UNITARY INTERPOLATION FOR VECTORS IN TRIDIAGONAL ALGEBRAS

  • Published : 2003.01.01

Abstract

Given vectors x and y in a Hilbert space, an interpolating operator is a bounded operator T such that Tx = y. An interpolating operator for n vectors satisfies the equation $Tx_i\;:\;y_i,\;for\;i\;=\;1,\;2,\;{\cdots},\;n$. In this article, we obtained the following : $Let\;x\;=\;\{x_i\}\;and\;y=\{y_\}$ be two vectors in a separable complex Hilbert space H such that $x_i\;\neq\;0$ for all $i\;=\;1,\;2;\cdots$. Let L be a commutative subspace lattice on H. Then the following statements are equivalent. (1) $sup\;\{\frac{\$\mid${\sum_{k=1}}^l\;\alpha_{\kappa}E_{\kappa}y\$\mid$}{\$\mid${\sum_{k=1}}^l\;\alpha_{\kappa}E_{\kappa}x\$\mid$}\;:\;l\;\in\;\mathbb{N},\;\alpha_{\kappa}\;\in\;\mathbb{C}\;and\;E_{\kappa}\;\in\;L\}\;<\;\infty\;and\;$\mid$y_n\$\mid$x_n$\mid$^{-1}\;=\;1\;for\;all\;n\;=\;1,\;2,\;\cdots$. (2) There exists an operator A in AlgL such that Ax = y, A is a unitary operator and every E in L reduces, A, where AlgL is a tridiagonal algebra.

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References

  1. J. Functional Analysis v.3 Interpolation problems in nest algebtas Arveson, W. B.
  2. Proc. Amer. Math. Soc. v.17 On majorization, and range inclusion of operators in Hilbert space Douglas, R. G.
  3. Operator Theory: Adv. appl. v.2 Commutants modulo the compact operators of certain CSL algebras Gilfeather, F.;Larson, D.
  4. Indiana University Math. J. v.29 The equation Tx- y in a reflexive operator algebra Hopenwasser, A.
  5. Illinois J. Math. v.33 no.4 Hilbert-Schmidt interpolation an CSL algebras Hopenwasser, A.
  6. Pacific Journal of Mathermatics v.140 no.1 Isometries of Tridiabonal Algebras Jo, Y. S.
  7. to appear in Rocky Monutain Journal of Math. Interpolation problems in CSL-Algebra AlgL Jo, Y. S.;Kang, J. H.
  8. Proc. Nat. Acad. Sci. Irreducible Operator Algebras Kadison, R.
  9. Korean J. Comput. and Appl. Math. Positive interpolation problems in trisiagonal algebra Kang, J. H.
  10. Proc. London Math. Soc. v.3 no.19 Some properties of nest algebras Lance, E. C.