the h calculus and sums of closed operators we develop a very general operator valued functional calcu lus for operators with an h 1 calculus we then apply this to the joint functional calculus of two commuting sectorial operators when one has an h 1 calculus using this we prove theorem of dore venni type on sums of commuting sectorial operators and apply our results to the problem of lp maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of ba nach spaces are essential here in the final section we exploit the special banach space structure of l1 spaces and c k spaces to obtain some more detailed results in this setting

on the inverse of the sum of two sectorial operators abstract we study an abstract linear operator equation on a banach space by using the inverse of the sum of two sectorial operators we prove that the boundedness of a special type of operator valued h calculus is sufficient for maximal regularity of the solution we apply the result to the abstract parabolic problem to give a maximal l p regularity condition we also study the abstract hyperbolic problem and give a sufficient condition for the existence of solution

the h infty calculus and sums of closed operators we develop a very general operator valued functional calculus for operators with an h infty calculus we then apply this to the joint functional calculus of two sectorial operators when one has an h infty calculus using this we prove theorem of dore venni type on sums of sectorial operators and apply our results to the problem of l p maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of banach spaces are essential here in the final section we exploit the special banach space structure of l 1 spaces and c k spaces to obtain some more detailed results in this setting

a cramer rule for least norm solutions of consistent linear equations abstract the least euclidean norm solution of a consistent linear equation ax b is given a determinantal form which reduces to cramer s rule if a is nonsingular

a cramer rule for minimum norm t least squares s solution of inconsistent linear equations abstract in this paper the minimum norm t least squares s solution of inconsistent linear equations ax y a c m n r y r a is given a determinantal form and reduces to common cramer s rule if a is nonsingular

a note on the representation and approximation of the outer inverse at s 2 of a matrix a in this note we present a unified representation and an approximation theorem for the outer inverse a t s 2 of a matrix a which is the matrix x satisfying xax x r x t n x s based on the new approximation theorem we can develop several expressions and computational procedures for the generalized inverse a t s 2 which improve the recent work on this topic

a cramer rule for the least norm least squared error solution of inconsistent linear equations abstract a recent cramer rule for obtaining the least norm solution of a consistent system of linear equations ax b is shown to also provide the least squared error solution in the inconsistent case i e the solution a b where a is the moore penrose inverse of a

condition numbers for the outer inverse and constrained singular linear system in this paper we investigate the condition number of the outer inverse at s 2 and outer inverse at s 2 solution of constrained linear system ax b x t where a is a real m n matrix b and x are real vectors t is a subspace let and be two positive real numbers when we consider the weighted frobenius norm a b qp f on the data we get the formula of condition number for the outer inverse at s 2 solution of constrained linear system for the normwise condition number the sensitivity of the relative condition number itself is studied and the componentwise perturbation is also investigated

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

the swift hohenberg equation on conic manifolds abstract we consider the swift hohenberg equation on manifolds with conical singularities and show existence uniqueness and maximal regularity of the short time solution in terms of mellin sobolev spaces moreover we give a necessary and sufficient condition so that the above solution exists for all times space asymptotic expansion of the solution near the singularity is also provided and its relation to the local geometry is shown the same problem is considered on closed manifolds and similar results are obtained by using the above singular analysis theory

the perturbation theory for the drazin inverse and its applications abstract let a and e be n n matrices and b a e denote the drazin inverse of a by ad we give an upper bound for the relative error b d a d a d under certain circumstances an error bound for the solution for the singular equations ax b b r a is also considered

index splitting for the drazin inverse and the singular linear system abstract an index splitting of a singular matrix a for the drazin inverse and its relative iterations for the minimal p norm solution of the singular linear system ax b b r a k k ind a are presented some convergence criteria are also given and a method is developed for constructing the index splitting of matrix a

the iterative methods for computing the generalized inverse at s 2 of the bounded linear operator between banach spaces in this paper we present the iterative methods for computing the generalized inverse a t s 2 over banach spaces and also for computing the generalized drazin inverses of banach algebra elements moreover we estimate the error bounds of the iterative methods for approximating a t s 2

the 2 inverse with applications in statistics an inverse g of a given matrix a which satisfies the property gag g is known as a 2 inverse this paper presents a three phase inversion procedure for which the 2 inverse is a special case we present the geometry of 2 inverses and show that starting from 2 inverses various types of generalized inverses can be derived two examples of the occurrence of 2 inverses in statistics are given one concerning the constrained least squares estimator the other concerning a necessary and sufficient condition for a quadratic form of singular multivariate normal variates to follow a chi square distribution

iterative methods for solving restricted linear equations in this paper the iterative methods for computing some solution of the general restricted linear equations ax b x et are established and the iterative methods for finding the solution of several special linear equations considered are constructed

