scholarly journals On the semigroup of bounded C1-mappings

1973 ◽  
Vol 15 (2) ◽  
pp. 129-137
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Upper Bound ◽  
Real Banach Space ◽  
Continuous Mappings ◽  
Image Position

Let E be a real Banach space. If f: E→E is (Fréchet-) differentiable at every point of E, the derivative of f at x is denoted by f'(x), which is an element of the Banach algebra ℒ=ℒ(E) of all linear continuous mappings of E into itself with the usual upper bound norm, and, if we put , we have .

scholarly journals Some inequalities for norm unitaries in Banach algebras

1978 ◽  
Vol 21 (1) ◽  
pp. 17-23 ◽  
Author(s):  
M. J. Crabb ◽  
J. Duncan
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Unit Circle ◽  
Banach Algebras ◽  
Bounded Operators ◽  
Image Position ◽  
Unital Banach Algebra

Let A be a complex unital Banach algebra. An element u∈A is a norm unitary if(For the algebra of all bounded operators on a Banach space, the norm unitaries arethe invertible isometries.) Given a norm unitary u∈A, we have Sp(u)⊃Γ, where Sp(u) denotes the spectrum of u and Γ denotes the unit circle in C. If Sp(u)≠Γ we may suppose, by replacing eiθu, that . Then there exists h ∈ A such that

scholarly journals A note on d-ideals in some near-algebras

1967 ◽  
Vol 7 (2) ◽  
pp. 129-134 ◽  
Author(s):  
Sadayuki Yamamuro
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Algebra ◽  
Real Banach Space ◽  
Continuous Linear ◽  
Algebraic Operations

Let E be a real Banach space. The set of all continuous linear mappings of E into E is a Banach algebra under the usual algebraic operations and the operator bound as norm. We denote this Banach algebra by ℒ, if E is a separate Hilbert space.

scholarly journals On the average distance property in finite dimensional real Banach spaces

1995 ◽  
Vol 51 (1) ◽  
pp. 87-101 ◽  
Author(s):  
Reinhard Wolf
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Positive Real Number ◽  
Real Banach Space ◽  
Average Distance ◽  
Positive Real ◽  
Finite Dimensional ◽  
Distance Property ◽  
Average Distance Property ◽  
Image Position

The average distance Theorem of Gross implies that for each N-dimensional real Banach space E (N ≥ 2) there is a unique positive real number r(E) with the following property: for each positive integer n and for all (not necessarily distinct) x1, x2, …, xn, in E with ‖x1‖ = ‖x2‖ = … = ‖xn‖ = 1, there exists an x in E with ‖x‖ = 1 such that.In this paper we prove that if E has a 1-unconditional basis then r(E)≤2−(l/N) and equality holds if and only if E is isometrically isomorphic to Rn equipped with the usual 1-norm.

ON A MEASURE ZERO STABILITY PROBLEM OF A CYCLIC EQUATION

2015 ◽  
Vol 93 (2) ◽  
pp. 272-282 ◽  
Author(s):  
JAEYOUNG CHUNG ◽  
JOHN MICHAEL RASSIAS
Keyword(s):  
Banach Space ◽  
Functional Equation ◽  
Lebesgue Measure ◽  
Stability Problem ◽  
Cyclic Subgroup ◽  
Real Banach Space ◽  
Measure Zero ◽  
Stability Theorem ◽  
Image Position ◽  
Dimensional Lebesgue Measure

Let $G$ be a commutative group, $Y$ a real Banach space and $f:G\rightarrow Y$. We prove the Ulam–Hyers stability theorem for the cyclic functional equation $$\begin{eqnarray}\displaystyle \frac{1}{|H|}\mathop{\sum }_{h\in H}f(x+h\cdot y)=f(x)+f(y) & & \displaystyle \nonumber\end{eqnarray}$$ for all $x,y\in {\rm\Omega}$, where $H$ is a finite cyclic subgroup of $\text{Aut}(G)$ and ${\rm\Omega}\subset G\times G$ satisfies a certain condition. As a consequence, we consider a measure zero stability problem of the functional equation $$\begin{eqnarray}\displaystyle \frac{1}{N}\mathop{\sum }_{k=1}^{N}f(z+{\it\omega}^{k}{\it\zeta})=f(z)+f({\it\zeta}) & & \displaystyle \nonumber\end{eqnarray}$$ for all $(z,{\it\zeta})\in {\rm\Omega}$, where $f:\mathbb{C}\rightarrow Y,\,{\it\omega}=e^{2{\it\pi}i/N}$ and ${\rm\Omega}\subset \mathbb{C}^{2}$ has four-dimensional Lebesgue measure $0$.

