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Harm Bart

Erasmus University Rotterdam, Erasmus School of Economics - Econometric Institute, Emeritus

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My name is Harm Bart. I am a mathematician who got his PhD at the Free University in Amsterdam. In 1984, I was appointed to a professorship at the Erasmus University in Rotterdam, from which I retired in 2004. My scientific work concentrates on analytic operator/matrix functions. I am an editor of the journal Lineair Algebra and its Applications. In the past three decades, I worked mainly on logarithmic residues and of anal ytic operator/matrix functions and sums of idempotents- jointly with Torsten Ehrhardt and Bernd Silbermann.

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Papers by Harm Bart

Operator Theory: Advances and Applications, Vol. 205, 5157 (C) 2009 Birkhauser Verlag Basel/Switzerland

Pseudo-Differential Operators: …, 2010

Spectral Invariance of SG Pseudo-Diﬀerential Operators on Lp (Rn) 53 where Lp (Rn) is the norm in... more

In memoriam Israel Gohberg August 23, 1928–October 12, 2009

Linear Algebra and its Applications, Oct 1, 2010

Linear Algebra, the M.G. Krein Prize (Ukrainian Academy of Sciences), the Landau Prize and the Ro... more Linear Algebra, the M.G. Krein Prize (Ukrainian Academy of Sciences), the Landau Prize and the Rothschild Prize (Israel), and the Alexander von Humboldt Prize (Germany). In 1970 he was elected a corresponding member of the Academy of Science of Moldova (removed 1974, reinstated 1996), and in 1985 he was elected Foreign Member of the Royal Netherlands Academy of Arts and Sciences. In 2009 he was elected SIAM Fellow. At this place it is appropriate to recall what was said in the report (1993) on the first Hans Schneider Prize, 1 written by Daniel Hershkowitz from Haifa: "Israel Gohberg was born in 1928 and began his mathematical studies in Frunze (Kirgizia), and then in Kishinev (Moldava). Later, he became a close friend and collaborator of M.G. Krein in Odessa (Ukraine) and began a justly famous partnership that lasted for 24 years. In 1964 he obtained a doctoral degree at

The Chemnitz connection

Linear Algebra and its Applications, 2013

It is described how my contacts with a group of functional analysts in Chemnitz (Germany) came al... more

Invariants for Wiener-Hopf Equivalence of Analytic Operator Functions

Necessary conditions for Wiener-Hopf equivalence are established in terms of the incoming and out... more Necessary conditions for Wiener-Hopf equivalence are established in terms of the incoming and outgoing subspaces associated with realizations of the given analytic operator functions. Other results about the behaviour of the incoming and outgoing subspaces under certain elementary operations are also included.

Factorization of Matrix and Operator Functions: The State Space Method

Motivating Problems, Systems and Realizations.- Motivating Problems.- Operator Nodes, Systems, an... more Motivating Problems, Systems and Realizations.- Motivating Problems.- Operator Nodes, Systems, and Operations on Systems.- Various Classes of Systems.- Realization and Linearization of Operator Functions.- Factorization and Riccati Equations.- Canonical Factorization and Applications.- Minimal Realization and Minimal Factorization.- Minimal Systems.- Minimal Realizations and Pole-Zero Structure.- Minimal Factorization of Rational Matrix Functions.- Degree One Factors, Companion Based Rational Matrix Functions, and Job Scheduling.- Factorization into Degree One Factors.- Complete Factorization of Companion Based Matrix Functions.- Quasicomplete Factorization and Job Scheduling.- Stability of Factorization and of Invariant Subspaces.- Stability of Spectral Divisors.- Stability of Divisors.- Factorization of Real Matrix Functions.

Transfer functions and operator theory

Linear Algebra and its Applications, 1986

The notion of a transfer function from systems theory serves as an important tool in dealing with... more The notion of a transfer function from systems theory serves as an important tool in dealing with certain problems in operator and matrix theory. This is illustrated with a variety of material taken from existing literature. A striking feature of the results is their high degree of explicitness. For recent developments concerning Wiener-Hopf factorization, see Constructioe Methods of Wiener-Hopf Factorization, Operator Theory: Advances and Applications (I. Gohberg and M. A. Kaashoek, Eds.), Birkhiuser, Basel, to appear.

