Invariant Subspaces for Linear Operators in Locally Convex Spaces

Chilana, Ajit Kaur

doi:10.1112/jlms/2.part_3.493

Cited by 13 publications

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“…(v) The sufficiency part is obvious. If T e <&% <= <g^ then by Proposition 1, (iv) and (v), ^ = ^T fl i and by [1], Proposition l(iv), c € T = c ti> and, therefore, the result follows from (iv). Now suppose that D[T] = E, T~x E ^f, and E has no norm.…”

Section: (I) An Operatormentioning

confidence: 84%

“…It follows directly from the definitions that if 33 c 28 then 'WS T < sa <= F^T j $ <= ^^ c ^^ and also if 33 = {<%} then # TfS = F^T a } = ^T )% . 'Following [1], an operator A in E is T-continuous if for any two neighbourhoods U and F of o (one is sufficient, in fact), there exists a neighbourhood W of o such that AW <= F + TC7. Let ^T denote the set of T-continuous operators in E.…”

mentioning

confidence: 99%

“…Following [1], an operator A in E is T-continuous if for any two neighbourhoods U and F of o (one is sufficient, in fact), there exists a neighbourhood W of o such that AW <= F + TC7. Let ^T denote the set of T-continuous operators in E.…”

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confidence: 99%

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Some Perturbation Results for Linear Operators in Locally Convex Spaces

Chilana

1973

Proceedings of the London Mathematical Society

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Proc. London Math. Soc. (3) 26 (1973) 320-338 PERTURBATION RESULTS FOR LINEAR OPERATORS 321 (ii) We define A to be 33-continuous with 33-norm \\A ||$ if A is (/, 33)continuous and ||^4 11$ = inf{||.4 || % : A is ^-continuous and % e 33}. DEFINITION 3. (i) If an operator A in E is [T, ^-continuous for each % e 33 we shall call A fully (T, ^-continuous.(ii) An operator A which is ^-continuous for each % e 33 will be called fully ^-continuous.Let ^ya* ^«> ^T,SB> ^SB» F^T.SB* an<^ -^JB denote the sets of operators defined above in their respective order. It follows directly from the definitions that if 33 c 28 then 'WS T < sa <= F^T j $ <= ^^ c ^^ and also if 33 = {<%} then # TfS = F^T a } = ^T )% .'Following [1], an operator A in E is T-continuous if for any two neighbourhoods U and F of o (one is sufficient, in fact), there exists a neighbourhood W of o such that AW <= F + TC7. Let ^T denote the set of T-continuous operators in E.

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Section: (I) An Operatormentioning

confidence: 84%

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confidence: 99%

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Some Perturbation Results for Linear Operators in Locally Convex Spaces

Chilana

1973

Proceedings of the London Mathematical Society

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“…REMARK 2. In [2] the first part of question (3) in [6] was answered in the negative and the following more general question was raised: If F is a BG space with dual F', then must there be a barrelled topology compatible with duality (F,F')Ί…”

Section: E Is An Hbg Spacementioning

confidence: 99%

The space of bounded sequences with the mixed topology

Chilana¹

1973

Pacific J. Math.

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“…If a locally convex Hausdorff space is the closed linear hull of a bounded set i.e. it is boundedly generated (in short, BG) in the terminology of [6] then it is BG under each topology compatible with duality (Remark 10 in [1]). Every normed linear space is BG and a product of BG spaces is again BG [6] (see also Remark 10 in [1] and [2]).…”

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Topological Algebras with a Given Dual

Chilana¹

1974

Proceedings of the American Mathematical Society

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Abstract.Given an algebra E and a total subspace E' of its algebraic dual, we obtain necessary and sufficient conditions in terms of E' for the existence of an /4-convex or a locally /n-convex topology on E compatible with duality (E, E'). It has also been proved that if E with the weak topology w(E, £') is the closed linear hull of a bounded set and has hypocontinuous multiplication then it is locally /«-convex.1. Introduction. Let £ be a complex (or real) algebra and £' be a total subspace of the algebraic dual £*. To avoid repetitions we use the notation, terminology and results in [3] and [4] without specifications. An algebra with a locally convex linear topology for which multiplication is separately continuous will be called a locally convex algebra. An absolutely convex set B in £ is called right (left) A-convex if it absorbs Bx (xB) for each x e E, it will be called A-convex if it is both right and left Aconvex. A locally convex algebra is called (right, left) A-convex if there exists a basis of (right, left) ^-convex neighbourhoods of zero. Multiplication in a locally convex algebra will be said to be right (left) hypocontinuous if given a neighbourhood U of o and a bounded set B there exists a neighbourhood V of o satisfying VB<^ U (BV<= U). We say that multiplication is hypocontinuous if it is both right and left hypocontinuous. Key words and phrases, (right, left) /í-convex algebra, locally /«-convex algebra, topology compatible with duality, (right, left) multiplicative translate of a functional, collectionwise multiplicative set of functionals, collectionwise (right, left) multiplicativetranslation invariant set of functionals, hypocontinuous and jointly continuous multiplication.

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Invariant Subspaces for Linear Operators in Locally Convex Spaces

Cited by 13 publications

References 0 publications

Some Perturbation Results for Linear Operators in Locally Convex Spaces

Some Perturbation Results for Linear Operators in Locally Convex Spaces

The space of bounded sequences with the mixed topology

Topological Algebras with a Given Dual

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