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let @xmath1 be a continuous linear operator defined on a separable banach space @xmath2 . let us say that @xmath1 is cyclic if @xmath3 such that @xmath4 is dense in @xmath2 . on the other hand , we said that @xmath1 is locally quasinilpotent at @xmath3 if @xmath5 the notion of local quasinilpotence was introduced in @xcite to obtain nontrivial invariant subspaces for positive operators . positive results about @xmath0-tuples of operators with a nontrivial common invariant subspace have been obtained in @xcite . in this article , we extend the results of abramovich _ et  al _ @xcite to the case of not - necessarily commuting @xmath0-tuples of continuous linear operators .    to extend these results it will be essential to introduce the notion of joint local quasinilpotence for @xmath0-tuples of operators ( not necessarily commuting ) . the results complement the results obtained in @xcite . the main result of this paper appears in  3 and is stated as follows :    let @xmath6 be a @xmath0-tuple of continuous linear operators . if @xmath1 is positive with respect to a cone @xmath7 and @xmath8 exists such that @xmath1 is joint locally quasinilpotent at @xmath9 , then there exists a common nontrivial invariant subspace for @xmath6 . moreover , using this theorem we deduce new results on nontrivial common invariant subspaces for @xmath0-tuples of operators ( see theorem  3.2 , corollary  3.3 ) . we will conclude this article with a section including open problems and further directions . it is easy to see that if @xmath10 are @xmath0 commuting operators and they are locally quasinilpotent at @xmath11 , then the compositions @xmath12 for all @xmath13 and all @xmath14 , are locally quasinilpotent at @xmath9 . in fact the intersection of the sets @xmath15 is a common invariant manifold . however if @xmath16 are not commuting , the problem becomes more complicated . let @xmath17 be two operators on the hilbert space @xmath18 defined in the following way : @xmath19 where @xmath20 is the canonical basis of @xmath18 . the operators @xmath21 and @xmath22 are locally quasinilpotent at @xmath23 for each @xmath24 , since @xmath25 for each @xmath26 , and therefore @xmath27 . on the other hand , @xmath28 , hence @xmath29 and therefore @xmath21 and @xmath22 are locally quasinilpotent at @xmath23 for each @xmath24 .    however , @xmath30 and @xmath31 are not locally quasinilpotent at @xmath23 for each @xmath24 . indeed , since @xmath32 , @xmath33 on the other hand , we know @xmath34 and hence @xmath35 . let @xmath36 be an @xmath0-tuple of continuous linear operators on a banach space @xmath2 not necessarily commuting . let us denote by @xmath37 the collection of all possible products of @xmath38 elements in @xmath1 . @xmath39    let @xmath36 be an @xmath0-tuple of continuous linear operators on a banach space @xmath2 not necessarily commuting . then , we will say that @xmath1 is uniform joint locally quasinilpotent at @xmath11 if @xmath40    the notion of uniform joint local quasinilpotence is closely related with the joint spectral radius defined by rota and strang @xcite . we can get more information about spectral theory in several variables in @xcite . although the results of this article are formulated under the hypothesis of uniform joint local quasinilpotence , nevertheless , sometimes it will be possible to replace the above - mentioned hypothesis by the following weaker property . @xmath39    let @xmath36 be an @xmath0-tuple of continuous linear operators on a banach space @xmath2 . then we will say that @xmath1 is joint locally quasinilpotent at @xmath11 if @xmath41 for all @xmath42 . the difference between the concept of uniform joint local quasinilpotence and joint local quasinilpotence is the lack of uniformity . next we see some properties of both concepts . @xmath39    let @xmath43 be an @xmath0-tuple of continuous linear operators on a banach space @xmath2 and let us suppose that @xmath1 is uniform joint locally quasinilpotent at @xmath44 . then for all polynomial @xmath45 of @xmath46 variables , such that @xmath47 we have that @xmath48 where @xmath49 , that is , the operator @xmath50 is locally quasinilpotent at @xmath51 . fix @xmath52 . let us suppose that @xmath53 is the number of summands of the polynomial @xmath45 , and let us denote by @xmath54 the maximum of the modulus of the coefficients of @xmath45 . then , since @xmath55 is uniform joint locally quasinilpotent at @xmath51 , there exists @xmath56 such that @xmath57 for all @xmath58 . now , taking into account that the polynomial @xmath45 