SELF-ADJOINT INTERPOLATION ON AX=Y IN A TRIDIAGONAL ALGEBRA ALG𝓛

• Journal title : Honam Mathematical Journal
• Volume 36, Issue 1,  2014, pp.29-32
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2014.36.1.29
Title & Authors
SELF-ADJOINT INTERPOLATION ON AX=Y IN A TRIDIAGONAL ALGEBRA ALG𝓛
Kang, Joo Ho; Lee, SangKi;

Abstract
Given operators X and Y acting on a separable Hilbert space $\small{\mathcal{H}}$, an interpolating operator is a bounded operator A such that AX = Y. In this article, we investigate self-adjoint interpolation problems for operators in a tridiagonal algebra : Let $\small{\mathcal{L}}$ be a subspace lattice acting on a separable complex Hilbert space $\small{\mathcal{H}}$ and let X = ($\small{x_{ij}}$) and Y = ($\small{y_{ij}}$) be operators acting on $\small{\mathcal{H}}$. Then the following are equivalent: (1) There exists a self-adjoint operator A = ($\small{a_{ij}}$) in $\small{Alg{\mathcal{L}}}$ such that AX = Y. (2) There is a bounded real sequence {$\small{{\alpha}_n}$} such that $\small{y_{ij}={\alpha}_ix_{ij}}$ for $\small{i,j{\in}\mathbb{N}}$.
Keywords
self-adjoint interpolation;CSL-algebra;tridiagonal algebra;Alg$\small{\mathcal{L}}$;
Language
English
Cited by
References
1.
Gilfeather, F. and Larson, D., Commutants modulo the compact operators of certain CSL algebras, Operator Theory: Adv. Appl. 2 (Birkhauser, Basel, 1981), 105-120.

2.
Hopenwasser, A., The equation Tx = y in a reflexive operator algebra, Indiana University Math. J. 29 (1980), 121-126.

3.
Hopenwasser, A., Hilbert-Schmidt interpolation in CSL algebras, Illinois J. Math. 33(4) (1989), 657-672.

4.
Kadison, R., Irreducible Operator Algebras, Proc. Nat. Acad. Sci. U.S.A. (1957), 273-276.

5.
Lance, E. C., Some properties of nest algebras, Proc. London Math. Soc., 19(3) (1969), 45-68.

6.
Munch, N., Compact causal data interpolation, J. Math. Anal. Appl. 140 (1989), 407-418.