additive mappings on von neumann algebras preserving absolute values abstract given hilbert spaces h and k and a von neumann algebra a b h let denote the class of all additive mappings a b k satisfying a a a a the paper shows that if a contains no nonzero abelian central projection then every preserves the operation the r linear combination and up to a commuting operator multiple i 0 the ring multiplication if a contains a nonzero abelian central projection p and if the dimension of k is at least 2 or 2 rank p according to whether or not p can be chosen to be minimal then there exists an additive mapping a b k such that i is a projection and a a for all a a but is neither multiplicative nor adjoint preserving in case a b h the result was proved by molnar bull aust math soc 53 1996 391 when a contained all finite rank operators and by radjabalipour et al linear algebra appl 327 2001 197 under the redundant restriction i i k i k

additive rank 1 preservers between hermitian matrix spaces and applications denote the set of n n complex hermitian matrices by hn we characterize the additive maps from hn into hm that preserve rank 1 matrices when the rank of the image of in is equal to n as applications the forms of the additive preserver of rank the additive preserver of rank additivity and the additive preserver of adjoint matrix between the spaces of hermitian matrices are also given

lie isomorphisms of nest algebras on banach spaces abstract we show that a lie isomorphism between nest algebras on banach spaces can be decomposed as where is an algebraic isomorphism or a negative of an algebraic anti isomorphism and is a linear map with image in the center vanishing on each commutator

additive mappings decreasing rank one abstract let b x be the algebra of bounded operators on a real or complex banach space x and f x a subalgebra of finite rank operators a complete description of additive mappings f x f x which map rank one operators to operators of rank at most one is given

admm based fast algorithm for multi group multicast beamforming in large scale wireless systems multi group multicast beamforming in wireless systems with large antenna arrays and massive audience is investigated in this paper multicast beamforming design is a well known non convex quadratically constrained quadratic programming qcqp problem a conventional method to tackle this problem is to approximate it as a semi definite programming problem via semi definite relaxation whose performance however deteriorates considerably as the number of per group users goes large a recent attempt is to apply convex concave procedure ccp to find a stationary solution by treating it as a difference of convex programming problem whose complexity however increases dramatically as the problem size increases in this paper we propose a low complexity high performance algorithm for multi group multicast beamforming design in large scale wireless systems by leveraging the alternating direction method of multipliers admm together with ccp in specific the original non convex qcqp problem is first approximated as a sequence of convex subproblems via ccp each convex subproblem is then reformulated as a novel admm form our admm reformulation enables that each updating step is performed by solving multiple small size subproblems with closed form solutions in parallel numerical results show that our fast algorithm maintains the same favorable performance as state of the art algorithms but reduces the complexity by orders of magnitude

rank one nonincreasing additive mappings on second symmetric product spaces in this note we describe those additive mappings from a second symmetric product space to another over a field of characteristic not 2 or 3 which preserve decomposable elements of the form u u where u is a vector and is a scalar this leads to the corresponding result concerning additive mappings from one vector space of symmetric matrices to another which preserve rank less than or equal to one we also discuss some consequences of this characterization theorem

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

additive jordan isomorphisms of nest algebras on normed spaces abstract let x be a real or complex banach space let alg n and alg m be two nest algebras on x suppose that is an additive bijective mapping from alg n onto alg m such that a2 a 2 for every a alg n then is either a ring isomorphism or a ring anti isomorphism moreover if x is a real space or an infinite dimensional complex space then there exists a continuous conjugate linear bijective mapping t such that either a tat 1 for every a alg n or a ta t 1 for every a alg n

additive rank one preservers on block triangular matrix spaces in this note we address a description of the general form of additive rank one preservers on block triangular matrix spaces over an arbitrary field as an application we characterize additive mappings preserving rank additivity on block triangular matrix algebras

unique decompositions faces and automorphisms of separable states let sk be the set of separable states on b cm cn admitting a representation as a convex combination of k pure product states or fewer if m 1 n 1 and k max m n we show that sk admits a subset vk such that vk is dense and open in sk and such that each state in vk has a unique decomposition as a convex combination of pure product states and we describe all possible convex decompositions for a set of separable states that properly contains vk in both cases we describe the associated faces of the space of separable states which in the first case are simplexes and in the second case are direct convex sums of faces that are isomorphic to state spaces of full matrix algebras as an application of these results we characterize all affine automorphisms of the convex set of separable states and all automorphisms of the state space of b cm cn that preserve entanglement and separability

