Linear transformations that are tridiagonal with respect to both eigenbases of a Leonard pair

Let V denote a vector space with finite positive dimension. We consider a pair of linear transformations A : V → V and A^∗ : V → V that satisfy (i) and (ii) below:

(i)

There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

We call such a pair a Leonard pair on V. Let X denote the set of linear transformations X : V → V such that the matrix representing X with respect to the basis (i) is tridiagonal and the matrix representing X with respect to the basis (ii) is tridiagonal. We show that X is spanned by

     

Balanced Leonard pairs

Let K denote a field, and let V denote a vector space over K with finite positive dimension. By a Leonard pair on V we mean an ordered pair of linear transformations A : V → V and A^∗ : V → V that satisfy the following two conditions:

(i)

There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

Let (respectively v₀, v₁, … , v_d) denote a basis for V that satisfies (i) (respectively (ii)). For 0 ? i ? d, let a_i denote the coefficient of , when we write as a linear combination of , and let denote the coefficient of v_i, when we write A^∗v_i as a linear combination of v₀, v₁, … , v_d.In this paper we show a₀ = a_d if and only if . Moreover we show that for d ? 1 the following are equivalent; (i) a₀ = a_d and a₁ = a_d−1; (ii) and ; (iii) a_i = a_d−i and for 0 ? i ? d. These give a proof of a conjecture by the second author. We say A, A^∗ is balanced whenever a_i = a_d−i and for 0 ? i ? d. We say A,A^∗ is essentially bipartite (respectively essentially dual bipartite) whenever a_i (respectively ) is independent of i for 0 ? i ? d. Observe that if A, A^∗ is essentially bipartite or dual bipartite, then A, A^∗ is balanced. For d ≠ 2, we show that if A, A^∗ is balanced then A, A^∗ is essentially bipartite or dual bipartite.     

The structure of a tridiagonal pair

Let K denote a field and let V denote a vector space over K with finite positive dimension.We consider a pair of K-linear transformations A:V→V and A^∗:V→V that satisfy the following conditions: (i) each of A,A^∗ is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A^∗V_i⊆V_i-1+V_i+V_i+1 for 0?i?d, where V_-1=0 and V_d+1=0; (iii) there exists an ordering of the eigenspaces of A^∗ such that for 0?i?δ, where and ; (iv) there is no subspace W of V such that AW⊆W,A^∗W⊆W,W≠0,W≠V.We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0?i?d the dimensions of coincide.In this paper we show that the following (i)-(iv) hold provided that K is algebraically closed: (i) Each of has dimension 1.(ii) There exists a nondegenerate symmetric bilinear form 〈,〉 on V such that 〈Au,v〉=〈u,Av〉 and 〈A^∗u,v〉=〈u,A^∗v〉 for all u,v∈V.(iii) There exists a unique anti-automorphism of End(V) that fixes each of A,A^∗.(iv) The pair A,A^∗ is determined up to isomorphism by the data , where θ_i (resp.) is the eigenvalue of A (resp.A^∗) on V_i (resp.), and is the split sequence of A,A^∗ corresponding to and .     

Some trace formulae involving the split sequences of a Leonard pair

Let K denote a field, and let V denote a vector space over K with finite positive dimension. We consider a pair of linear transformations A : V → V and A^∗ : V → V that satisfy (i) and (ii) below:

(i)

There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

We call such a pair a Leonard pair on V. Let diag(θ₀, θ₁, … , θ_d) denote the diagonal matrix referred to in (ii) above and let denote the diagonal matrix referred to in (i) above. It is known that there exists a basis u₀, u₁, … , u_d for V and there exist scalars ?₁, ?₂, … , ?_d in K such that Au_i = θ_iu_i + u_i+1 (0 ? i ? d − 1), Au_d = θ_du_d, , . The sequence ?₁, ?₂, … , ?_d is called the first split sequence of the Leonard pair. It is known that there exists a basis v₀, v₁, … , v_d for V and there exist scalars ?₁, ?₂, … , ?_d in K such that Av_i = θ_d−iv_i + v_i+1 (0 ? i ? d − 1),Av_d = θ₀v_d, , . The sequence ?₁, ?₂, … , ?_d is called the second split sequence of the Leonard pair. We display some attractive formulae for the first and second split sequence that involve the trace function.     

