Year: 2009
Type: article
Abstract: Let $\mathcal{L}$ be a subspace lattice on a Hilbert space $\mathcal{H}$ . Let x and y be vectors in $\mathcal{H}$ and let $P_x$ be the projection onto sp(x). If $P_xE$ = $EP_x$ for each E ${\in}\;\mathcal{L}$ , then the following are equivalent. (1) There exists an operator A in Alg $\mathcal{L}$ such that Ax = y, Af = 0 for all f in $sp(x)^{\perp}$ and A ${\geq}$ 0. (2) sup ${\frac{{\parallel}E^{\perp}y{\parallel}}{{\parallel}E^{\perp}x{\parallel}}:E{\in}\mathcal{L}}$ < ${\infty}$ < x, y > ${\geq}$ 0. Let X and Y be operators in $\mathcal{B}(\mathcal{H})$ . Let P be the projection onto $\overline{rangeX}$ . If PE = EP for each E ${\in}\;\mathcal{L}$ , then the following are equivalent: (1) sup ${\frac{{\parallel}E^{\perp}Yf{\parallel}}{{\parallel}E^{\perp}Xf{\parallel}}:f{\in}\mathcal{H},E{\in}\mathcal{L}}$ < ${\infty}$ and < Xf, Yf > ${\geq}$ 0 for all f in H. (2) There exists a positive operator A in Alg $\mathcal{L}$ such that AX = Y.
Source: Honam Mathematical Journal
Author: Joo-Ho Kang
Institution: Daegu University
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
Funder: Daegu University
OpenAlex ID: W834330302