$Then W(A) = 1+ C(\ — A)7!B is a minimal realization. Clearly BC = 0, and so A“ = A. Since the nilpotent Jordan block A is unicellular, it follows that W(A) is irreducible (regardless of how big one takes m).$

Meromorphic operator valued functions

Simultaneous reduction to triangular forms after extension with zeroes

Linear Algebra and its Applications, 1998

This paper is concerned with pairs of m x m matrices ,4. Z for which there exists an invertible m... more This paper is concerned with pairs of m x m matrices ,4. Z for which there exists an invertible m × m matrix S such that S ~AS is upper triangular and S ~ZS is lower triangular. Such pairs are said to admit simultaneous reduction Io complementary triangular forms. In particular, if such a similarity S does not exist lot the pair .4. Z, the following question is taken into consideration: Does there exist a positive integer r. such that the pair o1" (m + r) × (m + r) matrices A O Z O (o o)(o o) admits simultaneous reduction to complementary triangular forms, so that the pair ,4, Z obtains the property after extension with zeroes? It is shown that the answer to this question is mixed. An example of a pair is given which obtains the property alter extension with one zero. This example is put into contrast with a large number of results that answers the question negatively, i.e., that describes situations where the pair of extended matrices admits sim:,!ta,eou:; reduction to complementary triangular forms if and only if the original pair does.

Spectral Regularity of Banach Algebras and Non-commutative Gelfand Theory

A Panorama of Modern Operator Theory and Related Topics, 2012

Upper triangularization of matrices by lower triangular similarities

Linear Algebra and its Applications, 1988

An integer programming problem and rank decomposition of block upper triangular matrices

Linear Algebra and its Applications, 2000

A necessary and sufficient condition is given for a block upper triangular matrix A to be the sum... more A necessary and sufficient condition is given for a block upper triangular matrix A to be the sum of block upper rectangular matrices satisfying certain rank constraints. The condition is formulated in terms of the ranks of certain submatrices of A. The proof goes by reduction to an integer programming problem. This integer programming problem has a totally unimodular constraint matrix which makes it possible to utilize Farkas' Lemma.

Stability properties of finite meromorphic operator functions. II

Indagationes Mathematicae (Proceedings), 1974

Let A be a meromorphic function with values in the space of bounded linear operators between two ... more Let A be a meromorphic function with values in the space of bounded linear operators between two Banach spaces X and Y, and assume that the coefficients of the prinoipal part of the Laurent expansion of A at a certain point & are degenerate operators. In this paper it is shown that under rather general hypotheses the null spaces (resp. ranges) of A(1) converge in the gap topology to a certain subspace of X (reap. Y) as il approaches &. Further, under slightly stronger conditions, the null spaces (reap. ranges) of A(A) have E fixed complementary subspace in X (resp. Y) for all rl in some deleted neighbourhood of k. The hypotheses of these stability theorems are fulfilled if A is Fredholm at & or haa values in the set of degenerate operators.

The role of canonical factorization in solving convolution equations

A State Space Approach to Canonical Factorization with Applications, 2010

This chapter has a preparatory character. We review (without giving proofs) the role of canonical... more This chapter has a preparatory character. We review (without giving proofs) the role of canonical factorization in inverting systems of convolution equations. The chapter consists of three sections. Section 1.1 deals with Wiener-Hopf integral equations, Section 1.2 with block Toeplitz equations, and Section 1.3 with singular integral equations.

Factorization of positive definite rational matrix functions

A State Space Approach to Canonical Factorization with Applications, 2010

The central theme of this chapter is the state space analysis of rational matrix functions with H... more The central theme of this chapter is the state space analysis of rational matrix functions with Hermitian values either on the real line, on the imaginary axis, or on the unit circle. The main focus will be on rational matrix functions that take positive definite values on one of these contours. It will be shown that if W is such a function, then W admits a spectral factorization, i.e., a canonical factorization W=W−W+ with an additional symmetry between the corresponding factors, depending on the contour.

