A Stochastic Characterization of Arov-completable Matrix-valued Schur Functions
✍ Scribed by Bernd Fritzsche; Bernd Kirstein
- Publisher
- John Wiley and Sons
- Year
- 2006
- Tongue
- English
- Weight
- 658 KB
- Volume
- 169
- Category
- Article
- ISSN
- 0025-584X
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✦ Synopsis
Abstract
A (p + q) × (p + q) matrix‐valued inner function S in the unit disc 𝔻 is called (p, q)‐type Arov‐inner if in the block partition . the p × p diagonal block S~11~ and the q × q diagonal block S~22~ are outer matrix‐valued functions. A holomorphic p × q matrix‐valued function f in 𝔻 is called Arov‐completable if there is a (p, q)‐type Arov‐inner function S such that S~12~ = f Arov‐completability of a given p × q Schur function f is characterized in terms of a (p + q)‐variate stationary sequence (X~n) ϵ Z~) in Hilbert space which is naturally associated with f. The necessary and sufficient condition for Arov‐completability is an orthogonality condition for certain backward and forward innovation vectors generated by (X~n) ϵ Z~.