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We introduce the notion of one-sided mapping cones of positive linear maps between matrix algebras. These are convex cones of maps that are invariant under compositions by completely positive maps from either the left or right side. The duals of such convex cones can be characterized in terms of ampliation maps, which can also be used to characterize many notions from quantum information theory—such as separability, entanglement-breaking maps, Schmidt numbers, as well as decomposable maps and k -positive maps in functional analysis. In fact, such characterizations hold if and only if the involved cone is a one-sided mapping cone. Through this analysis, we obtain mapping properties for compositions of cones from which we also obtain several equivalent statements of the PPT (positive partial transpose) square conjecture. [ABSTRACT FROM AUTHOR]

Abstract If a continuous function f : M n (C) → M n (C) satisfies f (x) x = x f (x) for all x ∈ M n (C) , then there exist functions a 0 , a 1 , ... , a n − 1 : M n (C) → C such that f (x) = ∑ j = 0 n − 1 a j (x) x j for all x ∈ M n (C). Moreover, a j are continuous on the set of all non-derogatory matrices. [ABSTRACT FROM AUTHOR]

Abstract We study some "almost preserver" problems on von Neumann algebra modules. More precisely, we study (1) maps which "almost preserve" the right or left annihilator; (2) the "almost band preservers" – that is, maps which "almost preserve corners", and (3) "almost centralizers," which almost preserve module actions. Under certain conditions, we show that maps of these types are automatically continuous, and can be approximated by maps which precisely preserve these relations; often, the operators from the latter class are multiplication operators. [ABSTRACT FROM AUTHOR]

Our aim in this paper is to give a new formula of the joint spectral radius of commuting d -tuples of operators on a complex Hilbert space. Also we show that r ( T ) ≤ ω ( T ) for a commuting d -tuple of operators T , where r ( T ) and ω ( T ) denote respectively the joint spectral radius and the joint numerical radius of T . This generalizes the well known relation between the spectral and the numerical radii of Hilbert space operators which is proved in [10] . [ABSTRACT FROM AUTHOR]

We prove that the commutative nearly absolute valued division algebras have dimension ≤2. This extends a well-known theorem by H. Hopf asserting the same property for each commutative real division algebra of finite dimension. As an application we prove that R , C and C ⋆ are the unique commutative real nearly absolute valued division algebras with norm satisfying ‖ x 2 ‖ = ‖ x ‖ 2 . [ABSTRACT FROM AUTHOR]

This paper presents a new family of methods for evaluating matrix polynomials more efficiently than the state-of-the-art Paterson–Stockmeyer method. Examples of the application of the methods to the Taylor polynomial approximation of matrix functions like the matrix exponential and matrix cosine are given. Their efficiency is compared with that of the best existing evaluation schemes for general polynomial and rational approximations, and also with a recent method based on mixed rational and polynomial approximants. For many years, the Paterson–Stockmeyer method has been considered the most efficient general method for the evaluation of matrix polynomials. In this paper we show that this statement is no longer true. Moreover, for many years rational approximations have been considered more efficient than polynomial approximations, although recently it has been shown that often this is not the case in the computation of the matrix exponential and matrix cosine. In this paper we show that in fact polynomial approximations provide a higher order of approximation than the state-of-the-art computational methods for rational approximations for the same cost in terms of matrix products. [ABSTRACT FROM AUTHOR]

Based on Stokes' theorem we derive a non-holomorphic functional calculus for matrices, assuming sufficient smoothness near eigenvalues, corresponding to the size of related Jordan blocks. It is then applied to the complex conjugation function τ : z ↦ z ‾ . The resulting matrix agrees with the hermitian transpose if and only if the matrix is normal. Two other, as such elementary, approaches to define the complex conjugate of a matrix yield the same result. [ABSTRACT FROM AUTHOR]

The characterization of bounded essentially semi regular operators in terms of Kato decomposition of finite type was studied by several authors. In this paper, we extend this characterization to the case of essentially semi regular linear relations. We also give other characterization of such linear relations. Further, we apply the obtained results to analyse the stability of the class of essentially semi regular linear relations under additional operator perturbations. Finally, as an application, we get some useful connections between the Fredholm spectrum and the essentially semi regular spectrum of linear relations. [ABSTRACT FROM AUTHOR]

