Left and right Blaschke-Potapov products and Arov-singular matrix-valued functions

IfJ is an indefinite signature matrix, then there exists aJ contractive holomorphic matrix valued functionW(z) in the open unit disc which can be expressed as a left Blaschke-Potapov product:W(z)=B ^(l)(z), but not as a right Blaschke-Potapov product:W(z)=E(z)B ^(r)(z), whereB ^(r)(z) is a right Blaschke-Potapov product andE(z) is a so called Arov singular matrix function. In factB ^(l)(z) may be chosen to obtain any Arov singular matrix functionE(z) in the second representation. This phenomenon and multiplicative representations of Arov singular functions are discussed.This paper was written while the author was a guest of the Department of Theoretical Mathematics of The Weizmann Institute of Science, Rehovot. The author would like to thank the Department for its support and hospitality.