# Time Invariant Systems

## Abstract

Throughout this book the interconnection between interpolation and input-output systems in state space form plays a fundamental role. This chapter introduces discrete time-invariant input-output systems, and various notions connected with them. The chapter also serves as a further motivation for the interpolation problems considered earlier. An introduction to the state space theory is given. Moreover, it is shown that point and operator evaluation naturally occur in linear systems. State space techniques are used to compute the norm of certain Hankel operators, which is precisely the error in the corresponding Nehari interpolation problem. State space techniques are also used to give explicit state space formulas to connect the Nevanlinna-Pick problem to the Sarason problem, and the Nudelman problem to the two-sided Sarason problem. The last section presents some aspects of unitary systems that will be used later.

## Keywords

Toeplitz Operator Interpolation Problem Hankel Operator Lyapunov Equation Time Invariant System## Preview

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