Perturbation Formulas for Traces on $C^*$-algebras

Hansen, Frank; Pedersen, Gert K.

doi:10.2977/prims/1195164797

Cited by 20 publications

(14 citation statements)

References 8 publications

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“…A proof of the next result is given in [34], but analogous reasoning can be found in [57], where the editors point out that similar questions "were extensively investigated by Birman and Solomyak [12,13] within the very general scope of their theory of double operator integrals".…”

Section: The Fréchet Differentialmentioning

confidence: 99%

“…The author and Pedersen [34] proved the following generalization of results of Bernstein [11] and Brown and Kosaki [15].…”

Section: Theorem 63 If τ Is a Finite Trace On A C * -Algebramentioning

confidence: 99%

“…, λ n . The author and Pedersen [34] proved the following result which shows that the Fréchet differential under a trace acts like the usual derivative function.…”

Section: If T and S Are Hermitian N×n Matrices And Fmentioning

confidence: 99%

“…A complete and stringent formulation is found in Araki's paper [5], that "contains a powerful computational tool, which does not seem to be widely known among mathematicians." The following result is only a fraction of this tool, and a simple proof is given in the reference [34]. An interesting and non-trivial question is to specify conditions under which the map T → f (T ) is Fréchet differentiable, where f is a real function defined on an open interval I and T is restricted to self-adjoint operators acting on a Hilbert space with spectra in I.…”

Section: The Fréchet Differentialmentioning

confidence: 99%

“…where • is the Hadamard product and (6.17) is the Löwner matrix [34] defined from the (not necessarily distinct) eigenvalues λ 1 , . .…”

Section: If T and S Are Hermitian N×n Matrices And Fmentioning

confidence: 99%

See 4 more Smart Citations

Operator Inequalities Associated with Jensen’s Inequality

Hansen

2000

Survey on Classical Inequalities

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Abstract. We give a survey of various operator inequalities associated with Jensen's inequality and study the class of operator convex functions of several variables. Related questions are considered. Jensen's classical inequalityLet I be a real interval. A function f : I → R is said to be convex, iffor all t, s ∈ I and every λ ∈ [0, 1]. Notice that the definition, in order to be meaningful, requires that f can be evaluated in λt + (1 − λ)s, or equivalently that I is convex. But this is satisfied because the convex subsets of R are the intervals. If f satisfies (1.1) just for λ = 1/2, then f is said to be midpoint convex. It is easy to establish that a continuous and mid-point convex function is convex. The geometric interpretation of (1.1) is that the graph of f is below the chord and consequently above the extensions of the chord. This entails that a convex function defined on an open interval is continuous. Condition (1.1) can be reformulated asfor all mutually different numbers t, s, r ∈ I. If we for such numbers define the divided difference [ts] of f taken in the points t, s as

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Section: The Fréchet Differentialmentioning

confidence: 99%

“…The author and Pedersen [34] proved the following generalization of results of Bernstein [11] and Brown and Kosaki [15].…”

Section: Theorem 63 If τ Is a Finite Trace On A C * -Algebramentioning

confidence: 99%

“…, λ n . The author and Pedersen [34] proved the following result which shows that the Fréchet differential under a trace acts like the usual derivative function.…”

Section: If T and S Are Hermitian N×n Matrices And Fmentioning

confidence: 99%

Section: The Fréchet Differentialmentioning

confidence: 99%

“…where • is the Hadamard product and (6.17) is the Löwner matrix [34] defined from the (not necessarily distinct) eigenvalues λ 1 , . .…”

Section: If T and S Are Hermitian N×n Matrices And Fmentioning

confidence: 99%

See 3 more Smart Citations

Operator Inequalities Associated with Jensen’s Inequality

Hansen

2000

Survey on Classical Inequalities

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On the performance optimization in multiuser MIMO systems

Boche

Jorswieck

2007

Trans Emerging Tel Tech

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We maximize the performance of multiple-input multiple-output (MIMO) multiple access channels (MAC) and broadcast channels (BC) under individual and sum power constraints. We observe that the performance metrics sum capacity and sum mean-square error (MSE) belong to a general class of functions which are the trace of a matrix-monotone function. The sum capacity if successive interference cancellation (SIC) is applied in the uplink or if Costa Precoding is applied in the downlink, or the normalised sum MSE if a multiuser minimum mean-square error (MMSE) receiver is applied at the base, belong to that class of performance functions. We use Lowner's representation of matrix monotone functions in order to characterise the optimum transmission strategies. Using the Karush-Kuhn-Tucker optimality conditions, we show that the mutual covariance matrix optimisation can be decomposed into power allocation and covariance matrix optimisation under individual power constraints. The covariance matrix optimisation under individual power constraints in turn can be decomposed into a kind of modified single-user covariance matrix optimisation treating the other users as noise. The concrete structure of the single-user program depends on the performance metric used. The proposed algorithms efficiently solve the multi-user MIMO performance optimisation problem. These results generalise the results regarding the performance capacity optimisation recently reported for sum capacity optimisation and the results regarding the sum MSE optimisation. The optimal transmit strategy deconstructs into two independent parts: the coding and the signal processing part. The second part is adapted to the channel and performs multi-user power control and transmit covariance matrix optimisation

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Characterisation of Matrix Entropies

Hansen

Zhang

2015

Lett Math Phys

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The notion of matrix entropy was introduced by Tropp and Chen with the aim of measuring the fluctuations of random matrices. It is a certain entropy functional constructed from a representing function with prescribed properties, and Tropp and Chen gave some examples. We give several abstract characterisations of matrix entropies together with a sufficient condition in terms of the second derivative of their representing function.

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Perturbation Formulas for Traces on $C^*$-algebras

Cited by 20 publications

References 8 publications

Operator Inequalities Associated with Jensen’s Inequality

Operator Inequalities Associated with Jensen’s Inequality

On the performance optimization in multiuser MIMO systems

Characterisation of Matrix Entropies

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