preserving closedness of operators under summation we give a sufficient condition for the sum of two closed operators to be closed in particular we study the sum of two sectorial operators with the sum of their sectoriality angles greater than we show that if one of the operators admits bounded h calculus and the resolvent of the other operator satisfies a boundedness condition stronger than the standard sectoriality but weaker than the bounded imaginary powers property in the case of umd spaces then the sum is closed we apply the result to the abstract parabolic problem and give a sufficient condition for lp maximal regularity

on the convergence of splittings for semidefinite linear systems abstract recently lee et al young ju lee jinbiao wu jinchao xu ludmil zikatanov on the convergence of iterative methods for semidefinite linear systems siam j matrix anal appl 28 2006 634 641 introduce new criteria for the semi convergence of general iterative methods for semidefinite linear systems based on matrix splitting the new conditions generalize the classical notion of p regularity introduced by keller h b keller on the solution of singular and semidefinite linear systems by iterations siam j numer anal 2 1965 281 290 in view of their results we consider here stipulations on a splitting a m n which lead to fixed point systems such that the iterative scheme converges to a weighted moore penrose solution to the system ax b our results extend the result of lee et al to a more general case and we also show that it requires less restrictions on the splittings than keller s p regularity condition to ensure the convergence of iterative scheme

a characterization and representation of the generalized inverse a 2 t s and its applications abstract this paper presents an explicit expression for the generalized inverse a 2 t s based on this we established the characterization the representation theorem and the limiting process for a 2 t s as an application we estimate the error bound of the iterative method for approximating a 2 t s

on the solution of singular linear systems of algebraic equations by semiiterative methods for a square matrixt n n where i t is possibly singular we investigate the solution of the linear fixed point problemx tx c by applying semiiterative methods sim s to the basic iterationx0 n xk tck 1 c k 1 such problems arise if one splits the coefficient matrix of a linear systemax b of algebraic equations according toa m n m nonsingular which leads tox m 1nx m 1b tx c even ifx tx c is consistent there are cases where the basic iteration fails to converge namely ift possesses eigenvalues 1 with 1 or if 1 is an eigenvalue oft with nonlinear elementary divisors in these cases and also ifx tx c is incompatible we derive necessary and sufficient conditions implying that a sim tends to a vector hat x which can be described in terms of the drazin inverse of i t we further give conditions under which hat x is a solution or a least squares solution of i t x c

operators with equal projections related to their generalized inverses in this article we characterize operators on banach spaces which have the same projections related to their outer or inner generalized inverses as corollaries we obtain well known results for the drazin inverse of bounded operators

perturbation bounds for the generalized inverses at s 2 with application to constrained linear system in this paper the more general perturbation problem of the generalized inverse a t s 2 is considered we derive its perturbation bounds when the subspaces t and s also have perturbation and in special case when t and s have no perturbation we get some results on the condition numbers with respect to a t s 2 and constrained linear system ax b x t which are similar to the regular inverse a 1 and nonsingular linear system ax b

on continuity of the generalized inverse at s 2 this paper studies the continuity of the generalized inverse a t s 2 since common important generalized inverses such as a a m n a d a a d w a l 1 and a l etc are all generalized inverse a t s 2 in fact we give a unified continuity of such important generalized inverses

a joint functional calculus for sectorial operators with commuting resolvents 1 introduction in this paper we are concerned with h 1 functional calculus in the sense of the construction introduced by mcintosh 23 and developed recently by cowling doust mcintosh and yagi 8 given v in 0 p we will use the notation sv fz 2 c n f0g jarg z j vg to denote the open sector of angle 2v around the half line r let x be a complex banach space given a linear operator a on x we denote by d a n a and r a the domain the kernel and the range of a respectively we also denote by j a the spectrum of a and by r a the resolvent set of a

displacement structure of the generalized inverse a t s 2 it is well known that matrices with a uv displacement structure possess generalized inverse with a vu displacement structure estimation for the displacement rank of a t s 2 u va t s 2 are presented where a t s 2 is the 2 inverse of a we extend the results due to heinig and hellinger in 1994

condition number related to the outer inverse of a complex matrix in this paper we obtain the formula for computing the condition number of a complex matrix which is related to the outer generalized inverse of a given matrix we use the schur decomposition of a matrix we characterize the spectral norm and the frobenius norm of the relative condition number of the generalized inverse and the level 2 condition number of the generalized inverse the sensitivity for the generalized drazin inverse solution of linear systems is presented we also present the structured perturbation of the generalized inverse

a cramer rule for finding the solution of a class of singular equations a cramer rule for finding the unique solution x r ak of singular equations ax b ind a k b r ak is given and reduces to the common cramer rule if a is a nonsingular

