An equivalent formulation of the invariant subspace conjecture

Dyer, James A.; Pedersen, E. A.; Porcelli, Pasquale

doi:10.1090/s0002-9904-1972-13090-8

Cited by 24 publications

(22 citation statements)

References 6 publications

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“…Here we may decompose 77 by mutually orthogonal reducing subspaces 77, of F so that /7=770©//1©-• ■ , and the decomposition of F is T=T0(BTX@-■ • where p(T0, F*) = 0 and Ti (i^l) is irreducible with p(T¿, F*) also compact (Theorem 2, [3], [7]). From this, it follows that the reductive question is affirmative whenever F is polynomially compact [6], [12].…”

Section: F77 Then (I-e)e=(i-e)p=0mentioning

confidence: 94%

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A note on reductive operators

Gilfeather¹

1974

Proc. Amer. Math. Soc.

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Abstract.A bounded linear operator A on a Hubert space is called reductive if every invariant subspace of A reduces it. This paper gives examples of operators which give an affirmative answer to the reductive question: If A is reductive, then is A normal?The reductive question is significant since its affirmative answer for all bounded operators is an equivalent formulation of the well-known invariant subspace question [6]. Since the latter question remains unsettled in general, several attempts have been made to determine for which classes of operators the answer to the reductive question is affirmative. At present, there are essentially three types of operators which have been shown to give an affirmative answer to the reductive question. The first type involves compactness, another involves scalar type operators in the sense of N. Dunford, and the thrid involves combinations of the first two types.The purpose of this note is to consider these results and show that they (and generalizations) can be obtained from certain structure theorems of the author. The results involving compactness originally stem from the affirmative answer to the reduction question given by T. Ando [1] and based on the Aronszajn and Smith invariant subspace theorem [2]. We shall show that Ando's result, as well as generalizations of results combining compactness and scalar operator arguments, will follow from a structure theorem for an operator F for which a noncommutative polynomial p(T, F*) is compact [7]. We note that some form of a known invariant subspace theorem will also be used in this approach.A generalization of the result that a scalar type reductive operator is normal is that a spectral operator of finite type (that is, nilpotent quasi-nilpotent part) if reductive is normal. This easy generalization does lead to a new class of operators for which the reductive question is affirmative. Namely, we show that a root A of a locally nonzero abelian

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Section: F77 Then (I-e)e=(i-e)p=0mentioning

confidence: 94%

“…The reductive question is significant since its affirmative answer for all bounded operators is an equivalent formulation of the well-known invariant subspace question [6]. Since the latter question remains unsettled in general, several attempts have been made to determine for which classes of operators the answer to the reductive question is affirmative.…”

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confidence: 99%

A note on reductive operators

Gilfeather¹

1974

Proc. Amer. Math. Soc.

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“…We begin with the result of J. Dyer, E. Pedersen and P. Porcelli announced in [4] and [5]. A short proof using the methods presented here appears in [3].…”

Section: Proofmentioning

confidence: 98%

“…In fact, it was observed in [4] that a maximal decomposition of a reductive operator leads to transitive direct integrands-this allowed the authors of [4] to conclude that the transitive algebra problem for singly generated algebras (i.e., the invariant subspace problem) is equivalent to the reductive algebra problem for singly generated 352 E. A. AZOFF, C. K. FONG AND F. GILFEATHER algebras (i.e., the reductive operator problem). In a similar fashion, the reduction theory of the preceding paragraph allows us to express every reductive algebra as a direct integral of transitive algebras.…”

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confidence: 99%

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A Reduction Theory For Non-Self-Adjoint Operator Algebras

Azoff¹,

Fong²,

Gilfeather³

1976

Transactions of the American Mathematical Society

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ABSTRACT. It is shown that every strongly closed algebra of operators acting on a separable Hubert space can be expressed as a direct integral of irreducible algebras. In particular, every reductive algebra is the direct integral of transitive algebras. This decomposition is used to study the relationship between the transitive and reductive algebra problems. The final section of the paper shows how to view direct integrals of algebras as measurable algebra-valued functions.

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Invariant subspaces in the theory of operators and theory of functions

Никольский¹

1976

J Math Sci

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Invariant subspaces were defined by. von Neumann* in 1935 for natural problems of spectral theory, though their episodic use began much later. It soon became clear that this object can replace the fundamental concept of finite-dimensional linear analysis -the eigenvector -in a great variety of problems. Descriptive geometric investigations into invariant subspaces, however, did not yield any remarkable results for a long time. We may recall that even in finite-dimensional theory the study of the holomorphic functions --det(A--~,l). where A is a given matrix and ! is the unit matrix, has constituted the "analytic foundation" of all results. A similar analysis of invariant subspaces was expected in the 1950's and 1960's when the language of a functional model and of a characteristic fimction sufficient for finite-dimensional problems was found.On the other hand, the concept of an invariant subspace turned out to be directly connected to a number of disciplines that had completely formed by the 1950's. We recall the close relationships between invariant subspaces and approximation problems in the theory of functions, more precisely the investigation of the completeness of various functional spaces X of families of the form ~x = (Ax:AE~}, where x E X and where ~t is a given set of operators in X. These problems lead to special theorems about invariant subspaces ("individual n in the sense of Sec. 10), the first of which naturally had been proved by K. Weier- stress (Z----C[0, I], ~l={T~:n~0}, ~r ---I, (Tf)(t)=~f(t)).There exist other deeper relationships with the theory of approximations using the language and methods of harmonic analysis (Sees. 10-13).However, the close connection between the theory of operators and the theory of functions in our survey is not only due to these facts. Already by the 1950's the fundamental role of multiplication by an independent variable z in spaces of analytic functions had been discovered. This operator turned out to be universal in the theory of linear operators, so that modern branches of this theory form an independent ~spec-tral theory of functions TM that understands the study of nsingularities n (vector-valued) of analytic functions as the reduction of the functions with respect to their singularities, and so on. Here we should note the words of Wiener from 40 years ago (cf. Sec. 11, p. 191), whose true value can only now be estimated.The purpose of this survey is to provide a general presentation of works andresults inwhich invariant subspaces occur either as the subject of investigation or as its tool and also to discuss works usinginvariant subspace concepts in the theory of functions. The survey can provisionally be divided into four parts: I) descriptive geometric investigation of invariant subspaces (Secs. 1-4); II) analytic aspect; generalized spectral decompositions (Secs. 3-6, 8); m3 shift operator and functional models (Secs. 3, 6-9); IV) relation to approximations and problem of analysis and synthesis (Secs. 4, 6, 9-13). These parts do not intersect by accident; r...

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An equivalent formulation of the invariant subspace conjecture

Cited by 24 publications

References 6 publications

A note on reductive operators

A note on reductive operators

A Reduction Theory For Non-Self-Adjoint Operator Algebras

Invariant subspaces in the theory of operators and theory of functions

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