scholarly journals On the hereditary character of certain spectral properties and some applications

2021 ◽  
Author(s):  
Carlos Rafael Carpintero ◽  
Ennis Rafael Rosas Rodriguez ◽  
Orlando J. García Mojica ◽  
José Eduardo José Eduardo Sanabria ◽  
Andrés Malaver
Keyword(s):  
Banach Space ◽  
Invariant Subspace ◽  
Spectral Properties ◽  
Integral Operators ◽  
Complex Banach Space ◽  
Entire Space ◽  
Infinite Dimensional ◽  
Weyl Type Theorems ◽  
Hereditary Character

In this paper we study the behavior of certain spectral properties of an operator T on a proper closed and T-invariant subspace W ⊆ X such that Tn (X) ⊆ W, for some n ≥ 1, where T ∈ L(X) and X is an infinite-dimensional complex Banach space. We prove that for these subspaces a large number of spectral properties are transmitted from T to its restriction on W and vice-versa. As consequence of our results, we give conditions for which semiFredholm spectral properties, as well as Weyl type theorems, are equivalent for two given operators. Additionally, we give conditions under which an operator acting on a subspace can be extended on the entire space preserving the Weyl type theorems. In particular, we give some applications of these results for integral operators acting on certain functions spaces.

scholarly journals On the hereditary character of certain spectral properties and some applications

2021 ◽  
Author(s):  
Carlos Rafael Carpintero ◽  
Ennis Rafael Rosas Rodríguez ◽  
Orlando J. García Mojica ◽  
José Eduardo Sanabria ◽  
Andrés Malaver
Keyword(s):  
Banach Space ◽  
Invariant Subspace ◽  
Spectral Properties ◽  
Integral Operators ◽  
Complex Banach Space ◽  
Entire Space ◽  
Infinite Dimensional ◽  
Weyl Type Theorems ◽  
Hereditary Character

In this paper we study the behavior of certain spectral properties of an operator T on a proper closed and T-invariant subspace W ⊆ X such that Tn (X) ⊆ W, for some n ≥ 1, where T ∈ L(X) and X is an infinite-dimensional complex Banach space. We prove that for these subspaces a large number of spectral properties are transmitted from T to its restriction on W and vice-versa. As consequence of our results, we give conditions for which semiFredholm spectral properties, as well as Weyl type theorems, are equivalent for two given operators. Additionally, we give conditions under which an operator acting on a subspace can be extended on the entire space preserving the Weyl type theorems. In particular, we give some applications of these results for integral operators acting on certain functions spaces.

scholarly journals Upper triangular operators with SVEP: Spectral properties

Filomat ◽  
2007 ◽  
Vol 21 (1) ◽  
pp. 25-37 ◽  
Author(s):  
B.P. Duggal
Keyword(s):  
Banach Space ◽  
Spectral Properties ◽  
Extension Property ◽  
Complex Banach Space ◽  
Single Valued Extension Property ◽  
Infinite Dimensional

Spectral properties of upper triangular operators T = (Tij)1?i,j?n E B(?n) where ?n = ?ni=1?i and ?i is an infinite dimensional complex Banach space such that Tii - ? has the single-valued extension property, SVEP, for all complex ? are studied.

scholarly journals Characteristic Functions and Borel Exceptional Values ofE-Valued Meromorphic Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Zhaojun Wu ◽  
Zuxing Xuan
Keyword(s):  
Banach Space ◽  
Meromorphic Functions ◽  
Complex Banach Space ◽  
Schauder Basis ◽  
Characteristic Functions ◽  
Infinite Dimensional ◽  
Borel Exceptional Values

The main purpose of this paper is to investigate the characteristic functions and Borel exceptional values ofE-valued meromorphic functions from theℂR={z:|z|<R},  0<R≤+∞to an infinite-dimensional complex Banach spaceEwith a Schauder basis. Results obtained extend the relative results by Xuan, Wu and Yang, Bhoosnurmath, and Pujari.

Holomorphic functions with distinguished properties on infinite dimensional spaces

2018 ◽  
Vol 70 (3) ◽  
pp. 797-811
Author(s):  
Thiago R Alves ◽  
Geraldo Botelho
Keyword(s):  
Banach Space ◽  
Holomorphic Functions ◽  
Complex Banach Space ◽  
Algebraic Structures ◽  
Infinite Dimensional ◽  
Zero Function ◽  
Bounded Holomorphic

Abstract In this paper, we develop a method to construct holomorphic functions that exist only on infinite dimensional spaces. The following types of holomorphic functions f:U→ℂ on some open subsets U of an infinite dimensional complex Banach space are constructed: (1) f is bounded holomorphic on U and is continuously, but not uniformly continuously extended to U¯; (2) f is continuous on U¯ and holomorphic of bounded type on U, but f is unbounded on U; (3) f is holomorphic of bounded type on U and f cannot be continuously extended to U¯. The technique we develop is powerful enough to provide, in the cases (2) and (3) above, large algebraic structures formed by such functions (up to the zero function, of course).

