Unitary operators in Banach spaces

Panchapagesan, T. V.

doi:10.2140/pjm.1967.22.465

Cited by 6 publications

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“…or S is required to be imbedded in an appropriate topological group and the associated operator semigroup should have some continuity properties. As a sample of results in this direction we refer to [6], [14], [15], [16], [20], [21], [22], [24], [26], [27], [28], [31], [32], [36], [37] and the references therein, for example.…”

Section: Representation Of * -Semigroups Of Scalar Operatorsmentioning

confidence: 99%

Vector-valued positive definite functions, the Berg-Maserick theorem, and applications

Ressel¹,

Ricker²

2002

MATH. SCAND.

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Section: Representation Of * -Semigroups Of Scalar Operatorsmentioning

confidence: 99%

Vector-valued positive definite functions, the Berg-Maserick theorem, and applications

Ressel¹,

Ricker²

2002

MATH. SCAND.

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“…In [14] we gave the generalized notion of a unitary operator on a Banach space and obtained there an extension of Stone's theorem when the group was uniformly continuous and the underlying Banach space was reflexive. DEFINITION 11.…”

Section: Is An Increasing Sequence Of Borel Sets Of a Such That Q(\jnmentioning

confidence: 99%

Semi-groups of scalar type operators in Banach spaces

Panchapagesan¹

1969

Pacific J. Math.

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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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Unitary operators in Banach spaces

Cited by 6 publications

References 8 publications

Vector-valued positive definite functions, the Berg-Maserick theorem, and applications

Vector-valued positive definite functions, the Berg-Maserick theorem, and applications

Semi-groups of scalar type operators in Banach spaces

Invariant subspaces in the theory of operators and theory of functions

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