maps completely preserving commutativity and maps completely preserving jordan zero product abstract let x y be real or complex banach spaces with infinite dimension and let a b be standard operator algebras on x and y respectively in this paper we show that every map completely preserving commutativity from a onto b is a scalar multiple of either an isomorphism or in the complex case a conjugate isomorphism every map completely preserving jordan zero product from a onto b is a scalar multiple of either an isomorphism or in the complex case a conjugate isomorphism

maps on idempotent matrices over division rings abstract let d be an arbitrary division ring and p n d the set of all n n idempotent matrices over d under some mild conditions we give a complete description of maps on p n d that preserve either commutativity or order or orthogonality we give examples showing that our assumptions cannot be relaxed much further as an application we will prove a quaternionic analogue of ovchinnikov s result that is important in quantum mechanics other applications of our theorems include results on automorphisms of operator and matrix semigroups local automorphisms linear preserver problems and geometry of matrices and grassmannians

nonlinear maps preserving numerical radius of indefinite skew products of operators let h be a complex hilbert space of dimension greater than 2 and j b h be an invertible self adjoint operator denote by a j 1a j the indefinite conjugate of a b h with respect to j and denote by w a the numerical radius of a let w and v be subsets of b h which contain all rank one operators and let w v be a surjective map we show that satisfies w ab w a b and w a b w a b for all a b w if and only if there exist scalars i 1 1 i 1 2 unitary or conjugate unitary operators u v on h satisfying u u 1i v v 2i and a functional w c with a 1 such that a a uav for all a w satisfies w ab a w a b a for all a b w if and only if either there exist 1 1 a unitary or conjugate unitary operator u on h satisfying u u i and a functional w c with a 1 such that a a uau for all a w or there exist a nonzero real number b a unitary or conjugate unitary operator u on h satisfying u ju bj 1 and a functional w c with a 1 such that a a ua u for all a w

a variational iterative technique applied to quantum mechanical problems a technique for solving nonlinear operator equations iteratively is applied to quantum mechanical calculations it is shown that alternative methods of applying the technique lead to another way of approaching variational calculations and generalisations of perturbation theory respectively

ring isomorphisms and linear or additive maps preserving zero products on nest algebras it is shown that every ring isomorphism between norm closed standard subalgebras of nest algebras acting on infinite dimensional banach spaces is either a spatial isomorphism or a spatial conjugate isomorphism this is then used to characterize the linear or additive maps between nest algebras which preserve zero products

isometries of grassmann spaces abstract botelho jamison and molnar have recently described the general form of surjective isometries of grassmann spaces on complex hilbert spaces under certain dimensionality assumptions in this paper we provide a new approach to this problem which enables us first to give a shorter proof and second to remove dimensionality constraints completely in one of the low dimensional cases which was not covered by botelho jamison and molnar an exceptional possibility occurs as a byproduct we are able to handle the real case as well furthermore in finite dimensions we remove the surjectivity assumption a variety of tools is used in order to achieve our goal such as topological geometrical and linear algebra techniques the famous two projections theorem for two finite rank projections will be re proven using linear algebraic methods a theorem of gyory and the second author on orthogonality preservers on grassmann spaces will be strengthened as well this latter result will be obtained by using chow s fundamental theorem of geometry of grassmannians

interpolating the arithmetic geometric mean inequality and its operator version two families of means called heinz means and heron means that interpolate between the geometric and the arithmetic mean are considered comparison inequalities between them are established operator versions of these inequalities are obtained failure of such extensions in some cases is illustrated by a simple example

lieb thirring trace inequalities and a question of bourin in this paper we propose some conjectures in order to solve one of the questions posed by bourin regarding a special type of inequalities referred to as subadditivity inequalities in the case of the hilbert schmidt norm some of these conjectures are settled affirmatively for special cases in an algorithmic way by using some number theory tools

orthogonality preserving transformations on indefinite inner product spaces generalization of uhlhorn s version of wigner s theorem abstract we present an analogue of uhlhorn s version of wigner s theorem on symmetry transformations for the case of indefinite inner product spaces this significantly generalizes a result of van den broek the proof is based on our main theorem which describes the form of all bijective transformations on the set of all rank one idempotents of a banach space which preserve zero products in both directions

automorphisms of the poset of skew projections abstract any automorphism of the poset b of all idempotents of the algebra b h of all bounded linear operators in a hilbert space h dim h 3 is shown to be of the form p apa 1 or p ap a 1 with a c where c is the set of all semi linear bijections a h h if h is finite dimensional and the set of all continuous invertible linear or conjugate linear operators in h if h is infinite dimensional further applications of the method are outlined

