## On analytic semigroups and cosine functions in Banach spaces

• If $$A$$ generates a bounded cosine function on a Banach space $$X$$ then the negative square root $$B$$ of $$A$$ generates a holomorphic semigroup, and this semigroup is the conjugate potential transform of the cosine function. This connection is studied in detail, and it is used for a characterization of cosine function generators in terms of growth conditions on the semigroup generated by $$B$$. This characterization relies on new results on the inversion of the vector-valued conjugate potential transform.

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Author: Peter Vieten, Valentin Keyantuo urn:nbn:de:hbz:386-kluedo-7553 Preprint English 1999 1999 Technische Universität Kaiserslautern 2000/04/03 Analytic semigroup; Cosine function; Potential transform Studia Math. Fachbereich Mathematik 5 Naturwissenschaften und Mathematik / 510 Mathematik 44-XX INTEGRAL TRANSFORMS, OPERATIONAL CALCULUS (For fractional derivatives and integrals, see 26A33. For Fourier transforms, see 42A38, 42B10. For integral transforms in distribution spaces, see 46F12. For numerical methods, see 65R10) / 44Axx Integral transforms, operational calculus / 44A15 Special transforms (Legendre, Hilbert, etc.) 44-XX INTEGRAL TRANSFORMS, OPERATIONAL CALCULUS (For fractional derivatives and integrals, see 26A33. For Fourier transforms, see 42A38, 42B10. For integral transforms in distribution spaces, see 46F12. For numerical methods, see 65R10) / 44Axx Integral transforms, operational calculus / 44A35 Convolution 47-XX OPERATOR THEORY / 47Dxx Groups and semigroups of linear operators, their generalizations and applications / 47D03 Groups and semigroups of linear operators (For nonlinear operators, see 47H20; see also 20M20) 47-XX OPERATOR THEORY / 47Dxx Groups and semigroups of linear operators, their generalizations and applications / 47D09 Operator sine and cosine functions and higher-order Cauchy problems [See also 34G10] Standard gemäß KLUEDO-Leitlinien vor dem 27.05.2011