This_DT0 paper_NN1 presents_VVZ several_DT0 Lipschitz_NN1 implicit_AJ0 function_NN1 theorems_NN2 for_PRP maps_NN2 between_PRP Banach_NP0 spaces_NN2 ._SENT The_AT0 results_NN2 use_VVB semi-inner_AJ0 products_NN2 as_PRP a_AT0 means_NN0 for_PRP understanding_VVG the_AT0 geometry_NN1 of_PRF normed_AJ0 spaces_NN2 ._SENT The_AT0 arguments_NN2 rely_VVB on_PRP corresponding_AJ0 geometric_AJ0 properties_NN2 of_PRF operators_NN2 on_PRP normed_AJ0 spaces_NN2 ,_PUN and_CJC on_PRP Ekeland_NP0 's_POS variational_AJ0 principle_NN1 ,_PUN in_PRP contrast_NN1 with_PRP classic_AJ0 results_NN2 such_DT0 as_PRP the_AT0 inverse_AJ0 function_NN1 theorems_NN2 of_PRF Nash_NP0 and_CJC Moser_NP0 and_CJC their_DPS use_NN1 of_PRF Newton_NP0 iteration_NN1 procedures_NN2 (_PUL e.g._AV0 ,_PUN see_VVB Hamilton_NP0 )_PUR ._SENT 
The_AT0 literature_NN1 on_PRP implicit_AJ0 function_NN1 theorems_NN2 is_VBZ significant_AJ0 ,_PUN with_PRP many_DT0 important_AJ0 results_NN2 using_VVG variational_AJ0 analysis_NN1 to_TO0 extend_VVI standard_AJ0 theorems_NN2 ,_PUN including_PRP the_AT0 classic_AJ0 theorem_NN1 of_PRF Clarke_NP0 ._SENT Dontchev_NP0 and_CJC Rockafellar_NP0 provide_VVB a_AT0 comprehensive_AJ0 development_NN1 and_CJC extension_NN1 of_PRF much_DT0 of_PRF this_DT0 work_NN1 ._SENT Most_DT0 of_PRF these_DT0 results_NN2 focus_VVB on_PRP extending_VVG strong_AJ0 differentiability_NN1 notions_NN2 from_PRP functions_NN2 to_PRP set-valued_AJ0 maps_NN2 to_TO0 establish_VVI implicit_AJ0 mapping_NN1 theorems_NN2 for_PRP inclusions_NN2 ,_PUN or_CJC on_PRP using_VVG generalized_AJ0 derivatives_NN2 to_TO0 establish_VVI implicit_AJ0 function_NN1 theorems_NN2 for_PRP nonsmooth_AJ0 functions_NN2 ;_PUN for_PRP example_NN1 ,_PUN see_VVB [_PUL 13_CRD ]_PUR ,_PUN [_PUL 3_CRD ]_PUR ,_PUN [_PUL 10_CRD ]_PUR ,_PUN [_PUL 11_CRD ]_PUR ,_PUN [_PUL 12_CRD ]_PUR ,_PUN [_PUL 15_CRD ]_PUR ,_PUN [_PUL 16_CRD ]_PUR ,_PUN [_PUL 23_CRD ]_PUR ,_PUN among_PRP others_NN2 ._SENT In_PRP contrast_NN1 ,_PUN the_AT0 results_NN2 here_AV0 focus_VVB on_PRP the_AT0 standard_AJ0 implicit_AJ0 function_NN1 problem_NN1 ,_PUN and_CJC weaken_VVB differentiability_AJ0 properties_NN2 of_PRF these_DT0 functions_NN2 by_PRP using_VVG geometric_AJ0 properties_NN2 of_PRF operators_NN2 based_VVN on_PRP semi-inner_AJ0 products_NN2 ._SENT 
Semi-inner_NN1 products_NN2 have_VHB been_VBN important_AJ0 in_PRP many_DT0 problems_NN2 regarding_PRP operators_NN2 and_CJC geometry_NN1 in_PRP Banach_NP0 spaces_NN2 ,_PUN including_PRP characterizations_NN2 of_PRF metric_AJ0 projections_NN2 and_CJC best_AJS approximations_NN2 ,_PUN orthogonality_NN1 ,_PUN properties_NN2 of_PRF linear_AJ0 operators_NN2 ,_PUN differential_AJ0 equations_NN2 ,_PUN and_CJC machine_NN1 learning_NN1 and_CJC pattern_NN1 analysis_NN1 ,_PUN