scholarly journals Triangularizing semigroups of positive operators on an atomic normed Riesz Space

2000 ◽  
Vol 43 (1) ◽  
pp. 43-55 ◽  
Author(s):  
Roman Drnovšek
Keyword(s):  
Positive Operator ◽  
Riesz Space ◽  
Positive Operators ◽  
Closed Ideal ◽  
Bounded Operators ◽  
Order Continuous Norm ◽  
Continuous Norm ◽  
Multiplicative Semigroup ◽  
A Chain ◽  
Order Continuous

AbstractIn the first part of the paper we prove several results on the existence of invariant closed ideals for semigroups of bounded operators on a normed Riesz space (of dimension greater than 1) possessing an atom. For instance, if S is a multiplicative semigroup of positive operators on such space that are locally quasinilpotent at the same atom, then S has a non-trivial invariant closed ideal. Furthermore, if T is a non-zero positive operator that is quasinilpotent at an atom and if S is a multiplicative semigroup of positive operators such that TS ≤ ST for all S ∈ S, then S and T have a common non-trivial invariant closed ideal. We also give a simple example of a quasinilpotent compact positive operator on the Banach lattice l∞ with no non-trivial invariant band.The second part is devoted to the triangularizability of collections of operators on an atomic normed Riesz space L. For a semigroup S of quasinilpotent, order continuous, positive, bounded operators on L we determine a chain of invariant closed bands. If, in addition, L has order continuous norm, then this chain is maximal in the lattice of all closed subspaces of L.

M-embedded symmetric operator spaces and the derivation problem

2019 ◽  
Vol 169 (3) ◽  
pp. 607-622
Author(s):  
JINGHAO HUANG ◽  
GALINA LEVITINA ◽  
FEDOR SUKOCHEV
Keyword(s):  
Symmetric Spaces ◽  
Symmetric Operator ◽  
Von Neumann Algebra ◽  
Bounded Operators ◽  
Order Continuous Norm ◽  
Continuous Norm ◽  
Unbounded Operators ◽  
Von Neumann ◽  
Symmetric Operator Spaces ◽  
Order Continuous

AbstractLet ℳ be a semifinite von Neumann algebra with a faithful semifinite normal trace τ. Assume that E(0, ∞) is an M-embedded fully symmetric function space having order continuous norm and is not a superset of the set of all bounded vanishing functions on (0, ∞). In this paper, we prove that the corresponding operator space E(ℳ, τ) is also M-embedded. It extends earlier results by Werner [48, Proposition 4∙1] from the particular case of symmetric ideals of bounded operators on a separable Hilbert space to the case of symmetric spaces (consisting of possibly unbounded operators) on an arbitrary semifinite von Neumann algebra. Several applications are given, e.g., the derivation problem for noncommutative Lorentz spaces ℒp,1(ℳ, τ), 1 < p < ∞, has a positive answer.

Converses for the Dodds–Fremlin and Kalton–Saab theorems

1996 ◽  
Vol 120 (1) ◽  
pp. 175-179 ◽  
Author(s):  
A. W. Wickstead
Keyword(s):  
Positive Operator ◽  
Banach Lattices ◽  
Order Continuous Norm ◽  
Continuous Norm ◽  
Full Generality ◽  
Order Continuous

The two theorems in the title give conditions on Banach lattices E and F under which a positive operator from E into F, dominated by another positive operator with some property, must also have that property. The Dodds-Fremlin theorem says that this is true for compactness provided both E′ and F have order continuous norms, whilst the Kalton–Saab theorem establishes such a result for Dunford–Pettis operators provided F has an order continuous norm. These results were originally provided, in their full generality, in [3] and [5], respectively, whilst very readable proofs may be found in chapter 5 of [2] or §3·7 of [6].

scholarly journals Convex semigroups on $$L^p$$-like spaces

2021 ◽  
Author(s):  
Robert Denk ◽  
Michael Kupper ◽  
Max Nendel
Keyword(s):  
Semigroup Theory ◽  
Banach Lattices ◽  
Heat Equations ◽  
Well Posedness ◽  
Strong Continuity ◽  
Order Continuous Norm ◽  
Continuous Norm ◽  
Typical Application ◽  
Convex Case ◽  
Order Continuous

AbstractIn this paper, we investigate convex semigroups on Banach lattices with order continuous norm, having $$L^p$$ L p -spaces in mind as a typical application. We show that the basic results from linear $$C_0$$ C 0 -semigroup theory extend to the convex case. We prove that the generator of a convex $$C_0$$ C 0 -semigroup is closed and uniquely determines the semigroup whenever the domain is dense. Moreover, the domain of the generator is invariant under the semigroup, a result that leads to the well-posedness of the related Cauchy problem. In a last step, we provide conditions for the existence and strong continuity of semigroup envelopes for families of $$C_0$$ C 0 -semigroups. The results are discussed in several examples such as semilinear heat equations and nonlinear integro-differential equations.

