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1.
If the vector space of all regular operators between the vector lattices E and F is ordered by the collection of its positive operators, then the Dedekind completeness of F is a sufficient condition for to be a vector lattice. and some of its subspaces might be vector lattices also in a more general situation. In the paper we deal with ordered vector
spaces of linear operators and ask under which conditions are they vector lattices, lattice-subspaces of the ordered vector space
or, in the case that is a vector lattice, sublattices or even Banach lattices when equipped with the regular norm. The answer is affirmative for
many classes of operators such as compact, weakly compact, regular AM-compact, regular Dunford-Pettis operators and others if acting between appropriate Banach lattices. Then it is possible to
study the finite elements in such vector lattices , where F is not necessary Dedekind complete. In the last part of the paper there will be considered the question how the order structures
of E, F and are mutually related. It is also shown that those rank one and finite rank operators, which are constructed by means of finite
elements from E′ and F, are finite elements in . The paper contains also some generalization of results obtained for the case in [10].
相似文献
2.
Egor A. Alekhno 《Positivity》2009,13(1):3-20
Let T be a positive operator on a Banach lattice E. Some properties of Weyl essential spectrum σew(T), in particular, the equality , where is the set of all compact operators on E, are established. If r(T) does not belong to Fredholm essential spectrum σef(T), then for every a ≠ 0, where T−1 is a residue of the resolvent R(., T) at r(T). The new conditions for which implies , are derived. The question when the relation holds, where is Lozanovsky’s essential spectrum, will be considered. Lozanovsky’s order essential spectrum is introduced. A number of
auxiliary results are proved. Among them the following generalization of Nikol’sky’s theorem: if T is an operator of index zero, then T = R + K, where R is invertible, K ≥ 0 is of finite rank. Under the natural assumptions (one of them is ) a theorem about the Frobenius normal form is proved: there exist T-invariant bands such that if
, where , then an operator on Di is band irreducible.
相似文献
3.
On the Range of the Aluthge Transform 总被引:1,自引:0,他引:1
Let
be the algebra of all bounded linear operators on a complex separable Hilbert space
For an operator
let
be the Aluthge transform of T and we define
for all
where T = U|T| is a polar decomposition of T. In this short note, we consider an elementary property of the range
of Δ. We prove that R(Δ) is neither closed nor dense in
However R(Δ) is strongly dense if
is infinite dimensional.
An erratum to this article is available at . 相似文献
4.
A CDCSL algebra is a reflexive operator algebra with completely distributive and commutative subspace lattice. In this paper,
we show, for a weakly closed linear subspace
of a CDCSL algebra
, that
is a Lie ideal if and only if
for all invertibles A in
, and that
is a Jordan ideal if and only if it is an associative ideal. 相似文献
5.
Some Properties of Essential Spectra of a Positive Operator 总被引:1,自引:1,他引:0
Egor A. Alekhno 《Positivity》2007,11(3):375-386
Let E be a Banach lattice, T be a bounded operator on E. The Weyl essential spectrum σew(T) of the operator T is a set
, where
is a set of all compact operators on E. In particular for a positive operator T next subsets of the spectrum
are introduced in the article. The conditions by which
implies either
or
are investigated, where σef(T) is the Fredholm essential spectrum. By this reason, the relations between coefficients of the main part of the Laurent series
of the resolvent R(., T) of a positive operator T around of the point λ = r(T) are studied. The example of a positive integral operator T : L1→ L∞ which doesn’t dominate a non-zero compact operator, is adduced. Applications of results which are obtained, to the spectral
theory of band irreducible operators, are given. Namely, the criteria when the operator inequalities 0 ≤ S < T imply the spectral radius inequality r(S) < r(T), are established, where T is a band irreducible abstract integral operator. 相似文献
6.
If
is an initially hereditary family of finite subsets of positive integers (i.e., if
and G is initial segment of F then
) and M an infinite subset of positive integers then we define an ordinal index
. We prove that if
is a family of finite subsets of positive integers such that for every
the characteristic function χF is isolated point of the subspace
of { 0,1 }N with the product topology then
for every
infinite, where
is the set of all initial segments of the members of
and ω1 is the first uncountable ordinal. As a consequence of this result we prove that
is Ramsey, i.e., if
is a partition of
then there exists an infinite subset M of positive integers such that
where [M]< ω is the family of all finite subsets of M. 相似文献
7.
