Journal of Operator Theory

Volume 50, Issue 2, Fall 2003 pp. 249-261.

An intrinsic difficulty with interpolation on the bidisk

Authors: James P. Solazzo
Author institution: Department of Mathematics, University of Georgia, Athens, GA 30602, USA

Summary: The set of possible values $(w_1,\ldots,w_k)=(f(x_1),\ldots,f(x_k))$ arising from restricting contractive elements $f$ from some uniform algebra $A$ to a finite set $\{ x_1,\ldots,x_k \}$ in the domain is called an interpolation body. When the uniform algebra is the bidisk algebra, Cole and Wermer show that the associated interpolation body is a semi-algebraic set and it is in this sense that the interpolation body is ``computable''. Motivated by the work of Cole and Wermer, Paulsen introduced the notion of the Schur ideal which acts a natural ``dual'' object for these interpolation bodies. From this ``duality'' a stronger notion of ``computability'' follows which will allow us to discuss the intrinsic differences between interpolation on the bidisk and interpolation on the disk.

Keywords: Interpolation, $k$-idempotent operator algebras, Schur ideals

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