scholarly journals On the M-hypercyclicity of cosine function on Banach spaces

2019 ◽  
Vol 38 (3) ◽  
pp. 133-140
Author(s):  
Abdelaziz Tajmouati ◽  
Abdeslam El Bakkali ◽  
Ahmed Toukmati
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Complex Banach Space ◽  
Cosine Function ◽  
Infinite Dimensional ◽  
Uniformly Continuous

In this paper we introduce and study the M-hypercyclicity of strongly continuous cosine function on separable complex Banach space, and we give the criteria for cosine function to be M-hypercyclic. We also prove that every separable infinite dimensional complex Banach space admits a uniformly continuous cosine function.

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.

scholarly journals The Invariant Subspace Problem for Non-Archimedean Banach Spaces

2008 ◽  
Vol 51 (4) ◽  
pp. 604-617 ◽  
Author(s):  
Wiesław Śliwa
Keyword(s):  
Banach Space ◽  
Invariant Subspace ◽  
Banach Spaces ◽  
Continuous Operator ◽  
Linear Continuous Operator ◽  
Infinite Dimensional ◽  
Countable Type ◽  
Closed Invariant Subspace ◽  
Subspace Problem ◽  
Invariant Subspace Problem

AbstractIt is proved that every infinite-dimensional non-archimedean Banach space of countable type admits a linear continuous operator without a non-trivial closed invariant subspace. This solves a problem stated by A. C. M. van Rooij and W. H. Schikhof in 1992.

scholarly journals Approximate point spectrum and commuting compact perturbations

1986 ◽  
Vol 28 (2) ◽  
pp. 193-198 ◽  
Author(s):  
Vladimir Rakočević
Keyword(s):  
Banach Space ◽  
Essential Spectrum ◽  
Point Spectrum ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Complex Numbers ◽  
Approximate Point Spectrum ◽  
Infinite Dimensional ◽  
Compact Perturbations ◽  
Image Position

Let X be an infinite-dimensional complex Banach space and denote the set of bounded (compact) linear operators on X by B (X) (K(X)). Let σ(A) and σa(A) denote, respectively, the spectrum and approximate point spectrum of an element A of B(X). Setσem(A)and σeb(A) are respectively Schechter's and Browder's essential spectrum of A ([16], [9]). σea (A) is a non-empty compact subset of the set of complex numbers ℂ and it is called the essential approximate point spectrum of A ([13], [14]). In this note we characterize σab(A) and show that if f is a function analytic in a neighborhood of σ(A), then σab(f(A)) = f(σab(A)). The relation between σa(A) and σeb(A), that is exhibited in this paper, resembles the relation between the σ(A) and the σeb(A), and it is reasonable to call σab(A) Browder's essential approximate point spectrum of A.

scholarly journals Generalized spectrum and commuting compact perturbations

1993 ◽  
Vol 36 (2) ◽  
pp. 197-209 ◽  
Author(s):  
Vladimir Rakočević
Keyword(s):  
Banach Space ◽  
Null Space ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Range Space ◽  
Infinite Dimensional ◽  
Compact Perturbations ◽  
Generalized Spectrum

Let X be an infinite-dimensional complex Banach space and denote the set of bounded (compact) linear operators on X by B(X) (K(X)). Let N(A) and R(A) denote, respectively, the null space and the range space of an element A of B(X). Set R(A∞)=∩nR(An) and k(A)=dim N(A)/(N(A)∩R(A∞)). Let σg(A) = ℂ\{λ∈ℂ:R(A−λ) is closed and k(A−λ)=0} denote the generalized (regular) spectrum of A. In this paper we study the subset σgb(A) of σg(A) defined by σgb(A) = ℂ\{λ∈ℂ:R(A−λ) is closed and k(A−λ)<∞}. Among other things, we prove that if f is a function analytic in a neighborhood of σ(A), then σgb(f(A)) = f(σgb(A)).

scholarly journals On generalization of decomposability

1981 ◽  
Vol 22 (1) ◽  
pp. 77-81 ◽  
Author(s):  
Ridgley Lange
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Invariant Subspace ◽  
Bounded Linear Operator ◽  
Invariant Subspaces ◽  
Complex Banach Space ◽  
Open Cover ◽  
Bounded Linear ◽  
Image Position ◽  
Finite Open Cover

Let X be a complex Banach space and let T be a bounded linear operator on X. Then T is decomposable if for every finite open cover of σ(T) there are invariant subspaces Yi(i= 1, 2, …, n) such that(An invariant subspace Y is spectral maximal [for T] if it contains every invariant subspace Z for which σ(T|Z) ⊂ σ(T|Y).).

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