Compact Perturbations of Reflexive Algebras

1981 ◽  
Vol 33 (3) ◽  
pp. 685-700 ◽  
Author(s):  
Kenneth R. Davidson
Keyword(s):  
Unitary Operator ◽  
Operator Algebras ◽  
Invariant Subspaces ◽  
Compact Operators ◽  
Reverse Inclusion ◽  
Finite Dimensional ◽  
Compact Perturbations ◽  
Reflexive Algebras ◽  
The Ideal ◽  
Subspace Lattices

In this paper we study lattice properties of operator algebras which are invariant under compact perturbations. It is easy to see that if and are two operator algebras with contained in , then the reverse inclusion holds for their lattices of invariant subspaces. We will show that in certain cases, the assumption thats is contained in , where is the ideal of compact operators, implies that the lattice of is “approximately” contained in the lattice of . In particular, supposed and are reflexive and have commutative subspace lattices containing “enough” finite dimensional elements. We show (Corollary 2.8) that if is unitarily equivalent to a subalgebra of , then there is a unitary operator which carries all “sufficiently large” subspaces in lat into lat .

scholarly journals Some results about spectral continuity and compact perturbations

Filomat ◽  
2020 ◽  
Vol 34 (14) ◽  
pp. 4837-4845
Author(s):  
S. Sánchez-Perales ◽  
S.V. Djordjevic ◽  
S. Palafox
Keyword(s):  
Hilbert Spaces ◽  
Compact Operators ◽  
Compact Perturbations ◽  
Spectral Continuity ◽  
Continuity Of The Spectrum ◽  
The Ideal

In this paper, we are interested in the continuity of the spectrum and some of its parts in the setting of Hilbert spaces. We study the continuity of the spectrum in the class of operators {T}+K(H), where K(H) denote the ideal of compact operators. Also, we give conditions in order to transfer the continuity of spectrum from T to T + K, where K ? K(H). Then, we characterize those operators for which the continuity of spectrum is stable under compact perturbations.

scholarly journals Spatiality of Derivations of Operator Algebras in Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-13
Author(s):  
Quanyuan Chen ◽  
Xiaochun Fang
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Left Ideal ◽  
Operator Algebras ◽  
Reflexive Banach Space ◽  
Compact Operators ◽  
Alternative Proof ◽  
Additive Constant ◽  
Minimal Left Ideal ◽  
The Ideal

Suppose thatAis a transitive subalgebra ofB(X)and its norm closureA¯contains a nonzero minimal left idealI. It is shown that ifδis a bounded reflexive transitive derivation fromAintoB(X), thenδis spatial and implemented uniquely; that is, there existsT∈B(X)such thatδ(A)=TA−ATfor eachA∈A, and the implementationTofδis unique only up to an additive constant. This extends a result of E. Kissin that “ifA¯contains the idealC(H)of all compact operators inB(H), then a bounded reflexive transitive derivation fromAintoB(H)is spatial and implemented uniquely.” in an algebraic direction and provides an alternative proof of it. It is also shown that a bounded reflexive transitive derivation fromAintoB(X)is spatial and implemented uniquely, ifXis a reflexive Banach space andA¯contains a nonzero minimal right idealI.

scholarly journals Изометрии действительных подпространств самосопряженных операторов в банаховых симметричных идеалах

2019 ◽  
Author(s):  
B.R. Aminov ◽  
V.I. Chilin
Keyword(s):  
Hilbert Space ◽  
Unitary Operator ◽  
Hermitian Operator ◽  
Uniform Norm ◽  
Compact Operators ◽  
Linear Isometry ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Finite Dimensional ◽  
Adjoint Operators

Let (mathcal C_E, cdot_mathcal C_E) be a Banach symmetric ideal of compact operators, acting in a complex separable infinite-dimensional Hilbert space mathcal H. Let mathcal C_Ehxin mathcal C_E : xx be the real Banach subspace of self-adjoint operators in (mathcal C_E, cdot_mathcal C_E). We show that in the case when (mathcal C_E, cdot_mathcal C_E) is a separable or perfect Banach symmetric ideal (mathcal C_E eq mathcal C_2) any skew-Hermitian operator H: mathcal C_Ehto mathcal C_Eh has the following form H(x)i(xa - ax) for same aain mathcal B(mathcal H) and for all xin mathcal C_Eh. Using this description of skew-Hermitian operators, we obtain the following general form of surjective linear isometries V:mathcal C_Eh to mathcal C_Eh. Let (mathcal C_E, cdot_mathcal C_E) be a separable or a perfect Banach symmetric ideal with not uniform norm, that is p_mathcal C_E 1 for any finite dimensional projection p inmathcal C_E with dim p(mathcal H)1, let mathcal C_E eq mathcal C_2, and let V: mathcal C_Eh to mathcal C_Eh be a surjective linear isometry. Then there exists unitary or anti-unitary operator u on mathcal H such that V(x)uxu orV(x)-uxu for all x in mathcal C_Eh.