has no independent term , for all @xmath58 , @xmath59 which proves the desired result . @xmath60    in fact the condition on the polynomial @xmath61 is a necessary condition in the proof of proposition  2.3 . indeed , let @xmath62 be the forward shift defined on the sequences space @xmath63 by @xmath64 , for all @xmath65 . it is easy to see that the operator @xmath66 is not locally quasinilpotent at @xmath67 ( where @xmath68 denotes the identity operator ) . definitions  2.1 and 2.2 are the natural extensions of the notion of local quasinilpotence for @xmath0-tuples of continuous linear operators . in fact , let us denote @xmath69 and let us denote by @xmath70 the set of all uniform joint locally quasinilpotent vectors for @xmath71 . then , we have the following result .    @xmath39    let @xmath36 be an @xmath0-tuple of continuous linear operators on a banach space @xmath2 , then the sets @xmath70 and @xmath72 are common invariant manifolds by @xmath73 . clearly , @xmath74 implies that @xmath75 . now let @xmath76 , and fix @xmath52 . then , there exists some @xmath77 such that @xmath78 and @xmath79 for each @xmath80 and each @xmath81 . therefore , @xmath82 for all @xmath83 . therefore @xmath84 and so @xmath72 is a vector manifold . fix @xmath85 and let @xmath86 . then @xmath87 for each @xmath88 and for each @xmath89 . therefore @xmath72 is a common invariant manifold for @xmath90 . similar proof follows for the set @xmath70 , and hence we omit it . @xmath60    the above propositions show that if @xmath91 ( @xmath92 respectively ) and @xmath93 ( @xmath94 respectively ) , then @xmath95 ( @xmath96 respectively ) is a common nontrivial closed invariant subspace for @xmath16 . as far as the invariant subspace problem is concerned , we need only consider the two extreme cases @xmath97 and @xmath98 . let @xmath2 be a banach space with a schauder basis @xmath99 . we say that @xmath6 is positive with respect to the cone @xmath100 if @xmath101 for all @xmath102 . let us see the main result of this paper . let @xmath6 be an @xmath0-tuple of continuous linear operators on a banach space with a schauder basis such that @xmath6 is positive with respect to the cone @xmath7 , and let us suppose that @xmath103 exists such that @xmath36 is joint locally quasinilpotent at @xmath104 . then there exists a common nontrivial invariant subspace for @xmath105 . let @xmath99 be a schauder basis of the banach space @xmath2 and let @xmath106 be the sequence of coefficient functionals associated with the basis @xmath99 . assume that @xmath6 is joint locally quasinilpotent at some @xmath107 , i.e. , @xmath108 with @xmath109 for all @xmath110 . let us suppose that @xmath111 for all @xmath112 . then @xmath113 is a common nontrivial invariant subspace for each @xmath73 . thus , we can suppose that @xmath114 for some @xmath115 . by an appropriate scaling of @xmath104 , we can assume that @xmath116 and @xmath117 for some @xmath118 and some @xmath115 . now let @xmath119 denote the continuous projection onto the vector subspace generated by @xmath120 defined by @xmath121 . clearly , @xmath122 holds for each @xmath123 . we claim that @xmath124 for each @xmath125 . to see this , fix @xmath125 and let @xmath126 for some nonnegative scalar @xmath127 . since @xmath128 is a positive operator and the composition of positive operators is a positive operator , it follows that @xmath129 let us observe that the following inequality is not true because the norm is not monotone @xmath130 . however , if we use the fact that @xmath131 is a positive linear functional , we conclude that @xmath132 consequently , @xmath133 , and so @xmath134 from definition  2.2 we know @xmath135 . thus we deduce that @xmath136 , and condition ( [ uno ] ) must be true . now let us consider the linear subspace @xmath137 of @xmath2 generated by the set @xmath138 clearly , @xmath137 is invariant for each @xmath139 and , since @xmath140 for some @xmath141 , we see that @xmath142 . from ( [ uno ] ) it follows that @xmath143 , therefore @xmath144 for each @xmath145 . this implies that @xmath146 for each @xmath147 , and consequently @xmath146 for all @xmath148 , that is , @xmath149 . the latter shows that @xmath150 is a common nontrivial closed invariant subspace for the operators @xmath151 , and the proof is complete.