A characterization of Leonard pairs using the notion of a tail

Let V denote a vector space with finite positive dimension. We consider an ordered pair of linear transformations A:V→V and A^∗:V→V that satisfy (i) and (ii) below:

(i)

There exists a basis for V with respect to which the matrix representing A is irreducible tridiagonal and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A^∗ is irreducible tridiagonal and the matrix representing A is diagonal.

We call such a pair a Leonard pair on V. In this paper, we characterize the Leonard pairs using the notion of a tail. This notion is borrowed from algebraic graph theory.     

A classification of sharp tridiagonal pairs

Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A:V→V and A^∗:V→V that satisfy the following conditions: (i) each of A,A^∗ is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A^∗V_i⊆V_i-1+V_i+V_i+1 for 0?i?d, where V_-1=0 and V_d+1=0; (iii) there exists an ordering of the eigenspaces of A^∗ such that for 0?i?δ, where and ; (iv) there is no subspace W of V such that AW⊆W, A^∗W⊆W, W≠0, W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0?i?d the dimensions of coincide. The pair A,A^∗ is called sharp whenever . It is known that if F is algebraically closed then A,A^∗ is sharp. In this paper we classify up to isomorphism the sharp tridiagonal pairs. As a corollary, we classify up to isomorphism the tridiagonal pairs over an algebraically closed field. We obtain these classifications by proving the μ-conjecture.     

Normalized Leonard pairs and Askey-Wilson relations

Let V denote a vector space with finite positive dimension, and let (A, A^∗) denote a Leonard pair on V. As is known, the linear transformations A, A^∗ satisfy the Askey-Wilson relations

     

Thin Hessenberg pairs

A square matrix is called Hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero. Let V denote a nonzero finite-dimensional vector space over a field K. We consider an ordered pair of linear transformations A:V→V and A^∗:V→V which satisfy both (i) and (ii) below.

(i)

There exists a basis for V with respect to which the matrix representing A is Hessenberg and the matrix representing A^∗ is diagonal.

(ii)

There exists a basis for V with respect to which the matrix representing A is diagonal and the matrix representing A^∗ is Hessenberg.

We call such a pair a thin Hessenberg pair (or TH pair). This is a special case of a Hessenberg pair which was introduced by the author in an earlier paper. We investigate several bases for V with respect to which the matrices representing A and A^∗ are attractive. We display these matrices along with the transition matrices relating the bases. We introduce an “oriented” version of called a TH system. We classify the TH systems up to isomorphism.     

Some properties of sharp Hessenberg pairs

Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider an ordered pair of F-linear transformations A:V→V and A^∗:V→V that satisfy the following conditions: (i) each of A,A^∗ is diagonalizable on V; (ii) there exists an ordering of the eigenspaces of A such that A^∗V_i⊆V₀+V₁+?+V_i+1 for 0?i?d, where V_-1:=0 and V_d+1:=0; (iii) there exists an ordering of the eigenspaces of A^∗ such that for 0?i?δ, where and . We call such a pair a Hessenberg pair on V. It is known that if the Hessenberg pair A,A^∗ on V is irreducible then d=δ and for 0?i?d the dimensions of V_i and coincide. We say a Hessenberg pair A,A^∗ on V is sharp whenever it is irreducible and .In this paper, we give the definitions of a Hessenberg system and a sharp Hessenberg system. We discuss the connection between a Hessenberg pair and a Hessenberg system. We also define a finite sequence of scalars called the parameter array for a sharp Hessenberg system, which consists of the eigenvalue sequence, the dual eigenvalue sequence and the split sequence. We calculate the split sequence of a sharp Hessenberg system. We show that a sharp Hessenberg pair is a tridiagonal pair if and only if there exists a nonzero nondegenerate bilinear form on V that satisfies 〈Au,v〉=〈u,Av〉 and 〈A^∗u,v〉=〈u,A^∗v〉 for all u,v∈V.     

