linear maps on algebraic groups abstract let g be an irreducible simple classical algebraic subgroup of sl n c the n n matrices of determinant one we discuss the problem of classifying all invertible linear maps t on m n c such that t g g this has already been solved with two general exceptions we show how to eliminate one of these two cases in general t must have the form t x pxq or t x px t q for suitable invertible matrices p and q

linear preservers of weak majorization on l1 i when i is an infinite set abstract the necessary and sufficient conditions that a bounded linear map on the banach space l 1 i may be considered as a linear preserver of weak majorization on l 1 i where i is an arbitrary infinite set are given also we prove that the set of all linear preservers of weak majorization is closed under the norm topology

linear transformations preserving symmetric rank one matrices let 1 be the space of n x n symmetric matrices over a commutative ring r with unit and let y y r be the subset of rank 1 that is the matrices where all 2 x 2 minors are zero our purpose is to study the invertible linear transformations z on v preserving y the study of such maps falls into two parts the first of which has already been carried out we know explicitly all the invertible t that formally preserve the vanishing of 2 x 2 minors the basic maps of that kind are in fact just the ones that first come to mind t x 1uxu for invertible u and a however as i explained in s descent theory shows that these mappings for even n automatically produce certain others as well and we call that larger set the natural transformations preserving rank d 1 an explicit description of them is given in section 1 below more precisely the theorem already known is this

invertible preservers and algebraic groups abstract we consider groups of invertible linear transformations on matrices preserving some property let f be an infinite field our main result is that if an invertible transformation of m n f preserves a finite union of similarity classes then the transformation is essentially conjugation this was proved for f algebraically closed of characteristic zero by li and pierce our method relies heavily on results about maximal subgroups of algebraic groups these results were obtained by dynkin in characteristic zero and seitz in positive characteristic

additive preservers of rank additivity on matrix spaces abstract let f be a field let v denote the vector space of all m n matrices over f or the vector space of all n n symmetric matrices over f of characteristic not 2 or 3 for each fixed positive integer s 2 let q s denote the set of all matrix pairs a b in v such that rank a b rank a rank b s we characterize additive mappings on v such that a b q s whenever a b q s for a fixed s we also describe the structure of linear mappings from the space of n n matrices over f to the space of p q matrices over f that preserve rank additivity where char f 2

preservers of maximally entangled states abstract the linear structure of the real space spanned by maximally entangled states is investigated and used to completely characterize those linear maps preserving the set of maximally entangled states on m m m m where m m denotes the space of m m complex matrices aside from a degenerate rank one map such preservers are generated by a change of orthonormal basis in each tensor factor interchanging the two tensor factors and the transpose operator

linear preserver problems a brief introduction and some special techniques abstract linear preserver problems concern the characterization of linear operators on matrix spaces that leave certain functions subsets relations etc invariant the earliest papers on linear preserver problems date back to 1897 and a great deal of effort has been devoted to the study of this type of question since then we present a brief picture of the subject aiming at giving a gentle introduction to the reader then we describe some techniques used in our recent papers on this type of problem

operators preserving the strong birkhoff james orthogonality on b h abstract we characterize the class of surjective conjugate linear mappings b h b h that preserve the strong birkhoff james orthogonality in both directions we also give characterizations of rank one operators as well as coisometries in terms of the strong birkhoff james orthogonality

preservers of restricted unitary orbits of projections abstract given a pure state rank 1 self adjoint projection p and a group g of unitary operators acting on a complex hilbert space let the restricted unitary orbit of p relative to g be u g p u p u 1 u g we study linear transformations t which map u g p into pure states or more stringently back into u g p a complete characterization is obtained when g is an abelian group of unitary matrices equal to its bicommutant

additive maps preserving jordan zero products on nest algebras abstract let alg n and alg m be nest algebras associated with the nests n and m on banach spaces assume that n n and m m are complemented whenever n n and m m let alg n alg m be a unital additive surjection it is shown that preserves jordan zero products in both directions that is a b b a 0 ab ba 0 if and only if is either a ring isomorphism or a ring anti isomorphism particularly all unital additive surjective maps between hilbert space nest algebras which preserves jordan zero products are characterized completely

eigenvalue estimates and differential form laplacians on alexandrov spaces we give upper bounds on the eigenvalues of the differential form laplacian on a compact riemannian manifold the proof uses alexandrov spaces with curvature bounded below we also construct differential form laplacians on alexandrov spaces under a local bilipschitz assumption on the alexandrov space which is conjecturally always satisfied we show that the differential form laplacian has a compact resolvent we identify its kernel with an intersection homology group

