Adjoint Abelian Operators on Banach Space

1969 ◽  
Vol 21 ◽  
pp. 505-512 ◽  
Author(s):  
J. G. Stampfli
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Spectral Type ◽  
New Class ◽  
Definition Of ◽  
Adjoint Operators

I. In the first part of this paper we introduce a new class of operators, mentioned in the title. It is easy to say that these are a generalization of self-adjoint operators for Hilbert space. This is deceptive since it implies that the definition of self-adjointness is forced into the unnatural setting of a Banach space. We feel that the definition of adjoint abelian preserves the obvious distinction between a space and its dual. Certain attractive properties of self-adjoint operators have already been singled out and carried over to Banach space. Specifically, we mention the notion of hermitian (see 17; 11), and spectral type operators (see 4). There is some comparison of these concepts later.

scholarly journals On weak and strong convergence of an explicit iteration process for a total asymptotically quasi-I-nonexpansive mapping in Banach space

Filomat ◽  
2014 ◽  
Vol 28 (8) ◽  
pp. 1699-1710
Author(s):  
Hukmi Kiziltunc ◽  
Yunus Purtas
Keyword(s):  
Banach Space ◽  
Nonexpansive Mapping ◽  
Strong Convergence ◽  
Iterative Process ◽  
Iteration Process ◽  
Nonempty Closed Convex Subset ◽  
Weak And Strong Convergence ◽  
New Class ◽  
Convergence Results ◽  
Definition Of

In this paper, we introduce a new class of Lipschitzian maps and prove some weak and strong convergence results for explicit iterative process using a more satisfactory definition of self mappings. Our results approximate common fixed point of a total asymptotically quasi-I-nonexpansive mapping T and a total asymptotically quasi-nonexpansive mapping I, defined on a nonempty closed convex subset of a Banach space.

scholarly journals Fixed Point and Acute Point Theorems for New Mappings in a Banach Space

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Toshiharu Kawasaki
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Fixed Point ◽  
New Class ◽  
Space Concepts ◽  
Attractive Point

In a Hilbert space, concepts of attractive point and acute point are studied by many researchers. Moreover, these concepts extended to Banach space. In this paper, we introduce a new class of mappings on Banach space corresponding to the class of all widely more generalized hybrid mappings on Hilbert space. Moreover, we introduce some extensions of acute point and prove some acute point theorems.

scholarly journals On a Class of Generalized Nonexpansive Mappings

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1085 ◽  
Author(s):  
Simeon Reich ◽  
Alexander J. Zaslavski
Keyword(s):  
Hilbert Space ◽  
Nonexpansive Mapping ◽  
Fixed Points ◽  
Convex Set ◽  
Nonexpansive Mappings ◽  
General Function ◽  
New Class ◽  
Generalized Nonexpansive Mapping ◽  
Closed And Convex Set ◽  
Definition Of

In our recent work we have introduced and studied a notion of a generalized nonexpansive mapping. In the definition of this notion the norm has been replaced by a general function satisfying certain conditions. For this new class of mappings, we have established the existence of unique fixed points and the convergence of iterates. In the present paper we construct an example of a generalized nonexpansive self-mapping of a bounded, closed and convex set in a Hilbert space, which is not nonexpansive in the classical sense.

scholarly journals Commutativity, comonotonicity, and Choquet integration of self-adjoint operators

2018 ◽  
Vol 30 (10) ◽  
pp. 1850016 ◽  
Author(s):  
S. Cerreia-Vioglio ◽  
F. Maccheroni ◽  
M. Marinacci ◽  
L. Montrucchio
Keyword(s):  
Hilbert Space ◽  
Bounded Variation ◽  
Complex Hilbert Space ◽  
The Real ◽  
Choquet Expectation ◽  
Definition Of ◽  
Adjoint Operators

In this work, we propose a definition of comonotonicity for elements of [Formula: see text], i.e. bounded self-adjoint operators defined over a complex Hilbert space [Formula: see text]. We show that this notion of comonotonicity coincides with a form of commutativity. Intuitively, comonotonicity is to commutativity as monotonicity is to bounded variation. We also define a notion of Choquet expectation for elements of [Formula: see text] that generalizes quantum expectations. We characterize Choquet expectations as the real-valued functionals over [Formula: see text] which are comonotonic additive, [Formula: see text]-monotone, and normalized.

On Definition of Expansion Probabilistic Hilbert Space

2018 ◽  
Vol 14 (3) ◽  
pp. 59-73
Author(s):  
Ahmed Hasan Hamed ◽  
Keyword(s):  
Hilbert Space ◽  
Definition Of

scholarly journals A characterisation of Hilbert spaces via orthogonality and proximinality

2005 ◽  
Vol 71 (1) ◽  
pp. 107-111
Author(s):  
Fathi B. Saidi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Real Banach Space ◽  
Nonzero Element ◽  
Dimensional Subspace ◽  
Two Dimensional ◽  
Orthogonal Direction ◽  
Open Problems

In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.

Quasispectral subspaces of quasinilpotent operators

1984 ◽  
Vol 98 (3-4) ◽  
pp. 349-354 ◽  
Author(s):  
Matjaž Omladič
Keyword(s):  
Banach Space ◽  
Volterra Operator ◽  
Space Operator ◽  
Definition Of ◽  
Quasinilpotent Operators

SynopsisWe give a definition of “quasispectral maximalč subspaces for a quasinilpotent, but not nilpotent, bounded Banach space operator. The definition applies to a class of operators, close to the Volterra operator.

M. A. Krasnosel'skii Theorem and Iterative Methods for Solving Ill-Posed Linear Problems with a Self-Adjoint Operator

2015 ◽  
Vol 15 (3) ◽  
pp. 373-389
Author(s):  
Oleg Matysik ◽  
Petr Zabreiko
Keyword(s):  
Hilbert Space ◽  
Iterative Methods ◽  
Critical Case ◽  
Adjoint Operator ◽  
Successive Approximations ◽  
Operator Equations ◽  
Ill Posed ◽  
Linear Problems ◽  
Adjoint Operators ◽  
Linear Operator Equations

AbstractThe paper deals with iterative methods for solving linear operator equations ${x = Bx + f}$ and ${Ax = f}$ with self-adjoint operators in Hilbert space X in the critical case when ${\rho (B) = 1}$ and ${0 \in \operatorname{Sp} A}$. The results obtained are based on a theorem by M. A. Krasnosel'skii on the convergence of the successive approximations, their modifications and refinements.

scholarly journals An integral for a banach valued function

2009 ◽  
Vol 44 (1) ◽  
pp. 105-113
Author(s):  
Giuseppa Riccobono
Keyword(s):  
Banach Space ◽  
Bochner Integral ◽  
Definition Of

Abstract Using partitions of the unity ((PU)-partition), a new definition of an integral is given for a function f : [a, b] → X, where X is a Banach space, and it is proved that this integral is equivalent to the Bochner integral.

Sign in / Sign up

Export Citation Format

Share Document