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For a block upper triangular matrix, a necessary and sufficient condition has been given to let it be the sum of block upper rectangular matrices satisfying certain rank constraints; see [12] . The proof involves elements from Integer Programming (totally unimodular systems of equations playing a role in particular) and employs Farkas' Lemma. The linear space of block upper triangular matrices can be viewed as being determined by a special pattern of zeros. The present paper is concerned with the question whether the decomposition result can be extended to situations where other, less restrictive, zero patterns play a role. It is shown that such generalizations do indeed hold for certain directed graphs determining the pattern of zeros. The graphs in question are what will be called L -free. This notion is akin to other graph theoretical concepts available in the literature, among them the one of being N -free in the sense of [16] .

A zero pattern algebra is a matrix algebra determined by a pattern of zeros corresponding to a preorder (i.e., a reflexive transitive digraph). The issue studied in [6] is: under what conditions is a matrix in a zero pattern algebra A a sum of (rank one) idempotents in A or a logarithmic residue in A ? Here logarithmic residues are contour integrals of logarithmic derivatives of analytic A -valued functions. As has been established in [6] , there is a necessary condition involving certain rank/trace requirements. Algebras for which this necessary condition is also sufficient are said to be rank/trace complete. Several classes of rank/trace complete algebras are identified in [6] . Also operations are described there for producing rank/trace complete algebras out of given ones. A basic operation of that type is taking a disjoint union. In the present article this simple operation is generalized to a more involved one which preserves rank/trace completeness too. The new operation turns out to be a useful tool for establishing rank/trace completeness in situations that up to now could not be handled. One of the positive results is concerned with bouquets, a special type of rooted trees. It provides a partial answer to an issue raised in [7] .

The issue discussed in this paper is: how small can a sum of idempotents be? Here smallness is understood in terms of nilpotency or quasinilpotency. Thus the question is: given idempotents $$p_1,\ldots ,p_n$$ in a complex algebra or Banach algebra, is it possible that their sum $$p_1+\cdots +p_n$$ is quasinilpotent or (even) nilpotent (of a certain order)? The motivation for considering this problem comes from earlier work by the authors on the generalization of the logarithmic residue theorem from complex function theory to higher (possibly infinite) dimensions.

For a block upper triangular matrix, a necessary and sufficient condition has been given to let it be the sum of block upper rectangular matrices satisfying certain rank constraints; see H.Bart, A.P.M.Wagelmans (2000). The proof involves elements from integer programming and employs Farkas’ lemma. The algebra of block upper triangular matrices can be viewed as a matrix algebra determined by a pattern of zeros. The present note is concerned with the question whether the decomposition result referred to above can be extended to other zero pattern matrix algebras. It is shown that such a generalization does indeed hold for certain digraphs determining the pattern of zeros. The digraphs in question can be characterized in terms of forests, i.e., disjoint unions of rooted trees.

A zero pattern algebra is a matrix subalgebra of C n × n determined by a pattern of zeros. The issue in this paper is: under what conditions is a matrix in a zero pattern algebra A a sum of (rank one) idempotents in A or a logarithmic residue in A ? Here logarithmic residues are contour integrals of logarithmic derivatives of analytic A -valued functions. It turns out that there is a necessary condition involving certain rank/trace requirements. Although these requirements are generally not sufficient, there are several important cases where they are.

Let B be a unital Banach algebra, which can in a certain sense be approximated by finite dimensional algebras. For instance, AF C ⁎ -algebras belong to this class. Further, let f be an analytic function on some bounded Cauchy domain Δ with values in B and suppose that the contour integral of the logarithmic derivative f ′ ( λ ) f − 1 ( λ ) along the positively oriented boundary ∂Δ vanishes (or is even only quasinilpotent). We prove that then f takes invertible values on all of Δ. This means that such Banach algebras are spectrally regular.