on some inequalities for the matrix power and karcher means abstract we establish a useful inequality for matrix power means of positive definite matrices it eventually provides not only a new characterization of the karcher mean cartan centroid riemannian geometric mean but also a simple proof of the convergence of power means to the karcher mean some new matrix mean inequalities on the karcher and power means are presented

transition probability preserving maps on a grassmann space in a semifinite factor abstract in this paper the surjective transition probability preserving transformations on the grassmann space of infinite projections in an infinite semifinite factor are characterized as in the classical wigner s theorem all these maps are induced by isomorphisms or anti isomorphisms of the semifinite factor

matrix power means and the karcher mean we define a new family of matrix means pt a t 1 1 where and a vary over all positive probability vectors in rn and n tuples of positive definite matrices resp each of these means except t 0 arises as a unique positive definite solution of a non linear matrix equation satisfies all desirable properties of power means of positive real numbers and interpolates between the weighted harmonic and arithmetic means the main result is that the karcher mean coincides with the limit of power means as t 0 this provides not only a sequence of matrix means converging to the karcher mean but also a simple proof of the monotonicity of the karcher mean conjectured by bhatia and holbrook and other new properties which have recently been established by lawson and lim and also bhatia and karandikar using probabilistic methods on the metric structure of positive definite matrices equipped with the trace metric

transformations on the sets of states and density operators in this paper we consider two physically relevant numerical quantities namely the concurrence and fidelity and an operation called product defined on the set of all quantum states or density operators our main aim is to present the complete descriptions of all bijective transformations which preserve any of these quantities or operation

characterizing sequential isomorphisms on hilbert space effect algebras let be any sequential product on the hilbert space effect algebra with and be a bijective map we show that if satisfies ab a b for then there is either a unitary or an anti unitary operator u such that a uau for every let be a borel function satisfying g 0 0 g 1 1 and let us define a binary operation g on by a gb a1 2 g a bg a a1 2 where t denotes the conjugate of the operator t we also show that a bijective map satisfies a gb a g b for if and only if there is either a unitary or an anti unitary operator u such that a uau for every

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

applying projective geometry to transformations on rank one idempotents abstract we present a short proof of molnar s characterization of bijective transformations on the set of all rank one idempotent operators on a banach space which preserve zero products in both directions an improvement in the finite dimensional case is given we apply these results to describe automorphisms of standard operator semigroups and to improve uhlhorn s version of wigner s theorem

phase isometries on real normed spaces abstract we say that a map f from a normed space x to another normed space y is a phase isometry if the equality f x f y f x f y x y x y holds for all x y x a normed space x is said to have the wigner property if for any normed space y and every surjective phase isometry f x y there exists a phase function e x 1 1 such that e f is a linear isometry we show that all smooth normed spaces l type spaces and l 1 spaces enjoy this property

entropy preserving maps on quantum states abstract in the paper we give a characterization of surjective maps on quantum states preserving quantum entropy of convex combinations let s h be the set of all quantum states positive operators with the unit trace on a complex hilbert space h with dim h n and s the quantum entropy of the quantum state for arbitrary a surjective map s h s h we claim that preserves quantum entropy of convex combinations of quantum states i e satisfies s t 1 t s t 1 t for arbitrary s h and t 0 1 if and only if there exists a unitary or anti unitary operator u on h such that u u for all s h

maps on matrix spaces it is well known that every automorphism of the full matrix algebra is inner we give a short proof of this statement and discuss several extensions of this theorem including structural results for multiplicative maps on matrix algebras characterizations of monotone and orthogonality preserving maps on idempotent matrices some nonlinear preserver results and some recent theorems concerning geometry of matrices we show that all these topics are closely related and point out the connections with physics and geometry several open problems are posed

endomorphisms of the poset of idempotent matrices abstract let d be a division ring n 3 an integer and p n d the poset of all n n idempotent matrices over d with the partial order defined by p q if p q q p p let t m n d be an invertible matrix and d d an endomorphism d d an anti endomorphism for any p p n d we denote by p p the idempotent matrix obtained from p by applying entrywise the map p n d p n d defined by p t p t 1 p p n d p t t p t 1 p p n d is an injective map preserving order in both directions every such map is called a standard map it has been known before that if d is an eas division ring then every injective order preserving map p n d p n d is either standard or of a very special degenerate form in this paper we use some ideas from geometry of algebraic homogeneous spaces and elementary field theory to give examples showing that the eas assumption is indispensable then we define generalized standard maps and using them we describe the general form of injective order preservers on p n d for an arbitrary division ring d our proof is shorter than the original one for the special case of eas division rings under somewhat stronger assumptions we get two characterizations of standard maps all the results are optimal as shown by counterexamples