The theory of Generalized Locally Toeplitz (GLT) sequences of matrices has been developed in order to study the asymptotic behaviour of particular spectral distributions when the dimension of the matrices tends to infinity. Key concepts in this theory are the notions of Approximating Classes of Sequences (a.c.s.) and spectral symbols that lead to defining a metric structure on the space of matrix sequences and provide a link with the measurable functions. In this paper we prove additional results regarding theoretical aspects, such as the completeness of the matrix sequences space with respect to the metric a.c.s. and the identification of the space of GLT sequences with the space of measurable functions. [ABSTRACT FROM AUTHOR]

In this paper we study sufficient conditions for an operator to have an almost-invariant half-space. As a consequence, we show that if X is an infinite-dimensional complex Banach space then every operator T ∈ L ( X ) admits an essentially-invariant half-space. We also show that whenever a closed algebra of operators possesses a common AIHS, then it has a common invariant half-space as well. [ABSTRACT FROM AUTHOR]

Basing on recent results on the stability of collocation methods applied to Cauchy singular integral equations with additional fixed singularities we give necessary and sufficient conditions for the stability of collocation–quadrature methods for such equations. These methods have the advantage that the respective system of equations has a very simple structure and allows to apply fast summation methods which results in a fast algorithm with O ( n log ⁡ n ) complexity. We present numerical results of the application of the proposed collocation–quadrature methods to the notched half plane problem of two-dimensional elasticity theory. [ABSTRACT FROM AUTHOR]

In this paper we consider recent work linking “convex majorization” and isotonic operators on ℓ p ( I ) , where I is assumed to be an infinite set and p ∈ [ 1 , + ∞ ) . We proved that P c , the set of all isotonic operators on ℓ p ( I ) is closed. We also provided some other properties about isotonic operators. [ABSTRACT FROM AUTHOR]

In this paper we consider a simple form for pairs of matrices ( A , H ) , where H is a real symmetric invertible matrix, and A is a real H -expansive matrix, that is, A T H A − H is positive semidefinite. For such pairs we use the simple form to give an explicit construction of real A -invariant maximal H -semidefinite subspaces. [ABSTRACT FROM AUTHOR]

Abstract: In the present paper we study different conditions regarding the sum of operator ranges. In particular, we find conditions on operators A, B which imply that and derive some results concerning projections. [Copyright &y& Elsevier]

Abstract: We study the number of non-isomorphic functional graphs of affine-linear transformations from to itself, and we prove upper and lower bounds on this quantity as n→∞. As a corollary to our result, we prove bounds on the number of conjugacy classes in the symmetric group that intersect AGL n (q). [Copyright &y& Elsevier]

Abstract: 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. [Copyright &y& Elsevier]

Abstract: In this paper, we analyze the eigenfunctions of the edge-based Laplacian on a graph and the relationship of these functions to random walks on the graph. We commence by discussing the set of eigenfunctions supported at the vertices, and demonstrate the relationship of these eigenfunctions to the classical random walk on the graph. Then, from an analysis of functions supported only on the interior of edges, we develop a method for explicitly calculating the edge-interior eigenfunctions of the edge-based Laplacian. This reveals a connection between the edge-based Laplacian and the adjacency matrix of backtrackless random walk on the graph. The edge-based eigenfunctions therefore correspond to some eigenfunctions of the normalized Hashimoto matrix. [Copyright &y& Elsevier]

Abstract: A generalization of Powers–Størmer’s inequality for operator monotone functions on [0,+∞) and for positive linear functional on general C ∗-algebras will be proved. It also will be shown that the generalized Powers–Størmer inequality characterizes the tracial functionals on C ∗-algebras. [Copyright &y& Elsevier]

Abstract: Let be a commutative ring with identity, be unital algebras over and M be a unital -bimodule, which is faithful as a left A-module and also as a right B-module. Let be the triangular algebra consisting of and M. This work is motivated by some intensive works of Brešar , Cheung and Zhang et al. . Here, we study Lie triple derivations of . It is shown that under mild assumptions, every Lie triple derivation on is of standard form. That is, L can be expressed through an additive derivation and a linear functional vanishing on all second commutators of . Examples of Lie triple derivations on some classical triangular algebras are supplied. [Copyright &y& Elsevier]

