On the existence of Hilbert valued periodically correlated‎ autoregressive processes

ýIn this paper we provide sufficient condition for existence of aý ýunique Hilbert valued (H-valued) periodicallyý ýcorrelated solution to the first order autoregressive modelý ýXn=ρnXn−1ý, ýfor \ n∈Zý, ýandý ýformulate the existing solution and its autocovariance operatorý. ýAlso we specially investigate equivalent condition for theý ýcoordinate process ⟨Xn,v⟩ý, ýforý ýarbitrary element v in Hý, ýto satisfy in someý ýautoregressive modelý. ýFinallyý, ýwe extend our result to theý ýautoregressive process with finite orderý.