additive operators preserving idempotent matrices over fields and applications abstract suppose f is a field of characteristic not 2 let m n f be the algebra of all n n matrices over f and let a 2 a 3 a 1 and a be the semigroups of all additive operator on m n f that preserve idempotence preserve tripotence preserve inverses of matrices and preserve group inverses of matrices respectively the main result in this paper is that the semigroup a 2 is generated by transposition similarity the operators x x for fixed arbitrary injective endomorphisms on f and the operators x tr x for fixed arbitrary additive maps from f to m n f with 1 o as applications we determine the structures of a 3 a 1 and a when the characteristic of f is also not 3

the gauss map for surfaces part 2 the euclidean case we study smooth maps t m ci of a riemann surface m into the grassmannian gi of oriented 2 planes in e2 and determine necessary and sufficient conditons on t in order that it be the gauss map of a conformai immersion x m e2 we sometimes view as an oriented riemannian vector bundle it is a subbundle of ej the trivial bundle over m with fibre e2 l the necessary and sufficient conditions obtained for simply connected m involve the curvatures of t and tx the orthogonal complement of t in e f as well as certain components of the tension of viewed as a map t m sc 1 where sc l is a unit sphere of dimension c that contains ci as a submanifold in a natural fashion if t satisfies a particular necessary condition then the results take two different forms depending on whether or not t is the gauss map of a conformai minimal immersion the case t m gl is also studied in some additional detail

the reconstruction of a multidimensional sequence from the phase or magnitude of its fourier transform this paper addresses two fundamental issues involved in the reconstruction of a multidimensional sequence from either the phase or magnitude of its fourier transform the first issue relates to the uniqueness of a multidimensional sequence in terms of its phase or magnitude although phase or magnitude information alone is not sufficient in general to uniquely specify a sequence a large class of sequences are shown to be recoverable from their phase or magnitude the second issue which is addressed in this paper concerns the actual reconstruction of a multidimensional sequence from its phase or magnitude for those sequences which are uniquely specified by their phase several practical algorithms are described which may be used to reconstruct a sequence from its phase several examples of phase only reconstruction are also presented unfortunately however even for those sequences which are uniquely defined by their magnitude it appears that a practical algorithm is yet to be developed for reconstructing a sequence from only its magnitude nevertheless an iterative procedure which has been proposed is briefly discussed and evaluated

new results on hermitian matrix rank one decomposition in this paper we present several new rank one decomposition theorems for hermitian positive semidefinite matrices which generalize our previous results in huang and zhang math oper res 32 3 758 768 2007 ai and zhang siam j optim 19 4 1735 1756 2009 the new matrix rank one decomposition theorems appear to have wide applications in theory as well as in practice on the theoretical side for example we show how to further extend some of the classical results including a lemma due to yuan math program 47 53 63 1990 the classical results on the convexity of the joint numerical ranges pang and zhang in unpublished manuscript 2004 au yeung and poon in southeast asian bull math 3 85 92 1979 and the so called finsler s lemma bohnenblust in unpublished manuscript au yeung and poon in southeast asian bull math 3 85 92 1979 on the practical side we show that the new results can be applied to solve two typical problems in signal processing and communication one for radar code optimization and the other for robust beamforming the new matrix decomposition theorems are proven by construction in this paper and we demonstrate that the constructive procedures can be implemented efficiently stably and accurately the url of our matlab programs is given in this paper we strongly believe that the new decomposition procedures as a means to solve non convex quadratic optimization with a few quadratic constraints are useful for many other potential engineering applications

strong duality in nonconvex quadratic optimization with two quadratic constraints we consider the problem of minimizing an indefinite quadratic function subject to two quadratic inequality constraints when the problem is defined over the complex plane we show that strong duality holds and obtain necessary and sufficient optimality conditions we then develop a connection between the image of the real and complex spaces under a quadratic mapping which together with the results in the complex case lead to a condition that ensures strong duality in the real setting preliminary numerical simulations suggest that for random instances of the extended trust region subproblem the sufficient condition is satisfied with a high probability furthermore we show that the sufficient condition is always satisfied in two classes of nonconvex quadratic problems finally we discuss an application of our results to robust least squares problems

multiplicative mappings of operator algebras nest algebras are the natural analogues of upper triangular matix algebras in an infinite dimensional hilbert space in this paper we study multiplicative isomorphisms of subalgebras of nest algebras which contain all finite rank operators but might contain no idempotents we prove that such multiplicative mappings are automatically additive and linear or conjugate linear

