scholarly journals Integral decomposition of partial {$\ast$}-algebras of closed operators

1986 ◽  
Vol 22 (4) ◽  
pp. 689-724 ◽  
Author(s):  
Françoise D. Mathot
Keyword(s):  
Closed Operators ◽  
Integral Decomposition

Unbounded Linear Operators

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Hilbert Spaces ◽  
Linear Operators ◽  
Von Neumann ◽  
Limit Circle ◽  
Sectorial Operators ◽  
Closed Operators ◽  
The Stability ◽  
Symmetric Operators ◽  
Unbounded Linear Operators ◽  
Special Case

This chapter is concerned with closable and closed operators in Hilbert spaces, especially with the special classes of symmetric, J-symmetric, accretive and sectorial operators. The Stone–von Neumann theory of extensions of symmetric operators is treated as a special case of results for compatible adjoint pairs of closed operators. Also discussed in detail is the stability of closedness and self-adjointness under perturbations. The abstract results are applied to operators defined by second-order differential expressions, and Sims’ generalization of the Weyl limit-point, limit-circle characterization for symmetric expressions to J-symmetric expressions is proved.

Closed operators in semi-Hilbertian spaces

2021 ◽  
pp. 1-12
Author(s):  
Hamadi Baklouti ◽  
Sirine Namouri
Keyword(s):  
Closed Operators

scholarly journals Generalized Polar Decompositions for Closed Operators in Hilbert Spaces and Some Applications

2009 ◽  
Vol 64 (1) ◽  
pp. 83-113 ◽  
Author(s):  
Fritz Gesztesy ◽  
Mark Malamud ◽  
Marius Mitrea ◽  
Serguei Naboko
Keyword(s):  
Hilbert Spaces ◽  
Closed Operators

scholarly journals On the n-parameter abstract Cauchy problem

2004 ◽  
Vol 69 (3) ◽  
pp. 383-394
Author(s):  
M. Janfada ◽  
A. Niknam
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Unique Solution ◽  
Initial Value Problems ◽  
Abstract Cauchy Problem ◽  
Linear Operators ◽  
Uniqueness Of Solution ◽  
Initial Value ◽  
Bounded Linear ◽  
Closed Operators

Let Hi(i = 1, 2, …, n), be closed operators in a Banach space X. The generalised initialvalue problem of the abstract Cauchy problem is studied. We show that the uniqueness of solution u: [0, T1] × [0, T2] × … × [0, Tn] → X of this n-abstract Cauchy problem is closely related to C0-n-parameter semigroups of bounded linear operators on X. Also as another application of C0-n-parameter semigroups, we prove that many n-parameter initial value problems cannot have a unique solution for some initial values.

The concept of a filter

1967 ◽  
Vol 63 (1) ◽  
pp. 221-227 ◽  
Author(s):  
E. J. Hannan
Keyword(s):  
Linear Operator ◽  
Symmetric Space ◽  
Random Process ◽  
Response Function ◽  
Second Order ◽  
Closed Linear Operator ◽  
Direct Integral ◽  
Group Of Isometries ◽  
Direct Integral Decomposition ◽  
Integral Decomposition

AbstractIt is proved that for a second-order, homogeneous, random process on a globally symmetric space a filter, that is a closed linear operator which is invariant under a group of isometries of the space, may be fully described through a response function, that is that it has a direct integral decomposition into components which are scalar multiples of the identity.

The Asymptotic Ratio Set and Direct Integral Decompositions of a Von Neumann Algebra

1971 ◽  
Vol 23 (4) ◽  
pp. 598-607 ◽  
Author(s):  
Ole A. Nielsen
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Direct Integral ◽  
Von Neumann ◽  
Ratio Set ◽  
Almost All ◽  
Direct Integral Decomposition ◽  
Integral Decomposition ◽  
Remarkable Progress

The fact that any von Neumann algebra on a separable Hilbert space has an essentially unique direct integral decomposition into factors means that there is a global as well as a local aspect to any partial classification of von Neumann algebras. More precisely, suppose that J is a statement about von Neumann algebras which is either true or false for any given von Neumann algebra. Then a von Neumann algebra is said to satisfy J globally if it satisfies J, and to satsify J locally if almost all the factors appearing in some (and hence in any) central decomposition of it satisfy J . In a recent paper [3], H. Araki and E. J. Woods introduced the notion of the asymptotic ratio set of a factor, and by means of this they made remarkable progress in the classification of factors.

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