inequalities related to bourin and heinz means with a complex parameter abstract a conjecture posed by s hayajneh and f kittaneh claims that given a b positive matrices 0 t 1 and any unitarily invariant norm the following inequality holds a t b 1 t b t a 1 t a t b 1 t a 1 t b t recently r bhatia proved the inequality for the case of the frobenius norm and for t 1 4 3 4 in this paper using complex methods we extend this result to complex values of the parameter t z in the strip z c re z 1 4 3 4 we give an elementary proof of the fact that equality holds for some z in the strip if and only if a and b commute we also show a counterexample to the general conjecture by exhibiting a pair of positive matrices such that the claim does not hold for the uniform norm finally we give a counterexample for a related singular value inequality given by s j a t b 1 t b t a 1 t s j a b answering in the negative a question made by k audenaert and f kittaneh the methods of proof and examples can be adapted with no modifications to operator algebras infinite dimensional setting for instance it follows that the inequality above holds for hilbert schmidt operators with their banach algebra norm derived from the infinite trace of b h

twistor space minkowski space and the conformal group it is shown that the conformal group of compactified minkowski space is isomorphic to a group of rays of semilinear transformations of twistor space the action of the conformal group on twistor space is given by an explicit realisation of this isomorphism in this way we determine the transformation of twistor space under space inversion and time inversion

complementary variational principles and variational iterative principles complementary variational principles for the solution of certain linear equations are developed it is shown that these may be used iteratively for the solution of nonlinear equations examples are presented with applications in particle theory electromagnetic theory communication theory and the thomas fermi statistical theory for atoms

maps on states preserving generalized entropy of convex combinations abstract let s h be the set of all linear positive semidefinite self adjoint trace one operators states on h where h is an at least two dimensional finite dimensional real or complex hilbert space or at least three dimensional left quaternionic hilbert space of dimension n given a strictly convex function f 0 1 r for any s h we define f i f i where 1 2 n are the eigenvalues of counted with multiplicities in this note we completely describe maps s h s h having the property f t 1 t f t 1 t for all t 0 1 and every s h it turns out that u u s h where u is a real linear isometry of h note that there is no surjectivity assumption and that our result in particular improves the description of maps preserving the von neumann entropy of convex combinations of states in the complex hilbert space it can as well be applied to preserving schatten or some other strictly convex norms of convex combinations of states

transition probabilities of normal states determine the jordan structure of a quantum system let x1d516 m1 x1d516 m2 be a bijection not assumed affine nor continuous between the sets of normal states of two quantum systems modelled on the self adjoint parts of von neumann algebras m1 and m2 respectively this paper concerns with the situation when preserves or partially preserves one of the following three notions of transition probability on the normal state spaces the transition probability pu introduced by uhlmann rep math phys 9 273 279 1976 the transition probability pr introduced by raggio lett math phys 6 233 236 1982 and an asymmetric transition probability p0 as introduced in this article it is shown that the two systems are isomorphic i e m1 and m2 are jordan isomorphic if preserves all pairs with zero uhlmann respectively raggio or asymmetric transition probability in the sense that for any normal states and we have p 0 if and only if p 0 where p stands for pu respectively pr or p0 furthermore as an extension of wi

variational principles and variational iterative principles for a class of problems abstract complementary variational principles are developed for linear equations in a function space and a variational iterative scheme based on these principles is introduced for non linear equations

some log majorizations and an extension of a determinantal inequality abstract an eigenvalue inequality involving a matrix connection and its dual is established and some log majorization type results are obtained in particular some eigenvalues inequalities considered by f hiai and m lin 9 an associated conjecture and a singular values inequality by l zou 20 are revisited a reformulation of the inequality det a u b det a b for positive semidefinite matrices a b with u a unitary matrix that appears in the polar decomposition of b a is also extended using some known norm inequalities associated to furuta inequality and araki cordes inequality

generalized symmetry transformations on quaternionic indefinite inner product spaces an extension of quaternionic version of wigner s theorem uhlhorn s extension of wigner s unitary antiunitary theorem has been recently generalized by molnar to indefinite inner product spaces we present the quaternionic version of this result

maps leaving functional values of operator products invariant abstract let a be a standard operator algebra on a complex hilbert space h of dimension greater than 2 by invariants of certain functional values of operator products we characterize some surjective maps on a furthermore several kinds of general preserver problems on standard operator algebras are solved when we take respectively the functional as for example k numerical radius k 1 operator norm ky fan k norm schatten p norm 1 p and so on

wigner type theorem on transition probability preserving maps in semifinite factors abstract the wigner s theorem which is one of the cornerstones of the mathematical formulation of quantum mechanics asserts that every symmetry of quantum system is unitary or anti unitary this classical result was first given by wigner in 1931 thereafter it has been proved and generalized in various ways by many authors recently g p geher extended wigner s and molnar s theorems and characterized the transformations on the grassmann space of all rank n projections which preserve the transition probability the aim of this paper is to provide a new approach to describe the general form of the transition probability preserving not necessarily bijective maps between grassmann spaces as a byproduct we are able to generalize the results of molnar and g p geher

