Invariant Subspaces and Models for Linear Operators

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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Compact Operators in Reductive Algebras

Canadian Journal of Mathematics ◽

10.4153/cjm-1975-019-9 ◽

1975 ◽

Vol 27 (1) ◽

pp. 152-154

Author(s):

Edward A. Azoff

Keyword(s):

Hilbert Space ◽

Invariant Subspaces ◽

Compact Operators ◽

Linear Operators ◽

Bounded Linear Operators ◽

Bounded Linear ◽

Term Algebra

Let be a Hilbert space and denote the collection of (bounded, linear) operators on by . Throughout this paper, the term ‘algebra’ will refer to a subalgebra of ; unless otherwise stated, it will not be assumed to contain I or to be closed in any topology.An algebra is said to be transitive if it has no non-trivial invariant subspaces. The following lemma has revolutionized the study of transitive algebras. For a pr∞f and a general discussion of its implications, the reader is referred to [5].

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ON THE DECOMPOSITION OF OPERATORS WITH SEVERAL ALMOST-INVARIANT SUBSPACES

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972718001363 ◽

2019 ◽

Vol 99 (2) ◽

pp. 274-283

Author(s):

AMANOLLAH ASSADI ◽

MOHAMAD ALI FARZANEH ◽

HAJI MOHAMMAD MOHAMMADINEJAD

Keyword(s):

Banach Space ◽

Linear Operator ◽

Closed Subspace ◽

Invariant Subspaces ◽

Weak Limit ◽

Linear Operators ◽

Sufficient Condition ◽

Bounded Operators ◽

Bounded Linear ◽

Rank Operator

We seek a sufficient condition which preserves almost-invariant subspaces under the weak limit of bounded operators. We study the bounded linear operators which have a collection of almost-invariant subspaces and prove that a bounded linear operator on a Banach space, admitting each closed subspace as an almost-invariant subspace, can be decomposed into the sum of a multiple of the identity and a finite-rank operator.

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REFINEMENT EQUATIONS AND CORRESPONDING LINEAR OPERATORS

International Journal of Wavelets Multiresolution and Information Processing ◽

10.1142/s0219691306001385 ◽

2006 ◽

Vol 04 (03) ◽

pp. 461-474 ◽

Cited By ~ 2

Author(s):

VLADIMIR PROTASOV

Keyword(s):

Invariant Subspaces ◽

Complete Classification ◽

Linear Operators ◽

Refinable Functions ◽

Computer Design ◽

Hölder Exponent ◽

Refinement Equations ◽

Open Questions ◽

Compactly Supported Solutions

Refinement equations of the type [Formula: see text] play an exceptional role in the theory of wavelets, subdivision algorithms and computer design. It is known that the regularity of their compactly supported solutions (refinable functions) depends on the spectral properties of special N-dimensional linear operators T0, T1 constructed by the coefficients of the equation. In particular, the structure of kernels and of common invariant subspaces of these operators have been intensively studied in the literature. In this paper, we give a complete classification of the kernels and of all the root subspaces of T0 and T1, as well as of their common invariant subspaces. This result answers several open questions stated in the literature and clarifies the structure of the space spanned by the integer translates of refinable functions. This also leads to some results on the moduli of continuity of refinable functions and wavelets in various functional spaces. In particular, it is proved that the Hölder exponent of those functions is sharp whenever it is not an integer.

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