as_CJS illustrated_VVN in_PRP [_PUL 22_CRD ]_PUR ,_PUN [_PUL 24_CRD ]_PUR ,_PUN [_PUL 25_CRD ]_PUR ,_PUN [_PUL 9_CRD ]_PUR ,_PUN [_PUL 26_CRD ]_PUR ,_PUN [_PUL 29_CRD ]_PUR ,_PUN [_PUL 30_CRD ]_PUR ,_PUN among_PRP others_NN2 ._SENT Here_AV0 the_AT0 use_NN1 of_PRF semi-inner_NN1 products_NN2 allows_VVZ the_AT0 formulation_NN1 of_PRF geometric_AJ0 properties_NN2 of_PRF derivatives_NN2 that_CJT can_VM0 be_VBI used_VVN to_TO0 ensure_VVI Lipschitz_NN1 continuity_NN1 in_PRP place_NN1 of_PRF the_AT0 strong_AJ0 differentiability_NN1 conditions_NN2 required_VVN in_PRP standard_AJ0 results_NN2 ._SENT The_AT0 results_NN2 are_VBB illustrated_VVN with_PRP an_AT0 application_NN1 to_PRP a_AT0 central_AJ0 problem_NN1 in_PRP mathematical_AJ0 economics_NN1 ,_PUN characterizing_VVG Lipschitz_NP0 behavior_NN1 of_PRF Pareto_NP0 optimal_AJ0 allocations_NN2 in_PRP a_AT0 model_NN1 of_PRF trade_NN1 over_PRP an_AT0 infinite_AJ0 horizon_NN1 ,_PUN which_DTQ in_PRP turn_NN1 is_VBZ the_AT0 key_AJ0 property_NN1 ensuring_VVG generic_AJ0 determinacy_NN1 of_PRF equilibrium_NN1 in_PRP economies_NN2 with_PRP infinitely_AV0 many_DT0 commodities_NN2 ._SENT 
2._CRD Preliminaries_NN2 
Let_VVB X_ZZ0 be_VBI a_AT0 real_AJ0 Banach_NN1 space_NN1 and_CJC X_ZZ0 *_PUN be_VBI its_DPS norm_NN1 dual_AJ0 ._SENT 
The_AT0 main_AJ0 tools_NN2 in_PRP the_AT0 paper_NN1 are_VBB generalizations_NN2 of_PRF inner_AJ0 product_NN1 for_PRP Banach_NP0 spaces_NN2 ._SENT For_PRP the_AT0 real_AJ0 Banach_NN1 space_NN1 X_ZZ0 ,_PUN consider_VVB the_AT0 semi-inner_AJ0 products_NN2 between_PRP two_CRD elements_NN2 x_ZZ0 and_CJC y_ZZ0 defined_VVN by_PRP equation_NN1 and_CJC equation_NN1 ._SENT 
For_PRP Rn_NN0 or_CJC for_PRP any_DT0 Hilbert_NP0 space_NN1 ,_PUN these_DT0 semi-inner_AJ0 products_NN2 coincide_VVB with_PRP the_AT0 standard_AJ0 inner_AJ0 product_NN1 (_PUL x_ZZ0 ,_PUN y_ZZ0 )_PUR ._SENT In_PRP a_AT0 general_AJ0 Banach_NN1 space_NN1 ,_PUN these_DT0 share_VVB many_DT0 of_PRF the_AT0 important_AJ0 properties_NN2 of_PRF an_AT0 inner_AJ0 product_NN1 ;_PUN the_AT0 main_AJ0 difference_NN1 with_PRP an_AT0 inner_AJ0 product_NN1 is_VBZ that_CJT these_DT0 semi-inner_AJ0 products_NN2 are_VBB not_XX0 necessarily_AV0 bilinear_AJ0 or_CJC symmetric_AJ0 ._SENT See_VVB Deimling_AJ0 ,_PUN for_PRP example_NN1 ,_PUN for_PRP discussion_NN1 ._SENT Some_DT0 of_PRF the_AT0 main_AJ0 properties_NN2 of_PRF (_PUL ._SENT ,_PUN ._SENT )_PUR +-_NN1 are_VBB given_VVN in_PRP the_AT0 lemma_NN1 below_AV0 ._SENT 
Lemma_NN1 1_CRD 
Deimling_NN1 [_PUL 8_CRD ,_PUN Proposition_NN1 13.1_CRD ]_PUR 
