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

subgroups of containing let psu n be the image of the unitary group u n under the representation x axa 1 on the space mn c of n by n complex matrices we classify all connected lie subgroups of gl n2 c containing psu n we use this result to obtain a description of all abstract overgroups of psu n in gl n2 c we apply this classification to solve the problem of describing all invertible linear transformations of mn c which preserve the set of normal matrices our results can be applied to solve many other problems of similar nature

general theory of best variants choice some aspects the conventional mechanisms of making the best choice in particular optimization mechanisms make explicit or implicit use of the structural principle of pair comparability for variants and of the specific logic of the dominant rule for choice this paper argues for going beyond the framework of pairness dominance on the structures and rules of choice and for using nonconventional structures and rules this is shown to be possible if the notions of multitudinal influences of variants is formalized in determining the best choices

linear operators on matrices the invariance of the decomposable numerical range ii abstract the characterization of all linear operators on matrices which preserve the decomposable numerical radius under a mild condition is obtained the result is a refinement of marcus and filippenko s determination of all linear operators preserving the decomposable numerical range

linear maps preserving the spectral radius abstract let x be a complex banach space if b x b x is a surjective linear map such that a and a have the same spectral radius for every a b x then c where is either an algebra automorphism or an antiautomorphism of b x and c is a complex constant such that c 1

the decomposable numerical radius and numerical radius of a compound matrix let a be an n n complex matrix with singular values 1 2 n and eigenvalues 1 2 n where 1 2 n denote by cm a 1 m n the mth compound of a and by mcn the mth grassmann space over cn in which the elements are regarded as complex row vectors with nm coordinates we have the relation i 1m i rd cm a r cm a i 1m i where rd cm a max xcm a x x is decomposable in mcn xx 1 and rd cm a max xcm a x x mcn xx 1 are the decomposable numerical radius and numerical radius of cm a respectively in this note we classify those matrices for which mi 1 i rd cm a m i 1 i r cm a rd cm a r cm a etc some of these results answer the questions raised by marcus and andresen concerning the decomposable numerical radius of a complex matrix

growth conditions compact perturbations and operator subdecomposability with applications to generalized cesaro operators abstract we adapt recent results of albrecht and ricker to obtain conditions under which growth constraints on the left resolvent of a banach space operator are preserved under suitable perturbations as an application we establish bishop s property for certain generalized cesaro operators on the classical hardy spaces h p 1 p our methods also apply to unilateral weighted shifts whose weight sequence converges sufficiently rapidly as well as to perturbations of restrictions of a class of generalized scalar operators

hlawka popoviciu inequalities on positive definite tensors abstract we prove inequalities on symmetric tensor sums of positive definite operators in particular we prove multivariable operator inequalities inspired by generalizations to the well known hlawka and popoviciu inequalities as corollaries we obtain generalized hlawka and popoviciu inequalities for determinants permanents and generalized matrix functions the new operator inequalities and their corollaries contain a few recently published inequalities on positive definite matrices as special cases

some properties of choice functions based on valued binary relations abstract choice functions based on t norms of valued binary relations are introduced strict preference is also specified with the use of a t norm properties of the choice functions are investigated and rationality conditions are studied some classical particular cases are presented

operator properties of t and k t let v be an n dimensional inner product space over c a nd leth be a subgroup of the symmetric group on 1 m suppose h c is an irreducible character not necessarily linear let v m h denote the symmetry class of tensors over v associated with h and and let k t end v m h be the induced operator of t end v it is known that if t is normal unitary positive semi definite hermitian then k t has the corresponding property furthermore if t1 t 2 for some c with m 1 then k t1 k t2 the converse of these statements are not valid in general necessary and sufficient conditions on and the operators t t 1 t 2 ensuring the validity of the converses of the above statements are given these extend the results of those on linear characters by li and zaharia

induced operators on symmetry classes of tensors let v be an n dimensional hilbert space suppose h is a subgroup of the symmetric group of degree m and x h c is a character of degree 1 on h consider the symmetrizer on the tensor space m v s v 1 v m 1 h v 1 1 x v 1 m defined by h and x the vector space m v m x h s v is a subspace of m v called the symmetry class of tensors over v associated with h and x the elements in v m x h of the form s v 1 v m are called decomposable tensors and are denoted by v 1 v m for any linear operator t acting on v there is a unique induced operator k t acting on v m x h satisfying k t v 1 v m tv 1 tv m in this paper several basic problems on induced operators are studied