Wiener-Hopf factorization and factorization indices

A State Space Approach to Canonical Factorization with Applications, 2010

ABSTRACT This chapter concerns canonical as well as non-canonical Wiener-Hopf factorization of an... more ABSTRACT This chapter concerns canonical as well as non-canonical Wiener-Hopf factorization of an operator-valued function which is analytic on a Cauchy contour. Such an operator function is given by a realization with a possibly infinite dimensional Banach space as state space, and with a bounded state operator and with bounded input-output operators. The first main result is a generalization to operator-valued functions of the canonical factorization theorem for rational matrix functions presented earlier in Section 3.1. In terms of the given realization, necessary and sufficient conditions are also presented in order that the operator function involved admits a (possibly non-canonical) Wiener-Hopf factorization. The corresponding factorization indices are described in terms of certain spectral invariants which are defined in terms of the realization but do only depend on the operator function and not on the particular choice of the realization. The analysis of these spectral invariants is one of the main themes of this chapter.

Convolution equations and the transport equation

A State Space Approach to Canonical Factorization with Applications, 2010

In this chapter the factorization theory developed in the previous chapters is applied to solve a... more In this chapter the factorization theory developed in the previous chapters is applied to solve a linear transport equation. It is known that the transport equation may be transformed into a Wiener-Hopf integral equation with an operator-valued kernel function (see [40]). An equation of the latter type can be solved explicitly if a canonical factorization of its symbol is available (cf., Sections 1.1 and 3.2). In our case the symbol may be represented as a transfer function, and to make the factorization the general factorization theorem of the second chapter can be applied. This requires that one finds an appropriate pair of invariant subspaces. In the case of the transport equation the choice of the subspaces is evident, but to prove that their direct sum is the whole space takes some effort. The latter is related to a new difficulty that appears here. Namely, in this case the curve cuts through the spectra of the main operator and the associate main operator. Nevertheless, due to the special structure of the operators involved, the factorization can be made and explicit formulas are obtained.

Variants of the Two Machine Flow Shop Problem

Operations Research Proceedings, 1995

In this paper we consider a number of variants of the Two Machine Flow Shop Problem. In these var... more In this paper we consider a number of variants of the Two Machine Flow Shop Problem. In these variants the makespan is given and the problem is to find a schedule that meets this makespan, thereby minimizing the infeasibilities of the jobs in a prescribed sense: In the max-variant the maximum infeasibility of the jobs is to be minimized, whereas in the sum-variant the objective is to minimize the sum of the infeasibilities of the jobs. For both variants observations about the structure of the optimal schedules are presented. In particular, it is proved that every instance of these problems has an optimal permutation schedule. It is also shown that the max-variant can be solved by Johnson's Rule. For the sum-variant this is not the case: For solving this problem to optimality something quite different is necessary. Both variants are connected with factorization problems for certain rational matrix functions. The factorizations involved are optimal in some sense and generalize the notion of complete factorization. In this way a connection is established between job scheduling theory on one hand, and mathematical systems theory on the other.

Factorizations of Transfer Functions

SIAM Journal on Control and Optimization, 1980

[SIAM Journal on Control and Optimization 18, 675 (1980)]. H. Bart, I. Gohberg, MA Kaashoek, P. V... more

Spectral properties of locally holomorphic vector-valued functions

Pacific Journal of Mathematics, 1974

This paper deals with spectral properties of commutative locally holomorphic Banach algebra value... more This paper deals with spectral properties of commutative locally holomorphic Banach algebra valued functions. One of the main concepts is that of a spectral set of such a function. This concept, which is due to L. Mittenthal, extends that of a spectral set of a single Banach algebra element. It will be shown that the spectral idempotent associated with a nonvoid spectral set is nonzero. This result is a generalization of a well-known theorem in ordinary spectral theory. It will be used to prove a correctly stated but incorrectly proven theorem of L. Mittenthal.

