scholarly journals Multiplicative perturbations of local C-cosine functions

2021 ◽  
Vol 66 (3) ◽  
pp. 575-589
Author(s):  
Chung-Cheng Kuo ◽  
Nai-Sher Yeh
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Cosine Function ◽  
Additive Perturbation ◽  
Left And Right ◽  
Cosine Functions ◽  
Densely Defined

"We establish some left and right multiplicative perturbations of a local C-cosine function C(.) on a complex Banach space X with non-densely defined generator, which can be applied to obtain some new additive perturbation results concerning C(.)."

scholarly journals Perturbations of local C-cosine functions

2020 ◽  
Vol 65 (4) ◽  
pp. 585-597
Author(s):  
Chung-Cheng Kuo
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Complex Banach Space ◽  
Cosine Function ◽  
Bounded Linear ◽  
Cosine Functions

"We show that $\tA+\tB$ is a closed subgenerator of a local $\tC$-cosine function $\tT(\cdot)$ on a complex Banach space $\tX$ defined by $$\tT(t)x=\sum\limits_{n=0}^\infty \tB^n\int_0^tj_{n-1}(s)j_n(t-s)\tC(|t-2s|)xds$$ for all $x\in\tX$ and $0\leq t<T_0$, if $\tA$ is a closed subgenerator of a local $\tC$-cosine function $\tC(\cdot)$ on $\tX$ and one of the following cases holds: $(i)$ $\tC(\cdot)$ is exponentially bounded, and $\tB$ is a bounded linear operator on $\overline{\tD(\tA)}$ so that $\tB\tC=\tC\tB$ on $\overline{\tD(\tA)}$ and $\tB\tA\subset\tA\tB$; $(ii)$ $\tB$ is a bounded linear operator on $\overline{\tD(\tA)}$ which commutes with $\tC(\cdot)$ on $\overline{\tD(\tA)}$ and $\tB\tA\subset\tA\tB$; $(iii)$ $\tB$ is a bounded linear operator on $\tX$ which commutes with $\tC(\cdot)$ on $\tX$. Here $j_n(t)=\frac{t^n}{n!}$ for all $t\in\Bbb R$, and $$\int_0^tj_{-1}(s)j_0(t-s)\tC(|t-2s|)xds=\tC(t)x$$ for all $x\in\tX$ and $0\leq t<T_0$."

scholarly journals Integrated cosine functions

1996 ◽  
Vol 19 (3) ◽  
pp. 575-580 ◽  
Author(s):  
Quan Zheng
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Second Order ◽  
Basic Theory ◽  
Cosine Function ◽  
Integrated Semigroups ◽  
Approximation Theorems ◽  
The Relationship ◽  
Cosine Functions

In order to the second order Cauchy problem(CP2):x″(t)=Ax(t),x(0)=x∈D(An),x″(0)=y∈D(Am)on a Banach space, Arendt and Kellermann recently introduced the integrated cosine function. This paper is concerned with its basic theory, which contain some properties, perturbation and approximation theorems, the relationship to analytic integrated semigroups, interpolation and extrapolation theorems.

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.

An unbounded spectral mapping theorem

1968 ◽  
Vol 8 (1) ◽  
pp. 119-127 ◽  
Author(s):  
S. J. Bernau
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Closed Subset ◽  
Complex Banach Space ◽  
Complex Numbers ◽  
Spectral Mapping Theorem ◽  
Complex Polynomials ◽  
Spectral Mapping ◽  
Image Position ◽  
Densely Defined

Recall that the spectrum, σ(T), of a linear operator T in a complex Banach space is the set of complex numbers λ such that T—λI does not have a densely defined bounded inverse. It is known [7, § 5.1] that σ(T) is a closed subset of the complex plane C. If T is not bounded, σ(T) may be empty or the whole of C. If σ(T) ≠ C and T is closed the spectral mapping theorem, is valid for complex polynomials p(z) [7, §5.7]. Also, if T is closed and λ ∉ σ(T), (T–λI)−1 is everywhere defined.

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.

scholarly journals The Composition Operation on Spaces of Holomorphic Mappings

2020 ◽  
Vol 71 (2) ◽  
pp. 557-572
Author(s):  
María D Acosta ◽  
Pablo Galindo ◽  
Luiza A Moraes
Keyword(s):  
Banach Space ◽  
Complex Banach Space ◽  
Holomorphic Mappings ◽  
Fréchet Space ◽  
Frechet Space ◽  
Linear Subspaces ◽  
Left And Right ◽  
Relationship Of ◽  
The Relationship ◽  
Bounded Sets

Abstract We discuss the continuity of the composition on several spaces of holomorphic mappings on open subsets of a complex Banach space. On the Fréchet space of entire mappings that are bounded on bounded sets, the composition turns out to be even holomorphic. In such a space, we consider linear subspaces closed under left and right composition. We discuss the relationship of such subspaces with ideals of operators and give several examples of them. We also provide natural examples of spaces of holomorphic mappings where the composition is not continuous.

scholarly journals Semi-normal operators on uniformly smooth Banach spaces

1990 ◽  
Vol 32 (3) ◽  
pp. 273-276 ◽  
Author(s):  
Muneo Chō
Keyword(s):  
Banach Space ◽  
Linear Functional ◽  
Dual Space ◽  
Complex Banach Space ◽  
Linear Operators ◽  
Bounded Linear ◽  
Uniformly Smooth ◽  
Normal Operators ◽  
Uniformly Smooth Banach Spaces ◽  
The Relationship

In this paper we shall examine the relationship between the numerical ranges and the spectra for semi-normal operators on uniformly smooth spaces.Let X be a complex Banach space. We denote by X* the dual space of X and by B(X) the space of all bounded linear operators on X. A linear functional F on B(X) is called state if ∥F∥ = F(I) = 1. When x ε X with ∥x∥ = 1, we denoteD(x) = {f ε X*:∥f∥ = f(x) = l}.

The polynomial Daugavetian index of a complex Banach space

2018 ◽  
Vol 112 (4) ◽  
pp. 407-416
Author(s):  
Elisa R. Santos
Keyword(s):  
Banach Space ◽  
Complex Banach Space

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 Hypercyclic and chaotic integrated C-cosine functions

Filomat ◽  
2012 ◽  
Vol 26 (1) ◽  
pp. 1-44 ◽  
Author(s):  
Marko Kostic
Keyword(s):  
Structural Properties ◽  
Theoretical Approach ◽  
Cosine Function ◽  
Cosine Functions

The main purpose of the paper is to display the main structural properties of hypercyclic and chaotic integrated C-cosine functions. The notions of hypercyclicity, mixing and chaoticity of an ?-times integrated C-cosine function (??0) are defined by using distributional techniques. We provide several examples which justify our abstract theoretical approach.

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