exact solution of a 1d asymmetric exclusion model using a matrix formulation several recent works have shown that the one dimensional fully asymmetric exclusion model which describes a system of particles hopping in a preferred direction with hard core interactions can be solved exactly in the case of open boundaries here the authors present a new approach based on representing the weights of each configuration in the steady state as a product of noncommuting matrices with this approach the whole solution of the problem is reduced to finding two matrices and two vectors which satisfy very simple algebraic rules they obtain several explicit forms for these non commuting matrices which are in the general case infinite dimensional their approach allows exact expressions to be derived for the current and density profiles finally they discuss briefly two possible generalizations of their results the problem of partially asymmetric exclusion and the case of a mixture of two kinds of particles

limits of hypercyclic and supercyclic operators abstract an operator t on a complex separable infinite dimensional hilbert space h is hypercyclic if there is a vector y in h such that the orbit y ty t2y is dense in h it is shown that if t is hypercyclic then t does not have any eigenvalue and that the union of the spectrum of t and the unit circle is a connected set this result is used to obtain a spectral characterization of the norm closure of the class hc h of all hypercyclic operators acting on h as well as a formula for the distance from a given operator to hc h analogous results are obtained for the closely related class sc h of all supercyclic operators this paper includes some information about the structure of the set of all hypercyclic supercyclic vectors of a given hypercyclic supercyclic resp operator t

the point residual and continuous spectrum of an upper triangular operator matrix abstract in this paper for given operators a b h and b b k we completely describe the set of all c b k h such that m c is injective r m c is not dense in h k respectively and describe residual and continuous spectrum of the upper triangular operator matrix m c

a note on browder spectrum of operator matrices abstract when a b h and b b k are given we denote by m c the operator acting on the hilbert space h k of the form m c a c 0 b in this note it is shown that the following results in hai yan zhang hong ke du browder spectra of upper triangular operator matrices j math anal appl 323 2006 700 707 w 3 a b c w 1 a b c in line 17 on p 705 and c b k h b m c c b k h m c b a b b are not always true although the authors tried to fill the gap in their proofs by proposing an additional condition in h y zhang h k du corrigendum to browder spectra of upper triangular operator matrices j math anal appl 323 2006 700 707 j math anal appl 337 2007 751 752 a counterexample is given and then we show that under one of the following conditions i s u b b ii int p b iii a b iv a a a we have c b k h b m c l e a r e b w a b d a d b where w a b c n b and h r a are not isomorphic up to a finite dimensional subspace

browder spectra of upper triangular operator matrices abstract let m c a c 0 b be a 2 2 upper triangular operator matrix acting on the hilbert space h k in this paper for given operators a and b we prove that c b k h b m c c b k h m c b a b b where b t c b t denotes the browder resolvent of an operator t and c b k h m c has been determined in h k du p jin perturbation of spectrums of 2 2 operator matrices proc amer math soc 121 1994 761 776 moreover we explore the relations of a b m c b a b b b m c and w a w b w m c where a b a and w a denote the spectrum the browder spectrum and the weyl spectrum of a respectively

completions of upper triangular matrices to kato nonsingularity abstract in this paper we consider the problem of completion of the upper triangular operator matrix a 0 b where a b h and b b k to kato nonsingular operators and completely solve it in each of the following cases one of the operators a or b is kato nonsingular b is injective a is with dense range b is with finite ascent a is with finite descent 0 int p b 0 int c p a in particular the results generalize and complete those of 3 and 7 that concern the same problem

an analogue of hincin s characterization of infinite divisibility for operator valued free probability abstract let b be a finite separable von neumann algebra we prove that a b valued distribution that is the weak limit of an infinitesimal array is infinitely divisible the proof of this theorem utilizes the steinitz lemma and may be adapted to provide a nonstandard proof of this type of theorem for various other probabilistic categories we also develop weak topologies for this theory and prove the corresponding compactness and convergence results

drazin spectrum of operator matrices on the banach space abstract when a b x and b b y are given we denote by m c the operator acting on the banach space x y of the form m c a c 0 b in this paper it is concluded and proved that for a given pair a b of operators d a d b d m c w holds for every c b y x where w is the union of certain holes in d m c which happen to be subsets of d a d b moreover the set c b y x d m c is investigated and an example for it is considered

algebraic bethe ansatz for the six vertex model with upper triangular k matrices we consider a formulation of the algebraic bethe ansatz for the six vertex model with non diagonal open boundaries specifically we study the case where both left and right k matrices have an upper triangular form we show that the main difficulty entailed by those forms of the k matrices is the construction of the excited states however it is possible to treat this problem with the aid of an auxiliary transfer matrix and by means of a generalized creation operator

