self adjoint block operator matrices with non separated diagonal entries and their schur complements in this paper self adjoint 2 2 block operator matrices a in a hilbert space h1 h2 are considered for an interval which does not intersect the spectrum of at least one of the diagonal entries of a we prove angular operator representations for the corresponding spectral subspace l a of a and we study the supporting subspace in this angular operator representation of l a which is the orthogonal projection of l a to the corresponding component h1 or h2 our main result is a description of a special direct complement of this supporting subspace in its component in terms of spectral subspaces of the values of the corresponding schur complement of a in the endpoints of

non symmetric perturbations of self adjoint operators abstract we investigate the effect of non symmetric relatively bounded perturbations on the spectrum of self adjoint operators in particular we establish stability theorems for one or infinitely many spectral gaps along with corresponding resolvent estimates these results extend and improve classical perturbation results by kato and by gohberg kre n further we study essential spectral gaps and perturbations exhibiting additional structure with respect to the unperturbed operator in the latter case we can even allow for perturbations with relative bound 1 the generality of our results is illustrated by several applications massive and massless dirac operators point coupled periodic systems and two channel hamiltonians with dissipation

sharp eigenvalue enclosures for the perturbed angular kerr newman dirac operator we examine a certified strategy for determining sharp intervals of enclosure for the eigenvalues of matrix differential operators with singular coefficients the strategy relies on computing the second order spectrum relative to subspaces of continuous piecewise linear functions for smooth perturbations of the angular kerr newman dirac operator explicit rates of convergence linked to regularity of the eigenfunctions are established numerical tests which validate and sharpen by several orders of magnitude the existing benchmarks are also included

exchange leavitt path algebras and stable rank abstract we characterize those leavitt path algebras which are exchange rings in terms of intrinsic properties of the graph and show that the values of the stable rank for these algebras are 1 2 or concrete criteria in terms of properties of the underlying graph are given for each case

topological size of some subsets in certain calderon lozanowskiĭ spaces abstract for i 1 2 3 let i be young functions a topological measure space e an ideal of measurable complex valued functions defined on and e i be the corresponding calderon lozanowski space our aim in this paper is to give under mild conditions several results on topological size in the sense of baire of the sets f g e 1 e 2 f g e 3 and f g e 1 e 2 x v f g x is well defined where denotes the convolution or pointwise product of functions and v a compact neighborhood our results sharpen and unify the related results obtained in diverse areas during recent thirty years

graph c algebras with real rank zero given a row finite directed graph e a universal c algebra c e generated by a family of partial isometries and projections subject to the relations determined by e is associated to the graph e the cuntz krieger algebras are those graph c algebras associated to some finite graphs we prove that a graph c algebra c e has real rank zero in the sense that the set of invertible self adjoint elements is dense in the set of all self adjoint elements in c e or in the unitization of c e if c e is nonunital if and only if e satisfies a loop condition k that is analogous to the condition for a finite 0 1 matrix a under which cuntz analyzed the ideal structure of the cuntz krieger algebra oa

spectral inclusion for unbounded block operator matrices abstract in this paper we establish a new analytic enclosure for the spectrum of unbounded linear operators a admitting a block operator matrix representation for diagonally dominant and off diagonally dominant block operator matrices we show that the recently introduced quadratic numerical range w 2 a contains the eigenvalues of a and that the approximate point spectrum of a is contained in the closure of w 2 a this provides a new method to enclose the spectrum of unbounded block operator matrices by means of the non convex set w 2 a several examples illustrate that this spectral inclusion may be considerably tighter than the one by the usual numerical range or by perturbation theorems both in the non self adjoint case and in the self adjoint case applications to dirac operators and to two channel hamiltonians are given

graphs groupoids and cuntz krieger algebras we associate to each locally finite directed graphgtwo locally compact groupoidsgandg the unit space ofgis the space of one sided infinite paths ing andg is the reduction ofgto the space of paths emanating from a distinguished vertex we show that under certain conditions theirc algebras are morita equivalent the groupoidc algebrac g is the cuntz krieger algebra of an infinite 0 1 matrix defined byg and that the algebrasc g contain thec algebras used by doplicher and roberts in their duality theory for compact groups we then analyse the ideal structure of these groupoidc algebras using the general theory of renault and calculate theirk theory