A Generalization of Uniformly Rotund Banach Spaces

1979 ◽  
Vol 31 (3) ◽  
pp. 628-636 ◽  
Author(s):  
Francis Sullivan
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Boolean Algebra ◽  
Banach Spaces ◽  
Real Banach Space ◽  
Two Dimensional ◽  
Geometrical Characterization ◽  
Arbitrary Space ◽  
Image Position

Let X be a real Banach space. According to von Neumann's famous geometrical characterization X is a Hilbert space if and only if for all x, y ∈ XThus Hilbert space is distinguished among all real Banach spaces by a certain uniform behavior of the set of all two dimensional subspaces. A related characterization of real Lp spaces can be given in terms of uniform behavior of all two dimensional subspaces and a Boolean algebra of norm-1 projections [16]. For an arbitrary space X, one way of measuring the “uniformity” of the set of two dimensional subspaces is in terms of the real valued modulus of rotundity, i.e. for The space is said to be uniformly rotund if for each 0 we have .

A representation theorem for partially ordered Banach algebras

1968 ◽  
Vol 64 (1) ◽  
pp. 53-59 ◽  
Author(s):  
Kung-Fu Ng
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Representation Theorem ◽  
Banach Algebras ◽  
Positive Cone ◽  
Partially Ordered ◽  
Real Algebra ◽  
Empty Subset ◽  
Image Position ◽  
Ordered Banach Algebra

Let be a real algebra which is also a Banach space. Then is called a partially ordered Banach algebra if there is specified a non-empty subset of , called the positive cone, such thatand(A 5) is 1-normal and closed. (We recall that is said to be 1-normal if

Norm and spectral radius for algebraic elements of a Banach algebra

1980 ◽  
Vol 88 (1) ◽  
pp. 129-133 ◽  
Author(s):  
N. J. Young
Keyword(s):  
Banach Algebra ◽  
Spectral Radius ◽  
Upper Bound ◽  
Good Deal ◽  
Additional Information ◽  
Algebraic Element ◽  
Algebraic Elements ◽  
Image Position ◽  
Spectral Radius Formula ◽  
Unit Norm

The purpose of this note is to show that, for any algebraic element a of a Banach algebra and certain analytic functions f, one can give an upper bound for ‖f(a)‖ in terms of ‖a‖ and the spectral radius ρ(a) of a. To illustrate the nature of the result, consider the norms of powers of an element a of unit norm. In general, the spectral radius formulacontains all that can be said (that is, the limit ρ(a) can be approached arbitrarily slowly). If we have the additional information that a is algebraic of degree n we can say a good deal more. In the case of a C*-algebra we have the neat result that, if ‖a‖ ≤ 1,(see Theorem 2), while for a general Banach algebra we have at least

Nonlinear random equations with maximal monotone operators in Banach spaces

1985 ◽  
Vol 98 (3) ◽  
pp. 529-532 ◽  
Author(s):  
Dimitrios Kravvaritis
Keyword(s):  
Banach Space ◽  
Dual Space ◽  
Real Banach Space ◽  
Maximal Monotone ◽  
Maximal Monotone Operators ◽  
Monotone Operators ◽  
Measurable Space ◽  
Random Operator ◽  
Image Position ◽  
Monotone Type

Let X be a real Banach space, X* its dual space and ω a measurable space. Let D be a subset of X, L: Ω × D → X* a random operator and η:Ω →X* a measurable mapping. The random equation corresponding to the double [L, η] asks for a measurable mapping ξ: Ω → D such thatRandom equations with operators of monotone type have been studied recentely by Kannan and Salehi [7], Itoh [6] and Kravvarits [8].

scholarly journals A note on the modulus of convexity

1981 ◽  
Vol 22 (2) ◽  
pp. 157-158 ◽  
Author(s):  
Andrew T. Plant
Keyword(s):  
Banach Space ◽  
Global Minimum ◽  
Real Banach Space ◽  
Maximum Norm ◽  
Global Maximum ◽  
Modulus Of Convexity ◽  
Image Position

In [1, Corollary 5], Figiel gives an elegant demonstration that the modulus ofconvexity δ in real Banach space X is nondecreasing, whereIt is deduced from this that in fact δ(ɛ)/ɛ is nondecreasing [Proposition 3]. During the course of the proof [Lemma 4] it is stated that if v ∊ Sx is a local maximum on Sx of φ ∈Sx*, then v is a global maximum (φ(v) = 1). This is false; it could be that v is a global minimum. It is easy to construct such an example in R2 endowed with the maximum norm. What is true is that v is a global maximum of |φ|.

scholarly journals A Note on Finite-Dimensional Differentiable Mappings

1969 ◽  
Vol 9 (3-4) ◽  
pp. 405-408 ◽  
Author(s):  
K. J. Palmer ◽  
Sadayuki Yamamuro
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Continuous Linear ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Infinite Dimensional Banach Space ◽  
Image Position ◽  
Differentiable Mappings

Let E be a real infinite-dimensional Banach space. Let ℒ be the Banach algebra of all continuous linear mappings of E into itself with topology defined by the norm:

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