Let_VVB X_ZZ0 be_VBI a_AT0 real_AJ0 Banach_NN1 space_NN1 ._SENT For_PRP all_DT0 x_ZZ0 ,_PUN y_ZZ0 ,_PUN z_ZZ0 ∈_NN1 X_ZZ0 ,_PUN (_PUL i_ZZ0 )_PUR equation_NN1 and_CJC equation_NN1 ;_PUN (_PUL ii_CRD )_PUR equation_NN1 ;_PUN (_PUL iii_CRD )_PUR If_CJS equation_NN1 and_CJC equation_NN1 ,_PUN then_AV0 equation_NN1 ;_PUN (_PUL iv_CRD )_PUR (_PUL ._SENT ,_PUN ._SENT )_PUR +_NP0 is_VBZ upper_AJ0 semicontinuous_AJ0 ,_PUN (_PUL ._SENT ,_PUN ._SENT )_PUR -_PUN is_VBZ lower_AV0 semicontinuous_AJ0 ,_PUN and_CJC (_PUL ._SENT ,_PUN y_ZZ0 )_PUR +-_NN1 is_VBZ continuous_AJ0 for_PRP each_DT0 y_ZZ0 ∈_NN1 X_ZZ0 ;_PUN (_PUL v_ZZ0 )_PUR IfX∗_NP0 is_VBZ strictly_AV0 convex_AJ0 ,_PUN then_AV0 (_PUL ·_NN0 ,_PUN ·_NN0 )_PUR +_NP0 =_NP0 (_PUL ·_NN0 ,_PUN ·_NN0 )_PUR −_NN1 ._SENT 
In_PRP particular_AJ0 ,_PUN using_VVG (_PUL v_ZZ0 )_PUR and_CJC a_AT0 basic_AJ0 result_NN1 of_PRF Lindenstrauss_NP0 ,_PUN whenever_AVQ X_ZZ0 is_VBZ reflexive_NN1 then_AV0 it_PNP has_VHZ an_AT0 equivalent_AJ0 norm_NN1 with_PRP respect_NN1 to_PRP which_DTQ (_PUL ·_NN0 ,_PUN ·_NN0 )_PUR +_NP0 =_NP0 (_PUL ·_NN0 ,_PUN ·_NN0 )_PUR −_NN1 ._SENT More_AV0 generally_AV0 ,_PUN for_PRP any_DT0 x_ZZ0 ,_PUN y_ZZ0 ∈_NN1 X_ZZ0 ,_PUN (_PUL x_ZZ0 ,_PUN y_ZZ0 )_PUR +_NP0 =_NP0 −_NP0 (_PUL −x_NP0 ,_PUN y_ZZ0 )_PUR −_NN1 ._SENT 
Remark_NN1 ._SENT Alternatively_AV0 ,_PUN following_VVG e.g._AV0 ,_PUN Lumer_NP0 and_CJC Giles_NP0 ,_PUN a_AT0 semi-inner_NN1 product_NN1 can_VM0 be_VBI defined_VVN ab-_AJ0 stractly_AV0 to_TO0 be_VBI a_AT0 function_NN1 on_PRP X_ZZ0 ×_PRP X_ZZ0 satisfying_AJ0 key_AJ0 properties_NN2 of_PRF an_AT0 inner_AJ0 product_NN1 ,_PUN including_PRP linearity_NN1 in_PRP one_CRD component_NN1 and_CJC a_AT0 version_NN1 of_PRF the_AT0 Cauchy-Schwarz_NP0 inequality_NN1 ._SENT All_DT0 of_PRF the_AT0 definitions_NN2 below_AV0 could_VM0 be_VBI formulated_VVN analogously_AV0 for_PRP such_DT0 an_AT0 abstract_AJ0 semi-inner_NN1 product_NN1 ._SENT It_PNP is_VBZ straightforward_AJ0 to_TO0 see_VVI that_CJT all_DT0 of_PRF the_AT0 results_NN2 of_PRF the_AT0 paper_NN1 would_VM0 hold_VVI under_PRP these_DT0 more_DT0 general_AJ0 conditions_NN2 ,_PUN with_PRP the_AT0 modification_NN1 that_CJT the_AT0 norm_NN1 on_PRP X_ZZ0 is_VBZ given_VVN by_PRP the_AT0 norm_NN1 induced_VVN by_PRP the_AT0 semi-inner_AJ0 product_NN1 ∥x∥_NN1 (_PUL ·_NN0 ,_PUN ·_NN0 )_PUR =_NN1 (_PUL x_ZZ0 ,_PUN x_ZZ0 )_PUR 12_CRD ,_PUN which_DTQ need_VM0 not_XX0 coincide_VVI with_PRP the_AT0 original_AJ0 norm_NN1 on_PRP X_ZZ0 ._SENT We_PNP return_VVB to_TO0 discuss_VVI this_DT0 point_NN1 in_PRP more_DT0 detail_NN1 at_PRP the_AT0 end_NN1 of_PRF this_DT0 section_NN1 ._SENT 
