scholarly journals On commutative non-self-adjoint operator algebras

1974 ◽  
Vol 15 (1) ◽  
pp. 54-59 ◽  
Author(s):  
R. H. Kelly
Keyword(s):  
Hilbert Space ◽  
Operator Algebras ◽  
Adjoint Operator ◽  
Complex Hilbert Space ◽  
Special Case

A proof is given here of a theorem of Sarason [9, Theorem 2], the proof being valid in an arbitrary (non-separable) complex Hilbert space. Sarason's proof uses a theorem and lemma of Wermer which may both fail when the separability hypothesis is omitted [3]. By using a special case of Sarason's theorem and another result of Sarason [10, Lemma 1] a simplified and shortened proof is given of a result of Scroggs [11, Corollary 1].

On Operator Algebras and Invariant Subspaces

1969 ◽  
Vol 21 ◽  
pp. 1178-1181 ◽  
Author(s):  
Chandler Davis ◽  
Heydar Radjavi ◽  
Peter Rosenthal
Keyword(s):  
Hilbert Space ◽  
Elementary Proof ◽  
Operator Algebras ◽  
Equivalent Form ◽  
Invariant Subspaces ◽  
Complex Hilbert Space ◽  
Closed Algebra ◽  
Weakly Closed ◽  
Special Case

If is a collection of operators on the complex Hilbert space , then the lattice of all subspaces of which are invariant under every operator in is denoted by Lat . An algebra of operators on is defined (3; 4) to be reflexive if for every operator B on the inclusion Lat ⊆ Lat B implies .Arveson (1) has proved the following theorem. (The abbreviation “m.a.s.a.” stands for “maximal abelian self-adjoint algebra”.)ARVESON's THEOREM. Ifis a weakly closed algebra which contains an m.a.s.a.y and if Lat, then is the algebra of all operators on .A generalization of Arveson's Theorem was given in (3). Another generalization is Theorem 2 below, an equivalent form of which is Corollary 3. This theorem was motivated by the following very elementary proof of a special case of Arveson's Theorem.

On the von Neumann algebras associated to Yang–Baxter operators

2020 ◽  
pp. 1-24
Author(s):  
Panchugopal Bikram ◽  
Rahul Kumar ◽  
Rajeeb Mohanta ◽  
Kunal Mukherjee ◽  
Diptesh Saha
Keyword(s):  
Hilbert Space ◽  
Special Form ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Adjoint Operator ◽  
Complex Hilbert Space ◽  
Von Neumann ◽  
Finite Von Neumann Algebra

Bożejko and Speicher associated a finite von Neumann algebra M T to a self-adjoint operator T on a complex Hilbert space of the form $\mathcal {H}\otimes \mathcal {H}$ which satisfies the Yang–Baxter relation and $ \left\| T \right\| < 1$ . We show that if dim $(\mathcal {H})$ ⩾ 2, then M T is a factor when T admits an eigenvector of some special form.

scholarly journals Nonlinear *-Lie Higher Derivations of Standard Operator Algebras

2018 ◽  
Vol 26 (1) ◽  
pp. 15-29
Author(s):  
Mohammad Ashraf ◽  
Shakir Ali ◽  
Bilal Ahmad Wani
Keyword(s):  
Hilbert Space ◽  
Operator Algebra ◽  
Operator Algebras ◽  
Complex Hilbert Space ◽  
Standard Operator ◽  
Standard Operator Algebra ◽  
Finite Dimensional ◽  
Standard Operator Algebras ◽  
Higher Derivation ◽  
Adjoint Operation

Abstract Let ℌ be an in finite-dimensional complex Hilbert space and A be a standard operator algebra on ℌ which is closed under the adjoint operation. It is shown that every nonlinear *-Lie higher derivation D = {δn}gn∈N of A is automatically an additive higher derivation on A. Moreover, D = {δn}gn∈N is an inner *-higher derivation.

scholarly journals Operator algebras with a reduction property

2006 ◽  
Vol 80 (3) ◽  
pp. 297-315 ◽  
Author(s):  
James A. Gifford
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Invariant Subspace ◽  
Operator Algebras ◽  
Adjoint Operator ◽  
Compact Operators ◽  
Total Reduction ◽  
Reduction Property ◽  
Similarity Problem ◽  
Closed Invariant Subspace

