra of continuous functions with C(X) is deduced from the properties of the set of logarithms of moduli of invertible elements of the algebra.

Characterizations of 𝐶*-algebras

This note presents two related characterizations of those Banach algebras which are isometrically isomorphic to C*-algebras, i.e., to operator-norm closed, self-adjoint algebras of operators on a Hilbert space. The first characterization has evolved from a theorem of I. Vidav [7] and its extension by E. Berkson [l] and B. W. Glickfeld [2] (cf. [5]). The proof given below is considerably simpler than the proofs given in [l] and [2] for closely related, but weaker, results. It is based on Lemma 1 which refines a result of B. Russo and H. A. Dye [6]. All algebras considered here have complex scalars and an identity element I of norm one. In [6] it is shown that the closed unit ball 2ti of a C*-algebra SI is the norm closed convex hull clco £7(21) of the set U{%) of all unitary elements in SI. The set »« = {e: RE%,R = R*}> which can be defined in any Banach algebra with an involution, is a subset of U(%) in any C*-algebra. In a von Neumann algebra 2I« = Z7(2t), but in certain C*algebras 2Ie is a proper subset of £7(21). For instance if 21 is the usual Banach algebra of continuous functions on the unit circle in the complex plane, then multiplication by the complex variable belongs to £/(2I) but not to 2l«. Thus the following lemma strengthens Theorem 1 of [6].

Banach Algebras of Continuous Functions

Banach Algebras of Continuous Functions