SOLVING OPERATOR EQUATIONS Ax = Y AND Ax = y IN ALGL

LEE, SANG KI;KANG, JOO HO

• 투고 : 2014.12.20
• 심사 : 2015.03.09
• 발행 : 2015.05.30
• 8 5

초록

In this paper the following is proved: Let L be a subspace lattice on a Hilbert space H and X and Y be operators acting on a Hilbert space H. If XE = EX for each E ${\in}$ L, then there exists an operator A in AlgL such that AX = Y if and only if sup $\left{\frac{\parallel{XEf}\parallel}{\parallel{YEf}\parallel}\;:\;f{\in}H,\;E{\in}L\right}$ = K < $\infty$ and YE=EYE. Let x and y be non-zero vectors in H. Let Px be the orthogonal pro-jection on sp(x). If EPx = PxE for each E $\in$ L, then the following are equivalent. (1) There exists an operator A in AlgL such that Ax = y. (2) < f, Ey > y =< f, Ey > Ey for each E ${\in}$ L and f ${\in}$ H.

키워드

Interpolation Problem;Subspace Lattice;AlgL;CSL-AlgL

참고문헌

1. E.C. Lance, Some properties of nest algebras, Proc. London Math. Soc. 19(III) (1969), 45-68. https://doi.org/10.1112/plms/s3-19.1.45
2. Y.S. Jo, J.H. Kang, R.L. Moore and T.T. Trent, Interpolation in self-adjoint settings, Proc. Amer. Math. Soc. 130, 3269-3281. https://doi.org/10.1090/S0002-9939-02-06610-8
3. J.H. Kang, Compact interpolation on AX = Y in AlgL, J. Appl. Math. & Informatics 32 (2014), 441-446. https://doi.org/10.14317/jami.2014.441
4. R.G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413-415. https://doi.org/10.1090/S0002-9939-1966-0203464-1
5. Y.S. Jo and J.H. Kang, Interpolation problems in CSL-algebras AlgL, Rocky Mountain J. of Math. 33 (2003), 903-914. https://doi.org/10.1216/rmjm/1181069934
6. Y.S. Jo and J.H. Kang, Interpolation problems in AlgL, J. Appl. Math. & Computing 18 (2005), 513-524.
7. Y.S. Jo, J.H. Kang and K.S. Kim, On operator interpolation problems, J. Korean Math. Soc. 41 (2004), 423-433. https://doi.org/10.4134/JKMS.2004.41.3.423