Complementary Schur complements

Linear Algebra and its Applications, 1994

For square matrices, the relationship is discussed between the notion of Schur complement and the... more

Operator Theory: Advances and Applications, Vol. 205, 5157 (C) 2009 Birkhauser Verlag Basel/Switzerland

Pseudo-Differential Operators: …, 2010

Spectral Invariance of SG Pseudo-Diﬀerential Operators on Lp (Rn) 53 where Lp (Rn) is the norm in... more

In memoriam Israel Gohberg August 23, 1928–October 12, 2009

Linear Algebra and its Applications, Oct 1, 2010

Linear Algebra, the M.G. Krein Prize (Ukrainian Academy of Sciences), the Landau Prize and the Ro... more Linear Algebra, the M.G. Krein Prize (Ukrainian Academy of Sciences), the Landau Prize and the Rothschild Prize (Israel), and the Alexander von Humboldt Prize (Germany). In 1970 he was elected a corresponding member of the Academy of Science of Moldova (removed 1974, reinstated 1996), and in 1985 he was elected Foreign Member of the Royal Netherlands Academy of Arts and Sciences. In 2009 he was elected SIAM Fellow. At this place it is appropriate to recall what was said in the report (1993) on the first Hans Schneider Prize, 1 written by Daniel Hershkowitz from Haifa: "Israel Gohberg was born in 1928 and began his mathematical studies in Frunze (Kirgizia), and then in Kishinev (Moldava). Later, he became a close friend and collaborator of M.G. Krein in Odessa (Ukraine) and began a justly famous partnership that lasted for 24 years. In 1964 he obtained a doctoral degree at

The Chemnitz connection

Linear Algebra and its Applications, 2013

It is described how my contacts with a group of functional analysts in Chemnitz (Germany) came al... more

Invariants for Wiener-Hopf Equivalence of Analytic Operator Functions

Necessary conditions for Wiener-Hopf equivalence are established in terms of the incoming and out... more Necessary conditions for Wiener-Hopf equivalence are established in terms of the incoming and outgoing subspaces associated with realizations of the given analytic operator functions. Other results about the behaviour of the incoming and outgoing subspaces under certain elementary operations are also included.

Factorization of Matrix and Operator Functions: The State Space Method

Motivating Problems, Systems and Realizations.- Motivating Problems.- Operator Nodes, Systems, an... more Motivating Problems, Systems and Realizations.- Motivating Problems.- Operator Nodes, Systems, and Operations on Systems.- Various Classes of Systems.- Realization and Linearization of Operator Functions.- Factorization and Riccati Equations.- Canonical Factorization and Applications.- Minimal Realization and Minimal Factorization.- Minimal Systems.- Minimal Realizations and Pole-Zero Structure.- Minimal Factorization of Rational Matrix Functions.- Degree One Factors, Companion Based Rational Matrix Functions, and Job Scheduling.- Factorization into Degree One Factors.- Complete Factorization of Companion Based Matrix Functions.- Quasicomplete Factorization and Job Scheduling.- Stability of Factorization and of Invariant Subspaces.- Stability of Spectral Divisors.- Stability of Divisors.- Factorization of Real Matrix Functions.

Transfer functions and operator theory

Linear Algebra and its Applications, 1986

The notion of a transfer function from systems theory serves as an important tool in dealing with... more The notion of a transfer function from systems theory serves as an important tool in dealing with certain problems in operator and matrix theory. This is illustrated with a variety of material taken from existing literature. A striking feature of the results is their high degree of explicitness. For recent developments concerning Wiener-Hopf factorization, see Constructioe Methods of Wiener-Hopf Factorization, Operator Theory: Advances and Applications (I. Gohberg and M. A. Kaashoek, Eds.), Birkhiuser, Basel, to appear.

