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A survey of some norm inequalities. (English) Zbl 1514.47015

With 94 references cited, this paper gives a survey of some classical norm inequalities of Hardy, Kallman, Kato, Kolmogorov, Landau, Littlewood, and Rota of the type \[ \|Af\|_{\mathcal X}^2 \le C\|f\|_{\mathcal X}\|A^2f\|_{\mathcal X},\quad f\in \mathrm{dom}\, (A^2), \] where \(\mathcal X\) is a Banach space. The optimal constant \(C\) in these inequalities diminishes from \(4\) (e.g., when \(A\) is the generator of a \(C_0\) contraction semigroup on a Banach space \(\mathcal X\)) down to \(1\) (e.g., when \(A\) is a symmetric operator on a Hilbert space \(H\)). The authors also survey some results in connection with an extension of the Hardy-Littlewood inequality involving quadratic forms. Remarks and examples are given in the paper.
Reviewer: Tin Yau Tam (Reno)

MSC:

47A30 Norms (inequalities, more than one norm, etc.) of linear operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
47B25 Linear symmetric and selfadjoint operators (unbounded)
47B44 Linear accretive operators, dissipative operators, etc.
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
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References:

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