CONJUGATION-INVARIANT SUBSPACES AND LIE IDEALS IN NON-SELFADJOINT OPERATOR 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.

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On the topological stable rank of non-selfadjoint operator algebras

Mathematische Annalen ◽

10.1007/s00208-007-0180-5 ◽

2007 ◽

Vol 341 (2) ◽

pp. 239-253 ◽

Cited By ~ 5

Author(s):

K. R. Davidson ◽

R. H. Levene ◽

L. W. Marcoux ◽

H. Radjavi

Keyword(s):

Selfadjoint Operator ◽

Operator Algebras ◽

Stable Rank ◽

Topological Stable Rank

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Maximal invariant subspaces of strictly cyclic operator algebras

Pacific Journal of Mathematics ◽

10.2140/pjm.1973.49.45 ◽

1973 ◽

Vol 49 (1) ◽

pp. 45-50 ◽

Cited By ~ 2

Author(s):

Mary Embry

Keyword(s):

Operator Algebras ◽

Invariant Subspaces ◽

Cyclic Operator ◽

Maximal Invariant

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Interacting Fock spaces and subproduct systems

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025720500174 ◽

2020 ◽

Vol 23 (03) ◽

pp. 2050017

Author(s):

Malte Gerhold ◽

Michael Skeide

Keyword(s):

Selfadjoint Operator ◽

Operator Algebras ◽

Fock Space ◽

Fock Spaces ◽

Full Generality ◽

Necessary And Sufficient Criteria ◽

Definition Of ◽

Interacting Fock Space ◽

Necessary And Sufficient

We present a new more flexible definition of interacting Fock space that allows to resolve in full generality the problem of embeddability. We show that the same is not possible for regularity. We apply embeddability to classify interacting Fock spaces by squeezings. We give necessary and sufficient criteria for when an interacting Fock space has only bounded creators, giving thus rise to new classes of non-selfadjoint and selfadjoint operator algebras.

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On Operator Algebras and Invariant Subspaces

Canadian Journal of Mathematics ◽

10.4153/cjm-1969-129-9 ◽

1969 ◽

Vol 21 ◽

pp. 1178-1181 ◽

Cited By ~ 10

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.

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