Automatic Boundedness of Adjointable Operators on Barreled Vh-Spaces
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Abstract
We consider the space of adjointable operators on barreled VH (Vector Hilbert) spaces and show that such operators are automatically bounded. This generalizes the well known corresponding result for locally Hilbert C*-modules. We pick a consequence of this result in the dilation theory of VH-spaces and show that, when barreled VH-spaces are considered, a certain boundedness condition for the existence of VH-space linearisations, equivalently, of reproducing kernel VH-spaces, is automatically satisfied.
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Ordered *-space, Admissible space, VH-space, Barreled VH-space, Adjointable operators, Bounded operators, Positive semidefinite kernel, Invariant kernel, Linearisation, Reproducing kernel, *-Representation, Locally C*-algebra, Hilbert locally C*-module, Locally C∗-Algebra, Ordered ∗ -Space, ∗ -Representation, Hilbert Locally C∗-Module, Mathematics - Functional Analysis, FOS: Mathematics, Primary 46A08, 47L05, 47L10, Secondary 06F30, 43A35, 46L89, 47A20, Functional Analysis (math.FA), Hilbert locally \(C^*\)-module, positive semidefinite kernel, Other ``noncommutative'' mathematics based on \(C^*\)-algebra theory, locally \(C^*\)-algebra, barreled VH space, Barrelled spaces, bornological spaces, ordered \(*\)-space, Ordered topological structures, adjointable operator, Dilations, extensions, compressions of linear operators, Linear spaces of operators, Positive definite functions on groups, semigroups, etc., VH space
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16
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1
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