Reconstruction of a rational nonsquare matrix function from local data

In many problems the local zero-pole structure (i.e. locations of zeros and poles together with their orders) of a scalar rational functionw is a key piece of structure. Knowledge of the order of the pole or zero of the rational functionw at the point is equivalent to knowledge of the -module (where is the space of rational functions analytic at ). For the more intricate case of a rationalp×m matrix functionW, we consider the structure of the module as the appropriate analogue of zero-pole structure (location of zeros and poles together with directional information), where is the set of column vectors of heightm with entries equal to rational functions which are analytic at . Modules of the form in turn can be explicitly parametrized in terms of a collection of matrices (C ,A ,B ,B , ) together with a certain row-reduced(p–m)×m matrix polynomialP(z) (which is independent of ) which satisfy certain normalization and consistency conditions. We therefore define the collection (C ,A ,Z ,B , ,P(z)) to be the local spectral data set of the rational matrix functionW at . We discuss the direct problem of how to compute the local spectral data explicitly from a realizationW(z)=D+C(z–A) ^–1 B forW and solve the inverse problem of classifying which collections (C ,A ,Z ,B , ,P(z)) satisfying the local consistency and normalization conditions arise as the local spectral data sets of some rational matrix functionW. Earlier work in the literature handles the case whereW is square with nonzero determinant.     

Positive matrix functions on the bitorus with prescribed Fourier coefficients in a band

Let S be a band in Z² bordered by two parallel lines that are of equal distance to the origin. Given a positive definite ¹ sequence of matrices {c_j}_jS we prove that there is a positive definite matrix function f in the Wiener algebra on the bitorus such that the Fourier coefficients equal c_k for k S. A parameterization is obtained for the set of all positive extensions f of {c_j}_jS. We also prove that among all matrix functions with these properties, there exists a distinguished one that maximizes the entropy. A formula is given for this distinguished matrix function. The results are interpreted in the context of spectral estimation of ARMA processes.     

On canonical factorization of rational matrix functions

This paper concerns the problem of canonical factorization of a rational matrix functionW() which is analytic but may benot invertible at infinity. The factors are obtained explicitly in terms of the realization of the original matrix function. The cases of symmetric factorization for selfadjoint and positive rational matrix functions are considered separately.     

Quasicomplete factorizations of rational matrix functions

It is shown that within the class ofn×n rational matrix functions which are analytic at infinity with valueW()=I _n, any rational matrix functionW is the productW=W ₁...W _p of rational matrix functionsW ₁,...,W _p of McMillan degree one. Furthermore, such a factorization can be established with a number of factors not exceeding 2(W)–1, where (W) denotes the McMillan degree ofW.     

Towards a model theory for 2-hyponormal operators

We introduce the notion ofweak subnormality, which generalizes subnormality in the sense that for the extension ofT we only require that hold forf ; in this case we call a partially normal extension ofT. After establishing some basic results about weak subnormality (including those dealing with the notion of minimal partially normal extension), we proceed to characterize weak subnormality for weighted shifts and to prove that 2-hyponormal weighted shifts are weakly subnormal. Let { _n } _n=0 be a weight sequence and letW denote the associated unilateral weighted shift on . IfW is 2-hyponormal thenW is weakly subnormal. Moreover, there exists a partially normal extension on such that (i) is hyponormal; (ii) ; and (iii) . In particular, if is strictly increasing then can be obtained as

On the essential approximate point spectrum of operators

If belongs to the essential approximate point spectrum of a Banach space operatorTB(X) and is a sequence of positive numbers with lim ^j a _j =0, then there existsxX such that for every polynomialp. This result is the best possible — if for some constantc>0 thenT has already a non-trivial invariant subspace, which is not true in general.     

Factorization of staircase class operators in Hilbert space

A class of linear bounded staircase operators (H, G spaces) defined by (1) with two infinite sequences of orthogonal decompositions ofH and chain property (2) is considered. Necessary and sufficient conditions for the factorizationZ=XY are obtained, whereX, Y are block-diagonal, bounded, andY has a bounded inverse. All the pairs (X, Y) are explicitly constructed. These conditions are specialized for finite and infinite dimensions of the blocks ofX, Y and for differentX, Y. A direct application to bitriangular and biquasitriangular operators is indicated.     

