— —   Structural Analysis of Certain Linear Operators Representing Chemical Network Systems via the Existence and Uniqueness Theorems of Spectral Resolution. IV SHIGERU ARIMOTO, 1, * KENICHI FUKUI, 2 KEITH F. TAYLOR, 3 PAUL G. MEZEY 1 1 Department of Chemistry, University of Saskatchewan, Saskatoon, SK, Canada, S7N 5C9 2 Institute for Fundamental Chemistry, 34-4, Takano-Nishihiraki-cho, Sakyo-ku, Kyoto 606, Japan 3 Department of Mathematics, University of Saskatchewan, Saskatoon, SK, Canada, S7N 5E6 Received 11 July 1997; accepted 23 September 1997 ABSTRACT: Part IV of this series consists of two complementary subparts devoted to Ž. attain the following two goals: i By shifting from the previous setting of the Banach Ž . Ž . Ž . algebra B B  B B, B to a broader setting of the space B X, B of all bounded linear operators from a normed space X to a Banach space B, we extend our previous theoretical framework to incorporate part of the theory of additive correlation involving the Asymptotic Linearity Theorems, which have been developed for a study of correlation between structure and properties in molecules having many identical moieties, especially Ž. in macromolecules having repeating units. ii By reverting our focus to the special Ž . algebra B H with H being a Hilbert space, we develop a theorem which is useful for a structural analysis of spectral symmetry of linear operators representing physico-chemical network systems.  1998 John Wiley & Sons, Inc. Int J Quant Chem 67: 5769, 1998 Key words: linear operator; *-algebra; general topology; asymptotic analysis; chemical network systems Correspondence to:S. Arimoto. *On leave from Institute for Fundamental Chemistry, 34-4 Takano-Nishihiraki-cho, Sakyo-ku, Kyoto 606, Japan. ( ) International Journal of Quantum Chemistry, Vol. 67, 57 69 1998  1998 John Wiley & Sons, Inc. CCC 0020-7608 / 98 / 010057-13