AbstractGiven a representation θ: A → B(H) of a Banach algebra A on a Hilbert space H, H is said to have the reduction property as an A—module if every closed invariant subspace of H is complemented by a closed invariant subspace; A has the total reduction property if for every representation θ: A → B(H), H has the reduction property.We show that a C*—algebra has the total reduction property if and only if all its representations are similar to *—representations. The question of whether all C*-algebras have this property is the famous ‘similarity problem’ of Kadison.We conjecture that non-self-adjoint operator algebras with the total reduction property are always isomorphic to C*-algebras, and prove this result for operator algebras consisting of compact operators.

scholarly journals Best Approximations by Increasing Invariant Subspaces of Self-Adjoint Operators

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1918
Author(s):  
Oleh Lopushansky ◽  
Renata Tłuczek-Piȩciak
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Adjoint Operator ◽  
Invariant Subspaces ◽  
Accurate Estimate ◽  
Complex Hilbert Space ◽  
Best Approximations ◽  
Adjoint Operators ◽  
Increasing Sequences

The paper describes approximations properties of monotonically increasing sequences of invariant subspaces of a self-adjoint operator, as well as their symmetric generalizations in a complex Hilbert space, generated by its positive powers. It is established that the operator keeps its spectrum over the dense union of these subspaces, equipped with quasi-norms, and that it is contractive. The main result is an inequality that provides an accurate estimate of errors for the best approximations in Hilbert spaces by these invariant subspaces.

scholarly journals On zero-trace commutators

1986 ◽  
Vol 34 (1) ◽  
pp. 119-126 ◽  
Author(s):  
Fuad Kittaneh
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Adjoint Operator ◽  
Trace Class ◽  
Complex Hilbert Space ◽  
Trace Class Operator ◽  
Class Operator ◽  
Closed Disc ◽  
Schmidt Operator

We present some results concerning the trace of certain trace class commutators of operators acting on a separable, complex Hilbert space. It is shown, among other things, that if X is a Hilbert-Schmidt operator and A is an operator such that AX − XA is a trace class operator, then tr (AX − XA) = 0 provided one of the following conditions holds: (a) A is subnormal and A*A − AA* is a trace class operator, (b) A is a hyponormal contraction and 1 − AA* is a trace class operator, (c) A2 is normal and A*A − AA* is a trace class operator, (d) A2 and A3 are normal. It is also shown that if A is a self - adjoint operator, if f is a function that is analytic on some neighbourhood of the closed disc{z: |z| ≥ ||A||}, and if X is a compact operator such that f (A) X − Xf (A) is a trace class operator, then tr (f (A) X − Xf (A))=0.

scholarly journals On Derivations of Operator Algebras with Involution

2014 ◽  
Vol 47 (4) ◽  
Author(s):  
Nejc Širovnik ◽  
Joso Vukman
Keyword(s):  
Hilbert Space ◽  
Operator Algebra ◽  
Operator Algebras ◽  
Complex Hilbert Space ◽  
Linear Mapping ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Standard Operator Algebra ◽  
Adjoint Operation

AbstractThe purpose of this paper is to prove the following result. Let X be a complex Hilbert space, let L(X) be an algebra of all bounded linear operators on X and let A(X) ⊂ L(X) be a standard operator algebra, which is closed under the adjoint operation. Suppose there exists a linear mapping D : A(X) → L(X) satisfying the relation 2D(AA*A) = D(AA*)A + AA*D(A) + D(A)A*A + AD(A*A) for all A ∈ A(X). In this case, D is of the form D(A) = [A,B] for all A ∈ A(X) and some fixed B ∈ L(X), which means that D is a derivation.

Non-self-adjoint operator algebras in Hilbert space

1976 ◽  
Vol 5 (2) ◽  
pp. 250-278 ◽  
Author(s):  
M. A. Naimark ◽  
A. I. Loginov ◽  
V. S. Shul'man
Keyword(s):  
Hilbert Space ◽  
Operator Algebras ◽  
Adjoint Operator

A note on normal operators

1963 ◽  
Vol 59 (4) ◽  
pp. 727-729 ◽  
Author(s):  
S. J. Bernau ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Spectral Theorem ◽  
Unitary Operators ◽  
Unitary Space ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case

We recall that a bounded linear operator T in a Hilbert space or finite-dimensional unitary space is said to be normal if T commutes with its adjoint operator T*, i.e. TT* = T*T. Most of the proofs given in the literature for the spectral theorem for normal operators, even in the finite-dimensional case, appeal to the corresponding results for Hermitian or unitary operators.

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