multiplicative preservers and induced operators abstract let v be an n dimensional hilbert space suppose h is a subgroup of the symmetric group of degree m and h c is a character of degree 1 on h consider the symmetrizer on the tensor space m v s v 1 v m 1 h h v 1 1 v 1 m defined by h and the subspace v m h of m v spanned by s m v is called the symmetry class of tensors over v associated with h and the elements in v m h of the form s v 1 v m are called decomposable tensors and are denoted by v 1 v m for any linear operator t acting on v there is an unique induced operator k t or just k t for notational simplicity acting on v m h satisfying k t v 1 v m tv 1 tv m we characterize multiplicative maps such that f t f t for all operators t acting on v where f are various scalar or set valued functions including the spectral radius decomposable numerical radius spectral norm spectrum decomposable numerical range of t or k t

on the lattice structure of stable allocations in a two sided discrete concave market the stable allocation model is a many to many matching model in which each pair s partnership is represented by a nonnegative integer this paper establishes a link between two different formulations of this model the choice function model studied thoroughly by alkan and gale and the discrete concave m concave value function model introduced by eguchi fujishige and tamura we show that the choice functions induced from m concave value functions are endowed with consistency persistence and size monotonicity this implies by the result of alkan and gale that the stable allocations for m concave value functions form a distributive lattice with several significant properties such as polarity complementarity and uni size property furthermore we point out that these results can be extended for quasi m concave value functions

the numerical range of derivations abstract let p q n be integers satisfying 1 p q n the p q numerical range of an n n complex matrix a is defined by w p q a e p uau q u unitary where for an n n complex matrix x x q denotes its q q leading principal submatrix and e p x q denotes the p th elementary symmetric function of the eigenvalues of x q when 1 p q the set reduces to the classical numerical range of a which is well known to be convex many authors have used the concept of classical numerical range to study different classes of matrices in this note we extend the results to the generalized cases besides obtaining new results we collect existing ones and give alternative proofs for some of them we also study the p q numerical radius of a defined by r p q a max w p q a

tournament methods in choice theory choice procedures using the notion of tournament matrix are investigated in the framework of general choice theory tournament procedures of multicriterion choice are introduced and studied new characteristic conditions for describing some tournament and other essentially nonclassical choice functions are obtained the comparison of tournament and graph dominant choice mechanisms is also established

the constrained bilinear form and the c numerical range abstract let v be an n dimentional unitary space with inner product and s the set x v x x 1 for any a hom v v and q c with q 1 we define w a q ax y x y s x y q if q 1 then w a q is just the classical numerical range ax x x s the convexity of which is well known another generalization of the numerical range is the c numerical range which is defined to be the set w c a tr cu au u unitary where c hom v v in this note we prove that w a q is always convex and that wc a is convex for all a if rank c 1 or n 2

using fuzzy binary relations for identifying noninterior decision alternatives abstract a fuzzy noninferiority relation is introduced based on this relation fuzzy relations of preference strict preference indifference and incomparability are specified and their properties are investigated a fuzzy noninferior subset is defined and its basic characteristic properties are established an algorithm is presented for identifying a fuzzy set of noninferior alternatives in decision making situations that require the selection of a certain number of noninferior alternatives for more detailed study

weighted composition semigroups on hardy spaces abstract the operator semigroups of the title are studied on hardy spaces conditions are found for strong continuity and the infinitesimal generator and its point spectrum are identified a particular semigroup is used to study an averaging operator

linear operators preserving the p q numerical range abstract let bd m be h n or c n n the real linear space of all n n hermitian matrices and the complex linear space of all n n complex matrices respectively for 1 p q n the p q numerical radius of a m is defined and denoted by r p q a e p x ax x d n q x x i q where ep y denotes the pth elementary symmetric function on the eigenvalues of y we characterize the linear operators on m that satisfy rp q a rp q a for all a m

spectrum preserving linear maps abstract let x and y be banach spaces we show that a spectrum preserving surjective linear map from b x to b y is either of the form t ata 1 for an isomorphism a of x onto y or the form t bt b 1 for an isomorphism b of x onto y

linear transformations on matrices the invariance of sets of ranks abstract let r e denote the set of all m n matrices over an algebraically closed field f whose ranks lie in the set e where e is a subset of 1 2 m let t be a linear transformation which maps r e into itself under some restrictions on e or when t is nonsingular there are nonsingular matrices u and v such that t a uav for every m n matrix a

on a class of composition semigroups in hardy spaces the purpose of this note is to show that r 1 r is not always compact and then to give conditions for com pactness the separation of the semigroups giving rise to compact resolvent from those which do not seems to be very delicate our conditions are in terms of a certain associated univalent function to be introduced below

matrices with some extremal properties abstract for a complex matrix a there are many inequalities related to its eigenvalues diagonal elements and three types of singular values we investigate the conditions on the matrices for which those inequalities become equalities it is shown that in many cases the conditions are both necessary and sufficient for the matrices to be unitarily similar to matrices of lower orders moreover we get alternative proofs of various characterization theorems of different classes of matrices