existence and uniqueness of infty harmonic functions under assumption of infty poincare inequality given a complete metric measure space whose measure is doubling and supports an infty poincare inequality and a bounded domain omega in such a space together with a lipschitz function f partial omega rightarrow mathbb r we show the existence and uniqueness of an infty harmonic extension of f to omega to do so we show that there is a metric that is bi lipschitz equivalent to the original metric such that with respect to this new metric the metric space satisfies an infty weak fubini property and that a function which is infty harmonic in the original metric must also be infty harmonic with respect to the new metric we also show that if the metric on the metric space satisfies an infty weak fubini property then the notion of infty harmonic functions coincide with the notion of amles proposed by aronsson the notion of infty harmonicity is in general distinct from the notion of strongly absolutely minimizing lipschitz extensions found in crandall et al calc var partial differ equ 13 123 139 2001 juutinen ann acad sci fenn math 27 1 57 67 2002 juutinen and shanmugalingam math nachr 279 9 10 1083 1098 2006 but coincides when the metric space supports a p poincare inequality for some finite p ge 1

linear maps preserving reduced norms abstract if a linear map between central simple algebras preserves reduced norms it is an isomorphism or antiisomorphism followed by multiplication by an element of reduced norm 1

linear maps preserving products of positive or hermitian matrices abstract in this note we consider linear maps on the complex n n matrix algebra m n preserving products of either two positive matrices a positive matrix and a hermitian matrix or two hermitian matrices we give characterizations of all those injections

classifying nonsingular quadratic forms of rank three abstract the determinant on 2 by 2 matrices of trace zero gives a quadratic form linear algebra shows that its orthogonal group is the matrix algebra automorphisms together with scalars of square 1 as this holds over all commutative rings descent theory can be applied to it to get a quick classification of all nonsingular quadratic forms of rank 3 further descent arguments show how this classification yields the witt invariant and discriminant

c algebras of real rank zero abstract the concept of real rank of a c algebra is introduced as a non commutative analogue of dimension it is shown that real rank zero is equivalent to the previously defined conditions fs and hp and that it is invariant under strong morita equivalence in particular under stable isomorphism real rank zero is also invariant under inductive limits and split extensions and the class may well be regarded as the conceptual completion of the af algebras in some cases notably when the algebra is matroid it is shown that the multiplier algebra also has real rank zero although that is not true in general by a result of g j murphy this implies a weyl von neumann type result for self adjoint multiplier elements in these cases

linear preservers of controllability and or observability abstract consider a linear control differential equations system x ax bu y cx du where x c n u c m y c p and a b c d are matrices of appropriate sizes with entries in c this system or the matrix pair a b or the matrix 4 tuple a b c d is called controllable if rank a i b n for all 0 let be a linear transformation on c n n m the linear space of all matrix pairs a b then is said to preserve controllability if it maps controllable matrix pairs to controllable matrix pairs we prove that preserves controllability if and only if a b sas 1 sbf sbr f a b i 0 where is a nonzero scalar s r are nonsingular and f is a linear functional based on this result we also find all linear mappings on the linear space of all matrix 4 tuples a b c d which preserve controllability characterizations of linear preservers of observability a concept dual to controllability hence follow some variations of the above problems are also discussed

maps preserving semi fredholm operators on hilbert c modules abstract let m be a hilbert c module over the c algebra a and l a m the c algebra of all adjointable operators on m let l a m l a m be a unital linear map preserving semi a fredholm operators in both directions and assume that is surjective up to compact operators we will show that preserves compact operators and in certain cases the induced map l a m k a m l a m k a m is an automorphism or an anti automorphism

linear maps transforming the higher numerical ranges abstract let k 1 n the k numerical range of a m n is the set w k a tr x ax k x is n k x x i k and the k numerical radius of a is the quantity w k a max z z w k a suppose k 1 k 1 n and n c n k min k n k it is shown that there is a linear map m n m n satisfying w k a w k a for all a m n if and only if n n k k or n n k n k is a positive integer moreover if such a linear map exists then there are unitary matrix u m n and nonnegative integers p q with p q n n such that has the form a u a a p a t a t q u or a u a a p a t a t q u where m n m n has the form a tr a i n n k a k linear maps m n m n satisfying w k a w k a for all a m n are also studied furthermore results are extended to triangular matrices

