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

isometries between projection lattices of von neumann algebras abstract we investigate surjective isometries between projection lattices of two von neumann algebras we show that such a mapping is characterized by means of jordan isomorphisms in particular we prove that two von neumann algebras without type i 1 direct summands are jordan isomorphic if and only if their projection lattices are isometric our theorem extends the recent result for type i factors by g p geher and p semrl which is a generalization of wigner s theorem

higher traces noncommutative motives and the categorified chern character abstract we propose a categorification of the chern character that refines earlier work of toen and vezzosi and of ganter and kapranov if x is an algebraic stack our categorified chern character is a symmetric monoidal functor from a category of mixed noncommutative motives over x which we introduce to s 1 equivariant perfect complexes on the derived free loop stack l x as an application of the theory we show that toen and vezzosi s secondary chern character factors through secondary k theory our techniques depend on a careful investigation of the functoriality of traces in symmetric monoidal n categories which is of independent interest

relative quantum field theory we highlight the general notion of a relative quantum field theory which occurs in several contexts one is in gauge theory based on a compact lie algebra rather than a compact lie group this is relevant to the maximal superconformal theory in six dimensions

geometric version of wigner s theorem for hilbert grassmannians abstract let h be a complex hilbert space of dimension not less than 3 and let g k h be the grassmannian formed by k dimensional subspaces of h suppose that dim h 2 k 2 we show that the transformations of g k h induced by linear or conjugate linear isometries can be characterized as transformations preserving some of principal angles corresponding to the orthogonality adjacency and ortho adjacency relations as a consequence we get the following if the dimension of h is finite and greater than 2k then every transformation of g k h preserving the orthogonality relation in both directions is a bijection induced by a unitary or anti unitary operator

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

on isometric embeddings of wasserstein spaces the discrete case abstract the aim of this short paper is to offer a complete characterization of all not necessarily surjective isometric embeddings of the wasserstein space w p x where x is a countable discrete metric space and 0 p is any parameter value roughly speaking we will prove that any isometric embedding can be described by a special kind of x 0 1 indexed family of nonnegative finite measures our result implies that a typical non surjective isometric embedding of w p x splits mass and does not preserve the shape of measures in order to stress that the lack of surjectivity is what makes things challenging we will prove alternatively that w p x is isometrically rigid for all 0 p

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

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

a gentle guide to the basics of two projections theory abstract this paper is a survey of the basics of the theory of two projections it contains in particular the theorem by halmos on two orthogonal projections and roch silbermann gohberg and krupnik s theorem on two idempotents in banach algebras these two theorems which deliver the desired results usually very quickly and comfortably are missing or wrongly cited in many recent publications on the topic the paper is intended as a gentle guide to the field the basic theorems are precisely stated some of them are accompanied by full proofs others not but precise references are given in each case and many examples illustrate how to work with the theorems

on the continuity of generalized inverses of linear operators in hilbert spaces abstract let x and y be hilbert spaces and let t x y be a bounded linear operator with closed range we study the continuity problem of the generalized inverse of t and related least squares solutions to the operator equation tx y

on the perturbation of the least squares solutions in hilbert spaces abstract let h1 and h2 be hilbert spaces and let t h1 h2 be a bounded linear operator with closed range we present some results of the perturbation analysis for the least squares solutions to the operator equation tx y

on perturbations for oblique projection generalized inverses of closed linear operators in banach spaces the main concern of this paper is the perturbation problem for oblique projection generalized inverses of closed linear operators in banach spaces we provide a new stability characterization of oblique projection generalized inverses of closed linear operators under t bounded perturbations which improves some well known results in the case of the closed linear operators under the bounded perturbation or that the perturbation does not change the null space

optimal robustness in the gap metric the application of the gap metric to robust stabilization of feedback systems is considered in particular a solution to the problem of robustness optimization in the gap metric is presented the problem of robust stabilization under simultaneous plant controller perturbations is addressed and the least amount of combined plant controller uncertainty measured by the gap metric that can cause instability of a nominally stable feedback system is determined included are a detailed summary of the main properties of the gap metric and the introduction of a dual metric called the t gap metric a key contribution of this study is to show that the problem of robustness optimization in the gap metric is equivalent to robustness optimization for normalized coprime factor perturbations this settles the question as to whether maximizing allowable coprime factor uncertainty corresponds to tolerating the largest ball of uncertainty in a well defined metric

