EQUATIONS AX = Y AND Ax = y IN ALGL

Title & Authors
EQUATIONS AX = Y AND Ax = y IN ALGL
Jo, Young-Soo; Kang, Joo-Ho; Park, Dong-Wan;

Abstract
Let L be a subspace lattice on a Hilbert space H and X and Y be operators acting on a Hilbert space H. Let P be the projection onto $\small{\frac\;{R(X)}}$, where RX is the range of X. If PE = EP for each $\small{E\;\in\;L}$, then there exists an operator A in AlgL such that AX = Y if and only if $$sup\{{\parallel}E^{\bot}Yf{\parallel}/{\parallel}E^{\bot}Xf{\parallel}\;:\;f{\in}H,\; E{\in}L}=K\;<\;\infty$$ Moreover, if the necessary condition holds, then we may choose an operator A such that AX = Y and $\small{{\parallel}A{\parallel} = K.}$ Let x and y be vectors in H and let $\small{P_x}$ be the projection onto the singlely generated space by x. If $\small{P_xE = EP_x}$ for each $\small{E\inL}$, then the assertion that there exists an operator A in AlgL such that Ax = y is equivalent to the condition $$K_0\;:\;=\;sup\{{\parallel}E^{\bot}y{\parallel}/{\parallel}E^{\bot}x\;:\;E{\in}L}=<\;\infty$$ Moreover, we may choose an operator A such that $\small{{\parallel}A{\parallel} = K_0}$ whose norm is $\small{K_0}$ under this case.
Keywords
interpolation problem;subspace lattice;$\small{Alg{\punds}}$;$\small{CSL-alg{\punds}}$;
Language
English
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UNITARY INTERPOLATION ON Ax = y IN A TRIDIAGONAL ALGEBRA ALG𝓛, Honam Mathematical Journal, 2014, 36, 4, 907
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A NOTE ON QUASI-TOPOLOGICAL SPACES, Honam Mathematical Journal, 2011, 33, 1, 11
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