Askey-Wilson relations and Leonard pairs

It is known that if (A,A^*) is a Leonard pair, then the linear transformations A, A^* satisfy the Askey-Wilson relations

     

The asymptotics of the generalised Hermite-Bell polynomials

The Hermite-Bell polynomials are defined by for n=0,1,2,… and integer r≥2 and generalise the classical Hermite polynomials corresponding to r=2. We obtain an asymptotic expansion for as n→∞ using the method of steepest descents. For a certain value of x, two saddle points coalesce and a uniform approximation in terms of Airy functions is given to cover this situation. An asymptotic approximation for the largest positive zeros of is derived as n→∞. Numerical results are presented to illustrate the accuracy of the various expansions.     

A Morita type equivalence for dual operator algebras

We generalize the main theorem of Rieffel for Morita equivalence of W^∗-algebras to the case of unital dual operator algebras: two unital dual operator algebras A,B have completely isometric normal representations α,β such that α(A)=[M^∗β(B)M]^−w∗ and β(B)=[Mα(A)M^∗]^−w∗ for a ternary ring of operators M (i.e. a linear space M such that MM^∗M⊂M) if and only if there exists an equivalence functor which “extends” to a ∗-functor implementing an equivalence between the categories and . By we denote the category of normal representations of A and by the category with the same objects as and Δ(A)-module maps as morphisms (Δ(A)=A∩A^∗). We prove that this functor is equivalent to a functor “generated” by a B,A bimodule, and that it is normal and completely isometric.     

Spanning multi-paths in hypercubes

A set A of vertices of a hypercube is called balanced if . We prove that for every natural number n there exists a natural number π₁(n) such that for every hypercube Q with dim(Q)?π₁(n) there exists a family of pairwise vertex-disjoint paths P_i between A_i and B_i for i=1,2,…,n with if and only if {A_i,B_i∣i=1,2,…,n} is a balanced set.     

Tridiagonal matrices with nonnegative entries

In this paper, we characterize the nonnegative irreducible tridiagonal matrices and their permutations, using certain entries in their primitive idempotents. Our main result is summarized as follows. Let d denote a nonnegative integer. Let A denote a matrix in R and let denote the roots of the characteristic polynomial of A. We say A is multiplicity-free whenever these roots are mutually distinct and contained in R. In this case E_i will denote the primitive idempotent of A associated with theta_i(0?i?d). We say A is symmetrizable whenever there exists an invertible diagonal matrix Δ∈R such that ΔAΔ^-1 is symmetric. Let Γ(A) denote the directed graph with vertex set {0,1,…,d}, where i→j whenever i≠j and A_ij≠0.Theorem.Assume that each entry ofAis nonnegative. Then the following are equivalent for0≤s,t≤d.

(i)

The graphΓ(A)is a bidirected path with endpointss,t:s↔*↔*↔?↔*↔t.

(ii)

The matrixAis symmetrizable and multiplicity-free. Moreover the(s,t)-entry ofE_itimes(θ_i-θ₀)?(θ_i-θ_i-1)(θ_i-θ_i+1)?(θ_i-θ_d)is independent of i for0≤i≤d, and this common value is nonzero.

Recently Kurihara and Nozaki obtained a theorem that characterizes the Q-polynomial property for symmetric association schemes. We view the above result as a linear algebraic generalization of their theorem.     

Elementary maps on nest algebras

Let A₁, A₂ be algebras and let M:A₁→A₂, M^∗:A₂→A₁ be maps. An elementary map of A₁×A₂ is an ordered pair (M,M^∗) such that

     

The subconstituent algebra of a bipartite distance-regular graph; thin modules with endpoint two