bethe ansatz and q operator for the open asep in this paper we look at the asymmetric simple exclusion process with open boundaries with a current counting deformation we construct a two parameter family of transfer matrices which commute with the deformed markov matrix of the system we show that these transfer matrices can be factorized into two commuting matrices with one parameter each which can be identified with baxter s q operator and that for certain values of the product of those parameters they decompose into a sum of two commuting matrices one of which is the usual one parameter transfer matrix for a given dimension of the auxiliary space using this we find the t q equation for the open asep and through functional bethe ansatz techniques we obtain an exact expression for the dominant eigenvalue of the deformed markov matrix

stable laws and domains of attraction in free probability theory in this paper we determine the distributional behavior of sums of free in the sense of voiculescu identically distributed inflnitesimal random variables the theory is shown to parallel the classical theory of independent random variables though the limit laws are usually quite difierent our work subsumes all previously known instances of weak convergence of sums of free identically distributed random variables in particular we determine the domains of attraction of stable distributions in the free theory these freely stable distributions are studied in detail in the appendix where their unimodality and duality properties are demonstrated

slavnov and gaudin korepin formulas for models without u 1 symmetry the xxx chain on the segment we consider the isotropic spin heisenberg chain with the most general integrable boundaries the scalar product between the on shell bethe vector and its off shell dual obtained by means of the modified algebraic bethe ansatz is given by a modified slavnov formula the corresponding gaudin korepin formula i e the square of the norm is also obtained

browder spectra for upper triangular operator matrices abstract when a b h and b b k are given we denote by m c the operator acting on the infinite dimensional separable hilbert space h k of the form m c a c 0 b in this paper we prove that c b k h b m c a b a a b b c n a i n b i d a i d b i where b t a b t n t d t and t denote the browder spectrum browder essential approximate point spectrum nullity deficiency and conjugate of t respectively some related results are obtained

operators which do not have the single valued extension property abstract in this paper we shall consider the relationships between a local version of the single valued extension property of a bounded operator t l x on a banach space x and some quantities associated with t which play an important role in fredholm theory in particular we shall consider some conditions for which t does not have the single valued extension property at a point o c

browder essential approximate point spectra and hypercyclicity for operator matrices abstract when a b h and b b k are given we denote by m c the operator acting on the infinite dimensional separable hilbert space h k of the form m c a c 0 b in this paper it is shown that if a is upper semi fredholm of finite ascent and infinite codimension and if r b is closed of infinite kernel then m c is upper semi fredholm of finite ascent for some c b k h in addition we explore the hypercyclicity and supercyclicity for 2 2 upper triangular operator matrices on the hilbert space

matrix coordinate bethe ansatz applications to xxz and asep models we present the construction of the full set of eigenvectors of the open asymmetric simple exclusion process asep and xxz models with special constraints on the boundaries the method combines both recent constructions of coordinate bethe ansatz and the old method of matrix ansatz specific to the asep this matrix coordinate bethe ansatz can be viewed as a non commutative coordinate bethe ansatz the non commutative part being related to the algebra appearing in the matrix ansatz

boundary conditions for integrable quantum systems a new class of boundary conditions is described for quantum systems integrable by means of the quantum inverse scattering r matrix method the method proposed allows the author to treat open quantum chains with appropriate boundary terms in the hamiltonian the general considerations are applied to the xxz and xyz models the nonlinear schrodinger equation and toda chain

on the perturbations of spectra of upper triangular operator matrices abstract in this paper we investigate perturbations of the regular spectrum of an upper triangular operator matrix such as m c a c 0 b acting on a hilbert space h k

the representation and approximation of the drazin inverse of a linear operator in hilbert space we present a unified representation theorem for the drazin inverse of linear operators in hilbert space and a general error bound five specific expressions computational procedures and their error bounds for the drazin inverse are uniformly derived from the unified representation theorem

partially asymmetric exclusion process with open boundaries exclusive diffusion on a one dimensional lattice is studied in the model particles hop stochastically in both directions but with different rates at the ends of the lattice particles are injected and removed the exact stationary probability measure is represented in the form of a matrix product as a generalization of the solution given by derrida et al j phys a 26 1493 1993 for the fully asymmetric process the phase diagram of the current on the infinite lattice is obtained analytic expressions for the current in the different phases are derived the model is equivalent to an xxz heisenberg chain with a certain type of boundary terms the ground state of which corresponds to the stationary solution of the master equation