Given a bounded simply connected domain \({U} \subset {\mathbb{C}}\) having a Lyapunov curve as its boundary, let \(\mathcal{L}({L}^{2}(U))\) stand for the \((\mathbb{c}^\ast)\) -algebra of all bounded linear operators acting on the Hilbert space \(\mathcal{L}^{2}(U)\) with Lebesgue area measure. We show that the smallest C*-subalgebra \(\mathcal{A}\) of \(\mathcal{L}({L}^{2}(U))\) containing the singular integral operator $$(S_Uf)(z)\;=\;-\frac{1}{\pi}{\int\limits_{U}}\frac{f(w)}{(z-w)^2}dA(w),$$ along with its adjoint $$(S^*_Uf)(z)\;=\;-\frac{1}{\pi}{\int\limits_{U}}\frac{f(w)}{(z-w)^2}dA(w)$$ all multiplication operators \(aI, a \in\; C(\overline{U})\), and all compact operators on \(\mathcal{L}^{2}(U)\), is spectrally regular. Roughly speaking the latter means the following: if the contour integral of the logarithmic derivative of an analytic \(\mathcal{A}\)-valued function f is vanishing (or is quasi-nilpotent), then f takes invertible values on the inner domain of the contour in question.

It is common knowledge that matrices can be brought in echelon form by Gaussian elimination and that the reduced echelon form of a matrix is canonical in the sense that it is unique. In [4], working within the context of the algebra \( \mathbb{C}^{n\times n}_{\mathrm{upper}}\) of upper triangular n×n matrices, certain new canonical forms of echelon-type have been introduced. Subalgebras of \( \mathbb{C}^{n\times n}_{\mathrm{upper}}\) determined by a pattern of zeros have been considered too. The issue there is whether or not those subalgebras are echelon compatible in the sense that the new canonical forms belong to the subalgebras in question. In general they don’t, but affirmative answers were obtained under certain conditions on the given zero pattern. In the present paper these conditions are weakened. Even to the extent that a further relaxation is not possible because the conditions involved are not only sufficient but also necessary. The results are used to study equivalence classes in \( \mathbb{C}^{m\times n}\) associated with zero patterns. The analysis of the pattern of zeros referred to above is done in terms of graph theoretical notions.

It is common knowledge that matrices can be brought in echelon form by Gaussian elimination and that the reduced echelon form of a matrix is canonical (also) in the sense that it is unique. A crucial auxiliary result in [BW] suggests a generalization of the standard echelon form. For square matrices, some new canonical forms of echelon type are introduced. One of them (suggested by observations made in [Lay] and [SW]) has the important property of being an upper triangular idempotent. The others come up when working exclusively in the context of \( \mathbb{C}^{n \times n}_{upper} \), the algebra of upper triangular n × n matrices. Subalgebras of \( \mathbb{C}^{n \times n}_{upper} \) determined by a pattern of zeros are considered too. The issue there is whether or not the canonical forms referred to above belong to the subalgebras in question. In general they do not, but affirmative answers are obtained under certain conditions on the given preorder which allow for a large class of examples and that also came up in [BES4]. Similar results hold for canonical generalized diagonal forms involving matrices for which all columns and rows contain at most one nonzero entry. The new canonical forms are used to study left, right and left/right equivalence in zero pattern algebras. For the archetypical full upper triangular case a connection with the Stirling numbers (of the second kind) and with the Bell numbers is made.

It is described how my contacts with a group of functional analysts in Chemnitz (Germany) came along and resulted in a longstanding cooperation on logarithmic residues and spectral regularity in a Banach algebra setting.

Using families of irreducible Hilbert space representations as a tool, the theory of analytic Fredholm operator valued function is extended to a C ∗ -algebra setting. This includes a C ∗ -algebra version of Rouche’s Theorem known from complex function theory. Also, criteria for spectral regularity of C ∗ -algebras are developed. One of those, involving the (generalized) Calkin algebra, is applied to C ∗ -algebras generated by a non-unitary isometry.

This obituary for Israel Gohberg consists of a general introduction, separate contributions of the six authors, all of whom worked closely with him, and a final note. The material gives an impression of the life of this great mathematician, of his monumental impact in the areas he worked in, of how he cooperated with colleagues, and of his ability to stimulate people in their mathematical activities.

