Some Properties of Causal Linear Time-Invariant (LTI) Operators on the Weighted $\ell_2$ Space
Received:September 19, 2014  Revised:March 04, 2015
Key Word: causal LTI operators   weighted $\ell_2$ space   two-operator approach
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11271059).
 Author Name Affiliation Naifeng GAN School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China School of Mathematics and Information Science, Anshan Normal University, Liaoning 114007, P. R. China Yufeng LU School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China Liu LIU School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China
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We discuss the properties of causal LTI operators on weighted $\ell_2$ spaces for different choices of the weighting sequence $\{w(t)\}_{t\in \mathbb{Z}}$. Problems of closability of unstable causal LTI convolution operators are also discussed. We shall provide a new type of argument concerning causal LTI operators and robust design that can be applied to a large class of weighted $\ell_2$ spaces on $\mathbb{Z}$.