consensus admm for general quadratically constrained quadratic programming nonconvex quadratically constrained quadratic programming qcqp problems have numerous applications in signal processing machine learning and wireless communications albeit the general qcqp is np hard and several interesting special cases are np hard as well this paper proposes a new algorithm for general qcqp the problem is first reformulated in consensus optimization form to which the alternating direction method of multipliers can be applied the reformulation is done in such a way that each of the subproblems is a qcqp with only one constraint qcqp 1 which is efficiently solvable irrespective of non convexity the core components are carefully designed to make the overall algorithm more scalable including efficient methods for solving qcqp 1 memory efficient implementation parallel distributed implementation and smart initialization the proposed algorithm is then tested in two applications multicast beamforming and phase retrieval the results indicate superior performance over prior state of the art methods

isometries of a generalized numerical radius on compact operators abstract we describe all isometries of the q numerical radius on the space k h of compact operators on an infinite dimensional hilbert space h

additive spectrum compressors abstract additive bijections a b which compress the spectrum between two unital standard operator algebras are characterized applications to local approximate anti multiplications are also given

phase retrieval via wirtinger flow theory and algorithms we study the problem of recovering the phase from magnitude measurements specifically we wish to reconstruct a complex valued signal boldsymbol x in mathbb c n about which we have phaseless samples of the form y r left langle boldsymbol a r boldsymbol x rangle right 2 r 1 ldots m knowledge of the phase of these samples would yield a linear system this paper develops a nonconvex formulation of the phase retrieval problem as well as a concrete solution algorithm in a nutshell this algorithm starts with a careful initialization obtained by means of a spectral method and then refines this initial estimate by iteratively applying novel update rules which have low computational complexity much like in a gradient descent scheme the main contribution is that this algorithm is shown to rigorously allow the exact retrieval of phase information from a nearly minimal number of random measurements indeed the sequence of successive iterates provably converges to the solution at a geometric rate so that the proposed scheme is efficient both in terms of computational and data resources in theory a variation on this scheme leads to a near linear time algorithm for a physically realizable model based on coded diffraction patterns we illustrate the effectiveness of our methods with various experiments on image data underlying our analysis are insights for the analysis of nonconvex optimization schemes that may have implications for computational problems beyond phase retrieval

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

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

spectrum preservers revisited abstract we classify spectrum preserving additive bijections which map the set of linear possibly unbounded maps on a complex banach space x to the set of linear possibly unbounded maps on a complex banach space y

linear maps preserving the minimum and surjectivity moduli of hilbert space operators abstract let b h be the algebra of all bounded linear operators on a complex infinite dimensional hilbert space h for every t b h let m t and q t denote the minimum modulus and surjectivity modulus of t respectively let b h b h be a surjective linear map in this paper we prove that the following assertions are equivalent i m t m t for all t b h ii q t q t for all t b h iii there exist two unitary operators u v b h such that t u t v for all t b h this generalizes the result of mbekhta 7 theorem 3 1 to the non unital case

additive preservers of non zero decomposable tensors let t be an additive mapping from a tensor product of vector spaces over a field into itself we describe t for the following two cases i t is surjective and sends non zero decomposable elements to non zero decomposable elements and ii t a is a non zero decomposable element if and only if a is a non zero decomposable element

rank one preserving r linear maps on spaces of self adjoint operators on complex hilbert spaces abstract let h be a complex hilbert space and let s a h be the space of all self adjoint operators on h let denote the subset of s a h consisting of all rank one operators a map l from s a h to itself is called a rank one preserver on s a h if l and is said to be r linear if l a b l a l b and l ka kl a for any k r and a b s a h where r is the field of real numbers we give a complete classification of all r linear and weakly continuous rank one preservers on s a h moreover we also determine the general form of all r linear and weakly continuous rank preservers respectively rank one idempotence preservers on s a h

additive maps on standard operator algebras preserving invertibilities or zero divisors let a and b be standard operator algebras on infinite dimensional complex banach spaces x and y respectively and let be an additive surjection from a onto b we prove that if is unital and preserves any one of left right invertibility left right zero divisors left right topological divisors of zero quasi affinity injectivity surjectivity range density lower boundedness and left right maximal ideals in both directions then it has one of the following forms isomorphism conjugate isomorphism anti isomorphism and conjugate anti isomorphism we also show that preserves zero products in both directions if and only if is either an isomorphism or a conjugate isomorphism

