Some special characterisations of Fredholm operators in Banach space

A bounded linear operator which has a finite index and which is defined on a Banach space is often referred to in the literature as a Fredholm operator. Fredholm operators are important for a variety of reasons, one being the role that their index plays in global analysis. The aim of this paper is to prove the spectral theorem for compact operators in refined form and to describe some properties of the essential spectrum of general bounded operators by the use of the theorem of Fredholm operators. For this, we have analysed the Fredholm operator which is defined in a Banach space for some special characterisations. DOI: http://dx.doi.org/10.3126/bibechana.v11i0.10399 BIBECHANA 11(1) (2014) 169-174

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On Certain Classes of Bounded Linear Operators

Canadian Mathematical Bulletin ◽

10.4153/cmb-1970-086-6 ◽

1970 ◽

Vol 13 (4) ◽

pp. 469-473

Author(s):

C-S Lin

Keyword(s):

Banach Space ◽

Linear Operator ◽

Banach Algebra ◽

Fredholm Operator ◽

Bounded Linear Operator ◽

Complex Banach Space ◽

Linear Operators ◽

Complex Numbers ◽

Bounded Linear Operators ◽

Bounded Linear

Let T—c be a Fredholm operator, where T is a bounded linear operator on a complex Banach space and c is a scalar, the set of all such scalars is called the Φ-set of T [2] and was studied by many authors. In this connection, the purpose of the present paper is to investigate some classes Φ(V) of all such operators for any subset V of the complex plane.Let X be a Banach space over the field C of complex numbers with dim Z = ∞, unless otherwise stated, B(X) the Banach algebra of all bounded linear operators and K(X) the closed two-sided ideal of all compact operators on X.

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Projections on Tree-Like banach Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-1985-049-2 ◽

1985 ◽

Vol 37 (5) ◽

pp. 908-920

Author(s):

A. D. Andrew

Keyword(s):

Banach Space ◽

Linear Operator ◽

Banach Spaces ◽

Bounded Linear Operator ◽

Unconditional Basis ◽

Reflexive Banach Space ◽

Separable Space ◽

Bounded Linear ◽

Tree Space

1. In this paper, we investigate the ranges of projections on certain Banach spaces of functions defined on a diadic tree. The notion of a “tree-like” Banach space is due to James 4], who used it to construct the separable space JT which has nonseparable dual and yet does not contain l1. This idea has proved useful. In [3], Hagler constructed a hereditarily c0 tree space, HT, and Schechtman [6] constructed, for each 1 ≦ p ≦ ∞, a reflexive Banach space, STp with a 1-unconditional basis which does not contain lp yet is uniformly isomorphic to for each n.In [1] we showed that if U is a bounded linear operator on JT, then there exists a subspace W ⊂ JT, isomorphic to JT such that either U or (1 — U) acts as an isomorphism on W and UW or (1 — U)W is complemented in JT. In this paper, we establish this result for the Hagler and Schechtman tree spaces.

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A note on best approximation and invertibility of operators on uniformly convex Banach spaces

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171291000832 ◽

1991 ◽

Vol 14 (3) ◽

pp. 611-614 ◽

Cited By ~ 1

Author(s):

James R. Holub

Keyword(s):

Banach Space ◽

Linear Operator ◽

Banach Spaces ◽

Best Approximation ◽

Bounded Linear Operator ◽

Convex Banach Space ◽

Sufficient Condition ◽

Uniformly Convex ◽

Bounded Linear ◽

Uniformly Convex Banach Spaces

It is shown that ifXis a uniformly convex Banach space andSa bounded linear operator onXfor which‖I−S‖=1, thenSis invertible if and only if‖I−12S‖<1. From this it follows that ifSis invertible onXthen either (i)dist(I,[S])<1, or (ii)0is the unique best approximation toIfrom[S], a natural (partial) converse to the well-known sufficient condition for invertibility thatdist(I,[S])<1.

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BOUNDED LINEAR OPERATORS ON SPACES IN NORMED DUALITY

Glasgow Mathematical Journal ◽

10.1017/s0017089507003503 ◽

2007 ◽

Vol 49 (1) ◽

pp. 145-154

Author(s):

BRUCE A. BARNES

Keyword(s):

Banach Space ◽

Linear Operator ◽

Bounded Linear Operator ◽

Integral Operators ◽

Continuous Functions ◽

Linear Operators ◽

Bounded Linear Operators ◽

Closed Range ◽

Bounded Linear ◽

Spaces Of Continuous Functions

Abstract.LetTbe a bounded linear operator on a Banach spaceW, assumeWandYare in normed duality, and assume thatThas adjointT†relative toY. In this paper, conditions are given that imply that for all λ≠0, λ−Tand λ −T†maintain important standard operator relationships. For example, under the conditions given, λ −Thas closed range if, and only if, λ −T†has closed range.These general results are shown to apply to certain classes of integral operators acting on spaces of continuous functions.