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

surjective isometries on grassmann spaces let h be a complex hilbert space n a given positive integer and let pn h be the set of all projections on h with rank n under the condition dimh 4n we describe the surjective isometries of pn h with respect to the gap metric the metric induced by the operator norm

the characterizations of the stable perturbation of a closed operator by a linear operator in banach spaces in this paper we investigate the invertibility of iy tt when t is a closed operator from x to y with a generalized inverse t and t is a linear operator whose domain contains d t and range is contained in d t the characterizations of the stable perturbation t t of t by t in banach spaces are obtained the results extend the main results in 7 8

isometries of grassmann spaces ii abstract botelho jamison and molnar 1 and geher and semrl 4 have recently described the general form of surjective isometries of grassmann spaces of all projections of a fixed finite rank on a hilbert space h as a straightforward consequence one can characterize surjective isometries of grassmann spaces of projections of a fixed finite corank in this paper we solve the remaining structural problem for surjective isometries on the set p h of all projections of infinite rank and infinite corank when h is separable the proof technique is entirely different from the previous ones and is based on the study of geodesics in the grassmannian p h however the same method gives an alternative proof in the case of finite rank projections

perturbation of least squares problem in hilbert spaces let h 1 h 2 be two hilbert spaces over the same field and let t h 1 h 2 be a bounded linear operator with closed range we present some results of the perturbation analysis for the least squares solutions to the operator equation tx y

perturbation analysis of generalized inverses of linear operators in banach spaces abstract let x 1 x 2 be two banach spaces and t x 1 x 2 be a bounded linear operator with a bounded generalized inverse t we study the perturbation problem of generalized inverse t and provide two new stability characterizations of generalized inverse we also give a continuity characterization of moore penrose inverse in hilbert spaces our results are new even in the case of matrices

perturbation analysis of the least squares solution in hilbert spaces abstract let h 1 h 2 be two hilbert spaces over the same field and let t h 1 h 2 be a bounded linear operator with closed range we give a complete description of the perturbation analysis for the least squares solution to the operator equation t x y where x h 1 y h 2

commutativity preserving transformations on conjugacy classes of finite rank self adjoint operators abstract let h be a complex hilbert space and let c be a conjugacy class of rank k self adjoint operators on h with respect to the action of the group of unitary operators under the assumption that dim h 4 k we describe all bijective transformations of c preserving the commutativity in both directions in particular it follows from this description that every such transformation is induced by a unitary or anti unitary operator only in the case when for every operator from c the dimensions of eigenspaces are mutually distinct

subspace angles between arma models abstract we define a notion of subspace angles between two linear autoregressive moving average single input single output models by considering the principal angles between subspaces that are derived from these models we show how a recently defined metric for these models which is based on their cepstra relates to the subspace angles between the models

op lax natural transformations twisted quantum field theories and even higher morita categories abstract motivated by the challenge of defining twisted quantum field theories in the context of higher categories we develop a general framework for lax and oplax transformations and their higher analogs between strong n functors we construct a double n category built out of the target n category governing the desired diagrammatics we define op lax transformations as functors into parts thereof and an op lax twisted field theory to be a symmetric monoidal op lax natural transformation between field theories we verify that lax trivially twisted relative field theories are the same as absolute field theories as a second application we extend the higher morita category of e d algebras in a symmetric monoidal n category c to an n d category using the higher morphisms in c

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

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

perturbation analysis for the operator equationtx bin banach spaces letx1 x2be two banach spaces over the complex fieldcand lett x1 x2be a bounded linear operator with the generalized inverset lett t tbe a bounded linear operator with t t 1 suppose thatdimkert dimkert orr t kert 0 thenthas the generalized inverset i t t 1t with t t 1 t t this result generalizes theorem 3 9 ofna using this result we give some results of the perturbation analysis for the operator equationtx b

t hooft anomalies of discrete gauge theories and non abelian group cohomology we study discrete symmetries of dijkgraaf witten theories and their gauging in the framework of extended functorial quantum field theory non abelian group cohomology is used to describe discrete symmetries and we derive concrete conditions for such a symmetry to admit t hooft anomalies in terms of the lyndon hochschild serre spectral sequence we give an explicit realization of a discrete gauge theory with t hooft anomaly as a state on the boundary of a higher dimensional dijkgraaf witten theory this allows us to calculate the 2 cocycle twisting the projective representation of physical symmetries via transgression we present a general discussion of the bulk boundary correspondence at the level of partition functions and state spaces which we make explicit for discrete gauge theories

iterated spans and classical topological field theories we construct higher categories of iterated spans possibly equipped with extra structure in the form of higher categorical local systems and classify their fully dualizable objects by the cobordism hypothesis these give rise to framed topological quantum field theories which are the framed versions of the classical topological quantum field theories considered in the quantization programme of freed hopkins lurie teleman using this machinery we also construct an infty 1 category of symplectic derived algebraic stacks and lagrangian correspondences and show that all its objects are dualizable

higher algebraic structures and quantization we derive quasi quantum groups in 2 1 dimensional topological field theory directly from the classical action and the path integral detailed computations are carried out for the chern simons theory with finite gauge group the principles behind our computations are presumably more general we extend the classical action in ad 1 dimensional topological theory to manifolds of dimension less thand 1 we then construct a generalized path integral which ind 1 dimensions reduces to the standard one and ind dimensions reproduces the quantum hilbert space in a 2 1 dimensional topological theory the path integral over the circle is the category of representations of a quasi quantum group in this paper we only consider finite theories in which the generalized path integral reduces to a finite sum new ideas are needed to extend beyond the finite theories treated here

