## 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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### References:

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