DOI QR코드

DOI QR Code

SELF-ADJOINT INTERPOLATION ON AX = Y IN ALGL

  • Received : 2007.01.17
  • Published : 2007.03.25

Abstract

Given operators X and Y acting on a Hilbert space $\cal{H}$, an interpolating operator is a bounded operator A such that AX = Y. In this article, we showed the following : Let $\cal{L}$ be a subspace lattice acting on a Hilbert space $\cal{H}$ and let X and Y be operators in $\cal{B}(\cal{H})$. Let P be the projection onto $\bar{rangeX}$. If FE = EF for every $E\in\cal{L}$, then the following are equivalent: (1) $sup\{{{\parallel}E^{\perp}Yf\parallel\atop \parallel{E}^{\perp}Xf\parallel}\;:\;f{\in}\cal{H},\;E\in\cal{L}\}\$ < $\infty$, $\bar{range\;Y}\subset\bar{range\;X}$, and < Xf, Yg >=< Yf,Xg > for any f and g in $\cal{H}$. (2) There exists a self-adjoint operator A in Alg$\cal{L}$ such that AX = Y.

Keywords

Interpolation Problem;Self-Adjoint Interpolation Problem;Subspace Lattice;Alg $\cal{L}$.

References

  1. Anoussis, M. ; Katsoulis, E. ; Moore, R. L.; Trent, T. T., Interpolation problems for ideals in nest algebras, Math. Proc. Camb. Phil. Soc. 111 (1992), 151-160. https://doi.org/10.1017/S030500410007523X
  2. Douglas, R. G., 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
  3. Gilfeather, F. and Larson, D., Commutants modulo the compact operators of certain CSL algebras, Operator Theory: Adv. Appl. 2 (Birkhauser, Basel, 1981), 105-120. https://doi.org/10.1007/978-3-0348-5456-6_9
  4. Hopenwasser, A., Hilbert-Schmidt interpolation in CSL algebras, Illinois J. Math. (4), 33 (1989), 657-672.
  5. Jo, Y. S. and Kang, J. H., Equations AX = Y and Ax = y in AlgL, J. of K. M. S. 43 (2006), 399-411. https://doi.org/10.4134/JKMS.2006.43.2.399