Strong Reducibility of Operators

Gilfeather, Frank

doi:10.1512/iumj.1973.22.22032

Cited by 44 publications

(32 citation statements)

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“…Entirely reducible operators (i.e., containing a reducing part in each reducing subspace) were studied by Rosenthal [1017Rosenthal [ , 1018 (cf. also Gilfeather [670]). Operators lacking reducing subspaces form an everywhere dense subset of type G~ in L (H) (Halmos [695]) (cf.…”

Section: A) Reducing Subspaces Of An Operator T In the Hilbert Space mentioning

confidence: 96%

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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Section: A) Reducing Subspaces Of An Operator T In the Hilbert Space mentioning

confidence: 96%

Invariant subspaces in the theory of operators and theory of functions

Никольский¹

1976

J Math Sci

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“…As far as we know, Gilfeather [4] and Z. J. Jiang [15] have given the concept of strongly irreducible operators, respectively. The work of Herrero, Power and Jiang have answered a number of questions about operator structure of separable Hilbert spaces raised by Herrero and Z. J. Jiang (see [1,6,7,15,16]).…”

Section: Introductionmentioning

confidence: 99%

Strongly irreducible operators and Cowen-Douglas operators on c 0, l p (1 ⩽ p < ∞)

Zhang

Zhong

2011

Front. Math. China

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This paper gives some sufficient conditions for the strongly irreducibility of operators which have the forms of upper triangular operator matrices on Banach spaces. Based on these results, strongly irreducible Cowen-Douglas operators of index n are constructed on c 0 , l p (1 p < ∞) for all 1 n ∞.Keywords Banach space, strongly irreducible operator, Cowen-Douglas operator, upper triangular operator matrice MSC 46B20, 47A05

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“…It is easy to prove that every strongly irreducible operator on a finite dimensional space admits a Jordan block representation with respect to some basis. As far as we know, Gilfeather [4] and Jiang [17] gave the concept of strongly irreducible operators, respectively. Jiang further pointed out that the strongly irreducible operators can be considered as an approximate replacement of Jordan blocks on infinite dimensional spaces and he hoped that a theorem similar to the Jordan Standard Theorem can be set up with this replacement on infinite-dimensional spaces.…”

Section: Introductionmentioning

confidence: 99%

The approximate Jordan forms of operators on Σ e 1 type Banach spaces

2011

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This paper studies the structure of operators on Σ 1 e type Banach spaces. It solves the problem of the small compact perturbations of operators with connected spectra. Namely, it shows that every operator with a connected spectrum on separable Σ 1 e type Banach spaces is a small compact perturbation of a strongly irreducible operator. Based on this result, this paper establishes the approximate Jordan forms of operators on Σ 1 e type Banach spaces with Schauder bases. KeywordsΣ 1 e type Banach spaces, strongly irreducible operators, operators with connected spectra, approximate Jordan forms MSC(2000): 46B20, 47A05 Citation: Zhang Y N, Lin L Q, Zhong H J. The approximate Jordan forms of operators on Σ 1 e type Banach spaces.

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Strong Reducibility of Operators

Cited by 44 publications

References 3 publications

Invariant subspaces in the theory of operators and theory of functions

Invariant subspaces in the theory of operators and theory of functions

Strongly irreducible operators and Cowen-Douglas operators on c 0, l p (1 ⩽ p < ∞)

The approximate Jordan forms of operators on Σ e 1 type Banach spaces

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