$Then W(A) = 1+ C(\ — A)7!B is a minimal realization. Clearly BC = 0, and so A“ = A. Since the nilpotent Jordan block A is unicellular, it follows that W(A) is irreducible (regardless of how big one takes m).$

Meromorphic operator valued functions

Simultaneous reduction to triangular forms after extension with zeroes

Linear Algebra and its Applications, 1998

This paper is concerned with pairs of m x m matrices ,4. Z for which there exists an invertible m... more This paper is concerned with pairs of m x m matrices ,4. Z for which there exists an invertible m × m matrix S such that S ~AS is upper triangular and S ~ZS is lower triangular. Such pairs are said to admit simultaneous reduction Io complementary triangular forms. In particular, if such a similarity S does not exist lot the pair .4. Z, the following question is taken into consideration: Does there exist a positive integer r. such that the pair o1" (m + r) × (m + r) matrices A O Z O (o o)(o o) admits simultaneous reduction to complementary triangular forms, so that the pair ,4, Z obtains the property after extension with zeroes? It is shown that the answer to this question is mixed. An example of a pair is given which obtains the property alter extension with one zero. This example is put into contrast with a large number of results that answers the question negatively, i.e., that describes situations where the pair of extended matrices admits sim:,!ta,eou:; reduction to complementary triangular forms if and only if the original pair does.

Spectral Regularity of Banach Algebras and Non-commutative Gelfand Theory

A Panorama of Modern Operator Theory and Related Topics, 2012

Upper triangularization of matrices by lower triangular similarities

Linear Algebra and its Applications, 1988

An integer programming problem and rank decomposition of block upper triangular matrices

Linear Algebra and its Applications, 2000

A necessary and sufficient condition is given for a block upper triangular matrix A to be the sum... more A necessary and sufficient condition is given for a block upper triangular matrix A to be the sum of block upper rectangular matrices satisfying certain rank constraints. The condition is formulated in terms of the ranks of certain submatrices of A. The proof goes by reduction to an integer programming problem. This integer programming problem has a totally unimodular constraint matrix which makes it possible to utilize Farkas' Lemma.

Stability properties of finite meromorphic operator functions. II

Indagationes Mathematicae (Proceedings), 1974

Let A be a meromorphic function with values in the space of bounded linear operators between two ... more Let A be a meromorphic function with values in the space of bounded linear operators between two Banach spaces X and Y, and assume that the coefficients of the prinoipal part of the Laurent expansion of A at a certain point & are degenerate operators. In this paper it is shown that under rather general hypotheses the null spaces (resp. ranges) of A(1) converge in the gap topology to a certain subspace of X (reap. Y) as il approaches &. Further, under slightly stronger conditions, the null spaces (reap. ranges) of A(A) have E fixed complementary subspace in X (resp. Y) for all rl in some deleted neighbourhood of k. The hypotheses of these stability theorems are fulfilled if A is Fredholm at & or haa values in the set of degenerate operators.

The role of canonical factorization in solving convolution equations

A State Space Approach to Canonical Factorization with Applications, 2010

This chapter has a preparatory character. We review (without giving proofs) the role of canonical... more This chapter has a preparatory character. We review (without giving proofs) the role of canonical factorization in inverting systems of convolution equations. The chapter consists of three sections. Section 1.1 deals with Wiener-Hopf integral equations, Section 1.2 with block Toeplitz equations, and Section 1.3 with singular integral equations.

Factorization of positive definite rational matrix functions

A State Space Approach to Canonical Factorization with Applications, 2010

The central theme of this chapter is the state space analysis of rational matrix functions with H... more The central theme of this chapter is the state space analysis of rational matrix functions with Hermitian values either on the real line, on the imaginary axis, or on the unit circle. The main focus will be on rational matrix functions that take positive definite values on one of these contours. It will be shown that if W is such a function, then W admits a spectral factorization, i.e., a canonical factorization W=W−W+ with an additional symmetry between the corresponding factors, depending on the contour.