Throughout_AV0 let_VVB T_ZZ0 :_PUN X_ZZ0 →_NP0 X_ZZ0 be_VBI a_AT0 linear_AJ0 operator_NN1 on_PRP X_ZZ0 ,_PUN and_CJC given_VVN an_AT0 index_NN1 set_NN1 A_ZZ0 ,_PUN let_VVB T_ZZ0 :_PUN =_NP0 {_PUL T_ZZ0 (_PUL a_AT0 )_PUR :_PUN X_ZZ0 →_NN1 X_ZZ0 :_PUN a_AT0 ∈_NN1 A_ZZ0 }_PUR be_VBI a_AT0 family_NN1 of_PRF linear_AJ0 operators_NN2 on_PRP X_ZZ0 ._SENT We_PNP start_VVB with_PRP some_DT0 basic_AJ0 terminology_NN1 ._SENT The_AT0 operator_NN1 T_ZZ0 is_VBZ bounded_VVN below_AV0 if_CJS there_AV0 exists_VVZ b_ZZ0 >_NN1 0_CRD such_DT0 that_CJT ∥Tx∥_NP0 ≥_NP0 b∥x∥_NN1 for_PRP each_DT0 x_ZZ0 ∈_NP0 X_ZZ0 ._SENT The_AT0 family_NN1 T_ZZ0 is_VBZ uniformly_AV0 boundedbelowifthereexistsb_AJ0 >_NN1 0suchthat∥T_NN1 (_PUL a_AT0 )_PUR x∥≥b∥x∥foreachx∈Xandforeacha∈A_NN1 ,_PUN andT_NN1 is_VBZ uniformly_AV0 bounded_VVN if_CJS there_EX0 exists_VVZ M_NP0 >_NP0 0_CRD such_DT0 that_DT0 ∥T_NN1 (_PUL a_AT0 )_PUR x∥_NN1 ≤_NP0 M∥x∥_NP0 for_PRP each_DT0 x_ZZ0 ∈_NN1 X_ZZ0 and_CJC for_PRP each_DT0 a_AT0 ∈_NN1 A_ZZ0 ._SENT For_PRP a_AT0 subset_NN1 B_ZZ0 ⊆_NN1 X_ZZ0 ,_PUN co_NN1 B_ZZ0 denotes_VVZ the_AT0 convex_AJ0 hull_NN1 of_PRF B_ZZ0 ,_PUN and_CJC coB_NN1 denotes_VVZ the_AT0 closed_AJ0 convex_AJ0 hull_NN1 of_PRF B_ZZ0 ._SENT For_PRP r_ZZ0 >_NN1 0_CRD and_CJC x_ZZ0 ∈_NN1 X_ZZ0 ,_PUN Br_NP0 (_PUL x_ZZ0 )_PUR :_PUN =_NP0 {_PUL y_ZZ0 ∈_NN1 X_ZZ0 :_PUN ∥y_NP0 −_NP0 x∥_NN1 <_NN1 r_ZZ0 }_PUR ._SENT 
Of_PRF particular_AJ0 interest_NN1 in_PRP this_DT0 paper_NN1 are_VBB analogues_NN2 of_PRF negative_AJ0 definite_AJ0 linear_AJ0 operators_NN2 defined_VVN by_PRP use_NN1 of_PRF the_AT0 semi-inner_NN1 products_NN2 (_PUL ·_NN0 ,_PUN ·_NN0 )_PUR ±_NN0 ._SENT Say_VVB the_AT0 operator_NN1 T_ZZ0 :_PUN X_ZZ0 →_NN1 X_ZZ0 is_VBZ weakly_AV0 +_AJ0 negative_AJ0 definite_AJ0 if_CJS equation_NN1 and_CJC +_NN1 negative_NN1 definite_AJ0 if_CJS there_AV0 exists_VVZ c_ZZ0 >_NN1 0_CRD such_DT0 that_DT0 equation_NN1 ._SENT Similarly_AV0 ,_PUN T_PNP is_VBZ weakly_AV0 −_AJ0 negative_AJ0 definite_AJ0 if_CJS equation_NN1 and_CJC −_NN1 negative_NN1 definite_AJ0 if_CJS there_AV0 exists_VVZ c_ZZ0 >_NN1 0_CRD such_DT0 that_DT0 equation_NN1 ._SENT 
The_AT0 family_NN1 of_PRF linear_AJ0 operators_NN2 T_ZZ0 =_NP0 {_PUL T_ZZ0 (_PUL a_AT0 )_PUR :_PUN X_ZZ0 →_NN1 X_ZZ0 :_PUN a_AT0 ∈_NN1 A_ZZ0 }_PUR is_VBZ weakly_AV0 +_AJ0 negative_AJ0 definite_AJ0 if_CJS T_ZZ0 (_PUL a_AT0 )_PUR is_VBZ weakly_AV0 +_AJ0 negative_AJ0 definite_AJ0 for_PRP each_DT0 a_AT0 ∈_NN1 A_ZZ0 ,_PUN and_CJC uniformly_AV0 +_AJ0 negative_AJ0 definite_AJ0 if_CJS there_AV0 exists_VVZ c_ZZ0 >_NN1 0_CRD such_DT0 that_DT0 equation_NN1 ._SENT Similarly_AV0 ,_PUN T_PNP is_VBZ weakly_AV0 −_AJ0 negative_AJ0 definite_AJ0 if_CJS T_ZZ0 (_PUL a_AT0 )_PUR is_VBZ weakly_AV0 −_AJ0 negative_AJ0 definite_AJ0 for_PRP each_DT0 a_AT0 ∈_NN1 A_ZZ0 ,_PUN and_CJC uniformly_AV0 −_AJ0 negative_AJ0 definite_AJ0 if_CJS there_AV0 exists_VVZ c_ZZ0 >_NN1 0_CRD such_DT0 that_DT0 equation_NN1 ._SENT 
