Perturbation of semi-Browder linear relations by commuting Riesz operators

This paper is devoted to the investigation of the stability of upper semi-Browder, lower semi-Browder and Browder linear relations which satisfy the stabilization property, under commuting Riesz operator perturbations. As applications, we infer the invariance of the corresponding Browder’s essential approximate point spectrum, Browder’s essential defect spectrum and Browder’s essential spectrum under commuting Riesz operator perturbations.     

LEFT-RIGHT BROWDER LINEAR RELATIONS AND RIESZ PERTURBATIONS

A closed linear relation T in a Banach space X is called left(resp. right) Fredholm if it is upper(resp. lower) semi Fredholm and its range(resp. null space) is topologically complemented in X. We say that T is left(resp. right) Browder if it is left(resp. right)Fredholm and has a finite ascent(resp. descent). In this paper, we analyze the stability of the left(resp. right) Fredholm and the left(resp. right) Browder linear relations under commuting Riesz operator perturbations. Recent results of Zivkovic et al. to the case of bounded operators are covered.     

The Brezis–Browder Theorem in a general Banach space

During the 1970s Brezis and Browder presented a now classical characterization of maximal monotonicity of monotone linear relations in reflexive spaces. In this paper, we extend (and refine) their result to a general Banach space. We also provide an affirmative answer to a problem posed by Phelps and Simons.     

Coperturbation function and lower semi-Browder multivalued linear operators

In this article, we investigate the perturbation theory of lower semi-Browder and Browder linear relations. Our approach is based on the concept of a coperturbation function for linear relations in order to establish some perturbation theorems and deduce the stability under strictly cosingular operator perturbations. Furthermore, we apply the obtained results to study the invariance and the characterization of Browder's essential defect spectrum and Browder's essential spectrum.     

Deterministic and Stochastic Control of Navier—Stokes Equation with Linear, Monotone, and Hyperviscosities

This paper deals with the optimal control of space—time statistical behavior of turbulent fields. We provide a unified treatment of optimal control problems for the deterministic and stochastic Navier—Stokes equation with linear and nonlinear constitutive relations. Tonelli type ordinary controls as well as Young type chattering controls are analyzed. For the deterministic case with monotone viscosity we use the Minty—Browder technique to prove the existence of optimal controls. For the stochastic case with monotone viscosity, we combine the Minty—Browder technique with the martingale problem formulation of Stroock and Varadhan to establish existence of optimal controls. The deterministic models given in this paper also cover some simple eddy viscosity type turbulence closure models. Accepted 7 June 1999     

Equivalence of some contractivity properties over metrical structures

We establish an equivalence between eight contractive definitions. Next, we formulate a separation theorem for right upper semicontinuous functions. As its application, we give a complete characterization of relations between fixed point theorems of Boyd and Wong (1969), and Browder (1968).

     

Assembly maps and realization of splitting obstructions

In 1987 Kharshiladze introduced the concept of type for an element in a Wall group, and proved that the elements of first and second type cannot be realized by normal maps of closed manifolds. This approach is sufficiently easy for computing the assembly maps and sometimes very effective. Here we give a geometrical interpretation of this approach by using the Browder–Quinn surgery obstruction groups for filtered manifolds. To understand the obtained relations we give algebraic definitions of the element types which are based on the algebraic surgery theory of Ranicki. Our approach describes a level of indeterminate for the algebraic passing to surgery on a codimension k submanifold of a given Browder–Livesay filtration. Then we study the realization of splitting obstructions by simple homotopy equivalences of closed manifolds, and compute the assembly maps for some classes of groups.     

Fredholm theory in ordered Banach algebras

This paper illustrates some initial steps taken in the e?ort of unifying the theory of positivity in ordered Banach algebas (OBAs) with the general Fredholm theory in Banach algebras. We introduce here upper Weyl and upper Browder elements in an OBA relative to an arbitrary Banach algebra homomorphism and investigate the spectra corresponding to the sets of upper Weyl and upper Browder elements, which we shall refer to as the upper Weyl and upper Browder spectra, respectively.     