maps on matrix algebras preserving idempotents abstract let m n be the algebra of all n n complex matrices and p n the set of all idempotents in m n suppose m n m n is a surjective map satisfying a b p n if and only if a b p n a b m n c then either is of the form a tat 1 a m n or is of the form a ta t t 1 a m n where t m n is a nonsingular matrix

on operators preserving james orthogonality abstract we characterise isometries and coisometries in b h in terms of james orthogonality as a consequence we obtain a characterisation of surjective linear or conjugate linear mappings b h b h that preserve james orthogonality in both directions

additive idempotence preservers abstract let a be a local matrix algebra over a field of characteristic different from 2 b an arbitrary algebra and x y banach spaces additive mappings a b which preserve idempotents are characterized the result is then used in classifying additive surjections b x b y which preserve idempotents and do not annihilate finite rank operators

similarity preserving linear maps on upper triangular matrix algebras in this note we consider similarity preserving linear maps on the algebra of all n n complex upper triangular matrices tn we give characterizations of similarity invariant subspaces in tn and similarity preserving linear injections on tn furthermore we considered linear injections on tn preserving similarity in mn as well

a generalization of the motzkin taussky theorem abstract in this paper we generalize the motzkin taussky theorem to matrices with polynomial entries

similarity preserving linear maps on matrices abstract we determine all similarity preserving linear maps on the space of n x n complex matrices and all unitary equivalence preserving linear maps on the space of n x n hermitian matrices sub majorization preserving linear maps on hermitian matrices are also determined

on linear maps preserving complex symmetry abstract an operator t on a complex hilbert space h is said to be complex symmetric if there exists a conjugate linear isometric involution c h h so that c t c t this paper is devoted to describing which linear maps leave the class of complex symmetric operators invariant complete characterizations are obtained for several classes of linear maps including similarity transformations surjective linear isometries multiplication operators and certain completely positive maps

automorphism group schemes of basic matrix invariants for 3 0 if p does not divide n k 1 the invertible linear transformations on matrices preserving ek x up to scalars are in essence just the obvious ones arising from scaling similarities and transposition if the power ps dividing n k 1 is greater than k then we have these elements times maps of the form x x f x i when smaller powers ps divide n k 1 the group scheme is like the first with an infinitesimal part of the second one corollary is that every division algebra of finite dimension n2 4 over its center carries a canonical cubic form that determines it up to antiisomorphism on n x n matrices let ek be the homogeneous polynomial of degree k expressing the sum of all principal k x k subdeterminants thus e1 gives the trace en gives the determinant and we have det x 2i e ek x an k our goal in this paper is to determine the automorphisms and generalized automorphisms of each polynomial ek x over commutative rings r that is to find aut ek r q e gl mn r ek q x ek x and gaut ek r eb e gl mn r ek 4 x cek x for some constant c we shall succeed for all r that contain some field one interesting corollary of the main result is that the appropriate twisted form of ek defined on a central simple algebra determines that algebra up to isomorphism or anti isomorphism in particular every central simple algebra of dimension 4 is so determined by the cubic form that is the coefficient of n 3 in the reduced norm characteristic polynomial rn x a this is in general a much simpler invariant than the polynomial rn x of degree n that was already known to determine the algebra up to anti isomorphism 3 10 some cases of the problem are special or already familiar the function e1 is linear and e2 is a quadratic form these have exceptionally large automorphism groups 6 4 if k n then en is the determinant its automorphisms were determined over fields by frobenius and his result was extended to all received by the editors april 21 1994 1991 mathematics subject classification primary 14l1 5 secondary 15a1 5 this work was supported in part by the u s nsf nsa grant number dms9001223 1995 american mathematical society

linear transformations between matrix spaces that map one rank specific set into another we characterize in several instances those linear transformations from the linear space of m n matrices into the linear space of p q matrices that map the set of matrices having a fixed rank into the set of matrices having a fixed rank examples are given showing that in contrast with the case of linear transformations on the linear space of m n matrices mapping a rank specific set into itself in the more general case of linear transformations between two full matrix spaces often one cannot expect neat and predictable results

tests for the nonergodicity of multidimensional markov chains a markov chain is ergodic if it has a proper steady state distribution the stability of a queueing system often reduces to the ergodicity of an underlying multidimensional chain we gave criteria for the nonergodicity of a multidimensional markov chain and a generalization to continuous time markov chains we also give queueing model examples

