Singularities of the inverses of Fredholm operators

1986 ◽  
Vol 102 (1-2) ◽  
pp. 117-121 ◽  
Author(s):  
A. G. Ramm
Keyword(s):  
Banach Space ◽  
Complex Plane ◽  
Connected Domain ◽  
Inverse Operator ◽  
Fredholm Operators

SynopsisLet B(k) be a linear bounded mapping of a Banach space X into a Banach space Y meromorphic in the parameter k on a connected domain of the complex plane. Under certain assumptions on B(k), more general than previously considered, the singularities of the inverse operator are described.

scholarly journals The arc-length variation of analytic capacity and a conformal geometry

1992 ◽  
Vol 125 ◽  
pp. 151-216
Author(s):  
Takafumi Murai
Keyword(s):  
Banach Space ◽  
Complex Plane ◽  
Analytic Functions ◽  
Conformal Geometry ◽  
Supremum Norm ◽  
Length Variation ◽  
Analytic Capacity ◽  
Arc Length ◽  
Bounded Analytic Functions ◽  
Extended Complex

For a domain Ω in the extended complex plane C ∪{∞}, H∞(Ω) denotes the Banach space of bounded analytic functions in Ω with supremum norm ∥ · ∥H∞ For ζ ∈ Ω, we putwhere f′(∞) = lim,z→∞z{f (∞) = f(z)} if ζ = ∞.

The universal multiplicity theory for analytic operator-valued functions

1995 ◽  
Vol 118 (2) ◽  
pp. 315-320 ◽  
Author(s):  
Jón Arason ◽  
Robert Magnus
Keyword(s):  
Banach Space ◽  
Complex Plane ◽  
Open Subset ◽  
Closed Subset ◽  
Complex Banach Space ◽  
Singular Set ◽  
Analytic Operator ◽  
Set Of Points

An analytic operator-valued function A is an analytic map A: D → L(E, E), where D = D(A) is an open subset of the complex plane C and E = E(A) is a complex Banach space. For such a function A the singular set σ(A) of A is defined as the set of points z ∈ D such that A(z) is not invertible. It is a relatively closed subset of D.

scholarly journals Milloux Inequality ofE-Valued Meromorphic Function

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Zhaojun Wu ◽  
Zuxing Xuan
Keyword(s):  
Banach Space ◽  
Meromorphic Function ◽  
Complex Plane ◽  
Complex Banach Space ◽  
Schauder Basis ◽  
Infinite Dimensional ◽  
Borel Exceptional Values

The main purpose of this paper is to establish the Milloux inequality ofE-valued meromorphic function from the complex planeℂto an infinite dimensional complex Banach spaceEwith a Schauder basis. As an application, we study the Borel exceptional values of anE-valued meromorphic function and those of its derivatives; results are obtained to extend some related results for meromorphic scalar-valued function of Singh, Gopalakrishna, and Bhoosnurmath.

Some remarks on universal functions and Taylor series

2000 ◽  
Vol 128 (1) ◽  
pp. 157-175 ◽  
Author(s):  
G. COSTAKIS
Keyword(s):  
Complex Plane ◽  
Taylor Series ◽  
Connected Domain ◽  
Constructive Proof ◽  
Universal Functions ◽  
Baire’S Theorem ◽  
Simply Connected Domain ◽  
Universal Taylor Series ◽  
Simply Connected

We derive properties of universal functions and Taylor series in domains of the complex plane. For some of our results we use Baire's theorem. We also give a constructive proof, avoiding Baire's theorem, of the existence of universal Taylor series in any arbitrary simply connected domain.

scholarly journals THE CLOSED RANGE PROPERTY FOR BANACH SPACE OPERATORS

2008 ◽  
Vol 50 (1) ◽  
pp. 17-26 ◽  
Author(s):  
THOMAS L. MILLER ◽  
VLADIMIR MÜLLER
Keyword(s):  
Banach Space ◽  
Complex Plane ◽  
Open Subset ◽  
Weak Topology ◽  
Bounded Operator ◽  
Complex Banach Space ◽  
Fréchet Space ◽  
Closed Range ◽  
Open Set ◽  
Banach Space Operators