symmetry transformations in indefinite metric spaces a generalization of wigner s theorem we formulate and prove a generalization to indefinite metric spaces of uhlhorn s version of wigner s theorem

trace inequalities for products of positive definite matrices it is shown that for positive definite matrices a and b the inequality apbq bpaq 2 apbq aqbp 2 holds for all positive real number p and q for which p p q is in 1 4 3 4 this is a significant improvement on a recent result of hayajneh and kittaneh lieb thirring trace inequalities and a question of bourin j math phys 54 033504 2013

order preserving maps on the poset of upper triangular idempotent matrices abstract we obtain the general form of bijective order and orthogonality preserving maps on the poset of all n n upper triangular idempotent matrices over an arbitrary field f we also show that the assumption of surjectivity can be removed if every nonzero homomorphism g f f is surjective

additive mappings preserving operators of rank one abstract let b x be the algebra of all bounded linear operators on a nontrivial real or complex banach space and let f x be the subalgebra of all finite rank operators a characterization of additive mappings on f x which preserve operators of rank one or projections of rank one is given in the real case such mappings are automatically linear

some inequalities for the matrix heron mean abstract in this paper we prove some norm inequalities for the matrix heron mean of two positive definite matrices we also prove a determinant inequality for the t power mean of two positive definite matrices as a consequence we obtain a determinant inequality for the matrix heron mean

maps preserving the spectrum of generalized jordan product of operators abstract let a 1 a 2 be standard operator algebras on complex banach spaces x 1 x 2 respectively for k 2 let i 1 i m be a sequence with terms chosen from 1 k and define the generalized jordan product t 1 t k t i 1 t i m t i m t i 1 on elements in a i this includes the usual jordan product a 1 a 2 a 1 a 2 a 2 a 1 and the triple a 1 a 2 a 3 a 1 a 2 a 3 a 3 a 2 a 1 assume that at least one of the terms in i 1 i m appears exactly once let a map a 1 a 2 satisfy that a 1 a k a 1 a k whenever any one of a 1 a k has rank at most one it is shown in this paper that if the range of contains all operators of rank at most three then must be a jordan isomorphism multiplied by an m th root of unity similar results for maps between self adjoint operators acting on hilbert spaces are also obtained

maps preserving numerical radius and cross norms of operator products let h be a complex hilbert space with dim h 3 and x1d4b2 x1d4b1 h be subsets that contain all rank one idempotents let f stand for numerical radius w or a cross operator norm n all the general surjective maps x1d4b2 x1d4b1 satisfying f ab f a b resp f aba f a b a for all a b x1d4b2 are characterized

a generalization of wigner s unitary antiunitary theorem to hilbert modules let h be a hilbert c module over a matrix algebra a it is proved that any function t h h which preserves the absolute value of the generalized inner product is of the form tf f uf f h where is a phase function and u is an a linear isometry the result gives a natural extension of wigner s classical unitary antiunitary theorem for hilbert modules

wigner s theorem on grassmann spaces wigner s celebrated theorem which is particularly important in the mathematical foundations of quantum mechanics states that every bijective transformation on the set of all rank one projections of a complex hilbert space which preserves the transition probability is induced by a unitary or an antiunitary operator this vital theorem has been generalised in various ways by several scientists in 2001 molnar provided a natural generalisation namely he provided a characterisation of not necessarily bijective maps which act on the grassmann space of all rank n projections and leave the system of jordan principal angles invariant see 17 and 20 in this paper we give a very natural joint generalisation of wigner s and molnar s theorems namely we prove a characterisation of all not necessarily bijective transformations on the grassmann space which fix the quantity trpq i e the sum of the squares of cosines of principal angles for every pair of rank n projections p and q

generalization of wigner s unitary antiunitary theorem for indefinite inner product spaces we present a generalization of wigner s unitary antiunitary theorem for pairs of ray transformations as a particular case we get a new wigner type theorem for non hermitian indefinite inner product spaces

a lieb thirring inequality for singular values abstract let a and b be positive semidefinite matrices we investigate the conditions under which the lieb thirring inequality can be extended to singular values i e for which values of p the majorisation a p b p w ab p holds and for which values its reversed inequality ab p w a p b p

transformations on the set of all n dimensional subspaces of a hilbert space preserving principal angles wigner s classical theorem on symmetry transformations plays a fundamental role in quantum mechanics it can be formulated for example in the following way every bijective transformation on the set of all 1 dimensional subspaces of a hilbert space h which preserves the angle between the elements of is induced by either a unitary or an antiunitary operator on h the aim of this paper is to extend wigner s result from the 1 dimensional case to the case of n dimensional subspaces of h with n fixed