Wiener-Hopf factorization and factorization indices

A State Space Approach to Canonical Factorization with Applications, 2010

ABSTRACT This chapter concerns canonical as well as non-canonical Wiener-Hopf factorization of an... more ABSTRACT This chapter concerns canonical as well as non-canonical Wiener-Hopf factorization of an operator-valued function which is analytic on a Cauchy contour. Such an operator function is given by a realization with a possibly infinite dimensional Banach space as state space, and with a bounded state operator and with bounded input-output operators. The first main result is a generalization to operator-valued functions of the canonical factorization theorem for rational matrix functions presented earlier in Section 3.1. In terms of the given realization, necessary and sufficient conditions are also presented in order that the operator function involved admits a (possibly non-canonical) Wiener-Hopf factorization. The corresponding factorization indices are described in terms of certain spectral invariants which are defined in terms of the realization but do only depend on the operator function and not on the particular choice of the realization. The analysis of these spectral invariants is one of the main themes of this chapter.

Convolution equations and the transport equation

A State Space Approach to Canonical Factorization with Applications, 2010

In this chapter the factorization theory developed in the previous chapters is applied to solve a... more In this chapter the factorization theory developed in the previous chapters is applied to solve a linear transport equation. It is known that the transport equation may be transformed into a Wiener-Hopf integral equation with an operator-valued kernel function (see [40]). An equation of the latter type can be solved explicitly if a canonical factorization of its symbol is available (cf., Sections 1.1 and 3.2). In our case the symbol may be represented as a transfer function, and to make the factorization the general factorization theorem of the second chapter can be applied. This requires that one finds an appropriate pair of invariant subspaces. In the case of the transport equation the choice of the subspaces is evident, but to prove that their direct sum is the whole space takes some effort. The latter is related to a new difficulty that appears here. Namely, in this case the curve cuts through the spectra of the main operator and the associate main operator. Nevertheless, due to the special structure of the operators involved, the factorization can be made and explicit formulas are obtained.

Variants of the Two Machine Flow Shop Problem

Operations Research Proceedings, 1995

In this paper we consider a number of variants of the Two Machine Flow Shop Problem. In these var... more In this paper we consider a number of variants of the Two Machine Flow Shop Problem. In these variants the makespan is given and the problem is to find a schedule that meets this makespan, thereby minimizing the infeasibilities of the jobs in a prescribed sense: In the max-variant the maximum infeasibility of the jobs is to be minimized, whereas in the sum-variant the objective is to minimize the sum of the infeasibilities of the jobs. For both variants observations about the structure of the optimal schedules are presented. In particular, it is proved that every instance of these problems has an optimal permutation schedule. It is also shown that the max-variant can be solved by Johnson's Rule. For the sum-variant this is not the case: For solving this problem to optimality something quite different is necessary. Both variants are connected with factorization problems for certain rational matrix functions. The factorizations involved are optimal in some sense and generalize the notion of complete factorization. In this way a connection is established between job scheduling theory on one hand, and mathematical systems theory on the other.

Factorizations of Transfer Functions

SIAM Journal on Control and Optimization, 1980

[SIAM Journal on Control and Optimization 18, 675 (1980)]. H. Bart, I. Gohberg, MA Kaashoek, P. V... more

Spectral properties of locally holomorphic vector-valued functions

Pacific Journal of Mathematics, 1974

This paper deals with spectral properties of commutative locally holomorphic Banach algebra value... more This paper deals with spectral properties of commutative locally holomorphic Banach algebra valued functions. One of the main concepts is that of a spectral set of such a function. This concept, which is due to L. Mittenthal, extends that of a spectral set of a single Banach algebra element. It will be shown that the spectral idempotent associated with a nonvoid spectral set is nonzero. This result is a generalization of a well-known theorem in ordinary spectral theory. It will be used to prove a correctly stated but incorrectly proven theorem of L. Mittenthal.

Complementary Schur complements

Linear Algebra and its Applications, 1994

For square matrices, the relationship is discussed between the notion of Schur complement and the... more