factorization in lorentz spaces with an application to centralizers abstract we give an explicit formula to factorize functions in the lorentz spaces l p q some applications to centralizers twisted sums and interpolation theory are included

on the numerical computation of a structural decomposition in systems and control in this paper we develop a new numerical method for a special coordinate basis of a linear time invariant system such a special coordinate basis is essentially a structural decomposition which explicitly displays the finite and infinite zero structures as well as the invertibility structures of the given system the technique is playing important roles in numerous topics in system and control theory such as robust control h sub spl infin and h sub 2 optimal control almost disturbance decoupling and zero placement of linear systems just to name a few our method consists of three steps reduction by orthogonal transformations reduction by generalized sylvester equations and extraction of infinite zero structure the performance of our method is illustrated by some numerical examples

on blocking zeros in linear multivariable systems in this note the concept of a blocking zero is introduced and the relationship between blocking zeros and other zeros transmission invariant system is discussed it is shown that the existence of an uncontrollable and or unobservable mode in a given state space description of a multiinput multioutput system is equivalent to a pole blocking zero cancellation this can be regarded as an extension of some results which are well known for single input and or single output systems but cannot be extended to multiinput multioutput systems using the existing types of zeros

special coordinate basis for multivariable linear systems finite and infinite zero structure squaring down and decoupling abstract a special coordinate basis for multivariable linear systems is introduced here several fundamental properties of linear systems regarding controllability stabiliza bility observability detectability invariant zeros decoupling zeros infinite zero structure effect of feedback on zero structure squaring down diagonal and triangular decoupling etc can be directly displayed in terms of the special coordinate basis moreover connections between the special coordinate basis and the important invariant subspaces of the geometric theory of linear systems are established

the mackey topology and complemented subspaces of lorentz sequence spaces d w p for 0 p 1 in this paper we continue the study of lorentz sequence spaces d w p 0 p 1 moreover the sequence of unit vectors e is a symmetric basis of

the numerical range of 3 3 matrices abstract let a be an n n complex matrix then the numerical range of a w a is defined to be x ax x c n x x 1 in this article a series of tests is given allowing one to determine the shape of w a for 3 3 matrices reconstruction of a up to unitary similarity from w a is also examined

numerical ranges and gersgorin discs abstract for a complex matrix a a ij i j 1 n let w a be its numerical range and let g a be the convex hull of i 1 n z c z a ii i j a ij a ji 2 and g a g u au u n by n unitary it is known that w a is always contained in g a and hence in g a in this paper we consider conditions for w a to be equal to g a or g a we show that if w a g a then the boundary of w a consists only of circular arcs and line segments if moreover a is unitarily irreducible then w a is a circular disc almost complete characterizations of 2 by 2 and 3 by 3 matrices a for which w a g a are obtained we also give criteria for the equality of w a and g a in particular such a s among the permutationally irreducible ones must have even sizes we also characterize those a s with size 2 or 4 which satisfy w a g a

on flat portions on the boundary of the numerical range abstract the paper is devoted to matrices with flat portions on the boundary of their numerical range a constructive criterion for such portions to exist is obtained in case of tridiagonal matrices and a particular case of continuant matrices is considered as an application the cases of arbitrary 3 3 and 4 4 matrices are treated it is shown in particular that the sharp bound for the number of flat portions on the boundary of the numerical range for 4 4 matrices is four three if the matrices are assumed unitarily irreducible

basic sequences and subspaces in lorentz sequence spaces without local convexity after some preliminary results 1 we give in 2 another proof of the result of n j kalton 5 concerning the unicity of the unconditional bases of ip 0 1 there are subspaces of d w p 0 p in fact there are spaces d w p 0 p 1 which contain no complemented subspaces linearly homeomorphic to ip we conjecture that this is true for every d w p 0 p 1 the answer to the previous question seems to be important for example we can prove that a positive complemented sublattice e of d w p 0 p 1 with a symmetric basis is linearly homeomorphic either to ip or to d w p consequently a positive answer to this question implies that e is linearly homeomorphic to d w p in 5 we are able to characterise the sublattices of d w p p k however under a supplementary restriction concerning the sequence w l which are positive and contractive complemented as being the order ideals of d w p finally in 6 we characterise the mackey completion of d w p also in the case p k k e n 1 preliminary results let x be a real linear space and 0 p 1 a function denoted by 11 11 defined on x with the values in r is called a p norm or briefly a norm if the following conditions are verified 1 llxll 0 if and only if x 0 2 iiaxi i iaipiixi i forx e x and a e r 3 lix y 1 s lx 1 iiyii for x y e x then the subsets un x e x ilxii s n for n e n constitute a fundamental system of neighbourhoods of zero for a metric linear topology of x if x is complete with respect to this topology we say that x is ap banach space a sequence xn i in x is called a basis if for every x e x there is a unique sequence of scalars an i such that x i aixi the following lemma is essentially known see theorem 111 6 1 theorem 6 5 of 10 lemma 1 1 let xi i be a sequence in x the following assertions are equivalent 1 the series x converges for every permutation t of the integers received by the editors september 13 1979 and in revised form december 4 1979 a ms mos subject classifications 1 970 primary 46a45 46a 10 secondary 46a35

necessary and sufficient conditions for a nonminimum phase plant to have a recoverable target loop a stable compensator design for ltr abstract in connection with loop transfer recovery of nonminimum phase systems the purpose of this paper is two fold 1 to study the set of recoverable target loops and to establish necessary or and sufficient conditions for a given plant to have at least one recoverable target loop and 2 to show that the compensator structure developed earlier in chen saberi and sannuti 1991 automatica 27 257 280 for minimum phase systems can also recover any recoverable target loop for nonminimum phase systems as well while retaining all its advantages over conventional observer based controllers

