Effective intervals and regular Dirichlet subspaces

It is shown in Li and Ying (2019) that a regular and local Dirichlet form on an interval can be represented by so-called effective intervals with scale functions. This paper focuses on how to operate on effective intervals to obtain regular Dirichlet subspaces. The first result is a complete characterization for a Dirichlet form to be a regular subspace of such a Dirichlet form in terms of effective intervals. Then we give an explicit road map how to obtain all regular Dirichlet subspaces from a local and regular Dirichlet form on an interval, by a series of intuitive operations on the effective intervals in the representation above. Finally applying previous results, we shall prove that every regular and local Dirichlet form has a special standard core generated by a continuous and strictly increasing function.     

On the class of norm limits of nilpotents

It is known that every operator on a Hilbert space whose invariant subspace lattice is possibly is a norm-limit of a sequence of nilpotent operators. In this note we study properties of such approximating sequences.

     

One-particle subspaces in the stochasticXY model

We study the spectrum of the generatorH of the Glauber dynamics for a model of planar rotators on a lattice in the case of a high temperature 1/. We construct two so-called one-particle subspacesH _± forH and describe the spectrum of the generator in these subspaces. As a consequence we find time asymptotics of the correlations for the Glauber dynamics.     

Incomplete Delta Functions

By applying projection operators to state vectors of coordinates, we obtain subspaces in which these states are no longer normalized according to Dirac's delta function but normalized according to what we call incomplete delta functions. We show that this class of functions satisfy identities similar to those satisfied by the Dirac delta function. The incomplete delta functions may be employed advantageously in projected subspaces and in the link between functions defined on the whole space and the projected subspace. We apply a similar procedure to finite-dimensional vector spaces for which we define incomplete Kronecker deltas. Dispersion relations for the momenta are obtained and sums over poles are defined and obtained with the aid of differences of incomplete delta functions.     

正则Dirichlet子空间与Mosco收敛性

讨论了一维不可约强局部Dirichlet,型的正则子空间的Mosco收敛性.如果正则子空间的特征集是收敛的,那么相应的正则子空间在Mosco意义下也是收敛的.最后,用一些具体的例子说明了Mosco收敛不能保持Dirichlet型整体特性的稳定.     

On transitive algebras containing a standard finite von Neumann subalgebra

Let M be a finite von Neumann algebra acting on a Hilbert space H and A be a transitive algebra containing M^′. In this paper we prove that if A is 2-fold transitive, then A is strongly dense in B(H). This implies that if a transitive algebra containing a standard finite von Neumann algebra (in the sense of [U. Haagerup, The standard form of von Neumann algebras, Math. Scand. 37 (1975) 271-283]) is 2-fold transitive, then A is strongly dense in B(H). Non-selfadjoint algebras related to free products of finite von Neumann algebras, e.g., LFn and , are studied. Brown measures of certain operators in are explicitly computed.     

Sequences of 0’s and 1’s: Special sequence spaces with the separable Hahn property

As pointed out in [4] the paper [2], authored by G. Bennett, J. Boos and T. Leiger, contains a nontrivial gap in the argumentation of the proof of Theorem 5.2 which is one of main results of that paper and has been applied three times. Till now neither the gap is closed nor a counterexample has been stated. That is why the authors have examined in [4] the situation around the ‘gap’ aiming to a better understanding for the gap. The aim of this paper is to prove the mentioned applications of the theorem in doubt by using gliding hump arguments (quite similar to the classical proofs of the Theorems of Schur and Hahn in the first case (cf. [3]) and a very technical and artful construction, being of independent mathematical interest, in the second case). Research of T. Leiger supported by Estonian Science Foundation Grant 5376.