extensions of exchange rings we define non unital exchange rings and we prove that ifiis an ideal of a ringr thenris an exchange ring if and only ifiandr iare exchange rings and idempotents can be lifted moduloi we also show that we can replace the condition on liftability of idempotents with the condition that the canonical mapk0 r k0 r i be surjective

an extremal problem in fourier analysis with applications to operator theory a certain minimal extrapolation problem for fourier transforms is known to have consequences for the determination of best possible bounds in some problems in linear operator equations and in perturbation of operators in this paper we estimate the value of the constant in the fourier transform problem by an analytic reformulation

the leavitt path algebra of a graph abstract for any row finite graph e and any field k we construct the leavitt path algebra l e having coefficients in k when k is the field of complex numbers then l e is the algebraic analog of the cuntz krieger algebra c e described in i raeburn graph algebras in cbms reg conf ser math vol 103 amer math soc 2005 the matrix rings m n k and the leavitt algebras l 1 n appear as algebras of the form l e for various graphs e in our main result we give necessary and sufficient conditions on e which imply that l e is simple

analysis and modification of newton s method for algebraic riccati equations when newton s method is applied to find the maximal symmetric solution of an algebraic riccati equation convergence can be guaranteed under moderate conditions in particular the initial guess need not be close to the solution the convergence is quadratic if the frechet derivative is invertible at the solution in this paper we examine the behaviour of the newton iteration when the derivative is not invertible at the solution we find that a simple modification can improve the performance of the newton iteration dramatically

the block numerical range of matrix polynomials block numerical ranges of matrix polynomials especially the quadratic numerical range are considered the main results concern spectral inclusion boundedness of the block numerical range an estimate of the resolvent in terms of the quadratic numerical range geometrical properties of the quadratic numerical range and inclusion between block numerical ranges of the matrix polynomials for refined block decompositions as an application we connect the quadratic numerical range with the localization of the spectrum of matrix polynomials

closing the gap in a subspace perturbation bound abstract this paper sharpens an operator inequality from perturbation theory suppose x and y are self adjoint operators on a hilbert space kx x and ky y are two closed sets bounded apart by 0 e is the spectral projector for x belonging to kx and f is the spectral projector for y belonging to ky it is known that 2 x y ef for all operators as described and for any unitary invariant norm in this paper we show 2 is best possible in this inequality we then discuss an analogous bound which applies when x and y are normal

triple variational principles for self adjoint operator functions for a very general class of unbounded self adjoint operator function we prove upper bounds for eigenvalues which lie within arbitrary gaps of the essential spectrum these upper bounds are given by triple variations furthermore we find conditions which imply that a point is in the resolvent set for norm resolvent continuous operator functions we show that the variational inequality becomes an equality

the restricted grassmannian banach lie poisson spaces and coadjoint orbits we investigate some basic questions concerning the relationship between the restricted grassmannian and the theory of banach lie poisson spaces by using universal central extensions of lie algebras we find that the restricted grassmannian is symplectomorphic to symplectic leaves in certain banach lie poisson spaces and the underlying banach space can be chosen to be even a hilbert space smoothness of numerous adjoint and coadjoint orbits of the restricted unitary group is also established several pathological properties of the restricted algebra are pointed out

removal of the resolvent like energy dependence from interactions and invariant subspaces of a total hamiltonian the spectral problem a v z z is considered where the main hamiltonian a is a self adjoint operator of sufficiently arbitrary nature the perturbation v z b a z 1b depends on the energy z as resolvent of another self adjoint operator a the latter is usually interpreted as a hamiltonian describing an internal structure of physical system the operator b is assumed to have a finite hilbert schmidt norm the conditions are formulated when one can replace the perturbation v z with an energy independent potential w such that the hamiltonian h a w has the same spectrum more exactly a part of spectrum and the same eigenfunctions as the initial spectral problem the hamiltonian h is constructed as a solution of the non linear operator equation h a v h it is established that this equation is closely connected with the problem of searching for invariant subspaces of the hamiltonian h b aa b the orthogonality and expansion theorems are proved for eigenfunction systems of the hamiltonian h a w