AbstractLetTbe a bounded operator on a complex Banach spaceX. LetVbe an open subset of the complex plane. We give a condition sufficient for the mappingf(z)↦ (T−z)f(z) to have closed range in the Fréchet spaceH(V,X) of analyticX-valued functions onV. Moreover, we show that there is a largest open setUfor which the mapf(z)↦ (T−z)f(z) has closed range inH(V,X) for allV⊆U. Finally, we establish analogous results in the setting of the weak–* topology onH(V, X*).

scholarly journals Some special characterisations of Fredholm operators in Banach space

BIBECHANA ◽  
2014 ◽  
Vol 11 ◽  
pp. 169-174
Author(s):  
Mahendra Shahi
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Essential Spectrum ◽  
Fredholm Operator ◽  
Bounded Linear Operator ◽  
Global Analysis ◽  
Spectral Theorem ◽  
Bounded Operators ◽  
Fredholm Operators ◽  
Bounded Linear

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

Qp Spaces on Riemann Surfaces

1998 ◽  
Vol 50 (3) ◽  
pp. 449-464 ◽  
Author(s):  
Rauno Aulaskari ◽  
Yuzan He ◽  
Juha Ristioja ◽  
Ruhan Zhao
Keyword(s):  
Banach Space ◽  
Riemann Surface ◽  
Unit Disk ◽  
Complex Plane ◽  
Riemann Surfaces ◽  
Bloch Space ◽  
Dirichlet Space ◽  
Function Spaces ◽  
Hyperbolic Riemann Surfaces ◽  
Qp Spaces

AbstractWe study the function spaces Qp(R) defined on a Riemann surface R, which were earlier introduced in the unit disk of the complex plane. The nesting property Qp(R) ⊆Qq(R) for 0 < p < q < ∞ is shown in case of arbitrary hyperbolic Riemann surfaces. Further, it is proved that the classical Dirichlet space AD(R) ⊆ Qp(R) for any p, 0 < p < ∞, thus sharpening T. Metzger's well-known result AD(R) ⊆ BMOA(R). Also the first author's result AD(R) ⊆ VMOA(R) for a regular Riemann surface R is sharpened by showing that, in fact, AD(R) ⊆ Qp,0(R) for all p, 0 < p < ∞. The relationships between Qp(R) and various generalizations of the Bloch space on R are considered. Finally we show that Qp(R) is a Banach space for 0 < p < ∞.

About unbounded complex operators

2020 ◽  
pp. 57-67
Author(s):  
Vasiliy I. Fomin
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Explicit Form ◽  
Unbounded Operator ◽  
Inverse Operator ◽  
Complex Conjugate ◽  
Homogeneous Differential Equation ◽  
Linear Homogeneous Differential Equation

The concept of an unbounded complex operator as an operator acting in the pull-back of a Banach space is introduced. It is proved that each such operator is linear. Linear operations of addition and multiplication by a number and also the operation of multiplication are determined on the set of unbounded complex operators. The conditions for commutability of operators from this set are indicated. The product of complex conjugate operators and the properties of the conjugation operation are considered. Invertibility questions are studied: two contractions of an unbounded complex operator that have an inverse operator are proposed, and an explicit form of the inverse operator is found for one of these restrictions. It is noted that unbounded complex operators can find application in the study of a linear homogeneous differential equation with constant unbounded operator coefficients in a Banach space.

scholarly journals Range-kernel complementation

2018 ◽  
Vol 55 (3) ◽  
pp. 327-344
Author(s):  
Carlos. S. Kubrusly
Keyword(s):  
Banach Space ◽  
Normed Space ◽  
Compact Operators ◽  
Space Operator ◽  
Fredholm Operators ◽  
Compact Perturbations

If a Banach-space operator has a complemented range, then its normed-space adjoint has a complemented kernel and the converse holds on a re exive Banach space. It is also shown when complemented kernel for an operator is equivalent to complemented range for its normed-space adjoint. This is applied to compact operators and to compact perturbations. In particular, compact perturbations of semi-Fredholm operators have complemented range and kernel for both the perturbed operator and its normed-space adjoint.

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