parallel algorithms for constrained tensor factorization via alternating direction method of multipliers tensor factorization has proven useful in a wide range of applications from sensor array processing to communications speech and audio signal processing and machine learning with few recent exceptions all tensor factorization algorithms were originally developed for centralized in memory computation on a single machine and the few that break away from this mold do not easily incorporate practically important constraints such as non negativity a new constrained tensor factorization framework is proposed in this paper building upon the alternating direction method of multipliers admom it is shown that this simplifies computations bypassing the need to solve constrained optimization problems in each iteration and it naturally leads to distributed algorithms suitable for parallel implementation this opens the door for many emerging big data enabled applications the methodology is exemplified using non negativity as a baseline constraint but the proposed framework can incorporate many other types of constraints numerical experiments are encouraging indicating that admom based non negative tensor factorization ntf has high potential as an alternative to state of the art approaches

isometries for unitarily invariant norms a survey of linear isometries for unitarily invariant norms on real or complex rectangular matrices is given which includes some latest development on the topic a result on isometries for unitarily invariant norms without the linearity assumption is presented related results and problems are discussed

maps preserving the local spectrum of skew product of operators abstract let h and k be infinite dimensional complex hilbert spaces and let b h be the algebra of all bounded linear operators on h let t h denote the local spectrum of an operator t b h at any vector h h and fix two nonzero vectors h 0 h and k 0 k we show that if a map b h b k has a range containing all operators of rank at most two and satisfies t s h 0 t s k 0 for all t s b h then there exist two unitary operators u and v in b h k such that u h 0 k 0 for some nonzero c and t u t v for all t b h we also described maps b h b k satisfying t s t h 0 t s t k 0 for all t s b h and with the range containing all operators of rank at most four

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

a separability criterion for density operators we give a necessary and sufficient condition for a mixed quantum mechanical state to be separable the criterion is formulated as a boundedness condition in terms of the greatest cross norm on the tensor product of trace class operators

the concave convex procedure the concave convex procedure cccp is a way to construct discrete time iterative dynamical systems that are guaranteed to decrease global optimization and energy functions monotonically this procedure can be applied to almost any optimization problem and many existing algorithms can be interpreted in terms of it in particular we prove that all expectation maximization algorithms and classes of legendre minimization and variational bounding algorithms can be reexpressed in terms of cccp we show that many existing neural network and mean field theory algorithms are also examples of cccp the generalized iterative scaling algorithm and sinkhorn s algorithm can also be expressed as cccp by changing variables cccp can be used both as a new way to understand and prove the convergence of existing optimization algorithms and as a procedure for generating new algorithms

additive decomposability preservers and related results on tensor products of matrices abstract in this paper we characterize additive maps between tensor spaces that send decomposable tensors to decomposable tensors as an application we classify all additive maps from tensor products of spaces of rectangular matrices to spaces of rectangular matrices which do not increase the rank of tensor product of rank one matrices

linear additive preservers of rank 2 on spaces of alternate matrices over fields abstract let kn f be the linear space of all n n alternate matrices over a field f and let k n 2 f be its subset consisting of all rank 2 matrices an operator kn f kn f is said to be additive if a b a b for any a b kn f linear if is additive and aa af a for every a f and a kn f and a preserver of rank 2 on kn f if k n 2 f k n 2 f when n 4 we characterize all linear respectively additive preservers of rank 2 on kn f over any field respectively any field that is not isomorphic to a proper subfield of itself

a note on additive mappings decreasing rank one kuzma characterized additive mappings on the space of all finite rank bounded linear operators on a real or complex banach space that decreases operators of rank one in this note we give a short proof of his result in a slightly more general setting of the tensor product of a right and a left vector space over a division ring

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 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

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

rank k preservers and preservers of sets of ranks abstract let t be a linear transformation on the set of m n matrices with entries in an algebraically closed field if t maps the set of all matrices whose rank is k into itself and if n 3 k 2 then the rank of a is the rank of t a for every m n matrix