Let X, Y be Banach spaces. We say that a set
is uniformly p–summing if the series
is uniformly convergent for
whenever (xn) belongs to
. We consider uniformly summing sets of operators defined on a
-space and prove, in case X does not contain a copy of c0, that
is uniformly summing iff
is, where T (φ x) = (T#φ) x for all
and x∈X. We also characterize the sets
with the property that
is uniformly summing viewed in
.
Received: 1 July 2005 相似文献
8.
In this paper self-adjoint realizations in Hilbert and Pontryagin spaces of the formal expression
are discussed and compared. Here L is a positive self-adjoint operator in a Hilbert space
with inner product 〈·,·〉, α is a real parameter, and φ in the rank one perturbation is a singular element belonging to
with n ≥ 3, where
is the scale of Hilbert spaces associated with L in
相似文献
9.
We define the reduced minimum modulus
of a nonzero element a in a unital C
*-algebra
by
. We prove that
. Applying this result to
and its closed two side ideal
, we get that dist
,
and
for any
if RR
= 0, where
and
is the quotient homomorphism and
. These results generalize corresponding results in Hilbert spaces. 相似文献
10.
Alejandra Maestripieri Francisco Martínez Pería 《Integral Equations and Operator Theory》2007,59(2):207-221
The aim of this work is to generalize the notions of Schur complements and shorted operators to Krein spaces. Given a (bounded)
J-selfadjoint operator A (with the unique factorization property) acting on a Krein space
and a suitable closed subspace
of
, the Schur complement
of A to
is defined. The basic properties of
are developed and different characterizations are given, most of them resembling those of the shorted of (bounded) positive
operators on a Hilbert space.
To the memory of Professor Mischa Cotlar 相似文献
11.
We prove Tolokonnikov’s Lemma and the inner-outer factorization for the real Hardy space
, the space of bounded holomorphic (possibly operator-valued) functions on the unit disc all of whose matrix-entries (with
respect to fixed orthonormal bases) are functions having real Fourier coefficients, or equivalently, each matrix entry f satisfies
for all z ∈
.
Tolokonnikov’s Lemma for
means that if f is left-invertible, then f can be completed to an isomorphism; that is, there exists an F, invertible in
, such that F = [ f f
c
] for some f
c
in
. In control theory, Tolokonnikov’s Lemma implies that if a function has a right coprime factorization over
, then it has a doubly coprime factorization in
. We prove the lemma for the real disc algebra
as well. In particular,
and
are Hermite rings.
The work of the first author was supported by Magnus Ehrnrooth Foundation.
Received: December 5, 2006. Revised: February 4, 2007. 相似文献
12.
Let X be a nonempty measurable subset of and consider the restriction of the usual Lebesgue measure σ of to X. Under the assumption that the intersection of X with every open ball of has positive measure, we find necessary and sufficient conditions on a L2(X)-positive definite kernel in order that the associated integral operator be nuclear. Taken nuclearity for granted, formulas for the trace of the operator are derived. Some of the results are re-analyzed
when K is just an element of .
相似文献
13.
D. Jakubíková-Studenovská 《Algebra Universalis》2009,60(2):125-143
In the present paper we prove that the collection of all convexities of partial monounary algebras is finite; namely, it has exactly 23 elements. Further, we show that for
each element there exists a subset of such that is generated by and card .
This work was supported by the Science and Technology Assistance Agency under the contract No. APVT-20-004104. Supported by
Grant VEGA 1/3003/06. 相似文献
14.
Yury M. Arlinskiĭ Seppo Hassi Henk S. V. de Snoo 《Complex Analysis and Operator Theory》2009,3(1):19-56
Passive systems with and as an input and output space and as a state space are considered in the case that the main operator on the state space is normal. Basic properties are given
and a general unitary similarity result involving some spectral theoretic conditions on the main operator is established.
A passive system with is said to be quasi-selfadjoint if ran . The subclass of the Schur class is the class formed by all transfer functions of quasi-selfadjoint passive systems. The subclass is characterized and minimal passive quasi-selfadjoint realizations are studied. The connection between the transfer function
belonging to the subclass and the Q-function of T is given.
Received: December 16, 2007., Accepted: March 4, 2008. 相似文献
15.