additive rank one preserving surjections on symmetric matrix spaces we characterize additive rank one preserving surjections on symmetric matrix spaces over a field of characteristic not 2 or 3 as applications the invertibility preservers the determinant preservers and characteristic polynomial preservers have been characterized

linear preservers and representations with a 1 dimensional ring of invariants we determine the group of linear transformations on a vector space v that preserve a polynomial function f on v for several interesting pairs v f using the theory of semisimple algebraic groups in an 1897 paper fro97 frobenius proved that every linear transformation of the n by n real matrices that preserves the determinant is of the form x 7 axb or x 7 ax t b for some a b gln r such that det ab 1 this is the basic example of a linear preserver problem lpp one is given a finite dimensional vector space v over a field k and a polynomial function f v k and one wants to determine the linear transformations of v that preserve f since frobenius many such prob lems have been solved see for example the surveys lp01 pll 92 lt92 and mar62 we develop here a general method that solves several new problems see examples 3 14 and 3 15 and corollaries 7 10 8 3 8 7 8 5 8 8 and 8 11 our method is to introduce an auxiliary group g gl v that is semisimple and such that v is an irreducible representation or weyl module of g in section 2 we determine the normalizer ngl v g of g in gl v we prove in theorem 3 2 that this subgroup equals the stabilizer of a closed g orbit o in the projective space p v i e stabgl v o ngl v g using this result in sections 7 and 8 we solve two families of lpps by reducing the problems in each family to determining this stabilizer these two families consist of representations v with a 1 dimensional ring of g invariant functions generated by f and are examples of prehomogenous vector spaces of parabolic type the two families correspond to the cases where the unipotent radical u of the parabolic subgroup is abelian i e u u 0 or u u is 1 dimensional respectively and we use the general results on representations in these families from rrs92 roh93 and hel12 besides obtaining new results we also recover many known solutions to linear preserver problems the generality of our method is in contrast to many of the proofs in the literature which typically are highly dependent on the particular choice of v and f the arguments in pd95 and gur97 are notable exceptions further we require only very weak assumptions about the field k at most we require that the characteristic is 6 2 3 and determine the preserver precisely and not just its identity component or lie algebra

spaces of symmetric matrices of bounded rank abstract let i n denote the space of all n n symmetric matrices over a field f let t be a positive integer such that t a subspace w of i n f is said to be a t subspace provided that the rank of every matrix in w is bounded by t meshulam showed under the assumption f n 1 that the maximal dimension of a t subspace of i n f is given by max t 1 2 k 1 2 k n k if t 2k max t 1 2 k 1 2 k n k 1 if t 2k 1 provided that we also assume char f 2 we show here that any t subspace of i n f of maximal dimension is congruent to w 1 n t a i n f a ij 0 if i t or j t

zero product preserving additive maps on symmetric operator spaces and self adjoint operator spaces in this note we characterize the additive maps on the space of symmetric operators and the space of self adjoint operators which preserve zero products in both directions and the additive maps on the space of self adjoint operators which preserve jordan zero products in both directions we also give a complete classification of additive maps on the von neumann algebra of all bounded linear operators acting on a hilbert space which preserve square zero in both directions or preserve operators annihilated by a polynomial in both directions

c numerical radius isometries on matrix algebras and triangular matrix algebras abstract let c c 1 c n t r n and m n be the set of n n complex matrices for any a m n define the c numerical range and the c numerical radius of a by w c a i 1 n c i a x i x i x 1 x n is an orthonormal set in c n and w c a max z z w c a respectively let t n be the set of n n upper triangular matrices when w c is a norm on m n mappings t on m n or t n satisfying w c t a t b w c a b for all a b are characterized as an intermediate step we also characterize additive c numerical range preservers on m n or t n

technical note mean drifts and the non ergodicity of markov chains an important question in stochastic modeling is whether or not a denumerable state markov chain is ergodic we extend a result of kaplan giving a condition for the non ergodicity of a chain based on its drifts we also clarify the condition under which the mean drift in the stationary chain is zero

overgroups of some classical linear groups with applications to linear preserver problems abstract the study of linear operators on a matrix space that leave invariant certain functions subsets or relations is commonly referred to as the study of linear preserver problems and has attracted the attention of many mathematicians in the last few decades dynkin in his classic paper studied maximal subgroups of the classical groups and showed how his results may be used to study preserver problems the purpose of this paper is to further exploit this very powerful approach the subgroups g of the general linear group that we deal with are not necessarily maximal for the applications that we have in mind we obtained a description of all possible overgroups of g the results are then applied to various linear preserver problems shorter alternative proofs for various existing results are given and some open questions are answered