A new class of Banach algebra valued functions is identified for which the logarithmic residue with respect to a Cauchy domain Δ vanishes (if and) only if the functions take invertible values in Δ . Trace conditions and the extraction of elementary factors of the type e - p + ( λ - α ) p play an important role. The class contains the Fredholm operator valued functions and the Banach algebra valued functions possessing a simply meromorphic resolvent as special instances. An example is given to show that new ground is covered and a long standing open problem is discussed from a fresh angle.

A logarithmic residue is a contour integral of the (left or right) logarithmic derivative of an analytic Banach algebra valued function. Logarithmic residues are intimately related to sums of idempotents. The present paper is concerned with logarithmic residues in a specific Banach algebra, namely the one generated by the compact operators and the identity in the case when the underlying Banach space is infinite dimensional. The situation is more complex than encountered in previous investigations concerning logarithmic residues. Logarithmic derivatives may have essential singularities and the geometric properties of the Banach space play a role. Topological properties of the set of logarithmic residues are considered too. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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.

A connection is made between the Two Machine Flow Shop Problem (2MFSP) from job scheduling theory and the issue of quasicomplete factorization of rational matrix functions. A quasicomplete factorization is a factorization into elementary (i.e., degree one) factors such that the number of factors is minimal. For a companion based matrix function W, the number of factors in a quasicomplete factorization of W is related in a simple way to the minimum makespan of an instance J of 2MFSP which can be associated to W. As a consequence of this result, variants of the 2MFSP and other types of factorization can be related too.

In a commutative Banach algebraB the set of logarithmic residues (i.e., the elements that can be written as a contour integral of the logarithmic derivative of an analyticB-valued function), the set of generalized idempotents (i.e., the elements that are annihilated by a polynomial with non-negative integer simple zeros), and the set of sums of idempotents are all the same. Also, these (coinciding) sets consist of isolated points only and are closed under the operations of addition and multiplication. Counterexamples show that the commutativity condition onB is essential. The results extend to logarithmic residues of meromorphicB-valued functions.

A connection is made between two sets of problems. The first set involves factorization problems of specific rational matrix functions, the companion based matrix functions. The second set is concerned with variants of the two machine flow shop problem (2MFSP) from job scheduling theory. In particular, it is shown that with each companion based matrix function one can associate an instance of 2MFSP and vice versa. The latter can be done in such a way that the factorization properties of the companion based matrix function correspond to the combinatorial properties of the instance of 2MFSP.

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.

A large class of Banach algebras is identified allowing for non-trivial zero sums of idempotents, hence failing to be spectrally regular. Belonging to it are the C*-algebras known under the name Cuntz algebras. Other Banach algebras lying in the class are those of the form L(X) with X a (non-trivial) Banach space isomorphic to a (finite) direct sum of at least two copies of X. There do exist (somewhat exotic) Banach spaces for which L(X) is spectrally regular.

In non-commutative Gelfand theory, families of Banach algebra homomorphisms, and particularly families of matrix representations, play an important role.D epending on the properties imposed on them, they are called sufficient, weakly sufficient, partially weakly sufficient, radical-separating or separating.I n this paper these families are compared with one another. Conditions are given under which the defining properties amount to the same. Where applicable, examples are presented to show that they are genuinely different.

A new non-commutative Gelfand type criterion for spectrally regular behavior of vector-valued analytic functions is developed. Applications are given in situations that could not be handled with earlier methods. Some open problems are identified.

textabstractLet f be an analytic Banach algebra valued function and suppose that the contour integral of the logarithmic derivative f′f-1 around a Cauchy domain D vanishes. Does it follow that f takes invertible values on all of D? For important classes of Banach algebras, the answer is positive. In general, however, it is negative. The counterexample showing this involves a (nontrivial) zero sum of logarithmic residues (that are in fact idempotents). The analysis of such zero sums leads to results about the convex cone generated by the logarithmic residues.