Abstract: We present several extensions of Burnside’s well-known theorem which states that the only irreducible subalgebra of with algebraically closed field F is itself. We show, among some stronger results, that if F is quasi-algebraically closed (in particular, if F is finite), then the only irreducible subalgebra of that contains a linear basis of triangularizable matrices (a hypothesis that automatically holds in the classical case) is itself. We also consider the problem of “field of definition” for a semigroup in : If a linear functional on takes values in a smaller field F, is simultaneously similar to a semigroup in ? [Copyright &y& Elsevier]

Abstract: In this paper, we derive two versions of divided difference form of the Lagrange–Bürmann expansion formula. The first version expresses all coefficients by a form of a determinant. By means of the factorizations of the generalized Leibniz functional matrices with respect to a sequence, we obtain the second version with coefficients represented by a finite sum. Some new matrix inversions are also investigated. [Copyright &y& Elsevier]

Abstract: We consider reconstruction systems (RS’s), which are G-frames in a finite dimensional setting, and that includes the fusion frames as projective RS’s. We describe the spectral picture of the set of RS operators for the projective systems with fixed weights. We also introduce a functional defined on dual pairs of RS’s, called the joint potential, and study the structure of the minimizers of this functional. In the case of irreducible RS’s the minimizers are characterize as the tight systems. In the general case we give spectral and geometric characterizations of the minimizers of the joint potential. At the end of the paper we show several examples that illustrate our results. [Copyright &y& Elsevier]

Abstract: This paper addresses exponential stability problem for a class of linear systems with time-varying delay. The time delay is assumed to be a continuous function belonging to a given interval, but not necessary to be differentiable. By constructing a set of augmented Lyapunov–Krasovskii functional combined with the Newton–Leibniz formula technique, new delay-dependent sufficient conditions for the exponential stability of the systems are first established in terms of linear matrix inequalities (LMIs). An application to exponential stability of uncertain linear systems with interval time-varying delay is given. Numerical examples are given to show the effectiveness of the obtained results. [Copyright &y& Elsevier]

Abstract: We introduce a new invariant, the coronal of a graph, and use it to compute the spectrum of the corona of two graphs G and H. In particular, we show that this spectrum is completely determined by the spectra of G and H and the coronal of H. Previous work has computed the spectrum of a corona only in the case that H is regular. We then explicitly compute the coronals for several families of graphs, including regular graphs, complete n-partite graphs, and paths. Finally, we use the corona construction to generate many infinite families of pairs of cospectral graphs. [Copyright &y& Elsevier]

Abstract: Suppose T and S are bounded adjointable operators with close range between Hilbert -modules, then TS has closed range if and only if is an orthogonal summand, if and only if is an orthogonal summand. Moreover, if the Dixmier (or minimal) angle between and is positive and is an orthogonal summand then TS has closed range. [Copyright &y& Elsevier]

Abstract: We introduce a notion of chain of evolution algebras. The sequence of matrices of the structural constants for this chain of evolution algebras satisfies an analogue of Chapman–Kolmogorov equation. We give several examples (time homogenous, time non-homogenous, periodic, etc.) of such chains. For a periodic chain of evolution algebras we construct a continuum set of non-isomorphic evolution algebras and show that the corresponding discrete time chain of evolution algebras is dense in the set. We obtain a criteria for an evolution algebra to be baric and give a concept of a property transition. For several chains of evolution algebras we describe the behavior of the baric property depending on the time. For a chain of evolution algebras given by the matrix of a two-state evolution we define a baric property controller function and under some conditions on this controller we prove that the chain is not baric almost surely (with respect to Lebesgue measure). We also construct examples of the almost surely baric chains of evolution algebras. We show that there are chains of evolution algebras such that if it has a unique (resp. infinitely many) absolute nilpotent element at a fixed time, then it has unique (resp. infinitely many) absolute nilpotent element any time; also there are chains of evolution algebras which have not such property. For an example of two dimensional chain of evolution algebras we give the full set of idempotent elements and show that for some values of parameters the number of idempotent elements does not depend on time, but for other values of parameters there is a critical time such that the chain has only two idempotent elements if time and it has four idempotent elements if time . [Copyright &y& Elsevier]

