scholarly journals A NOTE ON LATTICE OPERATIONS OF RIESZ SPACES OF ORDER BOUNDED OPERATORS

1990 ◽  
Vol 21 (4) ◽  
pp. 395-398
Author(s):  
BORIS LAVRIČ
Keyword(s):  
Weak Form ◽  
Riesz Space ◽  
Linear Operators ◽  
Spectral Theorem ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Riesz Spaces

Let $L$, $M$ be Archimedean Riesz spaces with $M$ Dedekind complete, and let $\mathcal L_b(L,M )$ be the Riesz space of order bounded linear operators from $L$ into $M$. A theorem of Abramovic [1] on lattice operations of $\mathcal L_b(L,M )$ is generalized on Riesz spaces $L$ in which a weak form of Freudenthal's spectral theorem [4] holds.

scholarly journals Extension and inversion of extended orthomorphisms on Riesz spaces

1984 ◽  
Vol 37 (2) ◽  
pp. 223-242 ◽  
Author(s):  
Michel Duhoux ◽  
Mathieu Meyer
Keyword(s):  
Riesz Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Dense Ideal ◽  
Riesz Spaces ◽  
Dense Domain ◽  
And Inversion

AbstractLet E be an Archimedean Riesz space and let Orth∞(E) be the f-algebra consisting of all extended orthomorphisms on E, that is, of all order bounded linear operators T:D→E, with D an order dense ideal in E, such that T(B∩D) ⊆ B for every band B in E. We give conditions on E and on a Riesz subspace F of E insuring that every T ∈ Orth∞(F) can be extended to some ∈ Orth∞(E), and we also consider the problem of inversing an extended orthomorphism on its support. The same problems are also studied in the case of σ-orthomorphisms, that is, extended orthomorphisms with a super order dense domain. Furthermore, some applications are given.

scholarly journals A Study on Fuzzy Order Bounded Linear Operators in Fuzzy Riesz Spaces

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1512
Author(s):  
Juan Luis García Guirao ◽  
Mobashir Iqbal ◽  
Zia Bashir ◽  
Tabasam Rashid
Keyword(s):  
Riesz Space ◽  
Linear Operators ◽  
Separation Property ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Riesz Spaces ◽  
Fuzzy Order ◽  
Locally Convex ◽  
Special Case

This paper aims to study fuzzy order bounded linear operators between two fuzzy Riesz spaces. Two lattice operations are defined to make the set of all bounded linear operators as a fuzzy Riesz space when the codomain is fuzzy Dedekind complete. As a special case, separation property in fuzzy order dual is studied. Furthermore, we studied fuzzy norms compatible with fuzzy ordering (fuzzy norm Riesz space) and discussed the relation between the fuzzy order dual and topological dual of a locally convex solid fuzzy Riesz space.

Three Test Problems for Quasisimilarity

1987 ◽  
Vol 39 (4) ◽  
pp. 880-892 ◽  
Author(s):  
Hari Bercovici
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Structure Theory ◽  
Test Problems ◽  
Unitary Equivalence ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Densely Defined

Kaplansky proposed in [7] three problems with which to test the adequacy of a proposed structure theory of infinite abelian groups. These problems can be rephrased as test problems for a structure theory of operators on Hilbert space. Thus, R. Kadison and I. Singer answered in [6] these test problems for the unitary equivalence of operators. We propose here a study of these problems for quasisimilarity of operators on Hilbert space. We recall first that two (bounded, linear) operators T and T′ acting on the Hilbert spaces and , are said to be quasisimilar if there exist bounded operators and with densely defined inverses, satisfying the relations T′X = XT and TY = YT′. The fact that T and T′ are quasisimilar is indicated by T ∼ T′. The problems mentioned above can now be formulated as follows.

Ranges of Products of Operators

1974 ◽  
Vol 26 (6) ◽  
pp. 1430-1441 ◽  
Author(s):  
Sandy Grabiner
Keyword(s):  
Banach Space ◽  
Dual Problem ◽  
Bounded Operator ◽  
Linear Operators ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Closed Operators

Suppose that T and A are bounded linear operators. In this paper we examine the relation between the ranges of A and TA, under various additional hypotheses on T and A. We also consider the dual problem of the relation between the null-spaces of T and AT; and we consider some cases where T or A are only closed operators. Our major results about ranges of bounded operators are summarized in the following theorem.Theorem 1. Suppose that T is a bounded operator on a Banach space E and that A is a non-zero bounded operator from some Banach space to E.

scholarly journals The uniform boundedness principle for order bounded operators

1989 ◽  
Vol 12 (3) ◽  
pp. 487-492 ◽  
Author(s):  
Charles Swartz
Keyword(s):  
Uniform Boundedness ◽  
Linear Operators ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Uniform Boundedness Principle ◽  
Steinhaus Theorem

Under appropriate hypotheses on the spaces, it is shown that a sequence of order bounded linear operators which is pointwise order bounded is uniformly order bounded on order bounded subsets. This result is used to establish a Banach-Steinhaus Theorem for order bounded operators.

Commutators of Operators on Hilbert Space

1965 ◽  
Vol 17 ◽  
pp. 695-708 ◽  
Author(s):  
Arlen Brown ◽  
P. R. Halmos ◽  
Carl Pearcy
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Bilateral Shift

The purpose of this paper is to record some progress on the problem of determining which (bounded, linear) operators A on a separable Hilbert space H are commutators, in the sense that there exist bounded operators B and C on H satisfying A = BC — CB. It is thus natural to consider this paper as a continuation of the sequence (2; 3; 5). In §2 we show that many infinite diagonal matrices (with scalar entries) are commutators and that every weighted unilateral and bilateral shift is a commutator.

Strictly Singular and Cosingular Multiplications

2005 ◽  
Vol 57 (6) ◽  
pp. 1249-1278 ◽  
Author(s):  
Mikael Lindström ◽  
Eero Saksman ◽  
Hans-Olav Tylli
Keyword(s):  
Banach Space ◽  
Linear Operators ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Strictly Singular ◽  
Classical Banach Space

AbstractLet L(X) be the space of bounded linear operators on the Banach space X. We study the strict singularity and cosingularity of the two-sidedmultiplication operators S ↦ ASB on L(X), where A, B ∈ L(X) are fixed bounded operators and X is a classical Banach space. Let 1 < p < ∞and p ≠ 2. Our main result establishes that the multiplication S ↦ ASB is strictly singular on L(Lp(0, 1)) if and only if the non-zero operators A, B ∈ L(Lp(0, 1)) are strictly singular. We also discuss the case where X is a L1- or a L∞-space, as well as several other relevant examples.

Cline’s formula for g-Drazin inverses in a ring

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2249-2255
Author(s):  
Huanyin Chen ◽  
Marjan Abdolyousefi
Keyword(s):  
Operator Theory ◽  
Spectral Properties ◽  
Associative Ring ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Cline’S Formula

It is well known that for an associative ring R, if ab has g-Drazin inverse then ba has g-Drazin inverse. In this case, (ba)d = b((ab)d)2a. This formula is so-called Cline?s formula for g-Drazin inverse, which plays an elementary role in matrix and operator theory. In this paper, we generalize Cline?s formula to the wider case. In particular, as applications, we obtain new common spectral properties of bounded linear operators.

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