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Almost Surjective Epsilon-Isometry in The Reflexive Banach Spaces

CAUCHY ◽

10.18860/ca.v4i4.4100 ◽

2017 ◽

Vol 4 (4) ◽

pp. 167

Author(s):

Minanur Rohman

Keyword(s):

Banach Space ◽

Linear Operator ◽

Banach Spaces ◽

Lorentz Space ◽

Bounded Linear Operator ◽

Closed Subspace ◽

Reflexive Banach Space ◽

Reflexive Banach Spaces ◽

Bounded Linear ◽

Star Topology

In this paper, we will discuss some applications of almost surjective epsilon-isometry mapping, one of them is in Lorentz space ( L_(p,q)-space). Furthermore, using some classical theorems of w star-topology and concept of closed subspace -complemented, for every almost surjective epsilon-isometry mapping f : X to Y, where Y is a reflexive Banach space, then there exists a bounded linear operator T : Y to X with such that for every x in X.

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Separation problem for the Grushin differential operator in Banach spaces

Carpathian Journal of Mathematics ◽

10.37193/cjm.2014.01.05 ◽

2014 ◽

Vol 30 (1) ◽

pp. 31-37

Author(s):

H. A. ATIA ◽

◽

Keyword(s):

Banach Space ◽

Differential Operator ◽

Linear Operator ◽

Banach Spaces ◽

Bounded Linear Operator ◽

Separation Problem ◽

Bounded Linear

Our goal in this work is to study the separation problem for the Grushin differential operator formed by ... in the Banach space H1(R2), where the potential Q(x, y) ∈ L(1), is a bounded linear operator which transforms at 1 in value of (x, y).

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A Note on the Caradus Class of Bounded Linear Operators on a Complex Banach Space

Canadian Journal of Mathematics ◽

10.4153/cjm-1969-066-6 ◽

1969 ◽

Vol 21 ◽

pp. 592-594 ◽

Cited By ~ 4

Author(s):

A. F. Ruston

Keyword(s):

Banach Space ◽

Linear Operator ◽

Finite Number ◽

Bounded Linear Operator ◽

Present Note ◽

Complex Banach Space ◽

Linear Operators ◽

Bounded Linear Operators ◽

Bounded Linear ◽

The Subject

1. In a recent paper (1) on meromorphic operators, Caradus introduced the class of bounded linear operators on a complex Banach space X. A bounded linear operator T is put in the class if and only if its spectrum consists of a finite number of poles of the resolvent of T. Equivalently, T is in if and only if it has a rational resolvent (8, p. 314).Some ten years ago (in May, 1957), I discovered a property of the class g which may be of interest in connection with Caradus' work, and is the subject of the present note.2. THEOREM. Let X be a complex Banach space. If T belongs to the class, and the linear operator S commutes with every bounded linear operator which commutes with T, then there is a polynomial p such that S = p(T).

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(F,G)-operator frames for ℒ(ℋ,𝒦)

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691320500319 ◽

2020 ◽

Vol 18 (05) ◽

pp. 2050031

Author(s):

J. Sedghi Moghaddam ◽

A. Najati ◽

F. Ghobadzadeh

Keyword(s):

Linear Operator ◽

Hilbert Spaces ◽

Bounded Linear Operator ◽

Bounded Operators ◽

Closed Range ◽

Bounded Linear ◽

Atomic Systems ◽

Pseudo Inverse

The concept of [Formula: see text]-frames was recently introduced by Găvruta7 in Hilbert spaces to study atomic systems with respect to a bounded linear operator. Let [Formula: see text] be a unital [Formula: see text]-algebra, [Formula: see text] be finitely or countably generated Hilbert [Formula: see text]-modules, and [Formula: see text] be adjointable operators from [Formula: see text] to [Formula: see text]. In this paper, we study a class of [Formula: see text]-bounded operators and [Formula: see text]-operator frames for [Formula: see text]. We also prove that the pseudo-inverse of [Formula: see text] exists if and only if [Formula: see text] has closed range. We extend some known results about the pseudo-inverses acting on Hilbert spaces in the context of Hilbert [Formula: see text]-modules. Further, we also present some perturbation results for [Formula: see text]-operator frames in [Formula: see text].

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The () Property in Banach Spaces

Abstract and Applied Analysis ◽

10.1155/2012/754531 ◽

2012 ◽

Vol 2012 ◽

pp. 1-7

Author(s):

Danyal Soybaş

Keyword(s):

Banach Space ◽

Linear Operator ◽

Banach Spaces ◽

Bounded Linear Operator ◽

Isomorphic Copy ◽

Weakly Compact ◽

Geometric Properties ◽

Bounded Linear ◽

Real Functions

A Banach space is said to have (D) property if every bounded linear operator is weakly compact for every Banach space whose dual does not contain an isomorphic copy of . Studying this property in connection with other geometric properties, we show that every Banach space whose dual has (V∗) property of Pełczyński (and hence every Banach space with (V) property) has (D) property. We show that the space of real functions, which are integrable with respect to a measure with values in a Banach space , has (D) property. We give some other results concerning Banach spaces with (D) property.

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A certain fixed point theorem and its applications to integral-functional equations

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700011813 ◽

1992 ◽

Vol 46 (2) ◽

pp. 179-186 ◽

Cited By ~ 10

Author(s):

M. Zima

Keyword(s):

Banach Space ◽

Fixed Point ◽

Fixed Point Theorem ◽

Linear Operator ◽

Spectral Radius ◽

Bounded Linear Operator ◽

Functional Equations ◽

Integral Functional ◽

Bounded Linear ◽

Banach’S Contraction Principle

In this paper a variant of Banach's contraction principle is established. By using the properties of the spectral radius of a bounded linear operator A defined in a suitable Banach space, we conclude that another operator A has exactly one fixed point in this space. In the second part of this paper some applications are given.

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