the intersection of essential approximate point spectra of operator matrices abstract when a b h and b b k are given we denote by m c the operator acting on the infinite dimensional separable hilbert space h k of the form m c a c 0 b in this paper it is shown that there exists some operator c b k h such that m c is upper semi fredholm and ind m c 0 if and only if there exists some left invertible operator c b k h such that m c is upper semi fredholm and ind m c 0 a necessary and sufficient condition for m c to be upper semi fredholm and ind m c 0 for some c inv k h is given where inv k h denotes the set of all the invertible operators of b k h in addition we give a necessary and sufficient condition for m c to be upper semi fredholm and ind m c 0 for all c inv k h

an exact formula for the statistics of the current in the tasep with open boundaries we study the totally asymmetric exclusion process on a finite one dimensional lattice with open boundaries i e in contact with two reservoirs at different potentials the total time integrated current through the system is a random variable that scales linearly with time in the long time limit we conjecture a parametric representation for the generating function of the cumulants of the current which is related to the large deviation function by laplace transform this formula is valid for all system sizes and for all values of the boundary coupling parameters

inhomogeneous discrete time exclusion processes we study discrete time markov processes with periodic or open boundaryconditions and with inhomogeneous rates in the bulk the markov matrices aregiven by the inhomogeneous transfer matrices introduced previously to prove theintegrability of quantum spin chains we show that these processes have asimple graphical interpretation and correspond to a sequential update wecompute their stationary state using a matrix ansatz and express theirnormalization factors as schur polynomials a connection between bethe rootsand lee yang zeros is also pointed out

property b and property w for operator matrices abstract property w holds for t b x precisely when a t e a t 00 t by comparison property b holds for t precisely when a t e a t p 00 t in this paper we explore how property w and property b survive for 2 2 operator matrices on the banach space through local spectral theory also we consider property w for the 2 2 skew diagonal operator matrix of the form 0 a b 0 on the hilbert space

some results on fredholmness and boundedness below of an upper triangular operator matrix abstract in this paper for given operators a b h and b b k we completely describe the set of all c b k h such that m c is bounded below left right semi fredholm respectively

characterizations of perturbations of spectra of 2 2 upper triangular operator matrices abstract when a b h and b b k are given we denote by m c the operator acting on the infinite dimensional separable hilbert space h k of the form m c a c 0 b in this paper we first give some necessary and sufficient conditions for m c to be a left invertible operator an upper semi weyl upper semi fredholm operator for some c b k h which extend the corresponding results in cao et al 2006 4 cao and meng 2005 5 hwang and lee 2001 12 and li and du 2006 15 then we present some counter examples

weyl s theorem for upper triangular operator matrices abstract let ab t c t i is not an upper semi fredholm operator with finite ascent be the browder essential approximate point spectrum of t b h and let d t c t i is not surjective be the surjective spectrum of t in this paper it is shown that if m c a c 0 b is a 2 2 upper triangular operator matrix acting on the hilbert space h k then the passage from ab a ab b to ab m c is accomplished by removing certain open subsets of d a ab b from the former that is there is equality ab a ab b ab m c g where g is the union of certain of the holes in ab m c which happen to be subsets of d a ab b weyl s theorem and browder s theorem are liable to fail for 2 2 operator matrices in this paper it also explores how weyl s theorem browder s theorem a weyl s theorem and a browder s theorem survive for 2 2 upper triangular operator matrices on the hilbert space

an analogue to a result of takahashi ii abstract in this paper we address the question of the existence of an operator x such that a c x is of various appropriate types a closed range right invertible left invertible or a fredholm operator we point out to a connection between the existence of such x and the problem of completion of operator matrices of type a c b b h h and present necessary and sufficient conditions for the operator matrix a c b b h h to be completed to a fredholm operator also under certain conditions we solve the corresponding completion problem to a closed range operator

free products of completely positive maps and spectral sets let f be the free group on n generators and for each word w e f let 1 wi be its length haagerup lo showed that for each 0 r 1 the function h wp r is a positive definite function on f consider the positive definite function q n rinl on h then the free product function cpi cpr on ff z h coincides with h and the result was extended this way by de michele and fig talamanca 7 and by boiejko 4 51 in s it is proved that the free product of the unital positive defined functions ui gi z z is still positive defined on the free product group gi the correspondence between the positive defined functions on a discrete group g and the completely positive maps on the full c algebra c g and the isomorphism c g g2 n c g z c g suggested we consider amalgamated products of unital linear maps on the amalgamated product of a family of unital c algebras over a c subalgebra in section 3 we prove that the amalgamated product of a family of unital completely positive b bimodule maps qi a c is completely positive on the biggest amalgamated free product i a i as an application of our main result in section 2 we obtain some results concerning the dilation of noncommutative families of operators

