共查询到20条相似文献,搜索用时 31 毫秒
1.
Edward Hanson 《Linear algebra and its applications》2011,435(11):2961-2970
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.
2.
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.
3.
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.
4.
Bo Hou 《Linear algebra and its applications》2011,435(8):1987-1996
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∗Vi⊆V0+V1+?+Vi+1 for 0?i?d, where V-1:=0 and Vd+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 Vi 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. 相似文献
5.
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.
6.
7.
Vedran Šego 《Linear algebra and its applications》2010,433(7):1265-1275
In this paper, we propose the two-sided hyperbolic SVD (2HSVD) for square matrices, i.e., A=UΣV[∗], where U and V[∗] are J-unitary (J=diag(±1)) and Σ is a real diagonal matrix of “double-hyperbolic” singular values. We show that, with some natural conditions, such decomposition exists without the use of hyperexchange matrices. In other words, U and V[∗] are really J-unitary with regard to J and not some matrix which is permutationally similar to matrix J. We provide full characterization of 2HSVD and completely relate it to the semidefinite J-polar decomposition. 相似文献
8.
The following properties of the Holmes space H are established:
- (i)
- H has the Metric Approximation Property (MAP).
- (ii)
- The w∗-closure of the set of extreme points of the unit ball BH∗ of the dual space H∗ is the whole ball BH∗.
9.
Douglas Farenick Vyacheslav Futorny Tatiana G. Gerasimova Vladimir V. Sergeichuk Nadya Shvai 《Linear algebra and its applications》2011,435(6):1356-1369
Each square complex matrix is unitarily similar to an upper triangular matrix with diagonal entries in any prescribed order. Let A=[aij] and B=[bij] be upper triangular n×n matrices that
- •
- are not similar to direct sums of square matrices of smaller sizes, or
- •
- are in general position and have the same main diagonal.
10.
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.
11.
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∗Vi⊆Vi-1+Vi+Vi+1 for 0?i?d, where V-1=0 and Vd+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. 相似文献
12.
We prove the following: Let A and B be separable C*-algebras. Suppose that B is a type I C*-algebra such that
- (i)
- B has only infinite dimensional irreducible *-representations, and
- (ii)
- B has finite decomposition rank.
0→B→C→A→0 相似文献
13.
In this paper it is shown that if T∈L(H) satisfies
- (i)
- T is a pure hyponormal operator;
- (ii)
- [T∗,T] is of rank two; and
- (iii)
- ker[T∗,T] is invariant for T,
14.
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∗Vi⊆Vi-1+Vi+Vi+1 for 0?i?d, where V-1=0 and Vd+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 θi (resp.) is the eigenvalue of A (resp.A∗) on Vi (resp.), and is the split sequence of A,A∗ corresponding to and . 相似文献
15.
Tomasz Weiss 《Topology and its Applications》2008,156(1):138-141
We show that it is consistent with ZFC that there exist:
- (1)
- An unbounded (with respect to ?∗) and strongly measure zero subgroup of ZN, but without the Menger property.
- (2)
- An unbounded (with respect to ?∗) and strongly measure zero subgroup of ZN with the Menger property which does not have the Rothberger property.
16.
A square matrix is nonderogatory if its Jordan blocks have distinct eigenvalues. We give canonical forms for
- •
- nonderogatory complex matrices up to unitary similarity, and
- •
- pairs of complex matrices up to similarity, in which one matrix has distinct eigenvalues.
17.
The two main results are:
- A.
- If a Banach space X is complementably universal for all subspaces of c0 which have the bounded approximation property, then X∗ is non-separable (and hence X does not embed into c0).
- B.
- There is no separable Banach space X such that every compact operator (between Banach spaces) factors through X.
18.
Let denote a field and V denote a nonzero finite-dimensional vector space over . We consider an ordered pair of linear transformations A:V→V and A*:V→V that satisfy (i)–(iii) below.
- 1. [(i)]Each of A,A* is diagonalizable on V.
- 2. [(ii)]There exists an ordering of the eigenspaces of A such thatwhere V-1=0, Vd+1=0.
- 3. [(iii)]There exists an ordering of the eigenspaces of A* such thatwhere , .
Keywords: Leonard pair; Tridiagonal pair; q-Inverting pair; Split decomposition 相似文献
19.
Raimundas Vidūnas 《Linear algebra and its applications》2007,422(1):39-57
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
20.
Let G be a Hausdorff topological group. It is shown that there is a class C of subspaces of G, containing all (but not only) precompact subsets of G, for which the following result holds:Suppose that for every real-valued discontinuous function on G there is a set A∈C such that the restriction mapping f|A has no continuous extension to G; then the following are equivalent:
- (i)
- the left and right uniform structures of G are equivalent,
- (ii)
- every left uniformly continuous bounded real-valued function on G is right uniformly continuous,
- (iii)
- for every countable set A⊂G and every neighborhood V of the unit e of G, there is a neighborhood U of e in G such that AU⊂VA.