On Reflexivity of Algebras

1981 ◽  
Vol 33 (6) ◽  
pp. 1291-1308 ◽  
Author(s):  
Mehdi Radjabalipour
Keyword(s):  
Hilbert Space ◽  
Natural Number ◽  
Separable Hilbert Space ◽  
Invariant Subspaces ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Closed Algebra ◽  
Direct Sum ◽  
Weakly Closed

For each natural number n we define to be the class of all weakly closed algebras of (bounded linear) operators on a separable Hilbert space H such that the lattice of invariant subspaces of and (alg lat )(n) are the same. (If A is an operator, A(n) denotes the direct sum of n copies of A; if is a collection of operators,. Also, alg lat denotes the algebra of all operators leaving all invariant subspaces of invariant.) In the first section we show that . In Section 2 we prove that every weakly closed algebra containing a maximal abelian self adjoint algebra (m.a.s.a.) is , and that . It is also shown that certain algebras containing a m.a.s.a. are necessarily reflexive.

Generators of Nest Algebras

1974 ◽  
Vol 26 (3) ◽  
pp. 565-575 ◽  
Author(s):  
W. E. Longstaff
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Nest Algebra ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Closed Algebra ◽  
Nest Algebras ◽  
Weakly Closed

A collection of subspaces of a Hilbert space is called a nest if it is totally ordered by inclusion. The set of all bounded linear operators leaving invariant each member of a given nest forms a weakly-closed algebra, called a nest algebra. Nest algebras were introduced by J. R. Ringrose in [9]. The present paper is concerned with generating nest algebras as weakly-closed algebras, and in particular with the following question which was first raised by H. Radjavi and P. Rosenthal in [8], viz: Is every nest algebra on a separable Hilbert space generated, as a weakly-closed algebra, by two operators? That the answer to this question is affirmative is proved by first reducing the problem using the main result of [8] and then by using a characterization of nests due to J. A. Erdos [2].

scholarly journals Characterizations of woven frames

2020 ◽  
Vol 18 (05) ◽  
pp. 2050033
Author(s):  
A. Bhandari ◽  
S. Mukherjee
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Universal Bounds

In a separable Hilbert space [Formula: see text], two frames [Formula: see text] and [Formula: see text] are said to be woven if there are constants [Formula: see text] so that for every [Formula: see text], [Formula: see text] forms a frame for [Formula: see text] with the universal bounds [Formula: see text]. This paper provides methods of constructing woven frames. In particular, bounded linear operators are used to create woven frames from a given frame. Several examples are discussed to validate the results. Moreover, the notion of woven frame sequences is introduced and characterized.

Compact Operators in Reductive Algebras

1975 ◽  
Vol 27 (1) ◽  
pp. 152-154
Author(s):  
Edward A. Azoff
Keyword(s):  
Hilbert Space ◽  
Invariant Subspaces ◽  
Compact Operators ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Term Algebra

Let be a Hilbert space and denote the collection of (bounded, linear) operators on by . Throughout this paper, the term ‘algebra’ will refer to a subalgebra of ; unless otherwise stated, it will not be assumed to contain I or to be closed in any topology.An algebra is said to be transitive if it has no non-trivial invariant subspaces. The following lemma has revolutionized the study of transitive algebras. For a pr∞f and a general discussion of its implications, the reader is referred to [5].

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.

On the Calkin Algebra and the Covering Homotopy Property, II

1977 ◽  
Vol 29 (1) ◽  
pp. 210-215
Author(s):  
John B. Conway
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Compact Operators ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Calkin Algebra ◽  
Homotopy Property ◽  
The Ideal

For a separable Hilbert space is the algebra of bounded linear operators on is the ideal of compact operators, and Π is the natural map of onto the Calkin algebra .

Nonlinear ∗-Jordan triple derivations on von Neumann algebras

2018 ◽  
Vol 68 (1) ◽  
pp. 163-170 ◽  
Author(s):  
Fangfang Zhao ◽  
Changjing Li
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
New Product ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Von Neumann

AbstractLetB(H) be the algebra of all bounded linear operators on a complex Hilbert spaceHand 𝓐 ⊆B(H) be a von Neumann algebra with no central summands of typeI1. ForA,B∈ 𝓐, define byA∙B=AB+BA∗a new product ofAandB. In this article, it is proved that a map Φ: 𝓐 →B(H) satisfies Φ(A∙B∙C) = Φ(A) ∙B∙C+A∙ Φ(B) ∙C+A∙B∙Φ(C) for allA,B,C∈ 𝓐 if and only if Φ is an additive *-derivation.

Free E0-Semigroups

1995 ◽  
Vol 47 (4) ◽  
pp. 744-785 ◽  
Author(s):  
Neal J. Fowler
Keyword(s):  
Hilbert Space ◽  
Free Product ◽  
Continuous Semigroup ◽  
Fock Space ◽  
Linear Operators ◽  
Free Flow ◽  
Strongly Continuous Semigroup ◽  
Bounded Linear Operators ◽  
Bounded Linear

AbstractGiven a strongly continuous semigroup of isometries ∪ acting on a Hilbert space ℋ, we construct an E0-semigroup α∪, the free E0-semigroup over ∪, acting on the algebra of all bounded linear operators on full Fock space over ℋ. We show how the semigroup αU⊗V can be regarded as the free product of α∪ and αV. In the case where U is pure of multiplicity n, the semigroup au, called the Free flow of rank n, is shown to be completely spatial with Arveson index +∞. We conclude that each of the free flows is cocycle conjugate to the CAR/CCR flow of rank +∞.

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.

scholarly journals Multiparameter spectral theory and Taylor's joint spectrum in Hilbert space

1988 ◽  
Vol 31 (1) ◽  
pp. 127-144 ◽  
Author(s):  
B. P. Rynne
Keyword(s):  
Hilbert Space ◽  
Spectral Theory ◽  
Linear Operators ◽  
Coupled Systems ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Joint Spectrum ◽  
Image Position

Let n≧1 be an integer and suppose that for each i= 1,…,n, we have a Hilbert space Hi and a set of bounded linear operators Ti, Vij:Hi→Hi, j=1,…,n. We define the system of operatorswhere λ=(λ1,…,λn)∈ℂn. Coupled systems of the form (1.1) are called multiparameter systems and the spectral theory of such systems has been studied in many recent papers. Most of the literature on multiparameter theory deals with the case where the operators Ti and Vij are self-adjoint (see [14]). The non self-adjoint case, which has received relatively little attention, is discussed in [12] and [13].

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