on the perturbation and subproper splittings for the generalized inverse at s 2 of rectangular matrix a abstract in this paper the perturbation and subproper splittings for the generalized inverse at s 2 the unique matrix x such that xax x r x t and n x s are considered we present lower and upper bounds for the perturbation of at s 2 convergence of subproper splittings for computing the special solution at s 2 b of restricted rectangular linear system ax b x t are studied for the solution at s 2 b we develop a characterization therefore we give a unified treatment of the related problems considered in literature by ben israel berman hanke neumann plemmons etc

multi agent consensus with relative state dependent measurement noises in this note the distributed consensus corrupted by relative state dependent measurement noises is considered each agent can mea sure or receive its neighbors state information with random noises whose intensity is a vector function of agents relative states by investigating the structure of this interaction and the tools of stochastic differential equa tions we develop several small consensus gain theorems to give sufficient conditions in terms of the control gain the number of agents and the noise intensity function to ensure mean square m s and almost sure a s consensus and quantify the convergence rate and the steady state error especially for the case with homogeneous communication and control channels a necessary and sufficient condition to ensure m s consensus on the control gain is given and it is shown that the control gain is independent of the specific network topology but only depends on the number of nodes and the noise coefficient constant for symmetric measurement models the almost sure convergence rate is estimated by the iterated logarithm law of brownian motions index terms distributed consensus distributed coordination fading channel measurement noises multi agent system

on the perturbation of the group inverse and oblique projection for given matrices a and e which are n matrices and matrices and e we discuss upper bounds for b bb b a a and bb aa aa also a lower bound for b is derived

on the p q outer generalized inverse in banach algebra abstract in this note the p q outer generalized inverse in banach algebra is considered a new representation based on the 1 5 inverse is presented also we construct an iterative method for computing the p q outer generalized inverse in banach algebra

iterative methods for computing the generalized inverses a 2 t s of a matrix a abstract in this paper the iterative methods for computing the generalized inverse a t s 2 the unique matrix x such that xax x r x t and n x s are established and the iterative methods for computing many important generalized inverses of matrices such as a a mn a d a a d w a l 1 a l and ap l are constructed

on the closedness of the sum of two closed operators o t t 1 where e g a is the infinitesimal generator of an analytic semigroup in a complex banach space x we find that in general it is false that for every fe lp 0 t x problem 1 has a solution uewi p o t x c lp o t a usually it is necessary to impose some further regularity conditions on f consider for instance the way in which problem 1 is studied in the paper 8 and compare th 3 7 and 3 14 which are applied in w 4 of the same paper to solve problem 1 we find that in general we have only strong solutions while in the hilbert space setting we have strict solutions this corresponds to the fact that in general the operator a b where b u u is only closable while in a hilbert space it is closed this remark suggests that the better result may depend on the nicer geometric properties of a hilbert space with respect to a general banach space and it is natural to ask which conditions should be imposed on the banach space to force the sum of two suitable closed opera tors to be closed the above considerations motivate the manifold aim of the present paper a to find conditions on a banach space x and on closed linear operators a and b acting in x which ensure that a b is closed b to apply the result of a to problem 1 it seems that the convenient property for the banach space x is convexity or otherwise stated the umd condition for information about these equiva lent conditions we refer to 4 in which it is also proved that if x is convex and l p oe then the hilbert transform is continuous from lp r x into lp r x on the other hand it is also known see 2 that if there exists pe 1 oe such that the hilbert transform is continuous from lp r x into itself

preconditionings and splittings for rectangular systems recently eiermann marek and niethammer have shown how to applysemiiterative methods to a fixed point systemx tx c which isinconsistent or in whichthe powers of the fixed point operator t have no limit to obtain iterative methods which converge to some approximate solution to the fixed point system in view of their results we consider here stipulations on apreconditioning qax qb of the systemax b and separately on asplitting a m n which lead to fixed point systems such that with the aid of a semiiterative method the iterative scheme converges to a weighted moore penrose solution to the systemax b we show in several ways that to obtain a meaningful limit point from a semiiterative method requires less restrictions on the splittings or the reconditionings than those which have been required in the classical picard iterative method see e g the works of berman and plemmons berman and neumann and tanabe we pay special attention to the case when the weighted moore penrose solution which is sought is the minimal norm least squares solution toax b

generalized inverse a 2 t s and a rank equation in this paper the rank equationrankabcx rank a is considered in the sense that b and c are characterized when a 2 t s is solution to this equation as the special cases similar results for a a m n a d and a g are also derived this contributes to certain recent results in the literature including that obtained by grosz linear algebra appl 289 1999 127 and thome and wei appl math comput 141 2003 471

a class of numerical algorithms for computing outer inverses the aim of this study is to present a class of numerical algorithms for finding outer inverses with prescribed range and null space general convergence theorems are proven some particular cases of the general algorithm are introduced and demonstrated theoretical convergence speed and computational efficiency will further be supported by employing some numerical tests