Norm-Preserving Dilations and Their Applications to Optimal Error Bounds

Davis, Chandler; Kahan, W.; Weinberger, Hans F.

doi:10.1137/0719029

Cited by 189 publications

(85 citation statements)

References 7 publications

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“…To prove (2), we have to prove (Ã) for g m and the point i 0 Y j 0 pY q. But this follows from [DKW,Theoem 1.2].…”

Section: A Distance Formulamentioning

confidence: 99%

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Meet irreducible ideals in direct limit algebras

Donsig¹,

Hopenwasser²,

Hudson³

et al. 2000

MATH. SCAND.

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We study the meet irreducible ideals (ideals I so that I = J ∩ K implies I = J or I = K) in certain direct limit algebras. The direct limit algebras will generally be strongly maximal triangular subalgebras of AF C * -algebras, or briefly, strongly maximal TAF algebras. Of course, all ideals are closed and two-sided.These ideals have a description in terms of the coordinates, or spectrum, that is a natural extension of one description of meet irreducible ideals in the upper triangular matrices. Additional information is available if the limit algebra is an analytic subalgebra of its C * -envelope or if the analytic algebra is trivially analytic with an injective 0-cocycle. In the latter case, we obtain a complete description of the meet irreducible ideals, modeled on the description in the algebra of upper triangular matrices. This applies, in particular, to all full nest algebras.One reason for interest in the meet irreducible ideals of a strongly maximal TAF algebra is that each meet irreducible ideal is the kernel of a nest representation of the algebra (Theorem 2.4). A nest representation of an operator algebra A is a norm continuous representation of A acting on a Hilbert space with the property that the lattice of closed invariant subspaces for the representation is totally ordered. These representations were introduced in [L1] as analogues for a general operator algebra of the irreducible representations of a C * -algebra. The meet irreducible ideals seem analogous to the primitive ideals in a C * -algebra. Indeed, in a C * -algebra, the meet irreducible ideals are precisely the primitive ideals [L3, Theorem 2.1]. This analogy can be extended by noting that the meet irreducible ideals form a topological space under the hull-kernel topology and every ideal is the intersection of the meet * Partially supported by an NSF grant. † Partially supported by an NSERC grant.

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“…To prove (2), we have to prove (Ã) for g m and the point i 0 Y j 0 pY q. But this follows from [DKW,Theoem 1.2].…”

Section: A Distance Formulamentioning

confidence: 99%

“…sx ÀKxAx Ã Lx P C Then, by [DKW,Theorem 1.2], for every number w with jwj tx, the matrix Bx w sx Ax Cx has norm less than or equal to 1. In fact, this is also a necessary condition.…”

Section: A Distance Formulamentioning

confidence: 99%

Meet irreducible ideals in direct limit algebras

Donsig¹,

Hopenwasser²,

Hudson³

et al. 2000

MATH. SCAND.

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“…In [3] explicit solutions for problem (1.3) are given. Here we will give explicit solutions for problem (1.4).…”

Section: Introductionmentioning

confidence: 99%

“…Find X ∈ C N −n,N −m such that W (X) as in ( (1.4) Problem (1.3) is well known in dilation theory and was solved by Davis/Kahan/Weinberger, see [3] and the references therein. It has many applications in perturbation theory for eigenvalues, e.g.…”

Section: Introductionmentioning

confidence: 99%

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Minimization of the norm, the norm of the inverse and the condition number of a matrix by completion

Elsner

Mehrmann

1995

Numerical Linear Algebra App

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We study the problem of minimizing the norm, the norm of the inverse and the condition number with respect to the spectral norm, when a submatrix of a matrix can be chosen arbitrarily. For the norm minimization problem we give a different proof than that given by Davis/Kahan/Weinberger. This new approach can then also be used to characterize the completions that minimize the norm of the inverse. For the problem of optimizing the condition number we give a partial result.

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On ℋ︁_∞ model reduction for discrete‐time linear time‐invariant systems using linear matrix inequalities

Ebihara

Hirai

Hagiwara

2008

Asian Journal of Control

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In this paper, we address the H ∞ model reduction problem for linear time-invariant discrete-time systems. We revisit this problem by means of linear matrix inequality (LMI) approaches and first show a concise proof for the well-known lower bounds on the approximation error, which is given in terms of the Hankel singular values of the system to be reduced. In addition, when we reduce the system order by the multiplicity of the smallest Hankel singular value, we show that the H ∞ optimal reduced-order model can readily be constructed via LMI optimization. These results can be regarded as complete counterparts of those recently obtained in the continuous-time system setting.

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Norm-Preserving Dilations and Their Applications to Optimal Error Bounds

Cited by 189 publications

References 7 publications

Meet irreducible ideals in direct limit algebras

Meet irreducible ideals in direct limit algebras

Minimization of the norm, the norm of the inverse and the condition number of a matrix by completion

On ℋ︁_∞ model reduction for discrete‐time linear time‐invariant systems using linear matrix inequalities

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Norm-Preserving Dilations and Their Applications to Optimal Error Bounds

Cited by 189 publications

References 7 publications

Meet irreducible ideals in direct limit algebras

Meet irreducible ideals in direct limit algebras

Minimization of the norm, the norm of the inverse and the condition number of a matrix by completion

On ℋ︁∞ model reduction for discrete‐time linear time‐invariant systems using linear matrix inequalities

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Product

Resources

About

On ℋ︁_∞ model reduction for discrete‐time linear time‐invariant systems using linear matrix inequalities