wigner s type theorem in terms of linear operators which send projections of a fixed rank to projections of other fixed rank abstract let h be a complex hilbert space whose dimension is not less than 3 and let f s h be the real vector space formed by all self adjoint operators of finite rank on h for every non zero natural k dim h we denote by p k h the set of all rank k projections let h be other complex hilbert space of dimension not less than 3 and let l f s h f s h be a linear operator such that l p k h p m h for some natural k m and the restriction of l to p k h is injective if h h and k m then l is induced by a linear or conjugate linear isometry of h to itself except the case dim h 2 k when there is another one possibility we get a classical wigner s theorem if k m 1 if dim h 2 k then k m the main result describes all linear operators l satisfying the above conditions under the assumptions that h is infinite dimensional and for any p q p k h the dimension of the intersection of the images of l p and l q is not less than m k

a model for the homotopy theory of homotopy theory we describe a category the objects of which may be viewed as models for homotopy theories we show that for such models functors between two homotopy theories form a homotopy theory or more precisely that the category of such models has a well behaved internal hom object

perturbation analysis for oblique projection generalized inverses of closed linear operators in banach spaces in this paper we study the perturbation problem for oblique projection generalized inverses of closed linear operators in banach spaces by the method of the perturbation analysis of linear operators we obtain an explicit perturbation theorem and error estimates for the oblique projection generalized inverse of closed linear operators under the t bounded perturbation which extend the known results on the perturbation of the oblique projection generalized inverse of bounded linear operators in banach spaces

stacky dualities for the moduli of higgs bundles abstract the central result of this paper is an identification of the shifted cartier dual of the moduli stack m g c of g higgs bundles on c of arbitrary degree modulo shifts by z g with a quotient of the langlands dual stack m g l c via hyperkahler rotation this may equivalently be viewed as the identification of an syz fibration relating hitchin systems for arbitrary langlands dual semisimple groups coupled to nontrivial finite b fields as a corollary certain self dual stacks m g c are observed to exist which i conjecture to be the coulomb branches for the 3d reduction of the 4d n 2 theories of class s

higher dimensional algebra and topological quantum field theory the study of topological quantum field theories increasingly relies upon concepts from higher dimensional algebra such as n categories and n vector spaces we review progress towards a definition of n category suited for this purpose and outline a program in which n dimensional topological quantum field theories tqfts are to be described as n category representations first we describe a suspension operation on n categories and hypothesize that the k fold suspension of a weak n category stabilizes for k n 2 we give evidence for this hypothesis and describe its relation to stable homotopy theory we then propose a description of n dimensional unitary extended tqfts as weak n functors from the free stable weak n category with duals on one object to the n category of n hilbert spaces we conclude by describing n categorical generalizations of deformation quantization and the quantum double construction

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

enriched categories via non symmetric operads abstract we set up a general theory of weak or homotopy coherent enrichment in an arbitrary monoidal category our theory of enriched categories has many desirable properties for instance if the enriching category v is presentably symmetric monoidal then cat v is as well these features render the theory useful even when an category of enriched categories comes from a model category as is often the case in examples of interest e g dg categories spectral categories and n categories this is analogous to the advantages of categories over more rigid models such as simplicial categories for example the resulting categories of functors between enriched categories automatically have the correct homotopy type we construct the homotopy theory of v enriched categories as a certain full subcategory of the category of many object associative algebras in v the latter are defined using a non symmetric version of lurie s operads and we develop the basics of this theory closely following lurie s treatment of symmetric operads while we may regard these many object algebras as enriched categories we show that it is precisely the full subcategory of complete objects in the sense of rezk i e those whose spaces of objects are equivalent to their spaces of equivalences that are local with respect to the class of fully faithful and essentially surjective functors we also consider an alternative model of enriched categories as certain presheaves of spaces satisfying analogues of the segal condition for rezk s segal spaces lastly we present some applications of our theory most notably the identification of associative algebras in v as a coreflective subcategory of pointed v enriched categories as well as a proof of a strong version of the baez dolan stabilization hypothesis

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

convergence of newton like methods for singular operator equations using outer inverses we present a semilocal kantorovich type analysis for newton like methods for singular operator equations using outer inverses we establish sharp generalizations of the kantorovich theory and the mysovskii theory for operator equations when the derivative is not necessarily invertible the results reduce in the case of an invertible derivative to well known theorems of kantorovich and mysovskii with no additional assumptions unlike earlier theorems which impose strong conditions the strategy of the analysis is based on banach type lemmas and perturbation bounds for outer inverses which show that the set of outer inverses to a given bounded linear operator admits selections that behave like bounded linear inverses in contrast to inner inverses or generalized inverses which do not depend continuously on perturbations of the operator we give two examples to illustrate our results and compare them with earlier results and another numerical example to relate our results to computational issues

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