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

jordan zero product preserving additive maps on operator algebras let a b be an additive surjective map between some operator algebras such that ab ba 0 implies a b b a 0 we show that under some mild conditions is a jordan homomorphism multiplied by a central element such operator algebras include von neumann algebras c algebras and standard operator algebras etc particularly if h and k are infinite dimensional real or complex hilbert spaces and a b h and b b k then there exists a nonzero scalar c and an invertible linear or conjugate linear operator u h k such that either a cuau 1 for all a b h or a cua u 1 for all a b h

additive preservers of numerical range abstract we characterize all additive mappings preserving classical numerical range on various subalgebras of m n the subalgebra of all diagonal matrices d n the subalgebra of all upper triangular matrices t n and subalgebras of block diagonal and block triangular matrices characterizations obtained show that in some cases m n for example all such preservers are linear on the other hand we show that the group of all additive preservers on t n contains some additional mappings

similarity invariant real linear subspaces and similarity preserving additive maps let h be a separable complex infinite dimensional hilbert space all similarity invariant real linear subspaces of b h the banach algebra of all bounded linear operators on h are characterized as an application all continuous additive surjective maps on b h which preserving similarity in both directions are completely classified

convergence analysis of alternating direction method of multipliers for a family of nonconvex problems the alternating direction method of multipliers admm is widely used to solve large scale linearly constrained optimization problems convex or nonconvex in many engineering fields however there is a general lack of theoretical understanding of the algorithm when the objective function is nonconvex in this paper we analyze the convergence of the admm for solving certain nonconvex consensus and sharing problems we show that the classical admm converges to the set of stationary solutions provided that the penalty parameter in the augmented lagrangian is chosen to be sufficiently large for the sharing problems we show that the admm is convergent regardless of the number of variable blocks our analysis does not impose any assumptions on the iterates generated by the algorithm and is broadly applicable to many admm variants involving proximal update rules and various flexible block selection rules

phase retrieval using alternating minimization phase retrieval problems involve solving linear equations but with missing sign or phase for complex numbers information more than four decades after it was first proposed the seminal error reduction algorithm of gerchberg and saxton and fienup is still the popular choice for solving many variants of this problem the algorithm is based on alternating minimization i e it alternates between estimating the missing phase information and the candidate solution despite its wide usage in practice no global convergence guarantees for this algorithm are known in this paper we show that a resampling variant of this approach converges geometrically to the solution of one such problem finding a vector bf x from bf y bf a where bf y vert bf a t bf x vert and vert bf z vert denotes a vector of element wise magnitudes of bf z under the assumption that bf a is gaussian empirically we demonstrate that alternating minimization performs similar to recently proposed convex techniques for this problem which are based on lifting to a convex matrix problem in sample complexity and robustness to noise however it is much more efficient and can scale to large problems analytically for a resampling version of alternating minimization we show geometric convergence to the solution and sample complexity that is off by log factors from obvious lower bounds we also establish close to optimal scaling for the case when the unknown vector is sparse our work represents the first theoretical guarantee for alternating minimization albeit with resampling for any variant of phase retrieval problems in the non convex setting

high performance adaptive algorithms for single group multicast beamforming the single group multicast beamforming problem is np hard and the available approximations do not always achieve favorable performance complexity tradeoffs this paper introduces a new class of adaptive multicast beamforming algorithms that features guaranteed convergence and state of the art performance at low complexity each update takes a step in the direction of an inverse signal to noise ratio snr weighted linear combination of the snr gradient vectors of all the users convergence of this update to a karush kuhn tucker kkt point of proportionally fair beamforming is established simulations show that the proposed approach can enable better performance than the prior state of art in terms of multicast rate at considerably lower complexity this reveals an interesting link between max min fair and proportionally fair multicast beamforming formulations for cases where there is no initial channel state information at the transmitter an online algorithm is developed that simultaneously learns the user channel correlation matrices and adapts the beamforming vector to maximize the minimum long term average snr among the users using only periodic binary snr feedback from each receiver the online algorithm uses the analytic center cutting plane method to quickly learn the user correlation matrices with limited signaling overhead

on operators preserving commutativity abstract let l x be the algebra of all bounded operators on a non trivial complex banach space x and f l x l x a bijective linear operator such that f and f 1 both send commuting pairs of operators into commuting pairs then either f a uau 1 p a i or f a ua u 1 p a i where p is a linear functional on l x u is a bounded linear bijective operator between the appropriate two spaces is a complex constant and a is the adjoint of a the form of an operator f for which f and f 1 both send projections of rank one into projections of rank one is also determined

rank permutable additive mappings abstract let be a fixed non identical permutation on k elements additive bijections t on the matrix algebra m n f over a field f of characteristic zero with the property that rk a 1 a k rk a 1 a k implies the same condition on the t images are characterized it is also shown that the surjectivity assumption can be relaxed if this property is preserved in both directions

