Lattices of Invariant Subspaces for a Quasiaffine Transform of a Unilateral Shift of Finite Multiplicity

Let T be a contraction, let S be a unilateral shift of finite multiplicity, and let X be an operator with zero kernel, dense range, and such that . Then the mapping LatT, is an isomorphism between the lattices LatT and LatS of invariant subspaces of T and S. Bibliography: 8 titles.     

p-Frames and Shift Invariant Subspaces of Lp

In this article we investigate the frame properties and closedness for the shift invariant space V_p(F) = { ?_i=1^r ?_{j ? \Zd} d_i(j) f_i (·-j):  ( d_i(j) )_{j ? \Zd} ? l^p }, \q 1 ￡ p ￡ ￥ . \displaystyle V_p(\Phi) = \left\{ \sum_{i=1}^r \sum_{j\in \Zd} d_i(j) \phi_i (\cdot-j): \ \left( d_i(j) \right)_{j\in \Zd}\in \ell^p \right\}, \q 1\le p \le \infty~. We derive necessary and sufficient conditions for an indexed family {f_i(·-j): 1 ￡ i ￡ r, j ? \Zd}\{\phi_i(\cdot-j):\ 1\le i\le r, j\in \Zd\} to constitute a pp-frame for V_p(F)V_p(\Phi), and to generate a closed shift invariant subspace of L^pL^p. A function in the L^pL^p-closure of V_p(F)V_p(\Phi) is not necessarily generated by l^p\ell^p coefficients. Hence we often hope that V_p(F)V_p(\Phi) itself is closed, i.e., a Banach space. For p 1 2p\ne 2, this issue is complicated, but we show that under the appropriate conditions on the frame vectors, there is an equivalence between the concept of pp-frames, Banach frames, and the closedness of the space they generate. The relation between a function f ? V_p(F)f \in V_p(\Phi) and the coefficients of its representations is neither obvious, nor unique, in general. For the case of pp-frames, we are in the context of normed linear spaces, but we are still able to give a characterization of pp-frames in terms of the equivalence between the norm of ff and an l^p\ell^p-norm related to its representations. A Banach frame does not have a dual Banach frame in general, however, for the shift invariant spaces V_p(F)V_p(\Phi), dual Banach frames exist and can be constructed.     

On Invariant Graph Subspaces

In this paper we discuss the problem of decomposition for unbounded \({2\times2}\) operator matrices by a pair of complementary invariant graph subspaces. Under mild additional assumptions, we show that such a pair of subspaces decomposes the operator matrix if and only if its domain is invariant for the angular operators associated with the graphs. As a byproduct of our considerations, we suggest a new block diagonalization procedure that resolves related domain issues. In the case when only a single invariant graph subspace is available, we obtain block triangular representations for the operator matrices.     

关于压缩算子的不变子空间

证明关于压缩算子的如下不变子空间定理：如果T是Hilbert空间H上的压缩算子,且集合Z’={λ∈D;存在z∈H,使得‖z‖=1,且‖（λ-T）z‖＜1/3(1-‖λ‖}是开单位圆D的控制集,那么T有非平凡的不变子空间,这个定理包含了S.Brown,B.Chevreau,C.fPearcy和B.Beauzamy的两个重要结果作为特殊情况,特别是,为个定理包含了S.Brown等人的Hilbert空间上的每个具有厚谱的压缩算子都有平凡的不变子空间这个重要结果作为特殊情况。     

关于序列次可分解算子的不变子空间

得到了关于序列次可分解算子的一个不变子空间定理，推广了H.Mohebi和M.Rajiabalipour在1994年得到的一个不变子空间定理，并且举例说明存在l2上的有界线性算子T。它有无穷多个变子空间，但是它的不变子空间格Lat(T)不丰富。     

关于可分解算子的扰动的不变子空间(英文)

在文献［１］中,Ｊ．Ｅｓｃｈｍｅｉｅｒ和Ｂ．Ｐｒｕｎａｒｕ证明了（复）Ｂａｎａｃｈ空间上的每个具有Ｂｉｓｈｏｐ性质（β）和浓厚谱的有界线性算子有非平凡的不变子空间．在文献［２］中,Ｈ．Ｍｏｈｅｂｉ和Ｍ．Ｒａｄｊａｂａｌｉｐｏｕｒ在减弱算子的Ｂｉｓｈｏｐ性质（β）和加强谱的浓厚性条件的情况下得到了另外几个不变子空间定理．本文给出了一个更进一步的不变子空间定理（见定理１）．     