On Compact Perturbations of Locally Definitizable Selfadjoint Relations in Krein Spaces

The aim of this paper is to prove two perturbation results for a selfadjoint operator A in a Krein space which can roughly be described as follows: (1) If is an open subset of and all spectral subspaces for A corresponding to compact subsets of have finite rank of negativity, the same is true for a selfadjoint operator B in for which the difference of the resolvents of A and B is compact. (2) The property that there exists some neighbourhood of such that the restriction of A to a spectral subspace for A corresponding to is a nonnegative operator in is preserved under relative perturbations in form sense if the resulting operator is again selfadjoint. The assertion (1) is proved for selfadjoint relations A and B. (1) and (2) generalize some known results.     

Solution of the Truncated Hyperbolic Moment Problem

Let Q(x, y) = 0 be an hyperbola in the plane. Given real numbers β ≡ β ⁽²ⁿ⁾={ β _ij } _{i,j ≥ 0,i+j ≤ 2n} , with β₀₀ > 0, the truncated Q-hyperbolic moment problem for β entails finding necessary and sufficient conditions for the existence of a positive Borel measure μ, supported in Q(x, y) = 0, such that We prove that β admits a Q-representing measure μ (as above) if and only if the associated moment matrix is positive semidefinite, recursively generated, has a column relation Q(X,Y) = 0, and the algebraic variety associated to β satisfies card In this case, if then β admits a rank -atomic (minimal) Q-representing measure; if then β admits a Q-representing measure μ satisfying      

Visible shorelines inR d

Summary LetC be a closed set inR ^d and letj be a fixed integer,j 1. The setS R ^d ~C is said to have aj-partition relative toC if there existj or fewer pointsc ₁,, c _j ofC such that each point ofS sees via the complement ofC at least one pointc _i. For every triple of integersd, p, j withd 0, p d + 1, j 1, there exists a smallest integerf(d, p, j) such that the following is true: IfC is a convexd-polytope inR ^d havingp vertices and ifS R ^d ~C, S has aj-partition relative toC if and only if everyf(d, p, j)-member subset of S has such a partition.ForC a convex polytope inR ² andS R ² ~C, all points ofS see via the complement ofC a common neighborhood in the boundary ofC if and only if every three points ofS see via the complement ofC such a neighborhood.A weak analogue of this result holds for arbitrary compact convex sets inR ^d .     

Characterizing compact unions of two starshaped sets inR d

SetS inR ^d has propertyK ₂ if and only ifS is a finite union ofd-polytopes and for every finite setF in bdryS there exist points c₁,c₂ (depending onF) such that each point ofF is clearly visible viaS from at least one c_i,i = 1,2. The following characterization theorem is established: Let , d2. SetS is a compact union of two starshaped sets if and only if there is a sequence {S _j } converging toS (relative to the Hausdorff metric) such that each setS _j satisfies propertyK ₂. For , the sufficiency of the condition above still holds, although the necessity fails.     

j-partitions for visible shorelines

Summary LetC be a compact set inR ². A setS R ² C is said to have aj-partition relative toC if and only if there existj or fewer pointsc ₁,, c _j inC such that each point ofS sees somec _i via the complement ofC. Letm, j be fixed integers, 3 m, 2 j, and writem (uniquely) asm = qj + r, where 1 r j. Assume thatC is a convexm-gon in R², withS R ² C. Forq = 0 orq = 1, the setS has aj-partition relative toC. Forq 2,S has aj-partition relative toC if and only if every (qj + 1)-member subset ofS has aj-partition relative toC, and the Helly numberqj + 1 is best possible.IfC is a disk, no such Helly number exists.     

Isometric dilations in nest algebras

LetT be a contraction acting in a separable Hilbert space and leaving invariant a nest of subspaces of . We answer the question: when doesT have an isometric extension to which leaves invariant the nest = {N N :N ;}.     

Relations between convolution type operators on intervals and on the half-line

This paper is devoted to the question to obtain (algebraic and topologic) equivalence (after extension) relations between convolution type operators on unions of intervals and convolution type operators on the half-line. These operators are supposed to act between Bessel potential spaces,H ^s,p , which are the appropriate spaces in several applications. The present approach is based upon special properties of convenient projectors, decompositions and extension operators and the construction of certain homeomorphisms between the kernels of the projectors. The main advantage of the method is that it provides explicit operator matrix identities between the mentioned operators where the relations are constructed only by bounded invertible operators. So they are stronger than the (algebraic) Kuijper-Spitkovsky relation and the Bastos-dos Santos-Duduchava relation with respect to the transfer of properties on the prize that the relations depend on the orders of the spaces and hold only for non-critical orders:S – 1/p . For instance, (generalized) inverses of the operators are explicitly represented in terms of operator matrix factorization. Some applications are presented.This research was supported by Junta Nacional de Investigação Científica e Tecnológica (Portugal) and the Bundesminister für Forschung und Technologie (Germany) within the projectSingular Operators-new features and applications, and by a PRAXIS XXI project under the titleFactorization of Operators and Applications to Mathematical Physics.     

