Quasispectral subspaces of quasinilpotent operators

1984 ◽  
Vol 98 (3-4) ◽  
pp. 349-354 ◽  
Author(s):  
Matjaž Omladič
Keyword(s):  
Banach Space ◽  
Volterra Operator ◽  
Space Operator ◽  
Definition Of ◽  
Quasinilpotent Operators

SynopsisWe give a definition of “quasispectral maximalč subspaces for a quasinilpotent, but not nilpotent, bounded Banach space operator. The definition applies to a class of operators, close to the Volterra operator.

scholarly journals An integral for a banach valued function

2009 ◽  
Vol 44 (1) ◽  
pp. 105-113
Author(s):  
Giuseppa Riccobono
Keyword(s):  
Banach Space ◽  
Bochner Integral ◽  
Definition Of

Abstract Using partitions of the unity ((PU)-partition), a new definition of an integral is given for a function f : [a, b] → X, where X is a Banach space, and it is proved that this integral is equivalent to the Bochner integral.

scholarly journals The Linearity of Riemann Integral on Functions from ℝ into Real Banach Space

2013 ◽  
Vol 21 (3) ◽  
pp. 185-191
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Banach Space ◽  
Integral Operator ◽  
Real Banach Space ◽  
Closed Interval ◽  
Continuous Functions ◽  
Real Numbers ◽  
Riemann Integral ◽  
Basic Properties ◽  
Bounded Functions ◽  
Definition Of

Summary In this article, we described basic properties of Riemann integral on functions from R into Real Banach Space. We proved mainly the linearity of integral operator about the integral of continuous functions on closed interval of the set of real numbers. These theorems were based on the article [10] and we referred to the former articles about Riemann integral. We applied definitions and theorems introduced in the article [9] and the article [11] to the proof. Using the definition of the article [10], we also proved some theorems on bounded functions.

scholarly journals Quasi-regularity in Optimization

1986 ◽  
Vol 41 (2) ◽  
pp. 188-192 ◽  
Author(s):  
Kung-Fu Ng ◽  
David Yost
Keyword(s):  
Banach Space ◽  
Lagrange Multiplier ◽  
Optimization Problems ◽  
General Situation ◽  
Lagrange Multiplier Rule ◽  
Infinite Dimensional ◽  
Space Setting ◽  
Multiplier Rule ◽  
Dimensional Version ◽  
Definition Of

AbstractThe notion of quasi-regularity, defined for optimization problems in Rn, is extended to the Banach space setting. Examples are given to show that our definition of quasi-regularity is more natural than several other possibilities in the general situation. An infinite dimensional version of the Lagrange multiplier rule is established.

scholarly journals A note on range-kernel uncomplementation

2017 ◽  
Vol 50 (1) ◽  
pp. 252-260
Author(s):  
Carlos Kubrusly ◽  
Bhagwati Prashad Duggal
Keyword(s):  
Banach Space ◽  
Space Operator

Abstract This note exhibits a Banach-space operator such that neither the range nor the kernel is complemented both for the operator and its adjoint

scholarly journals SPACES OF SMALL METRIC COTYPE

2010 ◽  
Vol 02 (04) ◽  
pp. 581-597 ◽  
Author(s):  
E. VEOMETT ◽  
K. WILDRICK
Keyword(s):  
Banach Space ◽  
Metric Space ◽  
Metric Spaces ◽  
Ultrametric Space ◽  
Partial Converse ◽  
Definition Of ◽  
Hausdorff Limits

Mendel and Naor's definition of metric cotype extends the notion of the Rademacher cotype of a Banach space to all metric spaces. Every Banach space has metric cotype at least 2. We show that any metric space that is bi-Lipschitz is equivalent to an ultrametric space having infimal metric cotype 1. We discuss the invariance of metric cotype inequalities under snowflaking mappings and Gromov–Hausdorff limits, and use these facts to establish a partial converse of the main result.

scholarly journals Joint spectra of operators on Banach space

1986 ◽  
Vol 28 (1) ◽  
pp. 69-72 ◽  
Author(s):  
Muneo Chō
Keyword(s):  
Banach Space ◽  
Point Spectrum ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Joint Spectrum ◽  
Approximate Point Spectrum ◽  
Definition Of ◽  
Unit Vectors

