Hilbert space structure and positive operators

Drivaliaris, Dimosthenis; Yannakakis, Nikos

doi:10.1016/j.jmaa.2004.12.007

Cited by 9 publications

(7 citation statements)

References 3 publications

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“…for ∀ ∈ and elements of field. In (Casazza, Fickus, Mixon, & Peterson, 2013;Drivaliaris & Yannakakis, 2005), it is said that consequence of Theorem 3 is as follows A real Banach space is isomorphic to a Hilbert space if and only if there is an isomorphism : → ′ that is positive and symmetric.…”

Section: Resultsmentioning

confidence: 99%

“…The use of space ℓ is intended to have distinction on = 2 and > 2. Proceeding of ICSA 2019, p: 51-56 ISBN 978-979-19256-3-1 (PDF) 53 At (Drivaliaris & Yannakakis, 2005), it is said that let is a normed space with norm ‖•‖ and ′ is the dual space of the X and 〈•,•〉 is their duality product. The following definitions are explanations of the duality product, as follows: Definition 1.…”

Section: Methodsmentioning

confidence: 99%

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Completeness of P-Summable as Hilbert Space with Its Dual Space

Sanusi

Rusyaman

Chaerani

2021

ICSA

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Hilbert space is a complete inner product space, meaning that each Cauchy sequence converges to a point in that space. One of the vector spaces that will be examined as the inner product space is p-summable space. The inner product space is a subset of vector spaces that have special properties that must be fulfilled. One way to prove vector space is the inner product space is to use parallelogram equality theorems. After it is known that the vector space is the inner product space, the completeness of the space will be proven using the dual space. The space used is the p-summable space, data that can be changed in a sequence form will be usable in this study. The results of this study will be useful as another application in determining a Hilbert space by using a method that is different from the definition. The analysis used will show comparison of the speed of completion accuracy will be a benchmark in this study, so that will be a new reference in determining a space is Hilbert space.

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Section: Resultsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Completeness of P-Summable as Hilbert Space with Its Dual Space

Sanusi

Rusyaman

Chaerani

2021

ICSA

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“…It is well-known from [4] (see also [2]) that Theorem 1. A Banach space X has the property that there exists a strongly positive operator P from X onto X * such that P x, x ≥ θ x 2 for some θ > 0, if and only if, the Banach space X is isomorphic to a Hilbert space.…”

Section: Preliminariesmentioning

confidence: 99%

Exponential stability of c0-semigroup via Lyapunov inequality in Banach space

Madani,

Bendaoud

2021

Preprint

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We give a relation between the exponential stability of C 0 −semigroup T = {T (t)} t≥0 and the solutions of Lyapunov inequality QAx, x + Qx, Ax ≤ −||x|| 2 , in B + (X, X * ), with X is a Banach space. The solutions of this inequality characterizes, the boundedness of the resolvent R(λ, A) inside and outside of the left half-plane ℜλ ≥ 0, and also the left invertibility of the C 0 −semigroup T.

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“…(ii) This inequality is derived in passing in [10]. Consider ϕ ∈ X * , for any ε > 0 there is a normalized element g ∈ X such that ϕ, g ≥ (1 − ε) ϕ since ϕ = sup x∈X, x =1 ϕ, x .…”

Section: Topologies and Continuitymentioning

confidence: 99%

Positive semigroups and algebraic Riccati equations in Banach spaces

Koshkin

2015

Positivity

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We generalize Wonham's theorem on solvability of algebraic operator Riccati equations to Banach spaces, namely there is a unique stabilizing solution to A * P + P A − P B B * P + C * C = 0 when (A, B) is exponentially stabilizable and (C, A) is exponentially detectable. The proof is based on a new approach that treats the linear part of the equation as the generator of a positive semigroup on the space of symmetric operators from a Banach space to its dual, and the quadratic part as an order concave map. A direct analog of global Newton's iteration for concave functions is then used to approximate the solution, the approximations converge in the strong operator topology, and the convergence is monotone. The linearized equations are the well-known Lyapunov equations of the form A * P + P A = −Q, and semigroup stability criterion in terms of them is also generalized.

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Hilbert space structure and positive operators

Cited by 9 publications

References 3 publications

Completeness of P-Summable as Hilbert Space with Its Dual Space

Completeness of P-Summable as Hilbert Space with Its Dual Space

Exponential stability of c0-semigroup via Lyapunov inequality in Banach space

Positive semigroups and algebraic Riccati equations in Banach spaces

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