on the shape of numerical range of matrix polynomials abstract the numerical range of an n n matrix polynomial p a m m a 1 a 0 is defined by w p c x p x 0 x c n x 0 in this paper we investigate the shape of w p by using the notion of local dimension the numerical range of first order matrix polynomials is always simply connected the special cases of diagonal matrix polynomials and 2 2 matrix polynomials are also considered

weakly coupled bound state of 2 d schrodinger operator with potential measure abstract we consider a self adjoint two dimensional schrodinger operator h which corresponds to the formal differential expression where is a finite compactly supported positive radon measure on r 2 from the generalized kato class and 0 is the coupling constant it was proven earlier that ess h 0 we show that for sufficiently small the condition d h 1 holds and that the corresponding unique eigenvalue has the asymptotic expansion c o 1 exp 4 r 2 0 with a certain constant c 0 we also obtain a formula for the computation of c the asymptotic expansion of the corresponding eigenfunction is provided the statements of this paper extend the results of simon 41 to the case of potentials measures also for regular potentials our results are partially new

rational eigenvalue problems and applications to photonic crystals we establish new analytic results for a general class of rational spectral problems they arise e g in modelling photonic crystals whose capability to control the flow of light depends on specific

symmetry properties of higher numerical ranges abstract let a be a linear operator on a finite dimensional unitary space v of dimension n the k th higher numerical range of a denoted by w k a is the totality of complex numbers tr pap where p runs over all k dimensional orthogonal projections on v we show that w k a is a polygon with the real axis as a line of symmetry k 1 n if and only if a is normal with a real characteristic polynomial we also construct several nonnormal examples to investigate the extent to which the symmetry of all of the w k a is required

on the eigenvalues of operators with gaps application to dirac operators this paper is devoted to a general min max characterization of the eigenvalues in a gap of the essential spectrum of a self adjoint unbounded operator we prove an abstract theorem then we apply it to the case of dirac operators with a coulomb like potential the result is optimal for the coulomb potential

flow invariants in the classification of leavitt path algebras abstract we analyze in the context of leavitt path algebras some graph operations introduced in the context of symbolic dynamics by williams parry and sullivan and franks we show that these operations induce morita equivalence of the corresponding leavitt path algebras as a consequence we obtain our two main results the first gives sufficient conditions for which the leavitt path algebras in a certain class are morita equivalent while the second gives sufficient conditions which yield isomorphisms we discuss a possible approach to establishing whether or not these conditions are also in fact necessary in the final section we present many additional operations on graphs which preserve morita equivalence resp isomorphism of the corresponding leavitt path algebras

branching systems and representations of cohn leavitt path algebras of separated graphs abstract for each separated graph e c we construct a family of branching systems over a set x and show how each branching system induces a representation of the cohn leavitt path algebra associated with e c as homomorphisms over the module of functions in x we also prove that the abelianized cohn leavitt path algebra of a separated graph with no loops can be written as an amalgamated free product of abelianized cohn leavitt algebras that can be faithfully represented via branching systems

perturbation of spectra and spectral subspaces we consider the problem of variation of spectral subspaces for linear self adjoint operators with emphasis on the case of off diagonal perturbations we prove a number of new optimal results on the shift of the spectrum and obtain sharp estimates on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators respectively

variational principles for eigenvalues of the klein gordon equation in this paper variational principles for eigenvalues of an abstract model of the klein gordon equation with electromagnetic potential are established they are used to characterize and estimate eigenvalues in cases where the essential spectrum has a gap around 0 even in the presence of complex eigenvalues as a consequence a comparison between eigenvalues of the klein gordon equation in rd and eigenvalues of certain schrodinger operators is obtained the results are illustrated on examples including the klein gordon equation with coulomb and square well potential