Let X be a complex Banach space. We denote by B(X) the algebra of all bounded linear operators on X. Let = (T1, …, Tn) be a commuting n-tuple of operators on X. And let στ() and σ″() by Taylor's joint spectrum and the doubly commutant spectrum of , respectively. We refer the reader to Taylor [8] for the definition of στ() and σ″(), A point z = (z1,…, zn) of ℂn is in the joint approximate point spectrum σπ() of if there exists a sequence {xk} of unit vectors in X such that∥(Ti – zi)xk∥→0 as k → ∞ for i = 1, 2,…, n.

scholarly journals Non-weak compactness of the integration map for vector measures

1993 ◽  
Vol 54 (3) ◽  
pp. 287-303 ◽  
Author(s):  
S. Okada ◽  
W. J. Ricker
Keyword(s):  
Banach Space ◽  
Volterra Operator ◽  
Continuous Operator ◽  
Weakly Compact ◽  
Vector Measures ◽  
Mean Convergence ◽  
Integrable Functions ◽  
The Mean ◽  
The Fourier Transform ◽  
Adjoint Operators

AbstractLet m be a vector measure with values in a Banach space X. If L1(m) denotes the space of all m integrable functions then, with respect to the mean convergence topology, L1(m) is a Banach space. A natural operator associated with m is its integration map Im which sends each f of L1(m) to the element ∫fdm (of X). Many properties of the (continuous) operator Im are closely related to the nature of the space L1(m). In general, it is difficult to identify L1(m). We aim to exhibit non-trivial examples of measures m in (non-reflexive) spaces X for which L1(m) can be explicitly computed and such that Im is not weakly compact. The examples include some well known operators from analysis (the Fourier transform on L1 ([−π, π]), the Volterra operator on L1 ([0, 1]), compact self-adjoint operators in a Hilbert space); such operators can be identified with integration maps Im (or their restrictions) for suitable measures m.

The spectrum and eigenspaces of a meromorphic operator-valued function

1997 ◽  
Vol 127 (5) ◽  
pp. 1027-1051 ◽  
Author(s):  
Robert Magnus
Keyword(s):  
Banach Space ◽  
Riemann Surface ◽  
Spectral Theory ◽  
Approximation Theorem ◽  
Meromorphic Mapping ◽  
Bounded Operators ◽  
Single Operator ◽  
Definition Of ◽  
Vector Valued Functions ◽  
Vector Valued

SynopsisIt is shown how to associate eigenvectors with a meromorphic mapping defined on a Riemann surface with values in the algebra of bounded operators on a Banach space. This generalises the case of classical spectral theory of a single operator. The consequences of the definition of the eigenvectors are examined in detail. A theorem is obtained which asserts the completeness of the eigenvectors whenever the Riemann surface is compact. Two technical tools are discussed in detail: Cauchy-kernels and Runge's Approximation Theorem for vector-valued functions.

scholarly journals A Note on Property(gb)and Perturbations

2012 ◽  
Vol 2012 ◽  
pp. 1-10 ◽  
Author(s):  
Qingping Zeng ◽  
Huaijie Zhong
Keyword(s):  
Banach Space ◽  
Finite Rank ◽  
Point Spectrum ◽  
Bounded Linear ◽  
Approximate Point Spectrum ◽  
Weyl Spectrum ◽  
Nilpotent Operators ◽  
The Stability ◽  
Quasinilpotent Operators ◽  
Study Property

An operatorT∈ℬ(X)defined on a Banach spaceXsatisfies property(gb)if the complement in the approximate point spectrumσa(T)of the upper semi-B-Weyl spectrumσSBF+-(T)coincides with the setΠ(T)of all poles of the resolvent ofT. In this paper, we continue to study property(gb)and the stability of it, for a bounded linear operatorTacting on a Banach space, under perturbations by nilpotent operators, by finite rank operators, and by quasinilpotent operators commuting withT. Two counterexamples show that property(gb)in general is not preserved under commuting quasi-nilpotent perturbations or commuting finite rank perturbations.

scholarly journals New extensions of Cline’s formula for generalized inverses

Filomat ◽  
2017 ◽  
Vol 31 (7) ◽  
pp. 1973-1980 ◽  
Author(s):  
Qingping Zeng ◽  
Zhenying Wu ◽  
Yongxian Wen
Keyword(s):  
Banach Space ◽  
Generalized Inverses ◽  
Drazin Inverse ◽  
Space Operator ◽  
Cline’S Formula ◽  
Generalized Drazin Inverse

In this paper, Cline?s formula for the well-known generalized inverses such as Drazin inverse, pseudo Drazin inverse and generalized Drazin inverse is extended to the case when ( acd = dbd dba = aca. Also, applications are given to some interesting Banach space operator properties like algebraic, meromorphic, polaroidness and B-Fredholmness.

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