essential approximate point spectra and weyl s theorem for operator matrices abstract when a b h and b b k are given we denote by m c the operator acting on the infinite dimensional separable hilbert space h k of the form m c a c 0 b in this paper it is shown that a 2 2 operator matrix m c is upper semi fredholm and ind m c 0 for some c b k h if and only if a is upper semi fredholm and n b and n a n b d a d b or n b d a if r b is closed d a if r b is not closed we also give the necessary and sufficient conditions for which m c is weyl or m c is lower semi fredholm with nonnegative index for some c b k h in addition we explore how weyl s theorem browder s theorem a weyl s theorem and a browder s theorem survive for 2 2 upper triangular operator matrices on the hilbert space

matrix product solution to a 2 species tasep with open integrable boundaries we present an explicit representation for the matrix product ansatz for sometwo species tasep with open boundary conditions the construction relies on theintegrability of the models a property that constrains the possible rates atthe boundaries the realisation is built on a tensor product of copies of thedehp algebras using this explicit construction we are able to calculate thepartition function of the models the densities and currents in the stationarystate are also computed it leads to the phase diagram of the models dependingon the values of the boundary rates we obtain for each species shock waves maximal current or low high densities phases

quasitriangular small compact strongly irreducible let t be a bounded linear operator acting on a separable infinite dimensional hilbert space let be a positive number in this article we prove that the perturbation of t by a compact operator k with k 0 be given is there a compact operator k with k such that t k is strongly irreducible

the intersection of left right spectra of 2 2 upper triangular operator matrices when a b h and b b k are given we denote by mc the operator matrix acting on the infinite dimensional separable hilbert space h k of the form mc ac0b in this paper for given a and b the sets c bl k h l mc c inv k h l mc and c inv k h l mc are determined where l t bl k h and inv k h denote respectively the left spectrum of an operator t the set of all the left invertible operators and the set of all the invertible operators from k into h

essential point spectra of operator matrices trough local spectral theory abstract for a b x b b y and c b y x let m c be the operator defined on x y by a c 0 b in this paper we study defect set a b m c where is the browder spectrum the essential approximate point spectrum and browder essential approximate point spectrum we then give application for weyl s and browder s theorems

bethe ansatz solution of the open xxz chain with nondiagonal boundary terms we consider the integrable open xx quantum spin chain with non diagonal boundary terms we derive an exact inversion identity by which we obtain the eigenvalues of the transfer matrix and the bethe ansatz equations for generic values of the boundary parameters the bethe ansatz solution is formulated in terms of the jacobian elliptic functions

another note on weyl s theorem weyl s theorem holds for an operator t on a banach space x when the complement in the spectrum of the weyl spectrum coincides with the isolated points of spectrum which are eigenvalues of finite multiplicity this is close to but not quite the same as equality between the weyl spectrum and the browder spectrum which in turn ought to but does not guarantee the spectral mapping theorem for the weyl spectrum of polynomials in t in this note we try to explore these distinctions recall 2 4 6 that a bounded linear operator t e bl x x on a banach space x is fredholm if t x is closed and both t 1 0 and x cl tx are finite dimensional in this case we define the index of t by index t dim t 1 0 dim x t x an operator t e bl x x is called weyl if it is fredholm of index zero and is called browder if it is fredholm of finite ascent and descent equivalently 6 theorem 7 9 3 if t is fredholm and t ai is invertible for sufficiently small a 0 in c the fredholm essential spectrum uess t the weyl spectrum wess t and the browder spectrum wcmm t of t are defined by 0 1 jess t a e c t ai is not fredholm 0 2 wess t a e c t ai is not weyl and 0 3 wecssmm t e c tai is not browder evidently 0 4 jess t c wess t c wcomm t cess t u acc v t where we write acc k for the accumulation points of k c c and av t for the usual spectrum of t if we write iso k k acc k and 0 5 leftr t a e iso 5 t 0 dim tai 1 0 xo for the isolated eigenvalues of finite multiplicity and 6 9 8 3 4 0 6 troo t a5 t wcomm t for the riesz points of t then 6 theorem 9 8 4 with the help of the punctured neighbourhood theorem 0 7 iso v t jess t iso 5 t wess t troo t c 4left t received by the editors december 18 1995 1991 mathematics subject classification primary 47a10