Mixed Invariant Subspaces over the Bidisk

It is known that the structure of invariant subspaces I of the Hardy space H ² over the bidisk is extremely complicated. One reason is that it is difficult to describe infinite dimensional wandering spaces ${I\ominus zI}$ completely. In this paper, we study the structure of nontrivial closed subspaces N of H ² with ${T_zN\subset N}$ and ${T^*_wN\subset N}$ , which are called mixed invariant subspaces under T _z and ${T^*_w}$ . We know that the dimension of ${N\ominus zN}$ ranges from 1 to ??. If ${T^*_w(N\ominus zN)\subset N\ominus zN}$ , we may describe N completely. If ${T^*_w(N\ominus zN)\not\subset N\ominus zN}$ , it seems difficult to describe N generally. So we study N under the condition ${dim\,(N\ominus zN)=1}$ . Write ${M=H^2\ominus N}$ . We describe ${M\ominus wM}$ precisely. We give a characterization of N for which there is a nonzero function ${\varphi}$ in ${M\ominus wM}$ satisfying ${z^k\varphi\in M\ominus wM}$ for every k ?? 0. We also see that the space ${M\ominus wM}$ has a deep connection with the de Branges?CRovnyak spaces studied by Sarason.     

Sobolev圆盘代数的不变子空间

本文研究了Sobolev圆盘代数R(D)上的乘法算子M_z的不变子空间,完全刻画了余维有限的不变子空间的结构．     

Invariant Subspaces for Pairs of Projections

A simple geometrical argument shows that every pair of projectionson a finite-dimensional complex vector space has a common invariantsubspace of dimension 1 or 2. The idea extends to certain pairsof projections on an infinite-dimensional Hilbert space H. Inparticular every projection on H has a reducing subspace, althougha finite-dimensional one need not exist. In a final section,the results are extended to the existence of hyperinvariantsubspaces for pairs of projections.     

Invariant Subspaces of Three-Dimensional Periodic Lightguides

A three-dimensional periodic lightguide is considered. Oscillations of the lightguide are described by the reduced wave equation. The existence of bounded projections onto invariant subspaces of the waveguide operator corresponding to intervals of the continuous spectrum is established. Bibliography: 6 titles     

Sobolev圆盘代数的不变子空间

研究了Sobolev圆盘代数R(D)上乘自变量算子M_z的不变子空间,给出了M_z在任何不变子空间上限制的基本性质,证明了M_z分别限制在两个不变子空间上酉等价当且仅当这两个不变子空间相等,并描述了M_z的一类公共零点在边界的不变子空间的结构.     

多圆盘Hardy空间的不变子空间(英文)

设M是多圆盘Hardy空间H²（Tⁿ）的不变子空间.R_zi是以坐标函数z_i（i=1,2,…,n）为符号的乘法算子在M上的限制.本文作者证明了,存在一个内函数q,使得M=qH²（Tⁿ）当且仅当对i≠j,R_ziR_zj^*=R_zj^*R_zi.     

Stability of Invariant Subspaces of Quaternion Matrices

A quaternion invariant subspace of a quaternion matrix is said to be stable (in the sense of robustness) if every nearby matrix has an invariant subspace close to the original one. Under mild hypothesis, necessary and sufficient conditions are given for quaternion invariant subspaces to be stable. Other notions of stability of quaternion invariant subspaces are studied as well, and stability criteria developed. Applications to robustness of solutions of certain classes of quaternion matrix equations are given.     

Elliptic Operators in Subspaces and the Eta Invariant

In this paper we obtain a formula for the fractional part of the -invariant for elliptic self-adjoint operators in topological terms. The computation of the -invariant is based on the index theorem for elliptic operators in subspaces obtained by Savin and Sternin. We also apply the K-theory with coefficients _n . In particular, it is shown that the group K(T ^* M, _n ) is realized by elliptic operators (symbols) acting in appropriate subspaces.