The_AT0 lack_NN1 of_PRF symmetry_NN1 of_PRF the_AT0 semi-inner_NN1 products_NN2 (_PUL ·_NN0 ,_PUN ·_NN0 )_PUR ±_NN0 leads_VVZ to_PRP alternative_AJ0 definitions_NN2 of_PRF "_PUQ negative_AJ0 definite_AJ0 "_PUQ to_PRP the_AT0 ones_NN2 given_VVN above_AV0 ._SENT For_PRP a_AT0 linear_AJ0 operator_NN1 T_ZZ0 :_PUN X_ZZ0 →_NN1 X_ZZ0 ,_PUN say_VVB T_ZZ0 is_VBZ weakly_AV0 +_NN1 dissipative_AJ0 if_CJS equation_NN1 and_CJC +_NN1 dissipative_AJ0 if_CJS there_AV0 exists_VVZ c_ZZ0 >_NN1 0_CRD such_DT0 that_DT0 equation_NN1 ._SENT Similarly_AV0 ,_PUN T_PNP is_VBZ weakly_AV0 −_NN1 dissipative_AJ0 if_CJS equation_NN1 and_CJC −_NN1 dissipative_AJ0 if_CJS there_AV0 exists_VVZ c_ZZ0 >_NN1 0_CRD such_DT0 that_DT0 equation_NN1 ._SENT The_AT0 family_NN1 of_PRF linear_AJ0 operators_NN2 T_ZZ0 =_NP0 {_PUL T_ZZ0 (_PUL a_AT0 )_PUR :_PUN X_ZZ0 →_NN1 X_ZZ0 :_PUN a_AT0 ∈_NN1 A_ZZ0 }_PUR is_VBZ weakly_AV0 +_NN1 dissipative_AJ0 if_CJS T_ZZ0 (_PUL a_AT0 )_PUR is_VBZ weakly_AV0 +_NN1 dissipative_AJ0 for_PRP each_DT0 a_AT0 ∈_NN1 A_ZZ0 ,_PUN and_CJC uniformly_AV0 +_VVB dissipative_AJ0 if_CJS there_AV0 exists_VVZ c_ZZ0 >_NN1 0_CRD such_DT0 that_DT0 equation_NN1 ._SENT Similarly_AV0 ,_PUN T_PNP is_VBZ weakly_AV0 −_NN1 dissipative_AJ0 if_CJS T_ZZ0 (_PUL a_AT0 )_PUR is_VBZ weakly_AV0 −_NN1 dissipative_AJ0 for_PRP each_DT0 a_AT0 ∈_NN1 A_ZZ0 ,_PUN and_CJC uniformly_AV0 −_VVB dissipative_AJ0 if_CJS there_AV0 exists_VVZ c_ZZ0 >_NN1 0_CRD such_DT0 that_DT0 equation_NN1 ._SENT Throughout_PRP the_AT0 notation_NN1 ±_NN0 indicates_VVZ that_CJT the_AT0 associated_AJ0 condition_NN1 holds_VVZ with_PRP respect_NN1 to_PRP either_AV0 (_PUL ·_NN0 ,_PUN ·_NN0 )_PUR +_NN1 or_CJC (_PUL ·_NN0 ,_PUN ·_NN0 )_PUR −_NN1 ._SENT 
Remark_NN1 ._SENT When_CJS X∗_NP0 is_VBZ strictly_AV0 convex_AJ0 ,_PUN (_PUL ·_NN0 ,_PUN ·_NN0 )_PUR +_NP0 =_NP0 (_PUL ·_NN0 ,_PUN ·_NN0 )_PUR −_NN1 ,_PUN and_CJC the_AT0 qualifier_NN1 ±_NN0 can_VM0 be_VBI dropped_VVN everywhere_AV0 above_AV0 ._SENT Remark_NN1 ._SENT When_CJS X_ZZ0 is_VBZ a_AT0 Hilbert_NP0 space_NN1 ,_PUN a_AT0 linear_AJ0 operator_NN1 T_ZZ0 is_VBZ (_PUL weakly_AV0 )_PUR negative_AJ0 definite_AJ0 if_CJS and_CJC only_AV0 if_CJS it_PNP is_VBZ (_PUL weakly_AV0 )_PUR dissipative_AJ0 ._SENT 
The_AT0 following_AJ0 two_CRD lemmas_NN2 show_VVB that_CJT versions_NN2 of_PRF negative_AJ0 definite_AJ0 and_CJC dissipative_AJ0 operators_NN2 share_VVB many_DT0 of_PRF the_AT0 most_AV0 important_AJ0 properties_NN2 of_PRF standard_AJ0 negative_AJ0 definite_AJ0 operators_NN2 in_PRP Hilbert_NP0 spaces_NN2 ._SENT 
replaced-dns_NP0 X_ZZ0 be_VBI a_AT0 real_AJ0 Banach_NN1 space_NN1 ._SENT Let_VVB T_ZZ0 :_PUN X_ZZ0 →_NP0 X_ZZ0 be_VBI a_AT0 linear_AJ0 operator_NN1 ,_PUN and_CJC let_VVB T_ZZ0 =_NP0 {_PUL T_ZZ0 (_PUL a_AT0 )_PUR :_PUN X_ZZ0 →_NN1 X_ZZ0 :_PUN a_AT0 ∈_NN1 A_ZZ0 }_PUR be_VBI a_AT0 family_NN1 of_PRF linear_AJ0 operators_NN2 ._SENT 
(_PUL i_ZZ0 )_PUR If_CJS T_ZZ0 is_VBZ ±_NN0 negative_AJ0 definite_AJ0 or_CJC ±_NN0 dissipative_AJ0 ,_PUN then_AV0 T_ZZ0 is_VBZ bounded_VVN below_AV0 ._SENT 
