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A new view of some operators and their properties in terms of the Non-Newtonian Calculus

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Ünlüyol E., Salas S., Iscan I.
Topological Algebra and its Applications
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2017
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tom 5
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nr 1
49-54
EN
Differentiation and integration are basic operations of calculus and analysis. Indeed, they are in- finitesimal versions of substraction and addition operations on numbers, respectively. From 1967 till 1970 Michael Grossman and Robert Katz gave definitions of a new kind of derivative and integral, converting the roles of substraction and addition into division and multiplication, respectively and thus established a new calculus, called Non-Newtonian Calculus. So in this paper, it is investigated to a new view of some operators and their properties in terms of Non-Newtonian Calculus. Then we compare with the Newtonian and Non-Newtonian Calculus.
2

An infinite dimensional Banach algebra with all but one maximal abelian subalgebras of dimension two

100%
Żelazko W.
Commentationes Mathematicae
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2008
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tom 48
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nr 1
EN
I construct a unital closed subalgebra of \(L(H)\) with the property announced in the title. Moreover, for any two maxiamal abelian subalgebras of the algebra in question, their intersection consists only of scalar multiples of the unity.
3
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Maximal abelian subalgebras of \(B(\mathcal{X})\)

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Bračič J., Kuzma B.
Commentationes Mathematicae
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2008
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tom 48
|
nr 1
EN
Let \(\mathcal{X}\) be an infinite dimensional complex Banach space and \(B(\mathcal{X})\) be the Banach algebra of all bounded linear operators on \(\mathcal{X}\). Żelazko [1] posed the following question: Is it possible that some maximal abelian subalgebra of \(B(\mathcal{X})\) is finite dimensional? Interestingly, he was able to show that there does exist an infinite dimensional closed subalgebra of \(B(\mathcal{X})\) with all but one maximal abelian subalgebras of dimension two. The aim of this note is to give a negative answer to the original question and prove that there does not exist a finite dimensional maximal commutative subalgebra of \(B(\mathcal{X})\) if \(\text{dim} X = \infty\).
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