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In this work we develop a theory of Vessels. This object arises in the studyof overdetermined 2D systems invariant in one of the variables, which areusually called time invariant. To each overdetermined time invariant 2D systemsthere is associated a vessel, which is a collection of system operatorssatisfying certain relations and vise versa. Such an invariance forces thetheory of vessels to resemble a constant (classical) 1D case and as a resultmany notions are naturally redefined and most theorems are reproved in thissetting. The notion of transfer function and its connection to theoverdetermined 2D time invariant system (and the corresponding vessel) is oneof the topics of this work. It is well known that multiplicative structure of atransfer function of a 1D system is closely connected to the decomposition ofthe state space into invariant subspaces of the state operator and wegeneralize this result to a wider class of functions. This class (denoted by$\boldsymbol {\mathcal I}$) arises as a class of transfer functions, whichintertwine solutions of ODEs with spectral parameters. At the end we presentsolution of factorization problems for finite dimensional case.

Overdetermined 2D Systems Invariant in One Direction and Their Transfer Functions