on applications of schauderʼs fixed point theorem for the solution of the non symmetric algebraic riccati equation abstract in this article we give sufficient conditions for the existence of contractive solutions for the non symmetric algebraic riccati equation the conditions are given in terms of the matrix of the coefficients only we also provide sufficient conditions for unique contractive solution which is a unique minimal norm solution the results are generalized to contractive solutions with respect to generalized unit matrix balls

variational principles and eigenvalue estimates for unbounded block operator matrices and applications in this paper we establish variational principles eigenvalue estimates and asymptotic formulae for eigenvalues of three different classes of unbounded block operator matrices the results allow to characterise eigenvalues that are not necessarily located at the boundary of the spectrum applications to an example from magnetohydrodynamics and to dirac operators on certain manifolds are given

the regular algebra of a quiver let k be a fixed field we attach to each column finite quiver e a von neumann regular k algebra q e in a functorial way the algebra q e is a universal localization of the usual path algebra p e associated with e the monoid of isomorphism classes of finitely generated projective right q e modules is explicitly computed

riccati equation and the problem of decoherence the block operator matrix theory is used to investigate the problem of a single qubit we establish a connection between the riccati operator equation and the possibility of obtaining an exact reduced dynamics for the qubit in question the model of the half spin particle in the rotating magnetic field coupling with the external environment is discussed we show that the model defined in such a way can be reduced to a time independent problem

representing finitely generated refinement monoids as graph monoids abstract graph monoids arise naturally in the study of non stable k theory of graph c algebras and leavitt path algebras they play also an important role in the current approaches to the realization problem for von neumann regular rings in this paper we characterize when a finitely generated conical refinement monoid can be represented as a graph monoid the characterization is expressed in terms of the behavior of the structural maps of the associated i system at the free primes of the monoid

square free modules with the exchange property we prove that a square free module with finite exchange has full exchange more generally if r is an exchange ring with r j r abelian and r is endowed with a left linear hausdorff complete topology then r is a full exchange ring this provides an overarching framework for capturing many other results in the literature such as the fact that quasi continuous modules with finite exchange have full exchange we further show that square free modules with exchange satisfy an infinite version of the c3 property

multiplicativity factors for orlicz space function norms abstract let be a function norm defined by a young function with respect to a measure space t m and let l be the orlicz space determined by if l is an algebra then a constant 0 is called a multiplicativity factor for if fg f g for all f g l the main objective of this paper is to give conditions under which l is indeed an algebra and to obtain in this case the best least multiplicativity factor for the first of our principal results is that l is an algebra if and only if m inf inf m a 0 a 0 or x sup x 0 x our second main result states that if l is an algebra and t m is free of infinite atoms then the best multiplicativity factor for is 1 1 minf if minf 0 and x if minf 0

leavitt path algebras with bases consisting solely of units abstract an algebra is said to be an invertible algebra if it has a basis consisting solely of units given a field k and a finite graph e we give a condition on e that is equivalent to that of the leavitt path algebra l k e being an invertible k algebra we derive from this a necessary and sufficient condition for the cohn path algebra c k e to be an invertible k algebra in addition given a unital commutative ring r sufficient conditions on e for the leavitt path algebra l r e to be an invertible r algebra are given

on the connectedness of numerical range of matrix polynomials abstract an investigation on nonconnectedness of numerical range for monic matrix polynomials l is undertaking here the distribution of eigenvalues of l to the components of numerical range and some other algebraic properties are also presented

geometrical properties of numerical range of matrix polynomials abstract through the linearization of a matrix polynomial p the symmetry and the sharp points of the numerical range p are studied

a new concept for block operator matrices the quadratic numerical range abstract in this paper a new concept for 2 2 block operator matrices the quadratic numerical range is studied the main results are a spectral inclusion theorem an estimate of the resolvent in terms of the quadratic numerical range factorization theorems for the schur complements and a theorem about angular operator representations of spectral invariant subspaces which implies e g the existence of solutions of the corresponding riccati equations and a block diagonalization all results are new in the operator as well as in the matrix case

existence and uniqueness of contractive solutions of some riccati equations abstract in this paper we establish criteria for the existence and uniqueness of contractive solutions k of the riccati equation kbk ka dk c 0 under the assumption that the spectra of a and d are disjoint

