ON SOME CORRELATIONS IN THE THEORY OF DEGENERATE SEMIGROUPS OF OPERATORS

In the theory of degenerate operator semigroups the notions of $(L,p)$-radial and strongly $(L,p)$-radial operators have essential significances. In this work it is shown that in certain situations every of them means a generalisation in the case of degenerate strongly continouos semigroups of Hille -- Yosida conditions on an infinitesimal generator of $(C_0)$-continuous semigroup. Besides, sufficient conditions of equivalence of these notions are obtained. Analogous results are established for $(L,p)$-sectorial and strongly $(L,p)$-sectorial operators in the case of degenerate strongly holomorphic semigroups.

Keywords

degenerate operator semigroup, infinitesimal generator, Hille - Yosida theorem