nonequilibrium steady states of matrix product form a solver s guide we consider the general problem of determining the steady state of stochastic nonequilibrium systems such as those that have been used to model among other things biological transport and traffic flow we begin with a broad overview of this class of driven diffusive systems which includes exclusion processes focusing on interesting physical properties such as shocks and phase transitions we then turn our attention specifically to those models for which the exact distribution of microstates in the steady state can be expressed in a matrix product form in addition to a gentle introduction to this matrix product approach how it works and how it relates to similar constructions that arise in other physical contexts we present a unified pedagogical account of the various means by which the statistical mechanical calculations of macroscopic physical quantities are actually performed we also review a number of more advanced topics including nonequilibrium free energy functionals the classification of exclusion processes involving multiple particle species existence proofs of a matrix product state for a given model and more complicated variants of the matrix product state that allow various types of parallel dynamics to be handled we conclude with a brief discussion of open problems for future research

addition of certain non commuting random variables soit g le groupe libre non commutatif sur deux generateurs et soit u j j 1 2 les unites dans l 2 g correspondant a la translation a gauche par les generateurs et on considere l 2 g la fonction g g e soient x j des operateurs de la forme x j j u j les operateurs x j peuvent etre consideres comme des variables aleatoires de moments ou de facon equivalente avec des distributions donnees par les fonctionnelles analytiques j ou j f avec ces conventions la distribution de x 1 x 2 depend seulement des distributions de x 1 et x 2 on explique cette relation

a bispectral q hypergeometric basis for a class of quantum integrable models for the class of quantum integrable models generated from the q onsager algebra a basis of bispectral multivariable q orthogonal polynomials is exhibited in the first part it is shown that the multivariable askey wilson polynomials with n variables and n 3 parameters introduced by gasper and rahman dev math 13 209 2005 generate a family of infinite dimensional modules for the q onsager algebra whose fundamental generators are realized in terms of the multivariable q difference and difference operators proposed by iliev trans am math soc 363 1577 2011 raising and lowering operators extending those of sahi sigma 3 002 2007 are also constructed in the second part finite dimensional modules are constructed and studied for a certain class of parameters and if the n variables belong to a discrete support in this case the bispectral property finds a natural interpretation within the framework of tridiagonal pairs in the third part eigenfunctions of the q dolan grady hierarchy are considered in the polynomial basis in particular invariant subspaces are identified for certain conditions generalizing nepomechie s relations in the fourth part the analysis is extended to the special case q 1 this framework provides a q hypergeometric formulation of quantum integrable models such as the open xxz spin chain with generic integrable boundary conditions q 1 for the class of quantum integrable models generated from the q onsager algebra a basis of bispectral multivariable q orthogonal polynomials is exhibited in the first part it is shown that the multivariable askey wilson polynomials with n variables and n 3 parameters introduced by gasper and rahman dev math 13 209 2005 generate a family of infinite dimensional modules for the q onsager algebra whose fundamental generators are realized in terms of the multivariable q difference and difference operators proposed by iliev trans am math soc 363 1577 2011 raising and lowering operators extending those of sahi sigma 3 002 2007 are also constructed in the second part finite dimensional modules are constructed and studied for a certain

consistent invertibility of upper triangular operator matrices abstract let h and k be separable infinite dimensional hilbert spaces and let a b h and b b k be given operators a necessary and sufficient condition is obtained for a c 0 b to be a ci operator for some respectively every c b k h furthermore some related results are obtained

weyl type theorems of 2 2 upper triangular operator matrices abstract let m c a c 0 b be an upper triangular operator matrix on the hilbert space h h where h is a hilbert space in this paper necessary and sufficient conditions for m c a b are obtained where is the essential spectrum the weyl spectrum the browder spectrum the essential approximate point spectrum and the browder essential approximate point spectrum for m c we further investigate the relationship among weyl type theorems such as weyl s a weyl s browder s and a browder s theorems

algebraic bethe ansatz for the totally asymmetric simple exclusion process with boundaries we study the one dimensional totally asymmetric simple exclusion process in contact with two reservoirs including also a fugacity at one boundary the eigenvectors and the eigenvalues of the corresponding markov matrix are computed using the modified algebraic bethe ansatz a method introduced recently to study the spin chain with non diagonal boundaries we provide in this case a proof of this method