a variational principle for block operator matrices and its application to the angular part of the dirac operator in curved spacetime abstract the operator associated to the angular part of the dirac equation in the kerr newman background metric is a block operator matrix with bounded diagonal and unbounded off diagonal entries the aim of this paper is to establish a variational principle for block operator matrices of this type and to derive thereof upper and lower bounds for the angular operator mentioned above in the last section these analytic bounds are compared with numerical values from the literature

algebras of quotients of path algebras abstract leavitt path algebras are shown to be algebras of right quotients of their corresponding path algebras using this fact we obtain maximal algebras of right quotients from those leavitt path algebras whose associated graph satisfies that every vertex connects to a line point equivalently the leavitt path algebra has essential socle we also introduce and characterize the algebraic counterpart of toeplitz algebras

an infinite analogue of rings with stable rank one abstract replacing invertibility with quasi invertibility in bass first stable range condition we discover a new class of rings the qb rings these constitute a considerable enlargement of the class of rings with stable rank one b rings and include examples like end f v the ring of endomorphisms of a vector space v over some field f and b f the ring of all row and column finite matrices over f we show that the category of qb rings is stable under the formation of corners ideals and quotients as well as matrices and direct limits we also give necessary and sufficient conditions for an extension of qb rings to be a qb ring and show that extensions of b rings often lead to qb rings specializing to the category of exchange rings we characterize the subset of exchange qb rings as those in which every von neumann regular element extends to a maximal regular element i e a quasi invertible element finally we show that the c algebras that are qb rings are exactly the extremally rich c algebras studied by l g brown and the second author

perturbation of spectral subspaces and solution of linear operator equations abstract let a and b be normal operators on a hilbert space let k a and k b be subsets of the complex plane at distance at least from each other let e be the spectral projector for a belonging to k a and let f be the spectral projector for b belonging to k b our main results are estimates of the form ef c e a b f in some special situations the constant c is as low as 1 as an application we prove for an absolute constant d that if the space is finite dimensional and if a and b are normal with a b d then the spectrum of b can be obtained from that of a multiplicities counted by moving each eigenvalue by at most our main results have equivalent formulations as statements about the operator equation aq qb s let a and b be normal operators on perhaps different hilbert spaces assume a k a and b k b where k a k b and are as before then we give estimates of the forms q c aq qb

block diagonalisation of matrices and operators abstract in this short note we deal with a constructive scheme to decompose a continuous family of matrices a asymptotically as 0 into blocks corresponding to groups of eigenvalues of the limit matrix a 0 we also discuss the extension of the scheme to matrix families depending upon additional parameters and operators on hilbert spaces

on prime nonprimitive von neumann regular algebras let e be any directed graph and k any field we classify those graphs e for which the leavitt path algebra lk e is primitive as a consequence we obtain classes of examples of von neumann regular prime rings which are not primitive

on basis properties of selfadjoint operator functions for a selfadjoint operator function we study the existence of a riesz basis consisting of eigenvectors if not for the whole space then at least for the closed linear span of all the eigenvectors c 2000 academic press

g decompositions of matrices and related problems i in the present paper we introduce a notion of g decompositions of matrices main result of the paper is that a symmetric matrix am has a g decomposition in the class of stochastic resp substochastic matrices if and only if am belongs to the set um resp um to prove the main result we study extremal points and geometrical structures of the sets um um note that such kind of investigations enables to study birkhoff s problem for quadratic g doubly stochastic operators

birkhoff s theorem and multidimensional numerical range abstract we show that under certain conditions birkhoff s theorem on doubly stochastic matrices remains valid for countable families of discrete probability spaces which have nonempty intersections using this result we study the relation between the spectrum of a self adjoint operator a and its multidimensional numerical range it turns out that the multidimensional numerical range is a convex set whose extreme points are sequences of eigenvalues of the operator a every collection of eigenvalues which can be obtained by the rayleigh ritz formula generates an extreme point of the multidimensional numerical range however it may also have other extreme points