(_PUL ii_CRD )_PUR If_CJS T_ZZ0 is_VBZ uniformly_AV0 ±_NN0 negative_AJ0 definite_AJ0 or_CJC uniformly_AV0 ±_NN0 dissipative_AJ0 ,_PUN then_AV0 T_ZZ0 is_VBZ uniformly_AV0 bounded_VVN below_AV0 ._SENT 
Proof_NN1 ._SENT The_AT0 proofs_NN2 will_VM0 be_VBI for_PRP the_AT0 negative_AJ0 definite_AJ0 case_NN1 ;_PUN the_AT0 dissipative_AJ0 case_NN1 is_VBZ analogous_AJ0 ._SENT For_PRP part_NN1 (_PUL i_CRD )_PUR ,_PUN there_AV0 exists_VVZ c_ZZ0 >_NN1 0_CRD such_DT0 that_CJT (_PUL x_ZZ0 ,_PUN Tx_NP0 )_PUR ±_NN0 ≤_NP0 −c∥x∥2_CRD for_PRP all_DT0 x_ZZ0 ∈_NP0 X_ZZ0 ._SENT So_AV0 given_VVN x_ZZ0 ∈_NN1 X_ZZ0 ,_PUN equation_NN1 by_PRP Lemma_NN1 1_CRD (_PUL ii_CRD )_PUR ._SENT Then_AV0 ∥T_AJ0 x∥_NN1 ≥_NN1 c∥x∥_NN1 ._SENT For_PRP part_NN1 (_PUL ii_CRD )_PUR ,_PUN there_AV0 exists_VVZ c_ZZ0 >_NN1 0_CRD such_DT0 that_CJT (_PUL x_ZZ0 ,_PUN Tx_NP0 )_PUR ±_NN0 ≤_NP0 −c∥x∥2_CRD for_PRP all_DT0 T_ZZ0 ∈_NN1 T_ZZ0 and_CJC for_PRP all_DT0 x_ZZ0 ∈_NP0 X_ZZ0 ._SENT So_AV0 given_VVN T_ZZ0 ∈_NN1 T_ZZ0 and_CJC given_VVN x_PRP ∈_NN1 X_ZZ0 ,_PUN equation_NN1 by_PRP Lemma_NN1 1_CRD (_PUL ii_CRD )_PUR ._SENT Then_AV0 ∥T_AJ0 x∥_NN1 ≥_NN1 c∥x∥_NN1 ._SENT 
Lemma_NN1 3._CRD Let_VVB X_ZZ0 be_VBI a_AT0 real_AJ0 Banach_NN1 space_NN1 ._SENT Let_VVB T_ZZ0 :_PUN X_ZZ0 →_NP0 X_ZZ0 be_VBI a_AT0 continuous_AJ0 ,_PUN invertible_AJ0 linear_AJ0 operator_NN1 ._SENT Let_VVB T_ZZ0 =_NP0 {_PUL T_ZZ0 (_PUL a_AT0 )_PUR :_PUN X_ZZ0 →_NN1 X_ZZ0 :_PUN a_AT0 ∈_NN1 A_ZZ0 }_PUR be_VBI a_AT0 family_NN1 of_PRF continuous_AJ0 ,_PUN invertible_AJ0 linear_AJ0 operators_NN2 ,_PUN and_CJC let_VVB T_ZZ0 −1_CRD =_NN1 {_PUL T_ZZ0 (_PUL a_AT0 )_PUR −1_CRD :_PUN X_ZZ0 →_NN1 X_ZZ0 :_PUN a_AT0 ∈_NN1 A_ZZ0 }_PUR ._SENT 
(_PUL i_ZZ0 )_PUR If_CJS T_ZZ0 is_VBZ (_PUL weakly_AV0 )_PUR ±_NN0 negative_AJ0 definite_AJ0 ,_PUN then_AV0 T−1_NN1 is_VBZ (_PUL weakly_AV0 )_PUR ±_NN0 dissipative_AJ0 ._SENT 
(_PUL ii_CRD )_PUR If_CJS T_ZZ0 is_VBZ uniformly_AV0 ±_NN0 negative_AJ0 definite_AJ0 or_CJC uniformly_AV0 ±_NN0 dissipative_AJ0 ,_PUN then_AV0 T_ZZ0 −1_CRD is_VBZ uniformly_AV0 bounded_VVN ._SENT 
(_PUL iii_CRD )_PUR If_CJS T_ZZ0 is_VBZ uniformly_AV0 bounded_VVN and_CJC uniformly_AV0 ±_NN0 negative_AJ0 definite_AJ0 (_PUL uniformly_AV0 ±_NN0 dissipative_AJ0 )_PUR ,_PUN then_AV0 T_ZZ0 −1_CRD is_VBZ uniformly_AV0 ±_NN0 dissipative_AJ0 (_PUL uniformly_AV0 ±_NN0 negative_AJ0 definite_AJ0 )_PUR ._SENT 
Proof_NN1 ._SENT For_PRP part_NN1 (_PUL i_CRD )_PUR ,_PUN let_VVB y_ZZ0 ∈_NP0 X_ZZ0 and_CJC x_ZZ0 =_NN1 T−1y_NN1 ,_PUN so_CJS Tx_NP0 =_NP0 y_ZZ0 ._SENT Then_AV0 equation_NN1 ._SENT 