Generalized interpolation in finite maximal subdiagonal algebras

1995 ◽  
Vol 117 (1) ◽  
pp. 11-20 ◽  
Author(s):  
Kichi-Suke Saito
Keyword(s):  
Selfadjoint Operator ◽  
Operator Algebras ◽  
Invariant Subspaces ◽  
Factorization Theorem ◽  
Crossed Products ◽  
Analytic Operator ◽  
Nest Algebras ◽  
Function Algebras ◽  
Reflexive Algebras

Non-selfadjoint operator algebras have been studied since the paper of Kadison and Singer in 1960. In [1], Arveson introduced the notion of subdiagonal algebras as the generalization of weak *-Dirichlet algebras and studied the analyticity of operator algebras. After that, we have many papers about non-selfadjoint algebras in this direction: nest algebras, CSL algebras, reflexive algebras, analytic operator algebras, analytic crossed products and so on. Since the notion of subdiagonal algebras is the analogue of weak *-Dirichlet algebras, subdiagonal algebras have many fruitful properties from the theory of function algebras. Thus, we have several attempts in this direction: Beurling–Lax–Halmos theorem for invariant subspaces, maximality, factorization theorem and so on.

Completely Reducible Operator Algebras and Spectral Synthesis

1982 ◽  
Vol 34 (5) ◽  
pp. 1025-1035 ◽  
Author(s):  
Shlomo Rosenoer
Keyword(s):  
Operator Algebras ◽  
Spectral Synthesis ◽  
Invariant Subspaces ◽  
Bounded Operators ◽  
Von Neumann ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Completely Reducible ◽  
Weakly Closed ◽  
Reductive Algebra

An algebra of bounded operators on a Hilbert space H is said to be reductive if it is unital, weakly closed and has the property that if M ⊂ H is a (closed) subspace invariant for every operator in , then so is M⊥. Loginov and Šul'man [6] and Rosenthal [9] proved that if is an abelian reductive algebra which commutes with a compact operator K having a dense range, then is a von Neumann algebra. Note that in this case every invariant subspace of is spanned by one-dimensional invariant subspaces. Indeed, the operator KK* commutes with . Hence its eigenspaces are invariant for , so that H is an orthogonal sum of the finite-dimensional invariant subspaces of From this our claim easily follows.

scholarly journals Operator Algebras with Unique Preduals

2011 ◽  
Vol 54 (3) ◽  
pp. 411-421 ◽  
Author(s):  
Kenneth R. Davidson ◽  
Alex Wright
Keyword(s):  
Banach Space ◽  
Operator Algebra ◽  
Free Semigroup ◽  
Operator Algebras ◽  
Compact Operators ◽  
Semigroup Algebra ◽  
Dense Subalgebra

AbstractWe show that every free semigroup algebra has a (strongly) unique Banach space predual. We also provide a new simpler proof that a weak-∗ closed unital operator algebra containing a weak-∗ dense subalgebra of compact operators has a unique Banach space predual.

scholarly journals Conditional Lie-Bäcklund Symmetries and Reductions of the Nonlinear Diffusion Equations with Source

2014 ◽  
Vol 2014 ◽  
pp. 1-17
Author(s):  
Junquan Song ◽  
Yujian Ye ◽  
Danda Zhang ◽  
Jun Zhang
Keyword(s):  
Invariant Subspace ◽  
Dynamic Systems ◽  
Nonlinear Diffusion ◽  
Invariant Subspaces ◽  
Higher Order ◽  
Diffusion Equations ◽  
Canonical Forms ◽  
Nonlinear Diffusion Equations ◽  
Symmetry Approach ◽  
Finite Dimensional

Conditional Lie-Bäcklund symmetry approach is used to study the invariant subspace of the nonlinear diffusion equations with sourceut=e−qx(epxP(u)uxm)x+Q(x,u),m≠1. We obtain a complete list of canonical forms for such equations admit multidimensional invariant subspaces determined by higher order conditional Lie-Bäcklund symmetries. The resulting equations are either solved exactly or reduced to some finite-dimensional dynamic systems.

scholarly journals The ideal of Lipschitz classical p-compact operators and its injective hull

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Toufik Tiaiba ◽  
Dahmane Achour
Keyword(s):  
Banach Space ◽  
Compact Operator ◽  
Metric Space ◽  
Metric Spaces ◽  
Compact Operators ◽  
Operator Ideal ◽  
Linear Case ◽  
Injective Hull ◽  
Nuclear Operators ◽  
The Ideal

Abstract We introduce and investigate the injective hull of the strongly Lipschitz classical p-compact operator ideal defined between a pointed metric space and a Banach space. As an application we extend some characterizations of the injective hull of the strongly Lipschitz classical p-compact from the linear case to the Lipschitz case. Also, we introduce the ideal of Lipschitz unconditionally quasi p-nuclear operators between pointed metric spaces and show that it coincides with the Lipschitz injective hull of the ideal of Lipschitz classical p-compact operators.

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