关于零对合类算子

本文定义了ＮＩ算子和ＡＮＩ算子，给出共轭算子下有界时，一个算子为ＮＩ类算子的等价条件，并在一定条件下证明了ＮＩ类算子具有闭的数值值域．最后，本文研究了ＮＩ类算子对，尤其是Ｈ２空间上的Ｔｏｅｐｌｉｔｚ算子对，刻划了Ｔ＝（Ｔφ，Ｔ＊φ）的联合谱     

Reflexive linear transformations

A linear operator A is called reflexive if the only operators that leave invariant the invariant subspaces of A are the operators in the weak closure of the algebra of polynomials in A. In this note we completely characterize reflexive operators on finite-dimensional spaces.     

Generating all linear transformations

For an n×n Boolean matrix R, let A_R={n×n matrices A over a field F such that if r_ij=0 then a_ij=0}. We show that a collection A_R〈1〉,…,A_R〈k〉 generates all n×n matrices over F if and only if the matrix J all of whose entries are 1 can be expressed as a Boolean product of Hall matrices from the set {R〈1〉,…,R〈k〉}. We show that J can be expressed as a product of Hall matrices R〈i〉 if and only if ΣR〈i〉?R〈i〉 is primitive.     

Hessenberg pairs of linear transformations

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 that

where V_-1=0, V_d+1=0.

3. [(iii)]There exists an ordering of the eigenspaces of A^* such that

where , .

We call such a pair a Hessenberg pair on V. In this paper we obtain some characterizations of Hessenberg pairs. We also explain how Hessenberg pairs are related to tridiagonal pairs.

Countable linear transformations are clean

It is shown that every linear transformation on a vector space of countable dimension is the sum of a unit and an idempotent.

     

Algebraic reflexivity of linear transformations

Let be the set of all linear transformations from to , where and are vector spaces over a field . We show that every -dimensional subspace of is algebraically -reflexive, where denotes the largest integer not exceeding , provided is less than the cardinality of .

     

Harmonious orbits of linear transformations

We describe necessary and sufficient conditions for orbits of linear transformations onR ⁿ ,n1, and sets arising as sums of elements from orbits, to be harmonious subsets. This is done via a generalization of the notion of Pisot-Vijayaraghavan and Salem numbers.     

Extension of linear operator transformations

Acta Mathematica Hungarica -     

On orbits of fractional linear transformations

Let K be a field and let σ(t) be an automorphism of k(t) of infinite order. In his paper [1] Gerard Cauchon asked whether the transformation which is naturally defined on kU∞ by σ has infinitely many infinite orbits. This is indeed so.     

Reversible subsets of certain linear transformations

Summary Suppose U is a set,F is a field of subsets of U, p_AB is the set of all real-valued bounded finitely additive functions defined onF, and for each in p_AB, A()={: in p_AB, absolutely continuous with respect to }. SupposeM is a linear subspace of p_AB such that . A generalisation of a previously discussed collection of linear transformations (see J. London Math. Soc., vol. 44 (1969), pp. 385–396) is treated by letting C_M denote the set to which T belongs iff T is a linear transformation from M into p_AB such that for some K inR and all in M and V inF, . Certain theorems of the aforementioned reference are generalized, as well as one of Trans. Amer. Math. Soc., vol. 199, (1974), pp. 131–140. The principal result of the present paper is the following generalisation of a reversibility characterisation in the first mentioned reference: Theorem: If T is in C_M, then (, T()): in M A(T()) is the only reversible subset T⁰ if T such that: i) the domain M⁰ of T⁰ is a linear subspace of M and , and ii) the range of T⁰ is the range of T.     

Completely positive and Hermitian-preserving linear transformations

Characterization theorems for completely positive and hermitian-preserving linear transformations are given.     

Trace preserving linear transformations on matrix algebras

A characterization is given for all linear transformations on a matrix algebra which preserve (1) the trace (2) the unit matrix (3) the trace and the second elementary symmetric function of the eigenvalues.