For_PRP part_NN1 (_PUL ii_CRD )_PUR ,_PUN let_VVB T_ZZ0 be_VBI uniformly_AV0 ±_NN0 negative_AJ0 definite_AJ0 ;_PUN the_AT0 proof_NN1 for_PRP the_AT0 dissipative_AJ0 case_NN1 is_VBZ analogous_AJ0 ._SENT Let_VVB c_ZZ0 >_NP0 0_CRD such_DT0 that_CJT (_PUL x_ZZ0 ,_PUN Tx_NP0 )_PUR ±_NN0 ≤_NP0 −c∥x∥2_CRD for_PRP all_DT0 T_ZZ0 ∈_NP0 T_ZZ0 ._SENT Let_VVB T−1_NP0 ∈_NP0 T_ZZ0 −1_CRD and_CJC y_ZZ0 ∈_NN1 X_ZZ0 ._SENT Set_VVB x_ZZ0 =_NP0 T−1y_NP0 ,_PUN so_CJS Tx_NP0 =_NP0 y_ZZ0 ._SENT Then_AV0 by_PRP Lemma_NN1 2_CRD (_PUL ii_CRD )_PUR ,_PUN ∥T_NN1 x∥_NN1 ≥_NN1 c∥x∥_NN1 ._SENT So_AV0 equation_NN1 which_DTQ implies_VVZ equation_NN1 ._SENT 
For_PRP part_NN1 (_PUL iii_CRD )_PUR ,_PUN let_VVB T_ZZ0 be_VBI uniformly_AV0 ±_NN0 negative_AJ0 definite_AJ0 ;_PUN again_AV0 the_AT0 dissipative_AJ0 case_NN1 is_VBZ analogous_AJ0 ._SENT Let_VVB b_ZZ0 >_NP0 0_CRD such_DT0 ∥Tx∥_NP0 ≤_NN1 b∥x∥_NN1 for_PRP all_DT0 T_ZZ0 ∈_NN1 T_ZZ0 and_CJC all_DT0 x_ZZ0 ∈_NN1 X_ZZ0 ,_PUN and_CJC let_VVB c_ZZ0 >_NP0 0_CRD such_DT0 that_CJT (_PUL x_ZZ0 ,_PUN Tx_NP0 )_PUR ±_NN0 ≤_NP0 −c∥x∥2_CRD for_PRP all_DT0 T_ZZ0 ∈_NN1 T_ZZ0 and_CJC all_DT0 x_ZZ0 ∈_NP0 X_ZZ0 ._SENT Then_AV0 let_VVB T−1_CRD ∈_NP0 T−1_CRD and_CJC y_ZZ0 ∈_NN1 X_ZZ0 ._SENT Set_VVB x_ZZ0 =_NP0 T−1y_NP0 ,_PUN so_CJS Tx_NP0 =_NP0 y_ZZ0 ._SENT Then_AV0 equation_NN1 ._SENT Thus_AV0 T_ZZ0 −1_CRD is_VBZ uniformly_AV0 ±_NN0 dissipative_AJ0 ._SENT 
Say_VVB that_CJT T_ZZ0 =_NN1 {_PUL T_ZZ0 (_PUL a_AT0 )_PUR :_PUN X_ZZ0 →_NN1 X_ZZ0 :_PUN a_AT0 ∈_NN1 A_ZZ0 }_PUR is_VBZ convex_AJ0 ±_NN0 negative_AJ0 definite_AJ0 if_CJS co_NN1 T_ZZ0 is_VBZ ±_NN0 negative_AJ0 definite_AJ0 ,_PUN and_CJC uniformly_AV0 convex_AJ0 ±_NN0 negative_AJ0 definite_AJ0 if_CJS co_NN1 T_ZZ0 is_VBZ uniformly_AV0 ±_NN0 negative_AJ0 definite_AJ0 ._SENT Similarly_AV0 ,_PUN say_VVB that_CJT T_ZZ0 =_NN1 {_PUL T_ZZ0 (_PUL a_AT0 )_PUR :_PUN X_ZZ0 →_NN1 X_ZZ0 :_PUN a_AT0 ∈_NN1 A_ZZ0 }_PUR is_VBZ convex_AJ0 ±_NN0 dissipative_AJ0 if_CJS co_NN1 T_ZZ0 is_VBZ ±_NN0 dissipative_AJ0 ,_PUN and_CJC uniformly_AV0 convex_AJ0 ±_NN0 dissipative_AJ0 if_CJS co_NN1 T_ZZ0 is_VBZ uniformly_AV0 ±_NN0 dissipative_AJ0 ._SENT In_PRP the_AT0 +_AJ0 dissipative_AJ0 case_NN1 ,_PUN the_AT0 convex_AJ0 hull_NN1 of_PRF T_ZZ0 will_VM0 inherit_VVI the_AT0 properties_NN2 of_PRF T_ZZ0 ,_PUN using_VVG Lemma_NN1 1_CRD ,_PUN that_DT0 is_VBZ ,_PUN T_ZZ0 is_VBZ +_NN1 dissipative_AJ0 if_CJS and_CJC only_AV0 if_CJS co_NN1 T_ZZ0 is_VBZ +_NN1 dissipative_AJ0 ,_PUN and_CJC T_PNP is_VBZ uniformly_AV0 +_NN1 dissipative_AJ0 if_CJS and_CJC only_AV0 if_CJS co_NN1 T_ZZ0 is_VBZ uniformly_AV0 +_AJ0 dissipative_AJ0 ._SENT The_AT0 lemma_NN1 below_AV0 records_VVZ this_DT0 and_CJC related_AJ0 results_NN2 ._SENT 
Lemma_NN1 4._CRD Let_VVB X_ZZ0 be_VBI a_AT0 real_AJ0 Banach_NN1 space_NN1 ._SENT Let_VVB T1_CRD ,_PUN T2_CRD :_PUN X_ZZ0 →_NP0 X_ZZ0 be_VBI linear_AJ0 operators_NN2 ,_PUN and_CJC let_VVB T1_NP0 =_NP0 {_PUL T1_CRD (_PUL a_AT0 )_PUR :_PUN X_ZZ0 →_NN1 X_ZZ0 :_PUN a∈A_NP0 }_PUR and_CJC T2_NP0 =_NP0 {_PUL T2_CRD (_PUL a_AT0 )_PUR :_PUN X_ZZ0 →_NN1 X_ZZ0 :_PUN a_AT0 ∈_NN1 A_ZZ0 }_PUR be_VBI families_NN2 of_PRF linear_AJ0 operators_NN2 ._SENT 
