"We show that each power bounded operator with spectral radius equal to one a reflexive Banach space has a nonzero vector which is not supercyclic. Equivalently, the operator has a nontrivial closed invariant homogeneous subset. Moreover, the operator has a nontrivial closed invariant cone if 1 belongs to its spectrum. This generalizes the corresponding results for Hilbert space operators. For non-reflexive Banach spaces these results remain true; however, the non-supercyclic vector (inavariant cone, respectively) relates to the adjoint of the operator."@en . "2997;3004" . "We show that each power bounded operator with spectral radius equal to one a reflexive Banach space has a nonzero vector which is not supercyclic. Equivalently, the operator has a nontrivial closed invariant homogeneous subset. Moreover, the operator has a nontrivial closed invariant cone if 1 belongs to its spectrum. This generalizes the corresponding results for Hilbert space operators. For non-reflexive Banach spaces these results remain true; however, the non-supercyclic vector (inavariant cone, respectively) relates to the adjoint of the operator." . "supercyclic vectors; invariant subspace problem; positive operators"@en . . . "537403" . "Power bounded operators and supercyclic vectors II" . . . "Mocninov\u011B ohrani\u010Den\u00E9 oper\u00E1tory a supercyklick\u00E9 vektory II"@cs . "Proceedings of the American Mathematical Society" . . "Mocninov\u011B ohrani\u010Den\u00E9 oper\u00E1tory a supercyklick\u00E9 vektory II"@cs . "133" . "10" . "RIV/67985840:_____/05:00030787" . "US - Spojen\u00E9 st\u00E1ty americk\u00E9" . . . . . . "8"^^ . . . "Power bounded operators and supercyclic vectors II" . "Power bounded operators and supercyclic vectors II"@en . "M\u00FCller, Vladim\u00EDr" . . "1"^^ . "P(GA201/03/0041), Z(AV0Z10190503)" . "0002-9939" . "1"^^ . . "Power bounded operators and supercyclic vectors II"@en . "[B7E385EBFFDD]" . . . . "Ka\u017Ed\u00FD mocninov\u011B ohrani\u010Den\u00FD oper\u00E1tor v reflexivn\u00EDm Banachov\u011B prostoru, jeho\u017E spektrum obsahuje 1, m\u00E1 netrivi\u00E1ln\u00ED invariantn\u00ED ku\u017Eel."@cs . "RIV/67985840:_____/05:00030787!RIV06-AV0-67985840" .