Automorphic images of commutative subspace lattices
Automorphic images of commutative subspace lattices
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DOI：
10.1090/s0002-9947-1986-0837808-1
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发表时间：
1986
影响因子：
1.3
通讯作者：
K. Harrison;W. E. Longstaff
中科院分区：
文献类型：
--
作者：
K. Harrison;W. E. Longstaff
设C（H）表示复可分Hilbert空间H的所有（闭）子空间的格。设（AC）是子空间格1CC（H）可能满足或不满足的条件：对于C（H）的某个格自同构q5和某个交换子空间格LCC（H），（AC）r = O（L）.则1满足（AC）当且仅当1 C B对于某个布尔代数子空间格B C C（H）具有这样的性质：对每个K，L E B，向量和K + L是闭的。如果1是有限的，则1满足（AC）当且仅当1是分配的且K + L对每个K，L E S是闭的。在有限维中，r满足（AC）当且仅当r是分配的。每个满足（AC）的r都是自反的。对于这样的1 r，给定向量x，yEH，研究了方程Tx = y对TzAlg 1的可解性. 1.你的助手。在本文中，H将表示一个复的、可分的、非零的希尔伯特空间。H上的内积记为（。).设B（H）表示作用在H上的（有界线性）算子集，对于T ∈ B（H），设R（T）表示T的值域.设0（H）表示H的（闭）子空间的集合。H的子空间格是指在交和闭线性跨距下闭的子集Z C 0（H），即，M，N E Y意味着MnN E Y和MVN E F。一个子空间i的格是完备的，如果对于i的每个元素族{Ma，}，n M^，E i和V M，E Y;它是交换的，如果对于每个K，L E i，PKPL = PLPK（PK表示具有值域K的正交投影）。若{PK：KE 2}在强算子拓扑中闭，则子集ZCC（H）是强闭的. H上的子空间格是指包含（0）和H的子空间的完备格。通常，对于任何子集i C 0（H），我们令Alg：= {T E B（H）：TM C M对于每个M E i}，并且令LatA = {N E 0（H）：TN C N对于每个T E A}对于任何子集A C B（H）。如果Z = Lat Alg i，则称子集i C C（H）为自反的，如果Lat Alg Z = C（H），则称子集i C（H）为传递的。C（H）的自同构O是C（H）到自身上的满足M C N的双射当且仅当O（M）C O（N）。每个自同构o满足M（n m.）= nf（M.）和q$（VMa）= V？（M，），对于每个族{Mc}C C（H）。子空间格L在0（H）的自同构下的像O（C）显然是子空间格。若SE B（H）可逆，则M -）SM（ME 0（H））定义一个自同构.任何子集C 0（H）在这个自同构下的像将被记为ST。关于交换子空间格已经知道很多（例如，参见[1，6]）。很容易证明交换性不被自同构保持。(For例如，设H = K G K，= {（0），K（0），（0）EK，H}，设o是由可逆算子SE B（H）定义的自同构。1980年数学学科分类小学46 C10;中学06 A35。(?) 1986年美国数学学会0002-9947/86每页$1.00 + $.25
Let C(H) denote the lattice of all (closed) subspaces of a complex, separable Hilbert space H. Let (AC) be the following condition that a subspace lattice 1 C C(H) may or may not satisfy: (AC) r = O(L) for some lattice automorphism q5 of C(H) and some commutative subspace lattice L C C(H). Then 1 satisfies (AC) if and only if 1 C B for some Boolean algebra subspace lattice B C C(H) with the property that, for every K, L E B, the vector sum K + L is closed. If 1 is finite, then 1 satisfies (AC) if and only if 1 is distributive and K + L is closed for every K, L E S. In finite dimensions r satisfies (AC) if and only if r is distributive. Every r satisfying (AC) is reflexive. For such 1r, given vectors x, y E H, the solvability of the equation Tx = y for T z Alg 1 is investigated. 1. Preliminaries. Throughout this paper H will denote a complex, separable, nonzero Hilbert space. The inner product on H is denoted (. ). Let B(H) denote the set of (bounded, linear) operators acting on H and, for T E B(H), let R(T) denote the range of T. Let 0(H) denote the set of (closed) subspaces of H. By a lattice of subspaces of H is meant a subset Z C 0(H) which is closed under intersections and closed linear spans, i.e., M, N E Y implies MnN E Y and MVN E F. A lattice of subspaces i is complete if n M^, E i and V M, E Y for every family {Ma,} of elements of i; it is commutative if PKPL = PLPK for every K, L E i (PK denotes the orthogonal projection with range K). A subset Z C C(H) is strongly closed if {PK: K E 2} is closed in the strong operator topology. By a subspace lattice on H is meant a complete lattice of subspaces containing (0) and H. As usual, for any subset i C 0(H) we let Alg : = {T E B(H): TM C M for every M E i} and let LatA = {N E 0(H):TN C N for every T E A} for any subset A C B(H). The subset i C C(H) is called reflexive if Z = Lat Alg i and transitive if Lat Alg Z = C(H). An automorphism O of C(H) is a bijection of C(H) onto itself satisfying M C N if and only if O(M) C O(N). Every automorphism o satisfies M(n m.)= nf (M.) and q$(V Ma) = V?(M,) for every family {Mc}C C (H). The image O(C) of a subspace lattice L under an automorphism o of 0(H) is obviously a subspace lattice. If S E B(H) is invertible, M -) SM (M E 0(H)) defines an automorphism. The image of any subset C 0(H) under this automorphism will be denoted ST. Much is known about commutative subspace lattices (for example, see [1, 6]). It is easy to show that commutativity is not preserved by automorphisms. (For example, let H = K G K, = {(0), K (0), (0) EK, H}, and let o be the automorphism induced by the invertible operator S E B(H) defined Received by the editors June 12, 1985. 1980 Mathematics Subject Classification. Primary 46C10; Secondary 06A35. (?)1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page