| Co-Investigator(Kenkyū-buntansha) |
HAYASHI Mikihiro Hokkaido University, Faculty of Science, Associate Professor, 理学部, 助教授 (40007828)
KATUMATA Osamu Hokkaido University, Faculty of Science, Associate Professor, 理学部, 助教授 (40032825)
INOUE Junji Hokkaido University, Faculty of Science, Professor, 理学部, 教授 (40000856)
KOSHI Shozo Hokkaido University, Faculty of Science, Professor, 理学部, 教授 (40032792)
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| Research Abstract |
Let K be a Hilbert space, B a von Neumann algebra on K and A a weakly closed subalgebra of B. T = (T_<ij>) denotes 2 x 2 operator matrices on K <symmetry> K where T_<ij> epsilon B (i, j = 1, 2), T_<11> <greater than or equal> 0, T_<22> <greater than or equal> 0 and T^*_ = T_<12>. [B] denotes the set of such operator matrices T and [A]_0 denotes the subset of [B] such that T_<12> epsilon A and T_<11> = T_<22> = 0. Let F be the subset of lat A, the lattice of all A-invariant projections. We studied whether if T epsilon [B] is positive on PK <symmetry> (1 - P)K for every P epsilon F then there exists T^^- in T + [A]_0 which is positive on K <symmetry> K. We say that (B, A, F). has a lifting property when there exists such a T^^-.
When B is a commutative von Neumann algebra L^", if A is an abstract Hardy space H^", we could show that {B = L^", A = H^", F} has a lifting property even if F is very small. Using this result we could generalize a theorem of norms of Hankel operators and two theorems of weighted norm inequalities in the disc algebra to those in a general uniform algebra. When B is non-commutative, if A and F satisfy two conditions which are satisfied in many important examples, we could that {B, A, F} has a lifting property. Using the distanck formula of Arveson or the theorem of Neharils type. We could generalize two theorems of weighted norm inequalities in the vector-valued Hardy space to those in nest algebras and non-commutative Hardy spaces.
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