coherence invariant mappings on block triangular matrix spaces abstract in this paper we classify bijective mappings on the space of block upper triangular matrices such that both and 1 preserve pairs of matrices whose differences have rank one applications to rank one preservers and semi isomorphisms are considered

linear rank preservers of tensor products of rank one matrices abstract let n 1 n k be integers larger than or equal to 2 we characterize linear maps m n 1 n k m n 1 n k such that r a n k a 1 a k 1 whenever r a n k a 1 a k 1 for all a i m n i i 1 k applying this result we extend two recent results on linear maps that preserve the rank of special classes of matrices

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

additive rank one preserving mappings on triangular matrix algebras abstract we classify surjective additive maps on the space of block upper triangular matrices that preserve matrices of rank one as well as linear maps preserving matrices of rank one on fairly general subspaces of matrices

permanent polya problem for additive surjective maps abstract let m n f denote the set of square matrices of size n over a field f with characteristics different from two we say that the map f m n f m n f is additive if f a b f a f b for all a b m n f the main goal of this paper is to prove that for n 2 there are no additive surjective maps t m n f m n f such that per t a det a for all a m n f also we show that an arbitrary additive surjective map t m n f m n f which preserves permanent is linear and thus can be completely characterized

linear maps preserving numerical radius of tensor products of matrices abstract let m n 2 be positive integers denote by m m the set of m m complex matrices and by w x the numerical radius of a square matrix x motivated by the study of operations on bipartite systems of quantum states we show that a linear map m m n m m n satisfies w a b w a b for all a m m and b m n if and only if there is a unitary matrix u m m n and a complex unit such that a b u 1 a 2 b u for all a m m and b m n where k is the identity map or the transposition map x x t for k 1 2 and the maps 1 and 2 will be of the same type if m n 3 in particular if m n 3 the map corresponds to an evolution of a closed quantum system under a fixed unitary operator possibly followed by a transposition the results are extended to multipartite systems

new results on quadratic minimization in this paper we present several new results on minimizing an indefinite quadratic function under quadratic linear constraints the emphasis is placed on the case in which the constraints are two quadratic inequalities this formulation is termed the extended trust region subproblem in this paper to distinguish it from the ordinary trust region subproblem in which the constraint is a single ellipsoid the computational complexity of the extended trust region subproblem in general is still unknown in this paper we consider several interesting cases related to this problem and show that for those cases the corresponding semidefinite programming relaxation admits no gap with the true optimal value and consequently we obtain polynomial time procedures for solving those special cases of quadratic optimization for the extended trust region subproblem itself we introduce a parameterized problem and prove the existence of a trajectory that will lead to an optimal solution combining this with a result obtained in the first part of the paper we propose a polynomial time solution procedure for the extended trust region subproblem arising from solving nonlinear programs with a single equality constraint

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

phase recovery maxcut and complex semidefinite programming phase retrieval seeks to recover a signal x in mathbb c p x c p from the amplitude a x a x of linear measurements ax in mathbb c n a x c n we cast the phase retrieval problem as a non convex quadratic program over a complex phase vector and formulate a tractable relaxation called phasecut similar to the classical maxcut semidefinite program we solve this problem using a provably convergent block coordinate descent algorithm whose structure is similar to that of the original greedy algorithm in gerchberg and saxton optik 35 237 246 1972 where each iteration is a matrix vector product numerical results show the performance of this approach over three different phase retrieval problems in comparison with greedy phase retrieval algorithms and matrix completion formulations

linear preservers of tensor product of unitary orbits and product numerical range abstract it is shown that the linear group of automorphism of hermitian matrices which preserves the tensor product of unitary orbits 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 the result is then applied to show that automorphisms of the product numerical ranges have the same structure

linear rank and corank preserving maps on b h and an application to semigroup isomorphisms of operator ideals abstract we characterize linear rank k nonincreasing rank k preserving and corank k preserving maps on b h the algebra of all bounded linear operators on the hilbert space h this unifies and extends finite dimensional results and results on linear rank 1 non increasing and rank 1 preserving maps in the infinite dimensional case we conclude with an application to semigroup isomorphisms of operator ideals