bounds for eigenvalues of matrix polynomials upper and lower bounds are derived for the absolute values of the eigenvalues of a matrix polynomial or matrix the bounds are based on norms of the coefficient matrices and involve the inverses of the leading and trailing coefficient matrices they generalize various existing bounds for scalar polynomials and single matrices a variety of tools are used in the derivations including block companion matrices gershgorin s theorem the numerical radius and associated scalar polynomials numerical experiments show that the bounds can be surprisingly sharp on practical problems

non simple purely infinite rings in this paper we introduce the concept of purely infinite rings which in the simple case agrees with the already existing notion of pure infiniteness we establish various permanence proper ties of this notion with respect to passage to matrix rings corners and behaviour under extensions so being purely infinite is preserved under morita equivalence we show that a wealth of examples falls into this class including important analogues of constructions commonly found in operator al gebras in particular for any s unital k algebra having enough nonzero idempotents for example for a von neumann regular algebra its tensor product over k with many non simple leavitt path algebras is purely infinite introduction the notion of pure infiniteness has proved key in the the ory of operator algebras since its conception in the early eighties by j cuntz see 27 this was done for simple algebras and provided a huge list of exam ples one of the milestones of the theory became the classification of separable nuclear unital purely infinite simple algebras by means of k theoretic invariants due to kirchberg 41 and phillips 58

an estimation of the spectral radius of a product of block matrices let c r c ij r 1 2 r be block m x m matrices where c ij r are nonnegative n i x n j matrices for i j 1 2 m let be a consistent matrix norm denote for each r by b r c ij r an m m matrix the relation of the spectral radii r r 1 c r and r r 1 b r is studied in this paper it is shown with two proofs that r r 1 c r r r 1 b r as shown in one of the proofs r r 1 b r can be reduced so that it gives a better estimation of r r 1 c r

uniqueness theorems and ideal structure for leavitt path algebras abstract we prove leavitt path algebra versions of the two uniqueness theorems of graph c algebras we use these uniqueness theorems to analyze the ideal structure of leavitt path algebras and give necessary and sufficient conditions for their simplicity we also use these results to give a proof of the fact that for any graph e the leavitt path algebra l c e embeds as a dense subalgebra of the graph c algebra c e this embedding has consequences for graph c algebras and we discuss how we obtain new information concerning the construction of c e

the theory of prime ideals of leavitt path algebras over arbitrary graphs given an arbitrary graph e and a field k the prime ideals as well as the primitive ideals of the leavitt path algebra lk e are completely characterized in terms of their generators the stratification of the prime spectrum of lk e is indicated with information on its individual stratum necessary and sufficient conditions are given on the graph e under which every prime ideal of the leavitt path algebra lk e is primitive leavitt path algebras with krull dimension zero are characterized and those with various prescribed krull dimension are constructed the minimal prime ideals of lk e are described in terms of the graphical properties of e and using this complete descriptions of the height one as well as the co height one prime ideals of lk e are given

stable norms from theory to applications and back abstract the purpose of this survey paper is to give a brief review of certain aspects of stability of norms and subnorms acting on algebras over a field f either r or c a norm n on an associative algebra a over f shall be called stable if for some positive constant n a m n a m for all a a m 1 2 3 a norm shall be called strongly stable if the above inequality holds with 1 we begin the paper by discussing several results regarding norm stability including conditions under which norms on certain algebras are stable the second part of the paper is devoted to applications where we employ the notion of norm stability to obtain criteria for the convergence of a well known family of finite difference schemes for the initial value problem associated with the parabolic system u x t t 1 j k s a jk 2 u x t x j x k 1 j s b j u x t x j cu x t where ajk bj and c are constant matrices ajk being hermitian the third and last part of the paper deals with the question of stability for subnorms acting on subsets of power associative algebras that are closed under scalar multiplication and under raising to powers a subnorm f on such a set s is a real valued function satisfying f a 0 for all 0 a s and f a f a for all a s and f

a representation of all solutions of the control algebraic riccati equation for infinite dimensional systems we obtain a representation of all self adjoint solutions of the control algebraic riccati equation associated to the infinite dimensional state linear system a b c under the following assumptions a generates a c0 group the system is output stabilizable strongly detectable and the dual riccati equation has an invertible self adjoint non negative solution