Nondissipative curves in Hilbert spaces having a limit of the corresponding correlation function

In this work a class of nondissipative curves in Hilbert spaces whose correlation functions have a limit ast± is presented. These curves correspond to a class of nondissipative basic operators that are a coupling of a dissipative operator and an antidissipative one. The wave operators and the scattering operator for the couple (A ^*, A) ( ) are obtained. The present work is a continuation and a generalization of the investigations of K.Kirchev and V.Zolotarev [1, 2, 3] on the model representations of curves in Hilbert spaces where the respective semigroup generator is a dissipative operator. This article includes four parts. A new form of the triangular model of M.S. Livic ([4, 5]) for the considered operators is introduced in the first part by the help of a suitable representation of the selfadjoint operatorL. This allows us to describe the studied class of nondissipative curves. The second part studies some results concerning the application of the analogue for multiplicative integrals of the well-known Privalov's theorem ([6]) about the limit values in the scalar case. This analogue is a reconstruction of measure by limit values in Stieltjes-Perron's style and it is obtained by L.A. Sakhnovich ([7]). Another problem, considered in the second part is the analogue inC ^m of the classical gamma-function and several properties for further consideration. In the third part the asymptotics of the studied curves corresponding to the nondissipative operators-couplings of a dissipative and an antidissipative operator with absolutely continuous real spectra and the limits of their correlation functions are obtained In the fourth part a scattering theory of a couple (A ^*, A) with a nondissipative operatorA from is constructed as in the selfadjoint case ([8, 9, 10]) and in the dissipative case ([7]). These results show an interesting new effect: the studied nondissipative case is near to the dissipative one.Partially supported by Grant MM-810/98 of MESC     

On the essential norm of the Cauchy singular integral operator in weighted rearrangement-invariant spaces

In this paper we extend necessary conditions for Fredholmness of singular integral operators with piecewise continuous coefficients in rearrangement-invariant spaces [19] to the weighted caseX(,w). These conditions are formulated in terms of indices (Q _t w) and (Q _t w) of a submultiplicative functionQ _t w, which is associated with local properties of the space, of the curve, and of the weight at the pointt. Using these results we obtain a lower estimate for the essential norm |S| of the Cauchy singular integral operatorS in reflexive weighted rearrangement-invariant spacesX(,w) over arbitrary Carleson curves :

On the continuity of the spectrum in certain Banach algebras

In this paper we describe some classes of linear operatorsTL(H) (mainly Toeplitz, Wiener-Hopf and singular integral) on a Hilbert spacesH such that the spectrum (T, L(H)) is continuous at the pointsT from these classes. We also describe some subalgebras of the algebras for which the spectrum (x,) becomes continuous at the pointsx when (x,) is restricted to the subalgebra . In particular, we show that the spectrum (x,) is continuous in Banach algebras with polynomial identities. Examples of such algebras are given.This research was partially supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities.     

Compact and finite rank operators satisfying a Hankel type equationT 2X=XT 1 *

In 1997, V. Pták defined the notion of generalized Hankel operator as follows: Given two contractions and , an operatorX: is said to be a generalized Hankel operator ifT ₂ X=XT ₁ ^* andX satisfies a boundedness condition that depends on the unitary parts of the minimal isometric dilations ofT ₁ andT ₂. The purpose behind this kind of generalization is to study which properties of classical Hankel operators depend on their characteristic intertwining relation rather than on the theory of analytic functions. Following this spirit, we give appropriate versions of a number of results about compact and finite rank Hankel operators that hold within Pták's generalized framework. Namely, we extend Adamyan, Arov and Krein's estimates of the essential norm of a Hankel operator, Hartman's characterization of compact Hankel operators and Kronecker's characterization of finite rank Hankel operators.Dedicated to the memory of our master and friend Vlastimil Pták     

Totally P-posinormal operators are subscalar

LetH be a complex infinite-dimensional separable Hilbert space. An operatorT inL(H) is called totally P-posinormal (see [9]) iff there is a polynomialP with zero constant term such that for each , whereT _z =T–zI andM(z) is bounded on the compacts of C. In this paper we prove that every totally P-posinormal operator is subscalar, i.e. it is the restriction of a generalized scalar operator to an invariant subspace. Further, a list of some important corollaries about Bishop's property and the existence of invariant subspaces is presented.