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A survey of some norm inequalities. (English) Zbl 07352441

Summary: We survey some classical norm inequalities of Hardy, Kallman, Kato, Kolmogorov, Landau, Littlewood, and Rota of the type \[ \Vert A f\Vert_{\mathcal{X}}^2 \le C \Vert f\Vert_{\mathcal{X}} \left \Vert A^2 f\right \Vert_{\mathcal{X}}, \quad f \in\operatorname{dom}\left (A^2\right ), \] and recall that under exceedingly stronger hypotheses on the operator \(A\) and/or the Banach space \(\mathcal{X} \), 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 \(\mathcal{X})\) all the way down to 1 (e.g., when \(A\) is a symmetric operator on a Hilbert space \(\mathcal{H} \)). We also survey some results in connection with an extension of the Hardy-Littlewood inequality involving quadratic forms as initiated by Everitt.

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.
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