scholarly journals Generalized Derivations and Bilocal Jordan Derivations of Nest Algebras

2011 ◽  
Vol 2011 ◽  
pp. 1-6
Author(s):  
Dangui Yan ◽  
Chengchang Zhang
Keyword(s):  
Hilbert Space ◽  
Closed Subspace ◽  
Complex Hilbert Space ◽  
Generalized Derivations ◽  
Nest Algebra ◽  
Bounded Operators ◽  
Subspace Lattice ◽  
Nest Algebras

LetHbe a complex Hilbert space andB(H)the collection of all linear bounded operators,Ais the closed subspace lattice including 0 anH, thenAis a nest, accordingly algA={T∈B(H):TN⊆N,  ∀N∈A}is a nest algebra. It will be shown that of nest algebra, generalized derivations are generalized inner derivations, and bilocal Jordan derivations are inner derivations.

scholarly journals Nonlinear Isometries on Schatten-pClass in Atomic Nest Algebras

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Kan He ◽  
Qing Yuan
Keyword(s):  
Hilbert Space ◽  
Complete Characterization ◽  
Complex Hilbert Space ◽  
Nest Algebra ◽  
Nest Algebras ◽  
Surjective Isometry

LetHbe a complex Hilbert space; denote by Alg 𝒩and𝒞p(H)the atomic nest algebra associated with the atomic nest𝒩onHand the space of Schatten-pclass operators on,Hrespectively. Let𝒞p(H)∩Alg 𝒩be the space of Schatten-pclass operators in Alg 𝒩. When1≤p<+∞andp≠2, we give a complete characterization of nonlinear surjective isometries on𝒞p(H)∩Alg 𝒩. Ifp=2, we also prove that a nonlinear surjective isometry on𝒞2(H)∩Alg 𝒩is the translation of an orthogonality preserving map.

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 Generators of reflexive algebras

1975 ◽  
Vol 20 (2) ◽  
pp. 159-164
Author(s):  
W. E. Longstaff
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Complex Hilbert Space ◽  
Bounded Operators ◽  
Finitely Generated ◽  
Separable Space ◽  
Von Neumann ◽  
Underlying Space ◽  
Weakly Closed ◽  
Reflexive Algebras

For any collection of closed subspaces of a complex Hilbert space the set of bounded operators that leave invariant all the members of the collection is a weakly-closed algebra. The class of such algebras is precisely the class of reflexive algebras as defined for example in Radjavi and Rosenthal (1969) and contains the class of von Neumann algebras.In this paper we consider the problem of when such algebras are finitely generated as weakly-closed algebras. It is to be hoped that analysis of this problem may shed some light on the famous unsolved problem of whether every von Neumann algebra on a separable Hilbert space is finitely generated. The case where the underlying space is separable and the collection of subspaces is totally ordered is dealt with in Longstaff (1974). In the present paper the result of Longstaff (1974) is generalized to the case of a direct product of countably many totally ordered collections each on a separable space. Also a method of obtaining non-finitely generated reflexive algebras is given.

scholarly journals Invariant subspace lattices of Lambert's weighted shifts

1982 ◽  
Vol 33 (1) ◽  
pp. 135-142
Author(s):  
B. S. Yadav ◽  
S. Chatterjee
Keyword(s):  
Hilbert Space ◽  
Invariant Subspace ◽  
Bounded Sequence ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Weighted Shifts ◽  
Positive Real ◽  
Subspace Lattice ◽  
Bounded Linear ◽  
Infinite Dimensional

AbstractLet B(H) be the Banach algebra of all (bounded linear) operators on an infinite-dimensional separable complex Hilbert space H and let be a bounded sequence of positive real numbers. For a given injective operator A in B(H) and a non-zero vector f in H, we put We define a weighted shift Tw with the weight sequence on the Hilbert space 12 of all square-summable complex sequences by . The main object of this paper is to characterize the invariant subspace lattice of Tw under various nice conditions on the operator A and the sequence .

scholarly journals Hilbert space representations of decoherence functionals and quantum measures

2012 ◽  
Vol 62 (6) ◽  
Author(s):  
Stan Gudder
Keyword(s):  
Hilbert Space ◽  
Closed Subspace ◽  
Complex Hilbert Space ◽  
Space Representation ◽  
Quantum Operator ◽  
Quantum Measure ◽  
Usual Quantum ◽  
Vector Valued ◽  
Quantum Formulation ◽  
Decoherence Functional

AbstractWe show that any decoherence functional D can be represented by a spanning vector-valued measure on a complex Hilbert space. Moreover, this representation is unique up to an isomorphism when the system is finite. We consider the natural map U from the history Hilbert space K to the standard Hilbert space H of the usual quantum formulation. We show that U is an isomorphism from K onto a closed subspace of H and that U is an isomorphism from K onto H if and only if the representation is spanning. We then apply this work to show that a quantum measure has a Hilbert space representation if and only if it is strongly positive. We also discuss classical decoherence functionals, operator-valued measures and quantum operator measures.