the inverse of any two by two nonsingular partitioned matrix and three matrix inverse completion problems a formula for the inverse of any nonsingular matrix partitioned into two by two blocks is derived through a decomposition of the matrix itself and generalized inverses of the submatrices in the matrix the formula is then applied to three matrix inverse completion problems to obtain their complete solutions

self adjoint perturbations of spectra for upper triangular operator matrices abstract this paper is concerned with the self adjoint perturbations of the spectra for the upper triangular partial operator matrix with given diagonal entries a necessary and sufficient condition is given under which such operator matrix admits a weyl fredholm operator completion by choosing some bounded self adjoint operator it is shown that the self adjoint perturbation of the weyl essential spectrum can be the proper set of the general perturbation combining the spectral properties we further characterize the perturbation of the weyl essential spectrum for hamiltonian operators

the semi fredholmness of 2 2 operator matrices abstract let h 1 and h 2 be separable infinite dimensional hilbert spaces and let a b h 1 b b h 2 and c b h 2 h 1 be given operators a necessary and sufficient condition is obtained for a c x b to be a right left fredholm operator for some x b h 1 h 2

analytic subordination for bi free convolution in this paper we study some analytic properties of bi free additive convolution both scalar and operator valued we show that using properties of voiculescu s subordination functions associated to free additive convolution of operator valued distributions simpler formulas for bi free convolutions can be derived we use these formulas in order to prove a result about atoms of bi free additive convolutions

open two species exclusion processes with integrable boundaries we give a complete classification of integrable markovian boundary conditions for the asymmetric simple exclusion process with two species or classes of particles some of these boundary conditions lead to non vanishing particle currents for each species we explain how the stationary state of all these models can be expressed in a matrix product form starting from two key components the zamolodchikov faddeev and ghoshal zamolodchikov relations this statement is illustrated by studying in detail a specific example for which the matrix ansatz involving 9 generators is explicitly constructed and physical observables such as currents densities calculated

non commutative functions and the non commutative free levy hincin formula abstract the paper is discussing infinite divisibility in the setting of operator valued boolean free and more general c free independences particularly using hilbert bimodule and non commutative function techniques we obtain analogues of the levy hincin integral representation for infinitely divisible real measures

continuity of spectrum and approximate point spectrum on operator matrices abstract let x and y be given banach spaces for a b x b b y and c b y x let m c be the operator defined on x y by m c a c 0 b in this paper we give conditions for continuity of at m c through continuity of at a and b where can be equal to the spectrum or approximate point spectrum

applications of realizations aka linearizations to free probability abstract we show how the combination of new linearization ideas in free probability theory with the powerful realization machinery developed over the last 50 years in fields including systems engineering and automata theory allows solving the problem of determining the eigenvalue distribution or even the brown measure in the non selfadjoint case of noncommutative rational functions of random matrices when their size tends to infinity along the way we extend evaluations of noncommutative rational expressions from matrices to stably finite algebras e g type ii 1 von neumann algebras with a precise control of the domains of the rational expressions the paper provides sufficient background information with the intention that it should be accessible both to functional analysts and to algebraists

infinite divisibility and a non commutative boolean to free bercovici pata bijection abstract we use the theory of fully matricial or non commutative functions to investigate infinite divisibility and limit theorems in operator valued non commutative probability our main result is an operator valued analogue for the bercovici pata bijection an important tool is voiculescu s subordination property for operator valued free convolution

perturbation of spectra of operator matrices and local spectral theory abstract for a l x b l y and c l y x we denote by m c the operator defined on x y by a c 0 b in this article we study defect set d a b m c for different spectra including the spectrum the essential spectrum weyl spectrum and the approximate point spectrum we then apply the obtained results to the stability of such spectra d and the classes of operators c for which stability holds of m c using local spectral theory

eigenvalues of non hermitian random matrices and brown measure of non normal operators hermitian reduction and linearization method abstract we study the brown measure of certain non hermitian operators arising from voiculescu s free probability theory usually those operators appear as the limit in moments of certain ensembles of non hermitian random matrices and the brown measure gives then a canonical candidate for the limit eigenvalue distribution of the random matrices a prominent class for our operators is given by polynomials in free variables other explicit examples include r diagonal elements and elliptic elements for which the brown measure was already known and a new class of triangular elliptic elements our method for the calculation of the brown measure is based on a rigorous mathematical treatment of the hermitian reduction method as considered in the physical literature combined with subordination and the linearization trick