Invariant subspaces and weakly closed algebras

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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On Reflexivity of Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1981-098-5 ◽

1981 ◽

Vol 33 (6) ◽

pp. 1291-1308 ◽

Cited By ~ 2

Author(s):

Mehdi Radjabalipour

Keyword(s):

Hilbert Space ◽

Natural Number ◽

Separable Hilbert Space ◽

Invariant Subspaces ◽

Linear Operators ◽

Bounded Linear Operators ◽

Bounded Linear ◽

Closed Algebra ◽

Direct Sum ◽

Weakly Closed

For each natural number n we define to be the class of all weakly closed algebras of (bounded linear) operators on a separable Hilbert space H such that the lattice of invariant subspaces of and (alg lat )(n) are the same. (If A is an operator, A(n) denotes the direct sum of n copies of A; if is a collection of operators,. Also, alg lat denotes the algebra of all operators leaving all invariant subspaces of invariant.) In the first section we show that . In Section 2 we prove that every weakly closed algebra containing a maximal abelian self adjoint algebra (m.a.s.a.) is , and that . It is also shown that certain algebras containing a m.a.s.a. are necessarily reflexive.

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Completely Reducible Operator Algebras and Spectral Synthesis

Canadian Journal of Mathematics ◽

10.4153/cjm-1982-074-9 ◽

1982 ◽

Vol 34 (5) ◽

pp. 1025-1035 ◽

Cited By ~ 1

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.

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A Sufficient Condition that an Operator Algebra be Self-Adjoint

Canadian Journal of Mathematics ◽

10.4153/cjm-1971-066-7 ◽

1971 ◽

Vol 23 (4) ◽

pp. 588-597 ◽

Cited By ~ 8

Author(s):

Heydar Radjavi ◽

Peter Rosenthal

Keyword(s):

Invariant Subspace ◽

Operator Algebra ◽

Linear Manifold ◽

Invariant Subspaces ◽

Complex Hilbert Space ◽

List Type ◽

Type Number ◽

Bounded Linear ◽

Weakly Closed ◽

Algebra Of Operators

It is well-known, and easily verified, that each of the following assertions implies the preceding ones. (i)Every operator has a non-trivial invariant subspace.(ii)Every commutative operator algebra has a non-trivial invariant subspace,(iii)Every operator other than a multiple of the identity has a non-trivial hyperinvariant subspace.(iv)The only transitive operator algebra on is Note. Operator means bounded linear operator on a complex Hilbert space , operator algebra means weakly closed algebra of operators containing the identity, subspace means closed linear manifold, a non-trivial subspace is a subspace other than {0} and , a. hyperinvariant subspace for A is a subspace invariant under every operator which commutes with A, a transitive operator algebra is one without any non-trivial invariant subspaces and denotes the algebra of all operators on .

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