on maps preserving square zero matrices the problem of characterizing linear operators on matrix algebras that leave invariant certain functions subsets or relations has attracted the attention of many mathematicians see survey papers 20 22 27 for details for instance linear operators preserving zeroproduct of matrices are studied in 15 16 29 31 linear operators preserving idempotent matrices are studied in 1 11 13 14 linear operators preserving matrices annihilated by a fixed polynomial are studied in 12 13 18 linear operators preserving nilpotent matrices are studied in 9 and linear operators preserving square zero matrices are studied in 29 most linear preserver problems were investigated in the case of matrix algebras over fields in contrast not too much is known about matrix algebras over commutative rings to the best of our knowledge beside the papers by mcdonald 25 and waterhouse 30 only bresar and semrl 11 used elementary calculations to describe linear maps preserving idempotent matrices over commutative rings the reason why people seldom study matrix algebras over commutative rings is probably that one might encounter some difficulties that are not easy to overcome for example unlike matrix algebras over fields the automorphisms on matrix algebras over commutative rings can fail to be inner see 19 28

linear additive preservers of ranks 2 and 4 on alternate matrix spaces over fields abstract let kn f be the linear space of all n n alternate matrices over a field f an operator f kn f kn f is said to be additive if f a b f a f b for any a b kn f linear if f is additive and f aa af a for every a f and a kn f and a preserver of ranks 2 and 4 on kn f if rank f x rank x for every x kn f with rank x 2 or 4 when n 4 we characterize all linear respectively additive preservers of ranks 2 and 4 on kn f over any field respectively any field that is not isomorphic to a proper subfield of itself furthermore it is also shown that the condition f is not isomorphic to a proper subfield of itself is necessary for the obtained conclusion on additive preservers of ranks 2 and 4 on kn f

extending hua s theorem on the geometry of matrices to bezout domains abstract this paper extends hua s theorem on the geometry of rectangular matrices over a division ring to the case of bezout domains let m n m n be integers 2 r an r be two bezout domains assume that r m n r m n is an adjacency preserving bijective map in both directions further assume that r is a local ring or is an invertibility preserving map or is an additive map this paper obtains the algebraic formulas of as applications the ring semi isomorphisms from r m n to r m n are characterized and the group isomorphisms from gl n r to gl n r are discussed

linear operators on matrices the invariance of rank k matrices abstract a characterization is given of all nonsingular linear operators on the set of m n matrices over any field with at least four elements which map the set of rank k matrices into itself it is also shown that if l is any subspace ofm n matrices over any field with at least k 1 elements whose nonzero elements all have rank k then the dimension of l is at most max m n this fact is used to characterize all linear operators on the set of m n matrices over certain fields which map the set of rank k matrices into itself

hua s fundamental theorem of the geometry of matrices abstract let m n f be the space of all n n matrices over the field f n 2 two matrices a b m n f are adjacent if rank a b 1 hua s fundamental theorem of the geometry of square matrices characterizes bijective maps on m n f that preserve adjacency in both directions in this paper we treat a long standing open problem whether the result of hua holds true under the weaker assumption of preserving the adjacency in one direction only we answer this question in the affirmative in the case that every nonzero homomorphism f f f is surjective for example the field of real numbers has this property in order to prove this result we have to improve ovchinnikov s characterization of automorphisms of the poset of idempotent matrices

tests for nonergodicity of denumerable continuous time markov processes we provide nonergodicity criteria for denumerable continuous time markov processes in terms of test functions that satisfy kaplan s condition which resolves an open problem given by b d choi b kim nonergodicity criteria for denumerable continuous time markov processes and operations research letters 32 2004 575 580 we give two examples where the nonergodicity criteria are used

some general techniques on linear preserver problems abstract several general techniques on linear preserver problems are described the first one is based on a transfer principle in model theoretic algebra that allows one to extend linear preserver results on complex matrices to matrices over other algebraically closed fields of characteristic 0 the second one concerns the use of some simple geometric technique to reduce linear preserver problems to standard types so that known results can be applied the third one is about solving linear preserver problems on more general operator algebras by reducing the problems to idempotent preservers numerous examples will be given to demonstrate the proposed techniques