@xmath60    let @xmath73 be joint locally quasinilpotent operators at @xmath8 . then we can add arbitrary weights to each matrix representing the operators @xmath73 and still guarantee the existence of a common nontrivial closed invariant subspace . @xmath39    1 . first , let us observe that if @xmath71 is joint locally quasinilpotent at @xmath51 it is possible to obtain a closed invariant subspace @xmath62 ( nontrivial ) invariant also for every positive operator @xmath152 such that @xmath153 for all @xmath154 . indeed , the above proof follows considering the closed subspace generated by @xmath155 \hskip -1.25pc & \quad\ , \forall\ j\in \mathbb{n } , a\ \hbox{positive}\ at_{i}=t_{i}a ( \forall i)\}.\end{aligned}\ ] ] 2 .   on the other hand , let us mention that the subspace guaranteed in the above theorem is in fact an invariant nontrivial ideal .    in the following theorem , positivity is with respect to the cone generated by the schauder basis of the banach space . let @xmath2 be a banach space with a schauder basis . assume that the matrix @xmath156 defines a continuous operator @xmath157 for all @xmath158 , such that the @xmath0-tuple @xmath36 is joint locally quasinilpotent at a nonzero positive vector . let @xmath159 be @xmath0-double sequences of complex numbers . if the weighted matrices @xmath160 define continuous operators @xmath161 , then @xmath162 have a common nontrivial closed invariant subspace . let @xmath99 be a schauder basis of the banach space @xmath2 , and let @xmath106 be the sequence of functional coefficients associated with the basis @xmath99 . assume that the operators @xmath16 satisfy @xmath163 for some positive nonzero vector @xmath104 . an appropriate scaling of @xmath104 shows that there exists @xmath164 satisfying @xmath165 . let us suppose that @xmath166 for all @xmath89 , then an easy argument shows that @xmath167 for all @xmath89 , and @xmath168 is a nontrivial closed invariant subspace ( here we assume that @xmath169 for all @xmath89 ) . thus , we can suppose that @xmath170 for some @xmath115 . now , let us denote by @xmath119 , the positive projection defined by @xmath171 . arguing as in the proof of theorem  3.1 , we can establish that @xmath172 for each @xmath14 , where @xmath173 for all @xmath110 . in particular , we have @xmath174 for each @xmath145 . consequently , for each @xmath175 and for each positive operator @xmath176 satisfying @xmath177 , we have @xmath178 next , let us consider the vector subspace @xmath137 generated by the set @xmath179 clearly @xmath137 is invariant for each operator @xmath180 satisfying @xmath181 . also , from ( [ dos ] ) it follows that @xmath182 for all @xmath148 , that is , @xmath149 . the latter shows that @xmath150 is a nontrivial closed vector subspace of @xmath2 . let @xmath183 now be the operators defined by @xmath184 since @xmath185 satisfy @xmath186 for all @xmath89 , it follows that @xmath150 is invariant for each one of the operators @xmath185 . therefore , the vector subspace @xmath150 is invariant under the operators @xmath187 however , the sequence of operators @xmath188 converges in the strong operator topology to @xmath189 . therefore , @xmath190 and , thus , the operators @xmath191 , have a common nontrivial closed invariant subspace . @xmath60 @xmath39    let @xmath2 be a banach space with a schauder basis . assume that the positive matrices @xmath156 define continuous operators on @xmath2 , which are joint locally quasinilpotent at a nonzero positive vector . if the continuous operators @xmath192 are defined by the matrices @xmath193 satisfying @xmath194 whenever @xmath195 , then the operators have a common nontrivial closed invariant subspace . the notion of uniform joint local quasinilpotence is used extensively in @xcite to obtain common nontrivial invariant subspaces . both concepts , joint local quasinilpotence and uniform joint local quasinilpotence , play an important role in the search of common nontrivial invariant subspaces . it would be interesting to know something more on the sets @xmath196 and @xmath197 . our conjecture is that both sets are equal in majority of the cases .    on the other hand , it would be interesting to extend the results of theorems  3.1 and 3.3 for the case of @xmath0-tuples of positive operators defined on a hausdorff topological vector space , where the partial order is defined by means of a markushevish basis . the authors would like to thank the referee for his comments and suggestions that have improved this article .
in this article we obtain some positive results about the existence of a common nontrivial invariant subspace for @xmath0-tuples of not necessarily commuting operators on banach spaces with a schauder basis . 
 the concept of joint quasinilpotence plays a basic role . 
 our results complement recent work by kosiek @xcite and ptak @xcite . 
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