Asymptotically commuting families of operators

1990 ◽  
Vol 65 (1) ◽  
pp. 672-679
Author(s):  
Arthur Jaffe ◽  
Slawomir Klimek ◽  
Andrzej Lesniewski
Keyword(s):  
Families Of Operators

scholarly journals Existence of Mild Solutions to Nonlocal Fractional Cauchy Problems via Compactness

2016 ◽  
Vol 2016 ◽  
pp. 1-15 ◽  
Author(s):  
Rodrigo Ponce
Keyword(s):  
Banach Spaces ◽  
Fractional Derivatives ◽  
Mild Solutions ◽  
Cauchy Problems ◽  
Families Of Operators ◽  
Resolvent Families

We obtain characterizations of compactness for resolvent families of operators and as applications we study the existence of mild solutions to nonlocal Cauchy problems for fractional derivatives in Banach spaces. We discuss here simultaneously the Caputo and Riemann-Liouville fractional derivatives in the cases0<α<1and1<α<2.

Some families of operators

2016 ◽  
pp. 60-95
Author(s):  
Emmanuel Fricain ◽  
Javad Mashreghi
Keyword(s):  
Families Of Operators

scholarly journals On the invertibility of solutions of first order linear homogeneous differential equations in Banach algebras

2019 ◽  
Vol 21 (4) ◽  
pp. 430-442
Author(s):  
Oleg E. Galkin ◽  
Svetlana Y. Galkina
Keyword(s):  
Differential Equations ◽  
Banach Algebras ◽  
Foundations Of Quantum Mechanics ◽  
Continuous Functions ◽  
Heisenberg Equation ◽  
Complex Numbers ◽  
Bounded Operators ◽  
First Order ◽  
First Case ◽  
Families Of Operators

This work is devoted to the study of some properties of linear homogeneous differential equations of the first order in Banach algebras. It is found (for some types of Banach algebras), at what right-hand side of such an equation, from the invertibility of the initial condition it follows the invertibility of its solution at any given time. Associative Banach algebras over the field of real or complex numbers are considered. The right parts of the studied equations have the form [F(t)](x(t)), where {F(t)} is a family of bounded operators on the algebra, continuous with respect to t∈R. The problem is to find all continuous families of bounded operators on algebra, preserving the invertibility of elements from it, for a given Banach algebra. In the proposed article, this problem is solved for only three cases. In the first case, the algebra consists of all square matrices of a given order. For this algebra, it is shown that all continuous families of operators, preserving the invertibility of elements from the algebra at zero must be of the form [F(t)](y)=a(t)⋅y+y⋅b(t), where the families {a(t)} and {b(t)} are also continuous. In the second case, the algebra consists of all continuous functions on the segment. For this case, it is shown that all families of operators, preserving the invertibility of elements from the algebra at any time must be of the form [F(t)](y)=a(t)⋅y, where the family {a(t)} is also continuous. The third case concerns those Banach algebras in which all nonzero elements are invertible. For example, the algebra of complex numbers and the algebra of quaternions have this property. In this case, any continuous families of bounded operators preserves the invertibility of the elements from the algebra at any time. The proposed study is in contact with the research of the foundations of quantum mechanics. The dynamics of quantum observables is described by the Heisenberg equation. The obtained results are an indirect argument in favor of the fact, that the known form of the Heisenberg equation is the only correct one.

scholarly journals Approximation properties of Jain-Appell operators

2020 ◽  
pp. 44-44
Author(s):  
Mehmet Özarslan
Keyword(s):  
Approximation Property ◽  
Weighted Approximation ◽  
Lipschitz Class ◽  
Approximation Properties ◽  
Statistical Approximation ◽  
Special Cases ◽  
Families Of Operators

In this paper, we introduce the Jain-Appell operators by applying Gamma transform to the Jakimovski-Leviatan operators. In their special cases they include not only the Jain-Pethe operators, but also new families of operators, where we call them Appell-Baskakov and Appell-Lupa? operators, since their special cases contain Baskakov and Lupa? operators, respectively. We investigate their weighted approximation properties and compute the error of approximation by using certain Lipschitz class functions. Furthermore, we obtain their A-statistical approximation property.

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