Operator Algebras and Invariant Subspaces

1974 ◽  
Vol 100 (3) ◽  
pp. 433 ◽  
Author(s):  
William Arveson
Keyword(s):  
Operator Algebras ◽  
Invariant Subspaces

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.

Operator algebras and lattices of invariant subspaces. I

1992 ◽  
Vol 61 (2) ◽  
pp. 1963-1981
Author(s):  
V. V. Kapustin ◽  
A. V. Lipin
Keyword(s):  
Operator Algebras ◽  
Invariant Subspaces

Classes of Invariant Subspaces for Some Operator Algebras

2013 ◽  
Vol 53 (10) ◽  
pp. 3397-3408 ◽  
Author(s):  
Jan Hamhalter ◽  
Ekaterina Turilova
Keyword(s):  
Operator Algebras ◽  
Invariant Subspaces

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 .

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