2017
Том 69
№ 7

All Issues

On a criterion for the uniform boundedness of a C0-semigroup of operators in a Hilbert space

Gomilko A. M., Wróbel I., Zemanek J.

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Abstract

Let $T(t),\quad t ≥ 0$, be a $C_0$-semigroup of linear operators acting in a Hilbert space $H$ with norm $‖·‖$. We prove that $T(t)$ is uniformly bounded, i.e., $‖T(t)‖ ≤ M, \quad t ≥ 0$, if and only if the following condition is satisfied: $$\sup_{t > 0} \frac1t ∫_0^t∥(T(s)+T^{∗}(s))x ∥^2ds < ∞$$ forall $x ∈ H$, where $T^{*}$ is the adjoint operator.

English version (Springer): Ukrainian Mathematical Journal 59 (2007), no. 6, pp 938-944.

Citation Example: Gomilko A. M., Wróbel I., Zemanek J. On a criterion for the uniform boundedness of a C0-semigroup of operators in a Hilbert space // Ukr. Mat. Zh. - 2007. - 59, № 6. - pp. 853-858.

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