M. A. Bastos C. A. Fernandes Yu. I. Karlovich 《Integral Equations and Operator Theory》2006,55(1):19-67
We establish a symbol calculus for the C*-subalgebra
of
generated by the operators of multiplication by slowly oscillating and piecewise continuous functions and the operators
where
is the Cauchy singular integral operator and
The C*-algebra
is invariant under the transformations
where Uz is the rotation operator
Using the localtrajectory method, which is a natural generalization of the Allan-Douglas local principle to nonlocal type
operators, we construct symbol calculi and establish Fredholm criteria for the C*-algebra
generated by the operators
and
for the C*-algebra
generated by the operators
and
and for the C*-algebra
generated by the algebras
and
The C*-algebra
can be considered as an algebra of convolution type operators with piecewise slowly oscillating coefficients and shifts acting
freely. 相似文献
16.
María del Pilar Romero de la Rosa 《Positivity》2009,13(4):631-642
Let A be a bounded linear operator defined on a separable Banach space X. Then A is said to be supercyclic if there exists a vector x ∈ X (later called supercyclic for A), such that the projective orbit is dense in X. On the other hand, A is said to be positive supercyclic if for each supercyclic vector x, the positive projective orbit, is dense in X. Sometimes supercyclicity and positive supercyclicity are equivalent. The study of this relationship was initiated in [14]
by F. León and V. Müller. In this paper we study positive supercyclicity for operators A of the form , with , defined on . We will see that such a problem is related with the study of regular orbits. The notion of positive directions will be central
throughout the paper.
相似文献
17.
Christian Richter 《Journal of Geometry》2006,84(1-2):117-132
Let
be a group of affine transformations of the Euclidean plane
. Two topological discs D,
are called congruent by dissection with respect to
if D can be dissected into a finite number of subdiscs that can be rearranged by maps from
to a dissection of E.
Our main result says in particular that
admits congruence by dissection of any circular disc C with any square S if and only if
contains a contractive map and all orbits
,
, are dense in
. In this case any two discs D and E are congruent by dissection with respect to
and every disc D is congruent by dissection with n copies of D for every n ≥ 2.
Moreover, we give estimates on minimal numbers of pieces that are needed to realize congruences by dissection.
Dedicated to Irmtraud Stephani on the occasion of her 70th birthday 相似文献
18.
Let L and M be Archimedean vector lattices such that
and
are complex vector lattices. We constructively and intrinsically prove that if
is an order bounded disjointness preserving operator from
into
then the modulus
of
exists in the ordered vector space of all order bounded operators from L into M.
Received February 11, 2005; accepted in final form March 8, 2005. 相似文献
19.
We investigate R-bounded representations
, where X is a Banach space and G is a lca group. Observing that Ψ induces a (strongly continuous) group homomorphism
, we are then able to analyze certain classical homomorphisms U (e.g. translations in Lp (G)) from the viewpoint of R-boundedness and the theory of scalar-type spectral operators.
Dedicated to the memory of H. H. Schaefer 相似文献
20.
Let
be finite relational structure of finite type, and let CSP
denote the following decision problem: if
is a given structure of the same type as
, is there a homomorphism from
to
? To each relational structure
is associated naturally an algebra
whose structure determines the complexity of the associated decision problem. We investigate those finite algebras arising
from CSP’s of so-called bounded width, i.e., for which local consistency algorithms effectively decide the problem. We show that if a CSP has bounded width then
the variety generated by the associated algebra omits the Hobby-McKenzie types 1 and 2. This provides a method to prove that
certain CSP’s do not have bounded width. We give several applications, answering a question of Nešetřil and Zhu [26], by showing
that various graph homomorphism problems do not have bounded width. Feder and Vardi [17] have shown that every CSP is polynomial-time
equivalent to the retraction problem for a poset we call the Feder − Vardi poset of the structure. We show that, in the case where the structure has a single relation, if the retraction problem for the
Feder-Vardi poset has bounded width then the CSP for the structure also has bounded width. This is used to exhibit a finite
order-primal algebra whose variety admits type 2 but omits type 1 (provided P ≠ NP).
Presented by M. Valeriote.
Received January 8, 2005; accepted in final form April 3, 2006.
The first author’s research is supported by a grant from NSERC and the Centre de Recherches Mathématiques. The second author’s
research is supported by OTKA no. 034175 and 48809 and T 037877. Part of this research was conducted while the second author
was visiting Concordia University in Montréal and also when the first author was visiting the Bolyai Institute in Szeged.
The support of NSERC, OTKA and the Bolyai Institute is gratefully acknowledged. 相似文献