A remark on the Brézis–Browder theorem in finite dimensional spaces

Brézis and Browder gave an important characterization of maximal monotonicity for linear operators, in reflexive Banach spaces. In this paper, we show that this characterization also preserves the maximal p-monotonicity, in finite dimensional spaces.     

Around Browder’s fixed point theorem for contractions

We revisit a fixed point theorem for contractions established by Felix Browder in 1968. We show that many definitions of contractive mappings which appeared in the literature after 1968 turn out to be equivalent formulations or even particular cases of Browder’s definition. We also discuss the problem of the existence of approximate fixed points of continuous mappings; in particular, we settle it in the affirmative for Browder contractions. Finally, we recall three problems concerning Browder contractions which remain unsolved. With great respect for Professor Felix E. Browder     

Gustafson,weidmann, kato,wolf, schechter and browder essential spectra of some matrix operator and application to two‐group transport equation

In this paper, we investigate the Gustafson, Weidmann, Kato, Wolf, Schechter and Browder essential spectra of a 2 × 2 block matrix operator defined on a Banach space where entries are unbounded operators between Banach spaces and with domains consisting of vectors satisfying certain relations between their components under some properties. Furthermore, we give an application to two‐group transport equation. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

On the basic theorem of complementarity

Using a fixed point theorem of Browder, the basic existence theorem of Lemke in linear complementarity theory is generalized to the nonlinear case.This research was supported in part by the National Science Foundation Grand GK-5695 and by the Army Research Office — Durham, Contract DAHC 04-67-C-0028.     

The Taylor–Browder Spectrum on Prime C*-Algebras

We provide a formula for the Taylor–Browder spectrum of a pair (L _a , R _b ) of left and right multiplication operators acting on a prime C*-algebra with non-zero socle. We also compute ascent and descent for multiplication operators on a prime ring, characterise Browder elements in a prime C*-algebra and discuss upper semicontinuity for the Browder spectrum.     

Linear fractional mappings: invariant sets, semigroups and commutativity

We study commutativity and embeddability (into continuous semi-groups) properties of linear fractional self-mappings of the open unit disk in the complex plane. The common thread in our approach is the classical notion of the Kœnigs function which we use in each of the three possible cases (dilation, hyperbolic and parabolic). Since we are interested in a classical subject, the paper is written in the style of a survey, in order to make it accessible to a wider audience. Therefore it contains, in addition to our new results, an exposition of most relevant facts. Dedicated to Professor Felix E. Browder with admiration and respect     

SVEP and Generalized Weyl’s Theorem

We use localised single-valued extension property to prove generalized Weyl/Browder and a-generalized Weyl/Browder type theorems for Banach space operators.     

Comments on collectively fixed points in generalized convex spaces

Based on a Fan–Browder type fixed point theorem due to the author, we deduce new general collectively fixed point theorems for a family of Browder type multimaps defined on a product of generalized convex spaces.     

Remarks on u 0-positive operators

We show that the concept of a linear operator being u₀-positive, and refinements of some classical results of Krasnosel’skiĭ, can be used, in conjunction with the theory of fixed point index, to give some short proofs of existence and nonexistence of positive solutions for nonlinear maps. Dedicated to Felix Browder on the occasion of his 80th birthday     

Characterization of Closed Densely Defined Semi-Browder Linear Operators

In the present paper we characterize the closed densely defined semi-Browder operators through the Kato decomposition. Furthermore, we apply the obtained results to give a new characterization of Browder’s essential defect spectrum and Browder’s essential approximate point spectrum under finite rank operator perturbations.     

A Note on Generalized Contraction Theorems

Mathematical Notes - This work is devoted to estimates of the fixed point of a generalized contracting (in the sense of Browder’s and Krasnoselskii’s definitions) operator $$ G $$ in a...     

Essential,Weyl and Browder spectra of unbounded upper triangular operator matrices

This paper is concerned with the spectral properties of the unbounded upper triangular operator matrix with diagonal domain. Some sufficient and necessary conditions are given under which the essential spectrum, the Weyl spectrum and the Browder spectrum of such operator matrix, respectively, coincide with the union of the essential spectrum, the Weyl spectrum and the Browder spectrum of its diagonal entries.