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 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 M-hypercyclicity of C 0-semigroup and Svep of its generator

2021 ◽  
Vol 8 (1) ◽  
pp. 187-191
Author(s):  
A. Toukmati
Keyword(s):  
Banach Space ◽  
Closed Subspace ◽  
Extension Property ◽  
Single Valued Extension Property ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
The Relationship

Abstract Let 𝒯 = (Tt ) t ≥0 be a C 0-semigroup on a separable infinite dimensional Banach space X, with generator A. In this paper, we study the relationship between the single valued extension property of generator A, and the M-hypercyclicity of the C 0-semigroup. Specifically, we prove that if A does not have the single valued extension property at λ ∈ iℝ, then there exists a closed subspace M of X, such that the C 0-semigroup 𝒯 is M-hypercyclic. As a corollary, we get certain conditions of the generator A, for the C 0-semigroup to be M-hypercyclic.

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 Universal notions characterizing spectral decompositions

1992 ◽  
Vol 34 (1) ◽  
pp. 109-116
Author(s):  
Ridgley Lange ◽  
Shengwang Wang
Keyword(s):  
Banach Space ◽  
Linear Manifold ◽  
Sufficient Conditions ◽  
Extension Property ◽  
Complex Banach Space ◽  
Single Valued Extension Property ◽  
Closed Set ◽  
Bounded Linear ◽  
Image Position ◽  
Decomposable Operator

In this note we characterize certain types of spectral decomposition in terms of “universal” notions valid for any operator on a Banach space. To be precise, let X be a complex Banach space and let T be a bounded linear operator on X. If F is a closed set in the plane C, let X(T, F) consist of all y ∈ X satisfying thes identitywhere f:C\F → X is analytic. It is then easy to see that X(T, F) is a T-invariant linear manifold in X. Moreover, if y ∈ X thenis a compact subset of the spectrum σ(T). Our aim is to give necessary and sufficient conditions for a decomposable or strongly decomposable operator in terms of X(T, F) and γ(y, T). Recall that T is decomposable if whenever G1G2 are open and cover C there exist T-invariant closed linear manifolds M1, M2 with X= M1 + M2 and σ(T | M1) ⊂ Gi(i = 1,2) (equivalently, σ(T | Mi)⊂ Ḡi, see [4, p. 57]). In this case, X(T, F) is norm closed if Fis closed and each y in X has a unique maximally defined local resolvent satisfying (1.1) on C\Fy; Fy is called the local spectrum σ(y, T) and coincides with γ(y, T). Hence T has the single valued extension property (SVEP); i.e., zero is the only analytic function f:V → X satisfying (z − T)f(z) = 0 on V. If T is decomposable and the restriction T | X(T, F) is also decomposable for each closed F, then T is called strongly decomposable. We point out that Albrecht [2] has shown by example that not every decomposable operator is strongly decomposable, while Eschmeier [6]has given a simpler construction to show that this phenomenon occurs even in Hilbert space.

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 Propertiesandfor Bounded Linear Operators

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
M. H. M. Rashid
Keyword(s):  
Type Theorem ◽  
Main Tool ◽  
Extension Property ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Weyl’S Theorem ◽  
Bounded Linear Operators ◽  
Single Valued Extension Property ◽  
Bounded Linear ◽  
Weyl Type Theorems

We shall consider properties which are related to Weyl type theorem for bounded linear operators , defined on a complex Banach space . These properties, that we callproperty, means that the set of all poles of the resolvent of of finite rank in the usual spectrum are exactly those points of the spectrum for which is an upper semi-Fredholm with index less than or equal to 0 and we callproperty, means that the set of all poles of the resolvent of in the usual spectrum are exactly those points of the spectrum for which is an upper semi--Fredholm with index less than or equal to 0. Properties and are related to a strong variants of classical Weyl’s theorem, the so-called property and property We shall characterize properties and in several ways and we shall also describe the relationships of it with the other variants of Weyl type theorems. Our main tool is localized version of the single valued extension property. Also, we consider the properties and in the frame of polaroid type operators.

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