scholarly journals Strictly diagonal holomorphic functions on Banach spaces

2015 ◽  
Vol 7 (2) ◽  
pp. 254-258
Author(s):  
O.I. Fedak ◽  
A.V. Zagorodnyuk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Special Form ◽  
Holomorphic Functions ◽  
Fixed Radius ◽  
Uniformly Continuous

In this paper we investigate the boundedness of holomorphic functionals on a Banach space with a normalized basis $\{e_n\}$ which have a very special form $f(x)=f(0)+\sum_{n=1}^\infty c_nx_n^n$ and which we call strictly diagonal. We consider under which conditions strictly diagonal functions are entire and uniformly continuous on every ball of a fixed radius.

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.

APPROXIMATIONS OF HOLOMORPHIC FUNCTIONS IN CERTAIN BANACH SPACES

2004 ◽  
Vol 15 (05) ◽  
pp. 467-471 ◽  
Author(s):  
BENGT JOSEFSON
Keyword(s):  
Banach Space ◽  
Entire Function ◽  
Banach Spaces ◽  
Unconditional Basis ◽  
Holomorphic Functions ◽  
Open Ball

Let E be a Banach space and let B(R)⊂E denote the open ball with centre at 0 and radius R. The following problem is studied: given 0<r<R, ∊>0 and a function f holomorphic on B(R), does there always exist an entire function g on E such that |f-g|<∊ on B(r)? L. Lempert has proved that the answer is positive for Banach spaces having an unconditional basis with unconditional constant 1. In this paper a somewhat shorter proof of Lemperts result is given. In general it is not possible to approximate f by polynomials since f does not need to be bounded on B(r).

scholarly journals Composition operators on some Möbius invariant Banach spaces

2000 ◽  
Vol 62 (1) ◽  
pp. 1-19 ◽  
Author(s):  
Shamil Makhmutov ◽  
Maria Tjani
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Unit Disk ◽  
Composition Operator ◽  
Composition Operators ◽  
Bloch Space ◽  
Holomorphic Functions ◽  
Mean Oscillation

We characterise the compact composition operators from any Mobius invariant Banach space to VMOA, the space of holomorphic functions on the unit disk U that have vanishing mean oscillation. We use this to obtain a characterisation of the compact composition operators from the Bloch space to VMOA. Finally, we study some properties of hyperbolic VMOA functions. We show that a function is hyperbolic VMOA if and only if it is the symbol of a compact composition operator from the Bloch space to VMOA.

scholarly journals Approximation by functions with bounded derivative on Banach spaces

2004 ◽  
Vol 69 (1) ◽  
pp. 125-131 ◽  
Author(s):  
R. Fry
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Open Subset ◽  
Separable Banach Space ◽  
Smooth Functions ◽  
Continuous Map ◽  
Smooth Norm ◽  
Uniformly Continuous ◽  
Bounded Derivative

Let X be a separable Banach space which admits a C1-smooth norm, and let G ⊂ X be an open subset. Then any real-valued, bounded and uniformly continuous map on G can be uniformly approximated on G by C1-smooth functions with bounded derivative.

scholarly journals On approximation of homomorphisms of algebras of entire functions on Banach spaces

2019 ◽  
Vol 11 (1) ◽  
pp. 158-162
Author(s):  
H.M. Pryimak
Keyword(s):  
Banach Space ◽  
Unit Ball ◽  
Banach Algebra ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Commutative Banach Algebra ◽  
Uniformly Continuous ◽  
Positive Results

It is known due to R. Aron, B. Cole and T. Gamelin that every complex homomorphism of the algebra of entire functions of bounded type on a Banach space $X$ can be approximated in some sense by a net of point valued homomorphism. In this paper we consider the question about a generalization of this result for the case of homomorphisms to any commutative Banach algebra $A.$ We obtained some positive results if $A$ is the algebra of uniformly continuous analytic functions on the unit ball of $X.$

scholarly journals Weighted Vector-Valued Holomorphic Functions on Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Enrique Jordá
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Holomorphic Functions ◽  
Connected Subset ◽  
Weighted Banach Spaces ◽  
Vector Valued Functions ◽  
Vector Valued

We study the weighted Banach spaces of vector-valued holomorphic functions defined on an open and connected subset of a Banach space. We use linearization results on these spaces to get conditions which ensure that a functionfdefined in a subsetAof an open and connected subsetUof a Banach spaceX, with values in another Banach spaceE, and admitting certain weak extensions in a Banach space of holomorphic functions can be holomorphically extended in the corresponding Banach space of vector-valued functions.

scholarly journals Generalized Numerical Index and Denseness of Numerical Peak Holomorphic Functions on a Banach Space

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Sung Guen Kim ◽  
Han Ju Lee
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Holomorphic Functions ◽  
Strong Peak ◽  
Numerical Index

The generalized numerical index of a Banach space is introduced, and its properties on certain Banach spaces are studied. Ed-dari's theorem on the numerical index is extended to the generalized index and polynomial numerical index of a Banach space. The denseness of numerical strong peak holomorphic functions is also studied.

scholarly journals Functional Calculus for the Series of Semigroup Generators Via Transference

2019 ◽  
pp. 57-84
Author(s):  
Shawgy Hussein ◽  
Simon Joseph ◽  
Ahmed Sufyan ◽  
Murtada Amin ◽  
Ranya Tahire ◽  
...  
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Half Plane ◽  
Functional Calculus ◽  
Holomorphic Functions ◽  
Transference Principle ◽  
Semigroup Generators ◽  
Bounded Holomorphic ◽  
Semigroup Generator

In this paper, apply an established transference principle to obtain the boundedness of certain functional calculi for the sequence of semigroup generators. It is proved that if be the sequence generates 0- semigroups on a Hilbert space, then for each the sequence of operators has bounded calculus for the closed ideal of bounded holomorphic functions on right half–plane. The bounded of this calculus grows at most logarithmically as. As a consequence decay at ∞. Then showed that each sequence of semigroup generator has a so-called (strong) m-bounded calculus for all m∈ℕ, and that this property characterizes the sequence of semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so-called semigroups, the Hilbert space results actually hold in general Banach spaces.

scholarly journals Kneser's theorem for differential equations in Banach spaces

1986 ◽  
Vol 33 (3) ◽  
pp. 419-434 ◽  
Author(s):  
Nikolaos S. Papageorgiou
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Vector Field ◽  
Differential Equations ◽  
Banach Spaces ◽  
Measure Of Noncompactness ◽  
Solution Set ◽  
The Cauchy Problem ◽  
Uniformly Continuous

We consider the Cauchy problem x (t) = f (t,x (t)), x (O) = xO defined in a nonreflexive Banach space and with the vector field f: T × X → X being weakly uniformly continuous. Using a compactness hypothesis that involves the weak measure of noncompactness, we prove that the solution set of the above Cauchy problem is nonempty, connected and compact in .

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