The problem treated in this paper is the following.Let p 1,...,p k be idempotents in a Banach algebra B, and assume p 1+...+p k =0.Does it follow that p j =0,j=1,..., k? For important classes of Banach algebras the answer turns out to be positive; in general, however, it is negative. A counterexample is given involving five nonzero bounded projections on infinite-dimensional separable Hilbert space. The number five is critical here: in Banach algebras nontrivial zero sums of four idempotents are impossible. In a purely algebraic context (no norm), the situation is different. There the critical number is four.

As we know from the previous part there is an intimate connection between canonical factorization and Riccati equations. In this chapter this connection is developed further for the case when the rational matrix functions involved have Hermitian values on the imaginary axis. In this case the corresponding Riccati equation has additional symmetry properties too.

As we have seen in Chapter 1, canonical factorization serves as a tool to solve Wiener-Hopf integral equations, their discrete analogues, and the more general singular integral equations. In this chapter the state space factorization method developed in Chapter 2 is used to solve the problem of canonical factorization (necessary and sufficient conditions for its existence) and to derive explicit formulas for its factors. This is done in Section 3.1 for rational matrix functions and later in Section 7.1 for operator-valued transfer functions that are analytic on an open neighborhood of a curve. The results are applied to invert Wiener-Hopf integral equations with a rational matrix symbol (Section 3.2), block Toeplitz operators (Section 3.3) and singular integral equations (Section 3.4). The methods developed in this chapter also allow us to deal with the Riemann-Hilbert boundary value problem. This is done in the final section which also contains material on the homogeneous Wiener-Hopf equation.

In this chapter we consider rational matrix functions on a contour having values that are selfadjoint matrices, but not necessarily positive definite ones. Whereas in the previous chapter we studied spectral factorization, in the present chapter the focus will be on functions that have poles or zeros on the contour, and so we will consider pseudo-spectral factorization here.

This chapter describes in detail the elements of the state space method that are used throughout this book. The central notion is that of a realization of a matrix or operator function. The chapter consists of six sections. Section 2.1 presents preliminaries on realization, including the relevant definitions and the connection with systems theory. In the next two sections the realization problem is discussed. First for rational matrix functions in Section 2.2, and then for analytic operator functions in a possibly infinite dimensional setting in Section 2.3. The last three sections are devoted to the main operations on realizations that are needed in this book: inversion (Section 2.4), taking products (Section 2.5), and factorization (Section 2.6).

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.

In this chapter we present some background material on matrices in indefinite inner product spaces, and review the main results from this area that are used in this book. No proofs will be provided; we refer to the literature for more information. Good sources are [68] and [70]. The material is not only useful for understanding of the results of the preceding two chapters, but is also intended for use in subsequent chapters.

In this chapter rational matrix functions are studied of which the values on the imaginary axis or on the real line are contractive matrices. Included are solutions to spectral or canonical factorization problems for functions V of the form $$ V(\lambda ) = I - W( - \bar \lambda )*W(\lambda ) or V(\lambda ) = I + W(\lambda ), $$ where W is a rational matrix function which has contractive values on the imaginary axis or on the real line and, in addition, has a strictly contractive value at infinity.

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.

In this chapter the canonical factorization theorem from Section 7.1 is presented in a different way using the notion of an angular subspace and Riccati equations. In this case one has to look for solutions of the Riccati equation that have additional spectral properties. Section 12.1, which has a preliminary character, deals with angular subspaces, and in particular those that are also spectral subspaces. Section 12.2 deals with the connection between factorization and Riccati equations in general, while Section 12.3 contains the main result. It specifies further the main theorem of the second section for the case of canonical factorization. In Section 12.4, as an application, we solve in state space form the problem of obtaining a right canonical factorization when a left one is given (or reversely).

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.

In this chapter we gather together several results concerning minimal realizations and minimal factorizations that will play an important role in the sequel. Most of these results can also be found in Part II of the book [20]. For the reader’s convenience we have chosen to summarize them here (without proofs). Special attention is given to the notion of pseudo-canonical factorization, which is a generalization of canonical factorization by allowing singularities on the curve.