Abstract: In this paper we discuss Weyl matrix balls in the context of the matricial versions of the classical interpolation problems named after Carathéodory and Schur. Our particular focus will be on studying the monotonicity of suitably normalized semi-radii of the corresponding Weyl matrix balls. We, furthermore, devote a fair bit of attention to characterizing the case in which equality holds for particular matricial inequalities. Solving these problems will provide us with a new perspective on the role of the central functions for the classes of Carathéodory and Schur. [Copyright &y& Elsevier]

Abstract: Associated with an matrix polynomial of degree , are the eigenvalue problem and the linear system problem , where in the latter case is to be computed for many values of the parameter . Both problems can be solved by conversion to an equivalent problem or that is linear in the parameter or . This linearization process has received much attention in recent years for the eigenvalue problem, but it is less well understood for the linear system problem. We develop a framework in which more general versions of both problems can be analyzed, based on one-sided factorizations connecting a general nonlinear matrix function to a simpler function , typically a polynomial of degree 1 or 2. Our analysis relates the solutions of the original and lower degree problems and in the linear system case indicates how to choose the right-hand side and recover the solution from . For the eigenvalue problem this framework includes many special cases studied in the literature, including the vector spaces of pencils and recently introduced by Mackey, Mackey, Mehl, and Mehrmann and a class of rational problems. We use the framework to investigate the conditioning and stability of the parametrized linear system and thereby study the effect of scaling, both of the original polynomial and of the pencil . Our results identify situations in which scaling can potentially greatly improve the conditioning and stability and our numerical results show that dramatic improvements can be achieved in practice. [Copyright &y& Elsevier]

Abstract: In Markov chain models in finance and healthcare a transition matrix over a certain time interval is needed but only a transition matrix over a longer time interval may be available. The problem arises of determining a stochastic pth root of a stochastic matrix (the given transition matrix). By exploiting the theory of functions of matrices, we develop results on the existence and characterization of matrix pth roots, and in particular on the existence of stochastic pth roots of stochastic matrices. Our contributions include characterization of when a real matrix has a real pth root, a classification of pth roots of a possibly singular matrix, a sufficient condition for a pth root of a stochastic matrix to have unit row sums, and the identification of two classes of stochastic matrices that have stochastic pth roots for all p. We also delineate a wide variety of possible configurations as regards existence, nature (primary or nonprimary), and number of stochastic roots, and develop a necessary condition for existence of a stochastic root in terms of the spectrum of the given matrix. [Copyright &y& Elsevier]

Abstract: The problems of perturbation and expression for the generalized inverses of closed linear operators in Banach spaces and for the Moore–Penrose inverses of closed linear operators in Hilbert spaces are studied. We first provide some stability characterizations of generalized inverses of closed linear operators under -bounded perturbation in Banach spaces, which are exactly equivalent to that the generalized inverse of the perturbed operator has the simplest expression . Utilizing these results, we investigate the expression for the Moore–Penrose inverse of the perturbed operator in Hilbert spaces and provide a unified approach to deal with the range preserving or null space preserving perturbation. An explicit representation for the Moore–Penrose inverse of the perturbation is also given. Moreover, we give an equivalent condition for the Moore–Penrose inverse to have the simplest expression . The results obtained in this paper extend and improve many recent results in this area. [Copyright &y& Elsevier]

Abstract: Let and be an (A, B)-invariant subspace. In this paper the following results are presented: (i) If , necessary and sufficient conditions for the Lipschitz stability of are given. (ii) If contains the controllability subspace of the pair (A, B), sufficient conditions for the Lipschitz stability of the subspace are given. [Copyright &y& Elsevier]

Abstract: Decrease in absolute value of a symmetrically placed pair of off diagonal entries need not preserve positive definiteness of an matrix, A gangster operator is one that replaces some such pairs by 0s. Circumstances in which gangster operators preserve positive definiteness are investigated. Certain general circumstances are given, and graphs that ensure preservation are characterized. [ABSTRACT FROM AUTHOR]

Abstract: Our starting point is the proof of the following property of a particular class of matrices. Let be a non-negative matrix such that for each i. Suppose that for every pair of indices , there exists an index l such that . Then, there exists a real vector , such that, if . Then, we apply that property of matrices to probability theory. Let us consider an infinite sequence of linear functionals , corresponding to an infinite sequence of probability measures , on the Borel -algebra such that, . The property of matrices described above allows us to construct a real bounded one-to-one piecewise continuous and continuous from the left function f such that The relevance to quantum mechanics is showed. [Copyright &y& Elsevier]

