Gelfand Numbers of Identity Operators Between Symmetric Sequence Spaces

Hinrichs, Aicke; Michels, Carsten

doi:10.1007/s11117-005-0022-1

Cited by 8 publications

(7 citation statements)

References 21 publications

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“…The case 0 < q ≤ p ≤ ∞ is relatively easy and we remark that in the literature there are results and conjectures about the asymptotics formulas for a n (id : E m → F m ) and c n (id : E m → F m ), where E and F are symmetric sequence spaces and F is naturally embedded into E , see, e.g., [12,20,21,22].…”

Section: Gelfand Numbers In the Casementioning

confidence: 99%

Approximation, Gelfand, and Kolmogorov numbers of Schatten class embeddings

Prochno,

Strzelecki

2021

Preprint

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Let 0 < p, q ≤ ∞ and denote by S N p and S N q the corresponding Schatten classes of real N × N matrices. We study approximation quantities of natural identities S N p → S N q between Schatten classes and prove asymptotically sharp bounds up to constants only depending on p and q, showing how approximation numbers are intimately related to the Gelfand numbers and their duals, the Kolmogorov numbers. In particular, we obtain new bounds for those sequences of s-numbers. Our results improve and complement bounds previously obtained by B. Carl and A.

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Section: Gelfand Numbers In the Casementioning

confidence: 99%

Approximation, Gelfand, and Kolmogorov numbers of Schatten class embeddings

Prochno,

Strzelecki

2021

Preprint

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“…We also remark that the values of the outer radii of orthogonal cross-polytopes (and so the inner radii of orthogonal boxes) can be derived from [24,Theorem 11.11.7] via Theorem 2.2 (see [7,Proposition 4.3] and [13,Theorem 1]). Finally, we mention that the results from [20,21] can be used to compute (or to estimate, up to multiplicative constants) the successive radii of unit balls of symmetric n-dimensional normed spaces; in particular, this applies to unit balls of Lorentz and Orlicz sequence spaces.…”

Section: Successive Radii Of P-ballsmentioning

confidence: 99%

Successive Radii of Families of Convex Bodies

González¹,

Cifre²,

Hinrichs³

2014

Bull. Aust. Math. Soc.

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We study properties of the so-called inner and outer successive radii of special families of convex bodies. First we consider the balls of the $p$-norms, for which we show that the precise value of the outer (inner) radii when $p\geq 2$ ($1\leq p\leq 2$), as well as bounds in the contrary case $1\leq p\leq 2$ ($p\geq 2$), can be obtained as consequences of known results on Gelfand and Kolmogorov numbers of identity operators between finite-dimensional normed spaces. We also prove properties that successive radii satisfy when we restrict to the families of the constant width sets and the $p$-tangential bodies.

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“…The proof goes along the same lines as the proof of Corollary 5.3; in (iii), apply the submultiplicativity of the function to obtain the final estimates. The remaining estimates in (i) and (ii) as well as further results can be found in [20] (see also [19] for a general approach by interpolation). (ii) Let 1 < p < 2, 1 ≤ q ≤ 2 and 1 ≤ k ≤ n 2 .…”

Section: Singular Numbers Of Identities Between Unitary Idealsmentioning

confidence: 99%

Summing norms of identities between unitary ideals

2006

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Based on abstract interpolation, we prove asymptotic formulae for the (F, 2)-summing norm of inclusions id : S n E → S n 2 , where E and F are two Banach sequence spaces. Here, S n E stands for the unitary ideal of operators on the n-dimensional Hilbert space whose singular values belong to E, and S n 2 = S n 2 for the Hilbert-Schmidt operators. Our results are noncommutative analogues of results due to Bennett and Carl, as well as their recent generalizations to Banach sequence spaces. As an application, we give lower and upper estimates for certain s-numbers of the embeddings id : S n E → S n 2 and id : S n 2 → S n E . In the concluding section, we finally consider mixing norms.

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Gelfand Numbers of Identity Operators Between Symmetric Sequence Spaces

Cited by 8 publications

References 21 publications

Approximation, Gelfand, and Kolmogorov numbers of Schatten class embeddings

Approximation, Gelfand, and Kolmogorov numbers of Schatten class embeddings

Successive Radii of Families of Convex Bodies

Summing norms of identities between unitary ideals

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