scholarly journals Products of Positive Operators

2021 ◽  
Vol 15 (2) ◽  
Author(s):  
Maximiliano Contino ◽  
Michael A. Dritschel ◽  
Alejandra Maestripieri ◽  
Stefania Marcantognini
Keyword(s):  
Positive Operator ◽  
Hilbert Spaces ◽  
Spectral Properties ◽  
Compact Operators ◽  
Positive Operators ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Special Classes

AbstractOn finite dimensional spaces, it is apparent that an operator is the product of two positive operators if and only if it is similar to a positive operator. Here, the class $${\mathcal {L}^{+\,2}}$$ L + 2 of bounded operators on separable infinite dimensional Hilbert spaces which can be written as the product of two bounded positive operators is studied. The structure is much richer, and connects (but is not equivalent to) quasi-similarity and quasi-affinity to a positive operator. The spectral properties of operators in $${\mathcal {L}^{+\,2}}$$ L + 2 are developed, and membership in $${\mathcal {L}^{+\,2}}$$ L + 2 among special classes, including algebraic and compact operators, is examined.

scholarly journals Kadec–Klee properties of Orlicz–Lorentz sequence spaces equipped with the Orlicz norm

Positivity ◽  
2021 ◽  
Author(s):  
Yunan Cui ◽  
Paweł Foralewski ◽  
Henryk Hudzik ◽  
Radosław Kaczmarek
Keyword(s):  
Sufficient Conditions ◽  
Sequence Spaces ◽  
Order Continuous Norm ◽  
Continuous Norm ◽  
Orlicz Norm ◽  
Full Characterization ◽  
Orlicz Functions ◽  
Necessary And Sufficient ◽  
Order Continuous

AbstractThe necessary and sufficient conditions for both the Kadec–Klee property as well as the Kadec–Klee property with respect to the coordinatewise convergence in Orlicz–Lorentz sequence spaces equipped with the Orlicz norm and generated by arbitrary Orlicz functions as well as any non-increasing weight sequences are given. Moreover, for their subspaces of elements with an order continuous norm the full characterization of the Kadec–Klee property with respect to the coordinatewise convergence is presented. Some tools useful in the proofs of the main results are also provided.

scholarly journals On the Lattice Properties of Almost L-Weakly and Almost M-Weakly Compact Operators

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Barış Akay ◽  
Ömer Gök
Keyword(s):  
Linear Span ◽  
Compact Operators ◽  
Approximation Properties ◽  
Order Continuous Norm ◽  
Continuous Norm ◽  
Weakly Compact ◽  
Domination Property ◽  
Weakly Compact Operators ◽  
Lattice Approximation ◽  
Order Continuous

We establish the domination property and some lattice approximation properties for almost L-weakly and almost M-weakly compact operators. Then, we consider the linear span of positive almost L-weakly (resp., almost M-weakly) compact operators and give results about when they form a Banach lattice and have an order continuous norm.

scholarly journals Approximation of positive operators by analytic vectors

2020 ◽  
Vol 12 (2) ◽  
pp. 412-418
Author(s):  
M.I. Dmytryshyn
Keyword(s):  
Banach Space ◽  
Positive Operator ◽  
Invariant Subspaces ◽  
Positive Operators ◽  
Interpolation Spaces ◽  
Jackson Type ◽  
Normalization Factor ◽  
Approximation Spaces ◽  
Similar Role ◽  
Approximation Errors

We give the estimates of approximation errors while approximating of a positive operator $A$ in a Banach space by analytic vectors. Our main results are formulated in the form of Bernstein and Jackson type inequalities with explicitly calculated constants. We consider the classes of invariant subspaces ${\mathcal E}_{q,p}^{\nu,\alpha}(A)$ of analytic vectors of $A$ and the special scale of approximation spaces $\mathcal {B}_{q,p,\tau}^{s,\alpha}(A)$ associated with the complex degrees of positive operator. The approximation spaces are determined by $E$-functional, that plays a similar role as the module of smoothness. We show that the approximation spaces can be considered as interpolation spaces generated by $K$-method of real interpolation. The constants in the Bernstein and Jackson type inequalities are expressed using the normalization factor.

Preduals and Nuclear Operators Associated with Bounded, p-Convex, p-Concave and Positive p-Summing Operators

2007 ◽  
Vol 59 (3) ◽  
pp. 614-637 ◽  
Author(s):  
C. C. A. Labuschagne
Keyword(s):  
Banach Spaces ◽  
Positive Operators ◽  
Banach Lattices ◽  
Bounded Operators ◽  
Bounded Linear ◽  
Nuclear Operators ◽  
Linear Maps

AbstractWe use Krivine's form of the Grothendieck inequality to renorm the space of bounded linear maps acting between Banach lattices. We construct preduals and describe the nuclear operators associated with these preduals for this renormed space of bounded operators as well as for the spaces of p-convex, p-concave and positive p-summing operators acting between Banach lattices and Banach spaces. The nuclear operators obtained are described in terms of factorizations through classical Banach spaces via positive operators.

Order-continuous extension of a positive operator

1989 ◽  
Vol 29 (5) ◽  
pp. 707-716
Author(s):  
G. P. Akilov ◽  
E. V. Kolesnikov ◽  
A. G. Kusraev
Keyword(s):  
Positive Operator ◽  
Continuous Extension ◽  
Order Continuous
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