conditions for the generalized numerical range to be real abstract necessary and sufficient conditions are given for the c numerical range of a matrix a to be a subset of the real axis in particular it is shown that both a and c must be translates of hermitian matrices

the koebe semigroup and a class of averaging operators on hp d we study on the hardy space h p the operators t f given by formule where f z is analytic on the unit disc d and has re f z 0 each such operator is closely related to a strongly continuous semigroup of weighted composition operators by studying first an extremal such semigroup the koebe semigroup we are able to obtain the upper bound t f p 2p re 1 f 0 im 1 f 0 for the norm we also show that t f is compact on h p if and only if the measure in the herglotz representation of 1 f is continuous

choice functions and mechanisms for linguistic preference relations abstract the problem of finding a solution set of alternatives when a linguistic preference relation represents a collective preference is analyzed following two research lines choice functions and mechanisms four classical choice sets of alternatives for a linguistic preference relation are presented and some relations between them are pointed out the concept of linguistic choice function as a tool to derive linguistic choice sets of alternatives is introduced and some particular linguistic choice functions are presented together with a study of their rationality properties on the other hand two types of linguistic choice mechanisms to derive solution sets of alternatives from linguistic choice functions are introduced simple and composite ones the concept of linguistic covering relation is introduced with a view to allow us the design of consistent linguistic choice mechanisms which may achieve more precise and coherent solution sets of alternatives

on choosing rationally when preferences are fuzzy abstract in real life exact choices are induced by fuzzy preferences these choices eventually satisfy certain plausible rationality properties several alternative rules for generating exact choices from fuzzy weak preference relations are introduced and the extent to which exact choice sets generated by these rules satisfy fairly weak rationality conditions is studied max min transitivity is shown to be crucial for obtaining a positive result

logic of multicomponent systems and a combinatorial model of causal relationships abstract the effects which are typical for multicomponent systems i e for objects composed of parts are studied from the logical standpoint possible types of the formal relationship between a property of an object as a whole called here a hyperproperty and a property of its separate parts or elements are analyzed simple standard forms of such relationships are isolated separable hyperproperties the deviations from these standard forms are described in terms of intrasystem interactions between elements or parts the investigated formal types of the influence that an element property has on a system property are interpreted in causal terms this provides a basis for the formal analysis of the cause consequence dependence notion in the framework of a simple combinatorial model of the dependence between events

product numerical range in a space with tensor product structure we study operators acting on a tensor product hilbert space and investigate their product numerical range product numerical radius and separable numerical range concrete bounds for the product numerical range for hermitian operators are derived product numerical range of a non hermitian operator forms a subset of the standard numerical range containing the barycenter of the spectrum while the latter set is convex the product range needs not to be convex nor simply connected the product numerical range of a tensor product is equal to the minkowski product of numerical ranges of individual factors

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

the automorphism group of separable states in quantum information theory we show that the linear group of automorphism of hermitian matrices which preserves the set of separable states is generated by natural automorphisms change of an orthonormal basis in each tensor factor partial transpose in each tensor factor and interchanging two tensor factors of the same dimension we apply our results to the preservers of the product numerical range

decomposable numerical ranges on orthonormal tensors abstract let 1 m n and let h c be a degree 1 character on a subgroup h of the symmetric group of degree m the generalized matrix function on an m m matrix b b ij associated with is defined by d b h j 1 m b j j and the decomposable numerical range of an n n matrix a on orthonormal tensors associated with is defined by w a d x ax x is an n m matrix such that x x i m we study relations between the geometrical properties of w a and the algebraic properties of a and determine the structure of those linear operators l on n n complex matrices that satisfy w l a w a for all n n matrices a these results extend those of other researchers who treat the special cases of such as the principal or alternate character

the multilinear algebra of jose dias da silva and the portuguese school of mathematics abstract in portugal multilinear algebra has a strong group of researchers that achieved in the last years some impressive results in this area this paper surveys the results obtained by this group and the current state of knowledge on the different problems

compactness of resolvent operators generated by a class of composition semigroups on hp consider a semigroup of composition operators t tf f circ phi t t geq 0 acting on the standard hardy space h p 1 leq p infty of the unit disk d assuming that the phi t have a common fixed point at 0 a unique univalent function h can be found such that h phi t z e ct h z z in d t geq 0 where c is a constant which is related to the infinitesimal generator a of t t in this paper the compactness of the resolvent operator r lambda a is studied using the function h it is shown that r lambda a is compact if and only if h lies in h q for each q infty in the cases where r lambda a is not compact it is shown that the spectrum of r lambda a contains a nontrivial disk a condition under which compactness of r lambda a implies compactness of the operators in t t is also obtained the methods used are function theoretic in particular the notion of a spiral like function plays a key role less