In terms of the exactly nonzero partition,the reducible projection-system and correlation matrices,two characterizations for a rank three operator in a CSL algebra can be completely decomposed are given.
For finite rank operators in a commutative subspace lattice algebra alg(?)we introduce the concept of correlation matrices,basing on which we prove that a finite rank operator in alg(?)can be written as a finite sum o...For finite rank operators in a commutative subspace lattice algebra alg(?)we introduce the concept of correlation matrices,basing on which we prove that a finite rank operator in alg(?)can be written as a finite sum of rank-one operators in alg(?),if it has only finitely many different correlation matrices.Thus we can recapture the results of J.R.Ringrose,A.Hopenwasser and R.Moore as corollaries of our theorems.展开更多
Given two Banach spaces E, F, let B（E, F） be the set of all bounded linear operators from E into F, ∑r the set of all operators of finite rank r in B（E, F）, and ∑r^# the number of path connected components of ∑...Given two Banach spaces E, F, let B（E, F） be the set of all bounded linear operators from E into F, ∑r the set of all operators of finite rank r in B（E, F）, and ∑r^# the number of path connected components of ∑r. It is known that ∑r is a smooth Banach submanifold in B（E,F） with given expression of its tangent space at each A ∈ ∑r. In this paper, the equality ∑r^# = 1 is proved. Consequently, the following theorem is obtained： for any nonnegative integer r,∑r is a smooth and path connected Banach submanifold in B（E, F） with the tangent space TA∑r = {B E B（E,F） ： BN（A） belong to R（A）} at each A ∈ ∑r if dim F = ∞. Note that the routine method can hardly be applied here. So in addition to the nice topological and geometric property of ∑r the method presented in this paper is also interesting. As an application of this result, it is proved that if E = R^n and F = R^m, then ∑r is a smooth and path connected submanifold of B（R^n,R^m） and its dimension is dim ∑r = （m ＋ n）r- r^2 for each r, 0≤r 〈 min{n,m}.展开更多
Sparse sums of Lorentzians can give good approximations to functions consisting of linear combination of piecewise continuous functions. To each Lorentzian, two parameters are as- signed： translation and scale. These...Sparse sums of Lorentzians can give good approximations to functions consisting of linear combination of piecewise continuous functions. To each Lorentzian, two parameters are as- signed： translation and scale. These parameters can be found by using a method for complex fre- quency detection in the frequency domain. This method is based on an alternating projection scheme between Hankel matrices and finite rank operators, and have the advantage that it can be done in weighted spaces. The weighted spaces can be used to partially revoke the effect of finite band-width filters. Apart from frequency extrapolation the method provides a way of estimating discontinuity locations.展开更多
Using the technique of annihilators, we characterize the w^＊-closure of the finite-partition submodule R（N, ～） of a weakly closed nest algebra module U completely; and by virtue of this characterization, we obtain...Using the technique of annihilators, we characterize the w^＊-closure of the finite-partition submodule R（N, ～） of a weakly closed nest algebra module U completely; and by virtue of this characterization, we obtain a sufficient and necessary condition for U to have a maximal w^＊-closed submodule.展开更多
文摘In terms of the exactly nonzero partition,the reducible projection-system and correlation matrices,two characterizations for a rank three operator in a CSL algebra can be completely decomposed are given.
文摘For finite rank operators in a commutative subspace lattice algebra alg(?)we introduce the concept of correlation matrices,basing on which we prove that a finite rank operator in alg(?)can be written as a finite sum of rank-one operators in alg(?),if it has only finitely many different correlation matrices.Thus we can recapture the results of J.R.Ringrose,A.Hopenwasser and R.Moore as corollaries of our theorems.
基金Supported by the National Science Foundation of China (Grant No.10671049 and 10771101).
文摘Given two Banach spaces E, F, let B（E, F） be the set of all bounded linear operators from E into F, ∑r the set of all operators of finite rank r in B（E, F）, and ∑r^# the number of path connected components of ∑r. It is known that ∑r is a smooth Banach submanifold in B（E,F） with given expression of its tangent space at each A ∈ ∑r. In this paper, the equality ∑r^# = 1 is proved. Consequently, the following theorem is obtained： for any nonnegative integer r,∑r is a smooth and path connected Banach submanifold in B（E, F） with the tangent space TA∑r = {B E B（E,F） ： BN（A） belong to R（A）} at each A ∈ ∑r if dim F = ∞. Note that the routine method can hardly be applied here. So in addition to the nice topological and geometric property of ∑r the method presented in this paper is also interesting. As an application of this result, it is proved that if E = R^n and F = R^m, then ∑r is a smooth and path connected submanifold of B（R^n,R^m） and its dimension is dim ∑r = （m ＋ n）r- r^2 for each r, 0≤r 〈 min{n,m}.
文摘Sparse sums of Lorentzians can give good approximations to functions consisting of linear combination of piecewise continuous functions. To each Lorentzian, two parameters are as- signed： translation and scale. These parameters can be found by using a method for complex fre- quency detection in the frequency domain. This method is based on an alternating projection scheme between Hankel matrices and finite rank operators, and have the advantage that it can be done in weighted spaces. The weighted spaces can be used to partially revoke the effect of finite band-width filters. Apart from frequency extrapolation the method provides a way of estimating discontinuity locations.
基金Project partially supported by the National Natural Science Foundation of China(No.10401030)the Zhejiang Natural Science Foundation(No.M103044)
文摘Using the technique of annihilators, we characterize the w^＊-closure of the finite-partition submodule R（N, ～） of a weakly closed nest algebra module U completely; and by virtue of this characterization, we obtain a sufficient and necessary condition for U to have a maximal w^＊-closed submodule.