Remarks on Invariant Subspace Lattices

1969 ◽  
Vol 12 (5) ◽  
pp. 639-643 ◽  
Author(s):  
Peter Rosenthal
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Invariant Subspace ◽  
Complete Lattice ◽  
Bounded Linear Operator ◽  
Complex Hilbert Space ◽  
Subspace Lattice ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Subspace Lattices

If A is a bounded linear operator on an infinite-dimensional complex Hilbert space H, let lat A denote the collection of all subspaces of H that are invariant under A; i.e., all closed linear subspaces M such that x ∈ M implies (Ax) ∈ M. There is very little known about the question: which families F of subspaces are invariant subspace lattices in the sense that they satisfy F = lat A for some A? (See [5] for a summary of most of what is known in answer to this question.) Clearly, if F is an invariant subspace lattice, then {0} ∈ F, H ∈ F and F is closed under arbitrary intersections and spans. Thus, every invariant subspace lattice is a complete lattice.

Operators of Rank One in Reflexive Algebras

1976 ◽  
Vol 28 (1) ◽  
pp. 19-23 ◽  
Author(s):  
W. E. Longstaff
Keyword(s):  
Hilbert Space ◽  
Closed Operator ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Identity Operator ◽  
Subspace Lattice ◽  
Bounded Linear ◽  
Rank One ◽  
Weakly Closed ◽  
Reflexive Algebras

If H is a (complex) Hilbert space and is a collection of (closed linear) subspaces of H it is easily shown that the set of all (bounded linear) operators acting on H which leave every member of invariant is a weakly closed operator algebra containing the identity operator. This algebra is denoted by Alg . In the study of such algebras it may be supposed [4] that is a subspace lattice i.e. that is closed under the formation of arbitrary intersections and arbitrary (closed linear) spans and contains both the zero subspace (0) and H. The class of such algebras is precisely the class of reflexive algebras [3].

REPRESENTATIONS OF LIE ALGEBRAS BUILT OVER HILBERT SPACE

2011 ◽  
Vol 14 (03) ◽  
pp. 419-442
Author(s):  
PALLE E. T. JORGENSEN
Keyword(s):  
Hilbert Space ◽  
Lie Algebras ◽  
Fock Space ◽  
Quantum Statistical Mechanics ◽  
Complex Hilbert Space ◽  
Tensor Algebra ◽  
Bounded Operators ◽  
Inductive Limits ◽  
Tensor Algebras ◽  
Representations Of Lie Algebras

Starting with a complex Hilbert space, using inductive limits, we build Lie algebras, and find families of representations. They include those often studied in mathematical physics in order to model quantum statistical mechanics or quantum fields. We explore natural actions on infinite tensor algebras T(H) built with a functorial construction, starting with a fixed Hilbert space H. While our construction applies also when H is infinite-dimensional, the case with N ≔ dim H finite is of special interest as the symmetry group we consider is then a copy of the non-compact Lie group U(N, 1). We give the tensor algebra T(H) the structure of a Hilbert space, i.e. the unrestricted infinite tensor product Fock space [Formula: see text]. The tensor algebra T(H) is naturally represented as acting by bounded operators on [Formula: see text], and U (N, 1) as acting as a unitary representation. From this we built a covariant system, and we explore how the fermion, the boson, and the q on Hilbert spaces are reduced by the representations. In particular we display the decomposition into irreducible representations of the naturally defined U (N, 1) representation.

scholarly journals Decomposability of finite rank operators in certain subspaces and algebras

2001 ◽  
Vol 64 (2) ◽  
pp. 307-314
Author(s):  
Jiankui Li
Keyword(s):  
Hilbert Space ◽  
Finite Rank ◽  
Bounded Operators ◽  
Subspace Lattice ◽  
Finite Rank Operator ◽  
Rank Operator ◽  
Finite Rank Operators ◽  
Reflexive Algebra ◽  
Reflexive Subspace ◽  
Rank One

Let  be either a reflexive subspace or a bimodule of a reflexive algebra in B (H), the set of bounded operators on a Hilbert space H. We find some conditions such that a finite rank T ∈  has a rank one summand in  and  has strong decomposability. Let (ℒ) be the set of all operators on H that annihilate all the operators of rank at most one in alg ℒ. We construct an atomic Boolean subspace lattice ℒ on H such that there is a finite rank operator T in (ℒ) such that T does not have a rank one summand in (ℒ). We obtain some lattice-theoretic conditions on a subspace lattice ℒ which imply alg ℒ is strongly decomposable.

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