Abstract: Let and be two fixed positive integers such that . Let be a left vector space over a division ring with dimension at least . Let be the Grassmannian consisting of all -dimensional subspaces of . We characterize surjective mappings from onto itself such that for any in , the distance between and is not greater than if and only if the distance between and is not greater than . [Copyright &y& Elsevier]

Abstract: This article is a contribution to behavioral observer theory which was started by Valcher and Willems in 1999 and which was recently exposed by Fuhrmann in a comprehensive survey article. It is also a further development of the article on -observers by Oberst and the author. For a given continuous or discrete time linear time-invariant behavior we assume that a linear function of a trajectory (e.g., some components) can be measured. We are interested in estimating another linear function of this trajectory. We generalize the notions of -observability and -observers introduced by Oberst and the author. denotes a multiplicatively closed subset of the ring of operators. For different choices of -observability coincides with observability, reconstructibility, trackability, or detectability, a -observer is an exact, dead-beat, tracking, or asymptotic observer. We show the equivalence of -observability and the existence of -observers and give a constructive parametrization of all -observers. Corresponding results for proper -observers are also presented. Partial observation of the state of a Kalman state space system (compare e.g. Fuhrmann’s work) is a special case of our setting, and so are the observers of certain unknown components of a behavior studied by Bisiacco, Valcher, and Willems. The first result on functional observers in context with Rosenbrock equations or polynomial matrix descriptions is due to Wolovich (1974). [Copyright &y& Elsevier]

Abstract: A thorough theoretical explanation of the numerical behaviour of functional iteration methods for the computation of the minimal nonnegative solution of the matrix equation , arising in the numerical solution of type Markov chains, is given in Meini (1997) [2]. In this note we add some more results. In particular, we show that an upper bound of the mean asymptotic convergence rate of the best functional iteration method is given in terms of the second largest modulus eigenvalue of . [Copyright &y& Elsevier]

Abstract: The theme of the present paper is to introduce and study two different versions of tensor products of functional models, one over the underlying field and the other over the corresponding algebra of polynomials, as well as related models based on the Kronecker product of polynomial matrices. In the process, we study the Sylvester equation and its reduction to a polynomial matrix equation. We analyse the relation between the two tensor products and use this to elucidate the role of the Anderson-Jury generalized Bezoutians in this context. [Copyright &y& Elsevier]

Abstract: Let be a prime ring of characteristic not 2, with center and with involution . Let be the set of symmetric elements of . Suppose that is an additive map such that for all . Then unless is an order in a 4-dimensional central simple algebra, there exists an additive map such that for all or for all . [Copyright &y& Elsevier]

Abstract: The dumbbell graph, denoted by , is a bicyclic graph consisting of two vertex-disjoint cycles and joined by a path having only its end-vertices in common with the two cycles. By using a new cospectral invariant for -almost regular graphs, we will show that almost all dumbbell graphs (without cycle as a subgraph) are determined by the adjacency spectrum. [Copyright &y& Elsevier]

Abstract: A matrix is nilpotent of index 2 if . Let be a space of nilpotent matrices of index over a field where and suppose that is the maximum rank of any matrix in . The object of this paper is to give an elementary proof of the fact that . We show that the inequality is sharp and construct all such subspaces of maximum dimension. We use the result to find the maximum dimension of spaces of anti-commuting matrices and zero subalgebras of special Jordan Algebras. [Copyright &y& Elsevier]

Abstract: Let and be positive semidefinite matrices. We investigate the conditions under which the Lieb–Thirring inequality can be extended to singular values, i.e. for which values of the majorisation holds, and for which values its reversed inequality . [Copyright &y& Elsevier]

Abstract: Specific algorithms, such as those involving the supremal of the invariant subspaces contained in a suitable subspace, are known to be able to test whether a Disturbance Decoupling Problem (DDP) is solvable. Here, by reducing the system to its Molinari form, we obtain an alternative description of this supremal object and compute its dimension. Hence we have a general result for solving the decoupling provided that a Molinari basis is known. In particular, a necessary numerical condition for it is derived. The same technique is applied to the DDPS, that is, when stability of the decoupled closed loop system is required. [Copyright &y& Elsevier]

