Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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EQUATIONS AX = Y AND Ax = y IN ALGL

  • Published : 2006.03.01
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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 $\frac\;{R(X)}$, where RX is the range of X. If PE = EP for each $E\;\in\;L$, 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 ${\parallel}A{\parallel} = K.$ Let x and y be vectors in H and let $P_x$ be the projection onto the singlely generated space by x. If $P_xE = EP_x$ for each $E\inL$, 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 ${\parallel}A{\parallel} = K_0$ whose norm is $K_0$ under this case.

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References

  1. M. Anoussis, E. Katsoulis, R. L. Moore, and T. T. Trent, Interpolation problems for ideals in nest algebras, Math. Proc. Camb. Phil. Soc. 111 (1992), no. 1, 151-160
  2. R. G. Douglas, On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413-415
  3. A. Hopenwasser, The equation Tx = y in a reflexive operator algebra, Indiana University Math. J. 29 (1980), no. 1, 121-126 https://doi.org/10.1512/iumj.1980.29.29009
  4. A. Hopenwasser, Hilbert-Schmidt interpolation in CSL-algebras, Illinois J. Math. 33 (1989), no. 4, 657-672
  5. Y. S. Jo and J. H. Kang, Interpolation problems in AlgL, J. Appl. Math. comput. 18 (2005), 513-524
  6. Y. S. Jo, J. H. Kang, and K. S. Kim, On operator interpolation problems, J. Korean Math. Soc. 41 (2004), no. 3, 423-433 https://doi.org/10.4134/JKMS.2004.41.3.423
  7. R. Kadison, Irreducible Operator Algebras, Proc. Nat. Acad. Sci. U.S.A. (1957), 273-276
  8. E. Katsoulis, R. L. Moore, and T. T. Trent, Interpolation in nest algebras and applications to operator Corona theorems, J. Operator Theory 29 (1993), no. 1, 115-123
  9. E. C. Lance, Some properties of nest algebras, Proc. London Math. Soc. (3) 19 (1969), 45-68
  10. R. Moore and T. T. Trent, Linear equations in subspaces of operators, Proc. Amer. Math. Soc. 128 (2000), no. 3, 781-788
  11. R. Moore and T. T. Trent, Interpolation in in°ated Hilbert spaces, Proc. Amer. Math. Soc. 127 (1999), no. 2, 499-507
  12. N. Munch, Compact causal data interpolation, J. Math. Anal. Appl. 140 (1989), no. 2, 407-418 https://doi.org/10.1016/0022-247X(89)90074-7

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