In this chapter we treat the problem of factorizing a non-proper rational matrix function. The realization used in the earlier chapters is replaced by $$ V(\lambda ) = I + C(\lambda G - A)^{ - 1} B. $$ Here I=I m is the m × m identity matrix, A and G are square matrices of order n say, and the matrices C and B are of sizes m × n and n × m, respectively. Any rational m × m matrix function W, proper or non-proper, admits such a representation. The representation (4.1) allows us to extend the results obtained in the previous chapter to arbitrary rational matrix functions. As an application we treat the problem to invert a singular integral operator with a rational matrix symbol.

The focus of the chapter is on a part of control theory called H-infinity control. The problem involved is the general H-infinity control problem, the so-called standard problem. It concerns the construction of a stabilizing controller with additional constraints on the maximum of the norm of the closed loop transfer function, taken over the values of the argument on the imaginary line. In its simplest form the problem is equivalent to the rational matrix Nehari problem considered in Chapter 18. The label H-infinity is related to the fact that a proper rational matrix function is stable if and only if it is analytic and uniformly bounded in the open right half plane. A function with the latter properties is usually referred to as an H∞-function (on the right half plane).

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.

In this chapter the rational matrix version of the Nehari problem (relative to the imaginary axis) is solved using a J-spectral factorization approach. The data of the problem are given in realized form. This together with the state space results on J-spectral factorization derived in Chapter 14 allows us to solve the problem and to obtain an explicit linear fractional representation of all its solutions, again in realized form. The main attention is given to the so-called suboptimal case. The more general Nehari-Takagi problem is also solved using the J-spectral factorization method.

This chapter is concerned with canonical factorization (with respect to the real line) of rational matrix functions with a positive definite real part on the real line. Also the generalization to pseudo-canonical factorization for functions that have a nonnegative real part is developed. All factorizations are obtained explicitly using state space realizations of the functions involved. In Section 15.1 rational matrix functions that have a positive definite real part or a nonnegative real part on the real line are characterized in terms of realizations. Section 15.2 deals with canonical factorization, and Section 15.3, the final section of the chapter, with pseudo-canonical factorization.

In this chapter we continue the study of rational matrix functions that take Hermitian values on certain contours. In contrast to the previous chapters, the emphasis will not be on positive definite or nonnegative rational matrix functions, but rather on ones that have values for which the inertia is independent of the point on the contour. Such functions may still admit a symmetric canonical factorization, provided we allow for a constant Hermitian invertible matrix as a middle factor. Such a factorization is commonly known as a J-spectral factorization.We shall give necessary and sufficient conditions for its existence, and study the question when a function which admits a left J-spectral factorization also admits a right J-spectral factorization.

In this chapter a brief survey is given of a number of basic elements of control and mathematical systems theory. The main aim is to give the reader some understanding for the type of problems that will be treated in the final chapter.

This chapter deals with m × m matrix-valued functions of the form $$ W(\lambda ) = I - \int_{ - \infty }^\infty {e^{i\lambda t} k(t)dt,} $$ (5.1) where k is an m × m matrix-valued function with the property that for some ω < 0 the entries of e−ω|t|k(t) are Lebesgue integrable on the real line. In other words, k is of the form $$ k(t) = e^{\omega |t|} h(t) with h \in L_1^{m \times m} (\mathbb{R}). $$ (5.2) It follows that the function W is analytic in the strip \( \left| {\mathfrak{F}\lambda } \right| \), where τ=−ω. This strip contains the real line. The aim is to extend the canonical factorization theorem of Chapter 5 to functions of the type (5.1).

In this chapter realizations are used to study rational matrix functions of which the values on the imaginary axis are J-unitary matrices. Solutions to various factorization problems are given. Special attention is paid to factorization of J-unitary rational matrix functions into J-unitary factors. We also discuss the problem of embedding a contractive rational matrix function as the (1, 2) block in a unitary rational matrix function. The latter problem is related to the Darlington synthesis problem from network theory.