Algebras Intertwining Normal and Decomposable Operators

1979 ◽  
Vol 31 (6) ◽  
pp. 1339-1344 ◽  
Author(s):  
Ali A. Jafarian
Keyword(s):  
Banach Space ◽  
Compact Operator ◽  
Invariant Subspaces ◽  
Complex Banach Space ◽  
Bounded Operators ◽  
Zero Eigenvalue ◽  
Infinite Dimensional ◽  
Decomposable Operators ◽  
Closed Invariant Subspace ◽  
Injective Operator

The celebrated result of Lomonosov [6] on the existence of invariant subspaces for operators commuting with a compact operator have been generalized in different directions (for example see [2], [7], [8], [9]). The main result of [9] (see also [7]) is: If is a norm closed algebra of (bounded) operators on an infinite dimensional (complex) Banach space , if K is a nonzero compact operator on , and if then has a non-trivial (closed) invariant subspace. In [7], it is mentioned that the above result holds if instead of compactness for K we assume that K is a non-invertible injective operator with a non-zero eigenvalue belonging to the class of decomposable, hyponormal, or subspectral operators.

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 Multipliers on Vector Valued Bergman Spaces

2002 ◽  
Vol 54 (6) ◽  
pp. 1165-1186 ◽  
Author(s):  
Oscar Blasco ◽  
José Luis Arregui
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Bergman Space ◽  
Unit Disc ◽  
Bergman Spaces ◽  
Complex Banach Space ◽  
Bounded Operators ◽  
Taylor Coefficients ◽  
Vector Valued

AbstractLet X be a complex Banach space and let Bp(X) denote the vector-valued Bergman space on the unit disc for 1 ≤ p < ∞. A sequence (Tn)n of bounded operators between two Banach spaces X and Y defines a multiplier between Bp(X) and Bq(Y) (resp. Bp(X) and lq(Y)) if for any function we have that belongs to Bq(Y) (resp. (Tn(xn))n ∈ lq(Y)). Several results on these multipliers are obtained, some of them depending upon the Fourier or Rademacher type of the spaces X and Y. New properties defined by the vector-valued version of certain inequalities for Taylor coefficients of functions in Bp(X) are introduced.

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 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 Operadores de riesz en el Alglat(T)∩{T}

2021 ◽  
Vol 6 (1) ◽  
pp. 49
Author(s):  
Edixo Rosales
Keyword(s):  
Banach Space ◽  
Invariant Subspaces ◽  
Bounded Operators ◽  
Riesz Operator ◽  
Palabras Clave ◽  
Bounded Below

  En este trabajo X es un espacio de Banach y B(X) denota los operadores acotados. Si T∈B(X), por lat(T) entenderemos los subespacios invariantes por T. Se dice que T es lleno, si (T(M)) ̅=M, para todo M∈lat(T) (la barra indica la clausura en la topología inducida por la norma). Se prueba principalmente el siguiente resultado: Sean X un espacio de Banach y T ∈B(X) acotado por abajo. Sea K ∈Alglat(T)∩{T}' un operador de Riesz. Si K es lleno, entonces T es lleno. Aquí Alglat(T)={S∈B(X):M∈lat(T)⟾M∈lat(S)} y {T}^'={S∈B(X):S∘T=T∘S}.   Palabras clave: Operador lleno, operador de Riesz, operador acotado por abajo.   Abstract In this work X is a Banach space and B(X) denotes the bounded operators. If T ∈B(X), for lat(T) we will understand the invariant subspaces for T. An operator T is full, if (T(M)) ̅=M, for all M∈ latT (the bar indicates the closure in the topology induced by the norm). The following result is true: Let X be a Banach space, T ∈B(X) a bounded below operator and K ∈Alglat(T)∩{T}' a Riesz operator: If K is a full operator, then T is a full operator. Here Alglat(T)={S∈B(X):M∈lat(T)⟾M∈lat(S)} and {T}^'={S∈B(X):S∘T=T∘S}.   Keywords: full operator, Riesz operator, bounded below operator.  

Near Triangularizability Implies Triangularizability

2004 ◽  
Vol 47 (2) ◽  
pp. 298-313 ◽  
Author(s):  
Bamdad R. Yahaghi
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Compact Operators ◽  
Linear Operators ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Bounded Operators ◽  
Ground Field ◽  
Subspace Theorem ◽  
Finite Dimensional

AbstractIn this paper we consider collections of compact operators on a real or complex Banach space including linear operators on finite-dimensional vector spaces. We show that such a collection is simultaneously triangularizable if and only if it is arbitrarily close to a simultaneously triangularizable collection of compact operators. As an application of these results we obtain an invariant subspace theorem for certain bounded operators. We further prove that in finite dimensions near reducibility implies reducibility whenever the ground field is or .

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.

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