Abstract: We consider the class of normal complex matrices that commute with their complex conjugate. We show that such matrices are real orthogonally similar to a canonical direct sum of 1-by-1 and certain 2-by-2 matrices. A canonical form for quasi-real normal matrices is obtained as a special case. We also exhibit a special form of the spectral theorem for normal matrices that commute with their conjugate. [Copyright &y& Elsevier]

Abstract: Constacyclic codes are generalizations of the familiar linear cyclic codes. In this paper constacyclic codes over a finite field F are regarded as invariant subspaces of with respect to a suitable linear operator. By applying standard techniques from linear algebra one can derive properties of these codes which generalize several well-known results for cyclic codes, such as the various lower bounds for the minimum distance. [Copyright &y& Elsevier]

Abstract: We extend Liu’s fundamental theorem of the geometry of alternate matrices to the second exterior power of an infinite dimensional vector space and also use her theorem to characterize surjective mappings T from the vector space V of all alternate matrices over a field with at least three elements onto itself such that for any pair A, B in V, if and only if , where k is a fixed positive integer such that and . [Copyright &y& Elsevier]

Abstract: Let denote a field and let V denote a vector space over with finite positive dimension. We consider a pair of linear transformations and that satisfy the following conditions: (i) each of is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that for , where and ; (iii) there exists an ordering of the eigenspaces of such that for , where and ; (iv) there is no subspace W of V such that , . We call such a pair a tridiagonal pair on V. It is known that and for the dimensions of coincide. We say the pair is sharp whenever . It is known that if is algebraically closed then is sharp. A conjectured classification of the sharp tridiagonal pairs was recently introduced by T. Ito and the second author. We present a result which supports the conjecture. Given scalars in that satisfy the known constraints on the eigenvalues of a tridiagonal pair, we define an -algebra T by generators and relations. We consider the -algebra for a certain idempotent . Let denote the polynomial algebra over involving d mutually commuting indeterminates. We display a surjective -algebra homomorphism . We conjecture that is an isomorphism. We show that this -conjecture implies the classification conjecture, and that the -conjecture holds for . [Copyright &y& Elsevier]

Abstract: Under mild trace norm assumptions on the perturbing sequence, we extend a recent perturbation result based on a theorem by Mirsky. The analysis concerns the eigenvalue distribution and localization of a generic (non-Hermitian) complex perturbation of a bounded Hermitian sequence of matrices. Some examples of application are considered, ranging from the product of Toeplitz sequences to the approximation of PDEs with given boundary conditions. A final discussion on open questions and further lines of research ends the note. [Copyright &y& Elsevier]

Abstract: In this paper we study singular values of a matrix whose one entry varies while all other entries are prescribed. In particular, we find the possible pth singular value of such a matrix, and we define explicitly the unknown entry such that the completed matrix has the minimal possible pth singular value. This in turn determines possible pth singular value of a matrix under rank one perturbation. Moreover, we determine the possible value of pth singular value of a partially prescribed matrix whose set of unknown entries has a form of a Young diagram. In particular, we give a fast algorithm for defining the completion that minimizes the pth singular value of such matrix. [Copyright &y& Elsevier]

Abstract: Let be an -dimensional minimal locally linearly dependent space of operators acting between vector spaces and . We obtain the sharp lower bound and the sharp upper bound for and give a complete description of those minimal locally linearly dependent spaces at which the upper bound is attained. [Copyright &y& Elsevier]

Abstract: Let denote a field and let V denote a vector space over with finite positive dimension. Let denote the -algebra consisting of all -linear transformations from V to V. We consider a pair that satisfy (i)–(iv) below: [(i)] Each of is diagonalizable. [(ii)] There exists an ordering of the eigenspaces of A such that for , where and . [(iii)] There exists an ordering of the eigenspaces of such that for , where and . [(iv)] There is no subspace W of V such that . We call such a pair a tridiagonal pair on V. Let denote the element of such that and for . Let (resp. ) denote the -subalgebra of generated by A (resp. ). In this paper we prove that the span of equals the span of , and that the elements of mutually commute. We relate these results to some conjectures of Tatsuro Ito and the second author that are expected to play a role in the classification of tridiagonal pairs. [Copyright &y& Elsevier]