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 Some properties of shift operators on algebras generated by $*$-polynomials

2018 ◽  
Vol 10 (1) ◽  
pp. 206-212
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Banach Space ◽  
Holomorphic Functions ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Countable System ◽  
Shift Operators ◽  
Underlying Space ◽  
Algebra Of Functions ◽  
Algebras Of Holomorphic Functions ◽  
Proper Subalgebra

A $*$-polynomial is a function on a complex Banach space $X,$ which is a sum of so-called $(p,q)$-polynomials. In turn, for non-negative integers $p$ and $q,$ a $(p,q)$-polynomial is a function on $X,$ which is the restriction to the diagonal of some mapping, defined on the Cartesian power $X^{p+q},$ which is linear with respect to every of its first $p$ arguments and antilinear with respect to every of its other $q$ arguments. The set of all continuous $*$-polynomials on $X$ form an algebra, which contains the algebra of all continuous polynomials on $X$ as a proper subalgebra. So, completions of this algebra with respect to some natural norms are wider classes of functions than algebras of holomorphic functions. On the other hand, due to the similarity of structures of $*$-polynomials and polynomials, for the investigation of such completions one can use the technique, developed for the investigation of holomorphic functions on Banach spaces. We investigate the Frechet algebra of functions on a complex Banach space, which is the completion of the algebra of all continuous $*$-polynomials with respect to the countable system of norms, equivalent to norms of the uniform convergence on closed balls of the space. We establish some properties of shift operators (which act as the addition of some fixed element of the underlying space to the argument of a function) on this algebra. In particular, we show that shift operators are well-defined continuous linear operators. Also we prove some estimates for norms of values of shift operators. Using these results, we investigate one special class of functions from the algebra, which is important in the description of the spectrum (the set of all maximal ideals) of the algebra.

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.

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 The strict topology on spaces of bounded holomorphic functions

1994 ◽  
Vol 49 (2) ◽  
pp. 249-256 ◽  
Author(s):  
Juan Ferrera ◽  
Angeles Prieto
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Holomorphic Functions ◽  
Open Unit ◽  
Strict Topology ◽  
Open Unit Ball ◽  
Approximation By Polynomials ◽  
Bounded Holomorphic

We introduce in this paper the space of bounded holomorphic functions on the open unit ball of a Banach space endowed with the strict topology. Some good properties of this topology are obtained. As applications, we prove some results on approximation by polynomials and a description of the continuous homomorphisms.

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 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 Additive results for the generalized Drazin inverse

2002 ◽  
Vol 73 (1) ◽  
pp. 115-126 ◽  
Author(s):  
Dragan S. Djordjević ◽  
Yimin Wei
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Drazin Inverse ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Special Cases ◽  
Arbitrary Complex ◽  
Additive Perturbation ◽  
Generalized Drazin Inverse ◽  
Banach Space Operators

AbstractAdditive perturbation results for the generalized Drazin inverse of Banach space operators are presented. Precisely, if Ad denotes the generalized Drazin inverse of a bounded linear operator A on an arbitrary complex Banach space, then in some special cases (A + B)d is computed in terms of Ad and Bd. Thus, recent results of Hartwig, Wang and Wei (Linear Algebra Appl. 322 (2001), 207–217) are extended to infinite dimensional settings with simplified proofs.

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.

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