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

linear k power k potent preservers between matrix spaces abstract suppose f is a field let mn f be the linear space of all n n matrices over f and let sn f be its subspace consisting of all symmetric matrices let m n k be positive integers with k 2 and let v m n f s n f and w m m f s m f a linear map f v w is called a k power preserver if f a k f ak for every a v and a k potent preserver if f a k f a for any a v with ak a we characterize i linear k power preservers from mn f to mm f when ch f k or ch f 0 ii linear k potent preservers from mn f to mm f when f is an algebraically closed field with ch f 0 iii linear k power preservers from sn f to mm f respectively sm f when ch f k 5 or ch f 0 and iv linear k potent preservers from sn f to mm f respectively sm f when f is an algebraically closed field with ch f 0

linear maps and additive maps that preserve operators annihilated by a polynomial abstract let h k be infinite dimensional complex hilbert spaces and let p t be a complex polynomial with deg p 2 in this paper we give a sufficient and necessary condition for a surjective linear or additive map b h b k to preserve operators annihilated by p in both directions and answer a question raised by semrl affirmatively

rank constrained separable semidefinite programming with applications to optimal beamforming consider a downlink communication system where multiantenna base stations transmit independent data streams to decentralized single antenna users over a common frequency band the goal of the base stations is to jointly adjust the beamforming vectors to minimize the transmission powers while ensuring the signal to interference noise ratio requirement of each user within the system at the same time it may be necessary to keep the interference generated on other coexisting systems under a certain tolerable level in addition one may want to include general individual shaping constraints on the beamforming vectors this beamforming problem is a separable homogeneous quadratically constrained quadratic program and it is difficult to solve in general in this paper we give conditions under which strong duality holds and propose efficient algorithms for the optimal beamforming problem first we study rank constrained solutions of general separable semidefinite programs sdps and propose rank reduction procedures to achieve a lower rank solution then we show that the sdp relaxation of three classes of optimal beamforming problem always has a rank one solution which can be obtained by invoking the rank reduction procedures

additive mappings between hermitian matrix spaces preserving rank not exceeding one abstract let k be a field of characteristic not two or three with an automorphism of order two let f a k a a we characterize additive mappings t from one f vector space of hermitian matrices over k to another that preserve rank less than or equal to one for the following cases i the image of t contains a matrix of rank at least three ii every nonzero endomorphism of f is surjective

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

additive mappings that preserve rank one nilpotent operators let b x be the algebra of bounded operators on a complex infinite dimensional banach space x and f x be the subalgebra of all finite rank operators in b x a characterization of additive mappings on f x which preserve rank one nilpotent operators in both directions is given as applications of this result some additive preservers are described

isometries of a generalized numerical radius abstract for 0 q 1 the q numerical range is defined on the algebra m n of all n n complex matrices by w q a x ay x y c n x y 1 y x q the q numerical radius is defined by r q a max w q a we characterize isometries of the metric space m n r q i e the maps m n m n that satisfy r q a b r q a b we also characterize maps on m n that preserves the q numerical range the maps are not assumed to be linear

some linear preserver problems onb h concerning rank and corank abstract as a continuation of the work of linear maps between operator algebras which preserve certain subsets of operators with finite rank or finite corank here we consider the problem inbetween that is we treat the question of preserving operators with infinite rank and infinite corank since as it turns out in this generality our preservers cannot be written in a nice form what we have got used to when dealing with linear preserver problems hence we restrict our attention to certain important classes of operators like idempotents or projections or partial isometries we conclude the paper with a result on the form of linear maps which preserve the left ideals in b h

approximation bounds for quadratic optimization with homogeneous quadratic constraints we consider the np hard problem of finding a minimum norm vector in n dimensional real or complex euclidean space subject to m concave homogeneous quadratic constraints we show that a semidefinite programming sdp relaxation for this nonconvex quadratically constrained quadratic program qp provides an o m 2 approximation in the real case and an o m approximation in the complex case moreover we show that these bounds are tight up to a constant factor when the hessian of each constraint function is of rank 1 namely outer products of some given so called steering vectors and the phase spread of the entries of these steering vectors are bounded away from pi 2 we establish a certain constant factor approximation depending on the phase spread but independent of m and n for both the sdp relaxation and a convex qp restriction of the original np hard problem finally we consider a related problem of finding a maximum norm vector subject to m convex homogeneous quadratic constraints we show that an sdp relaxation for this nonconvex qp provides an o 1 ln m approximation which is analogous to a result of nemirovski e math program 86 1999 pp 463 473 for the real case

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