We consider a bipartite distance-regular graph Γ with diameter D?4, valency k?3, intersection numbers b_i,c_i, distance matrices A_i, and eigenvalues θ₀>θ₁>?>θ_D. Let X denote the vertex set of Γ and fix x∈X. Let T=T(x) denote the subalgebra of Mat_X(C) generated by , where A=A₁ and denotes the projection onto the ith subconstituent of Γ with respect to x. T is called the subconstituent algebra (or Terwilliger algebra) of Γ with respect to x. An irreducible T-module W is said to be thin whenever for 0?i?D. By the endpoint of W we mean . Assume W is thin with endpoint 2. Observe is a one-dimensional eigenspace for ; let η denote the corresponding eigenvalue. It is known where , and d=⌊D/2⌋. To describe the structure of W we distinguish four cases: (i) ; (ii) D is odd and ; (iii) D is even and ; (iv) . We investigated cases (i), (ii) in MacLean and Terwilliger [Taut distance-regular graphs and the subconstituent algebra, Discrete Math. 306 (2006) 1694-1721]. Here we investigate cases (iii), (iv) and obtain the following results. We show the dimension of W is D-1-e where e=1 in case (iii) and e=0 in case (iv). Let v denote a nonzero vector in . We show W has a basis , where E_i denotes the primitive idempotent of A associated with θ_i and where the set S is {1,2,…,d-1}∪{d+1,d+2,…,D-1} in case (iii) and {1,2,…,D-1} in case (iv). We show this basis is orthogonal (with respect to the Hermitian dot product) and we compute the square-norm of each basis vector. We show W has a basis , and we find the matrix representing A with respect to this basis. We show this basis is orthogonal and we compute the square-norm of each basis vector. We find the transition matrix relating our two bases for W.     

Local cohomology modules with respect to an ideal containing the irrelevant ideal

Let R=?_n≥0R_n be a homogeneous Noetherian ring, let M be a finitely generated graded R-module and let R₊=?_n>0R_n. Let b?b₀+R₊, where b₀ is an ideal of R₀. In this paper, we first study the finiteness and vanishing of the n-th graded component of the i-th local cohomology module of M with respect to b. Then, among other things, we show that the set becomes ultimately constant, as n→−∞, in the following cases:

(i)

and (R₀,m₀) is a local ring;

(ii)

dim(R₀)≤1 and R₀ is either a finite integral extension of a domain or essentially of finite type over a field;

(iii)

i≤g_b(M), where g_b(M) denotes the cohomological finite length dimension of M with respect to b.

Also, we establish some results about the Artinian property of certain submodules and quotient modules of .     

On transitive Lie bialgebroids and Poisson groupoids

We prove that, for any transitive Lie bialgebroid (A, A^∗), the differential associated to the Lie algebroid structure on A^∗ has the form d_∗=A[Λ,⋅]+Ω, where Λ is a section of ^∧2A and Ω is a Lie algebroid 1-cocycle for the adjoint representation of A. Globally, for any transitive Poisson groupoid (Γ,Π), the Poisson structure has the form , where ΠF is a bivector field on Γ associated to a Lie groupoid 1-cocycle.     

Singular-value-like decomposition for complex matrix triples

The classical singular value decomposition for a matrix A∈C^m×n is a canonical form for A that also displays the eigenvalues of the Hermitian matrices AA^∗ and A^∗A. In this paper, we develop a corresponding decomposition for A that provides the Jordan canonical forms for the complex symmetric matrices and . More generally, we consider the matrix triple , where are invertible and either complex symmetric or complex skew-symmetric, and we provide a canonical form under transformations of the form , where X,Y are nonsingular.     

The extension of the Krein-Šmulian theorem for Orlicz sequence spaces and convex sets

If X is a Banach space and C⊂X∗∗ a convex subset, for x∗∗∈X∗∗ and A⊂X∗∗ let be the distance from x∗∗ to C and . In this paper we prove that if φ is an Orlicz function, I an infinite set and X=?φ(I) the corresponding Orlicz space, equipped with either the Luxemburg or the Orlicz norm, then for every w^∗-compact subset K⊂X∗∗ we have if and only if φ satisfies the Δ₂-condition at 0. We also prove that for every Banach space X, every nonempty convex subset C⊂X and every w^∗-compact subset K⊂X∗∗ then and, if K∩C is w^∗-dense in K, then .