circularity of the numerical range abstract an equivalent condition on a 3 square complex or a 4 square real upper triangular matrix is found for its numerical range to be a circular disk centered at the origin sufficient conditions for the circularity of the numerical range of certain sparse matrices are also given in terms of graphs

on blocking zeros and strong stabilizability of linear multivariable systems abstract motivated by a crucial role blocking zeros play in deciphering the strong stabilizability of a given system a careful study of blocking zeros is undertaken here after developing certain properties of blocking zeros and based on the multiplicity structure of invariant zeros we identify what kind of invariant zeros are blocking zeros for controllable and observable systems an invariant zero is a blocking zero if and only if its geometric multiplicity is equal to the normal rank of the transfer function of the given system this result leads to delineation of the class of controllable and observable time invariant linear systems into two subclasses 1 simply siso systems whose normal rank is unity and 2 truly mimo systems whose normal rank is greater than unity in a simply siso system every invariant zero is a blocking zero and hence a simply siso system is not necessarily strongly stabilizable on the other hand a truly mimo system with distinct invariant zeros does not have any blocking zeros and hence is always strongly stabilizable also given any truly mimo system there always exists an arbitrarily small perturbation of its dynamic matrix such that the perturbed system has no blocking zeros and hence is strongly stabilizable in this sense one can say that a mimo system almost always has no blocking zeros and hence is almost always strongly stabilizable

comments on on blocking zeros in linear multivariable systems with reply reference is made to a recently published correspondence by p g ferreira in which concern was expressed about the name blocking zeros given in a paper by r v patel see ibid vol ac 31 p 239 41 mar 1986 to a newly defined class of zeros for multivariable linear systems the commenter refers to a few remarks about some well known results by ferreira and objects to the lack of precision in the wording of two of these remarks counterexamples to these well known results are presented in replying the original author claims that example 1 is not a counterexample to his statement and objects to some lack of precision in martin s words

when is the numerical range of a nilpotent matrix circular the problem formulated in the title is investigated the case of nilpotent matrices of size at most 4 allows a unitary treatment the numerical range of a nilpotent matrix m of size at most 4 is circular if and only if the traces trm m 2 and trm m 3 are null the situation becomes more complicated as soon as the size is 5 the conditions under which a 5x5 nilpotent matrix has circular numerical range are thoroughly discussed

uniqueness of symmetric basis in quasi banach spaces we show that if x is a nonlocally convex natural quasi banach space with symmetric basis whose banach envelope is isomorphic to l1 then all symmetric bases of x are equivalent the scope of this result is quite ample since the banach envelopes of natural quasi banach spaces with basis always exhibit an l1 like behavior in the sense that they contain copies of l1n s uniformly complemented

squaring down of non strictly proper systems in an earlier work a procedure for squaring down proper systems by static and dynamic compensators has been developed the compensators designed there have the following properties a they are asymptotically stable b the additional finite zeros induced by them are assignable to the open left half complex plane and c they preserve the fundamental properties such as stabilizability detectability infinite zero structure and minimum phase nature of the original system the earlier design procedure is extended to non strictly proper systems while retaining all the above mentioned properties

theory of ltr for non minimum phase systems recoverable target loops and recovery in a subspace part 1 analysis a complete analysis of loop transfer recovery problem using full order observer based controllers for general not necessarily left invertible and not necessarily minimum phase systems is considered the analysis here while showing that neither exact nor asymptotic loop transfer recovery is in general possible focuses on three fundamental issues the first issue is concerned with what can and what canot be achieved for a given system and for an arbitrarily specified target loop transfer function while the second issue is concerned with the development of necessary and or sufficient conditions a target loop has to satisfy so that it can be either exactly or asymptotically be recovered for a given system the third issue deals with the development of method s to test whether recovery is possible in a given subspace of the control space or not i e to test whether projections of target and achievable sensitivity and complimentary sensitivity functions onto a given subspace match each other or not such an

geometry of the numerical range of matrices abstract some techniques for the study of the algebraic curve c a which generates the numerical range w a of an n n matrix a as its convex hull are developed these enable one to give an explicit point equation of c a and a formula for the curvature of c a at a boundary point of w a applied to the case of a nonnegative matrix a a simple relation is found between the curvature of the function a p 1 a a t p being the perron root at 1 2 and the curvature of w a at the perron root of 1 2 a a t a connection with 2 dimensional pencils of hermitian matrices is mentioned and a conjecture formulated

research on system zeros a survey this report provides a review of notable publications regarding the topic of zeros of multivariable systems in the period from 1970 into 1989 particular attention is paid to relationships between definitions of such zeros and their theoretical development transmission invariant system module infinite and finite zeros are addressed the latter subdivided into a hierarchical structure with respect to information considered in the definition process

on a conjecture of kippenhahn about the characteristic polynomial of a pencil generated by two hermitian matrices i abstract let a be an n n matrix write a h ik where i 2 1 and h and k are hermitian let f x y z det zi xh yk we first show that a pair of matrices over an algebraically closed field which satisfy quadratic polynomials can be put into block upper triangular form with diagonal blocks of size 1 1 or 2 2 via a simultaneous similarity this is used to prove that if f x y z g x y z n 2 where g has degree 2 then for some unitary matrix u the matrix u au is the direct sum of n 2 copies of a 2 2 matrix a 1 where a 1 is determined up to unitary similarity by the polynomial g x y z we use the connection between f x y z and the numerical range of a to investigate the case where f x y z has the form z ax y r g x y z s where g x y z is irreducible of degree 2