(_PUL i_ZZ0 )_PUR If_CJS T1_CRD and_CJC T2_CRD are_VBB weakly_AV0 +_AJ0 dissipative_AJ0 ,_PUN then_AV0 αT1_AJ0 +_NN1 βT2_NN1 is_VBZ weakly_AV0 +_NN1 dissipative_AJ0 for_PRP any_DT0 α_NN1 ,_PUN β_NP0 ∈_NP0 R_NP0 with_PRP α_NP0 ,_PUN β_NP0 >_NP0 0._CRD 
(_PUL ii_CRD )_PUR If_CJS T1_NN1 is_VBZ +_NN1 dissipative_AJ0 and_CJC T2_NP0 is_VBZ weakly_AV0 +_AJ0 dissipative_AJ0 ,_PUN then_AV0 αT1_AJ0 +_NN1 βT2_NN1 is_VBZ +_NN1 dissipative_AJ0 for_PRP any_DT0 α_NN1 ,_PUN β_NP0 ∈_NP0 R_NP0 with_PRP α_NP0 ,_PUN β_NP0 >_NP0 0._CRD 
(_PUL iii_CRD )_PUR If_CJS T1_NN1 is_VBZ uniformly_AV0 +_AJ0 dissipative_AJ0 and_CJC T2_NP0 is_VBZ weakly_AV0 +_AJ0 dissipative_AJ0 ,_PUN then_AV0 T1_CRD +_NP0 T2_CRD =_NP0 {_PUL T1_CRD (_PUL a_AT0 )_PUR +_NN1 T2_CRD (_PUL a_AT0 )_PUR :_PUN a_AT0 ∈_NN1 A_ZZ0 }_PUR is_VBZ uniformly_AV0 +_AJ0 dissipative_AJ0 ._SENT 
(_PUL iv_CRD )_PUR If_CJS T_ZZ0 is_VBZ uniformly_AV0 +_AJ0 dissipative_AJ0 ,_PUN then_AV0 co_NN1 T_ZZ0 is_VBZ uniformly_AV0 +_AJ0 dissipative_AJ0 ._SENT Similarly_AV0 ,_PUN if_CJS T_ZZ0 is_VBZ weakly_AV0 +_AJ0 dissipative_AJ0 ,_PUN then_AV0 co_NN1 T_ZZ0 is_VBZ weakly_AV0 +_AJ0 dissipative_AJ0 ._SENT 
Proof_NN1 ._SENT For_PRP part_NN1 (_PUL i_CRD )_PUR ,_PUN let_VVB α_NP0 ,_PUN β_NP0 >_NP0 0._CRD Then_AV0 by_PRP Lemma_NN1 1_CRD (_PUL i_CRD )_PUR ,_PUN (_PUL iii_CRD )_PUR ,_PUN equation_NN1 ._SENT For_PRP part_NN1 (_PUL ii_CRD )_PUR ,_PUN let_VVB c1_CRD >_NP0 0_CRD such_DT0 that_CJT (_PUL T1x_NP0 ,_PUN x_ZZ0 )_PUR +_NN1 ≤_NN1 −c1∥x∥2_NN1 for_PRP all_DT0 x_ZZ0 ∈_NP0 X_ZZ0 ._SENT Let_VVB α_NP0 ,_PUN β_NP0 >_NP0 0._CRD Then_AV0 equation_NN1 ._SENT 
The_AT0 argument_NN1 for_PRP part_NN1 (_PUL iii_CRD )_PUR is_VBZ analogous_AJ0 to_PRP the_AT0 proof_NN1 of_PRF part_NN1 (_PUL ii_CRD )_PUR ,_PUN noting_VVG that_CJT c1_CRD >_NN1 0_CRD can_VM0 be_VBI chosen_VVN independent_AJ0 of_PRF T1_CRD ∈_NP0 T1_CRD above_AV0 in_PRP this_DT0 case_NN1 ._SENT 
For_PRP part_NN1 (_PUL iv_CRD )_PUR ,_PUN suppose_VVB T_ZZ0 is_VBZ uniformly_AV0 +_AJ0 dissipative_AJ0 ,_PUN and_CJC let_VVB c_ZZ0 >_NP0 0_CRD such_DT0 that_CJT (_PUL Tx_NP0 ,_PUN x_ZZ0 )_PUR +_NN1 ≤_NN1 −c∥x∥2_NN1 for_PRP all_DT0 T_ZZ0 ∈_NP0 T_ZZ0 ._SENT Then_AV0 let_VVB T_ZZ0 ∈_NP0 co_NN1 T_ZZ0 ,_PUN so_AV0 T_ZZ0 =_NP0 αiTi_NN1 for_PRP some_DT0 Ti_NP0 ∈_NP0 T_ZZ0 and_CJC some_DT0 αi_AJ0 ≥_NN1 0_CRD with_PRP αi_NP0 =_NP0 1._CRD Then_AV0 equation_NN1 ._SENT The_AT0 argument_NN1 for_PRP the_AT0 weakly_AV0 +_AJ0 dissipative_AJ0 case_NN1 is_VBZ analogous_AJ0 ._SENT The_AT0 final_AJ0 lemmas_NN2 below_AV0 record_VVB another_DT0 useful_AJ0 property_NN1 of_PRF +_AJ0 dissipative_AJ0 operators_NN2 on_PRP product_NN1 spaces_NN2 ._SENT 
